Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210) 9780691218526

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Annals of Mathematics Studies Number 210

Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Sergiu Klainerman J´er´emie Szeftel

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2020

c 2020 by Princeton University Press Copyright Princeton University Press is committed to the protection of copyright and the intellectual property our authors entrust to us. Copyright promotes the progress and integrity of knowledge. Thank you for supporting free speech and the global exchange of ideas by purchasing an authorized edition of this book. If you wish to reproduce or distribute any part of it in any form, please obtain permission. Requests for permission to reproduce material from this work should be sent to [email protected] Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu All Rights Reserved Library of Congress Control Number: 2020948579 ISBN 9780691212432 ISBN (pbk.) 9780691212425 ISBN (e-book) 9780691218526 British Library Cataloging-in-Publication Data is available Editorial: Susannah Shoemaker and Kristen Hop Production Editorial: Nathan Carr Production: Brigid Ackerman Publicity: Matthew Taylor and Amy Stewart The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. This book has been composed in LATEX Printed on acid-free paper. ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

List of Figures

xiii

Acknowledgments

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1 Introduction 1.1 Basic notions in general relativity . . . . . . . . . . . . . . . . . . . 1.1.1 Spacetime and causality . . . . . . . . . . . . . . . . . . . . 1.1.2 The initial value formulation for Einstein equations . . . . . 1.1.3 Special solutions . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Stability of Minkowski space . . . . . . . . . . . . . . . . . 1.1.5 Cosmic censorship . . . . . . . . . . . . . . . . . . . . . . . 1.2 Stability of Kerr conjecture . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Formal mode analysis . . . . . . . . . . . . . . . . . . . . . 1.2.2 Vectorfield method . . . . . . . . . . . . . . . . . . . . . . . 1.3 Nonlinear stability of Schwarzschild under polarized perturbations 1.3.1 Bare-bones version of our theorem . . . . . . . . . . . . . . 1.3.2 Linear stability of the Schwarzschild spacetime . . . . . . . 1.3.3 Main ideas in the proof of Theorem 1.6 . . . . . . . . . . . 1.3.4 Beyond polarization . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Note added in proof . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 2 3 10 11 13 15 16 17 17 17 18 21 22 22

2 Preliminaries 2.1 Axially symmetric polarized spacetimes . . . . . . . . . 2.1.1 Axial symmetry . . . . . . . . . . . . . . . . . . 2.1.2 Z-frames . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Axis of symmetry . . . . . . . . . . . . . . . . . . 2.1.4 Z-polarized S-surfaces . . . . . . . . . . . . . . . 2.1.5 Invariant S-foliations . . . . . . . . . . . . . . . . 2.1.6 Schwarzschild spacetime . . . . . . . . . . . . . . 2.2 Main equations . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Main equations for general S-foliations . . . . . . 2.2.2 Null Bianchi identities . . . . . . . . . . . . . . . 2.2.3 Hawking mass . . . . . . . . . . . . . . . . . . . 2.2.4 Outgoing geodesic foliations . . . . . . . . . . . . 2.2.5 Additional equations . . . . . . . . . . . . . . . . 2.2.6 Ingoing geodesic foliation . . . . . . . . . . . . . 2.2.7 Adapted coordinates systems . . . . . . . . . . . 2.3 Perturbations of Schwarzschild and invariant quantities 2.3.1 Null frame transformations . . . . . . . . . . . . 2.3.2 Schematic notation Γg and Γb . . . . . . . . . . .

24 24 24 25 26 28 45 50 51 51 54 56 57 70 71 71 78 78 82

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3 Main Theorem 3.1 General covariant modulated admissible spacetimes . . . . . . . . 3.1.1 Initial data layer . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Main definition . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Renormalized curvature components and Ricci coefficients 3.2 Main norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Main norms in (ext) M . . . . . . . . . . . . . . . . . . . . 3.2.2 Main norms in (int) M . . . . . . . . . . . . . . . . . . . . 3.2.3 Combined norms . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Initial layer norm . . . . . . . . . . . . . . . . . . . . . . . 3.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Smallness constants . . . . . . . . . . . . . . . . . . . . . 3.3.2 Statement of the main theorem . . . . . . . . . . . . . . . 3.4 Bootstrap assumptions and first consequences . . . . . . . . . . . 3.4.1 Main bootstrap assumptions . . . . . . . . . . . . . . . . 3.4.2 Control of the initial data . . . . . . . . . . . . . . . . . . 3.4.3 Control of averages and of the Hawking mass . . . . . . . 3.4.4 Control of coordinates system . . . . . . . . . . . . . . . . 3.4.5 Pointwise bounds for higher order derivatives . . . . . . . 3.4.6 Construction of a second frame in (ext) M . . . . . . . . . 3.5 Global null frames . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Extension of frames . . . . . . . . . . . . . . . . . . . . . 3.5.2 Construction of the first global frame . . . . . . . . . . . 3.5.3 Construction of the second global frame . . . . . . . . . . 3.6 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Main intermediate results . . . . . . . . . . . . . . . . . . 3.6.2 End of the proof of the main theorem . . . . . . . . . . . 3.6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 The general covariant modulation procedure . . . . . . . . . . . . 3.7.1 Spacetime assumptions for the GCM procedure . . . . . . 3.7.2 Deformations of surfaces . . . . . . . . . . . . . . . . . . . 3.7.3 Adapted frame transformations . . . . . . . . . . . . . . . 3.7.4 GCM results . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.5 Main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Overview of the proof of Theorems M0–M8 . . . . . . . . . . . . 3.8.1 Discussion of Theorem M0 . . . . . . . . . . . . . . . . . 3.8.2 Discussion of Theorem M1 . . . . . . . . . . . . . . . . . 3.8.3 Discussion of Theorem M2 . . . . . . . . . . . . . . . . . 3.8.4 Discussion of Theorem M3 . . . . . . . . . . . . . . . . . 3.8.5 Discussion of Theorem M4 . . . . . . . . . . . . . . . . . 3.8.6 Discussion of Theorem M5 . . . . . . . . . . . . . . . . . 3.8.7 Discussion of Theorem M6 . . . . . . . . . . . . . . . . . 3.8.8 Discussion of Theorem M7 . . . . . . . . . . . . . . . . . 3.8.9 Discussion of Theorem M8 . . . . . . . . . . . . . . . . .

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89 89 89 91 95 96 96 99 100 100 101 101 102 105 105 105 106 107 109 109 111 111 112 113 114 114 115 116 125 125 128 128 129 131 133 133 134 135 136 137 138 138 139 140

2.4

2.3.3 The invariant quantity q . . . . 2.3.4 Several identities for q . . . . . Invariant wave equations . . . . . . . . 2.4.1 Preliminaries . . . . . . . . . . 2.4.2 Wave equations for α, α, and q

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3.9

Structure of the rest of the book . . . . . . . . . . . . . . . . . . .

4 Consequences of the Bootstrap Assumptions 4.1 Proof of Theorem M0 . . . . . . . . . . . . . 4.2 Control of averages and of the Hawking mass 4.2.1 Proof of Lemma 3.15 . . . . . . . . . . 4.2.2 Proof of Lemma 3.16 . . . . . . . . . . 4.3 Control of coordinates systems . . . . . . . . 4.4 Pointwise bounds for higher order derivatives 4.5 Proof of Proposition 3.20 . . . . . . . . . . . 4.6 Existence and control of the global frames . . 4.6.1 Proof of Proposition 3.23 . . . . . . . 4.6.2 Proof of Lemma 4.16 . . . . . . . . . . 4.6.3 Proof of Proposition 3.26 . . . . . . .

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145 145 164 164 172 174 183 188 197 197 200 208

5 Decay Estimates for q (Theorem M1) 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The foliation of M by τ . . . . . . . . . . . . . . . . . . 5.1.2 Assumptions for Ricci coefficients and curvature . . . . 5.1.3 Structure of nonlinear terms . . . . . . . . . . . . . . . . 5.1.4 Main quantities . . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Theorem M1 . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Flux decay estimates for q . . . . . . . . . . . . . . . . . 5.2.2 Proof of Theorem M1 . . . . . . . . . . . . . . . . . . . 5.2.3 Proof of Proposition 5.10 . . . . . . . . . . . . . . . . . 5.3 Improved weighted estimates . . . . . . . . . . . . . . . . . . . 5.3.1 Basic and higher weighted estimates for wave equations 5.3.2 Proof of Theorem 5.14 . . . . . . . . . . . . . . . . . . . 5.3.3 Proof of Theorem 5.15 . . . . . . . . . . . . . . . . . . . 5.4 Decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 First flux decay estimates . . . . . . . . . . . . . . . . . 5.4.2 Flux decay estimates for ˇq . . . . . . . . . . . . . . . . . 5.4.3 Proof of Theorem 5.9 . . . . . . . . . . . . . . . . . . . 5.4.4 Proof of Proposition 5.12 . . . . . . . . . . . . . . . . . 5.4.5 Proof of Proposition 5.13 . . . . . . . . . . . . . . . . .

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213 213 214 215 216 218 223 223 224 226 230 231 233 244 249 249 253 255 259 260

6 Decay Estimates for α and α (Theorems M2, M3) 6.1 Proof of Theorem M2 . . . . . . . . . . . . . . . . . 6.1.1 A renormalized frame on (ext) M . . . . . . . 6.1.2 A transport equation for α . . . . . . . . . . 6.1.3 Estimates for transport equations in e3 . . . 6.1.4 Decay estimates for α . . . . . . . . . . . . . 6.1.5 End of the proof of Theorem M2 . . . . . . . 6.2 Proof of Theorem M3 . . . . . . . . . . . . . . . . . 6.2.1 Estimate for α in (int) M . . . . . . . . . . . 6.2.2 Estimate for α on Σ∗ . . . . . . . . . . . . . . 6.2.3 Proof of Proposition 6.10 . . . . . . . . . . . 6.2.4 Proof of Lemma 6.12 . . . . . . . . . . . . . . 6.2.5 Proof of Proposition 6.14 . . . . . . . . . . . 6.2.6 Proof of Lemma 6.16 . . . . . . . . . . . . . .

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7 Decay Estimates (Theorems M4, M5) 7.1 Preliminaries to the proof of Theorem M4 . . . . . . . . . 7.1.1 Geometric structure of Σ∗ . . . . . . . . . . . . . . 7.1.2 Main assumptions . . . . . . . . . . . . . . . . . . 7.1.3 Basic lemmas . . . . . . . . . . . . . . . . . . . . . 7.1.4 Main equations . . . . . . . . . . . . . . . . . . . . 7.1.5 Equations involving q . . . . . . . . . . . . . . . . 7.1.6 Additional equations . . . . . . . . . . . . . . . . . 7.2 Structure of the proof of Theorem M4 . . . . . . . . . . . 7.3 Decay estimates on the last slice Σ∗ . . . . . . . . . . . . 7.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 7.3.2 Differential identities involving GCM conditions on 7.3.3 Control of the flux of some quantities on Σ∗ . . . . 7.3.4 Estimates for some ` = 1 modes on Σ∗ . . . . . . . 7.3.5 Decay of Ricci and curvature components on Σ∗ . 7.4 Control in (ext) M, Part I . . . . . . . . . . . . . . . . . . 7.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 7.4.2 Proposition 7.33 . . . . . . . . . . . . . . . . . . . 7.4.3 Estimates for κ ˇ, µ ˇ in (ext) M . . . . . . . . . . . . 7.4.4 Estimates for the ` = 1 modes in (ext) M . . . . . 7.4.5 Completion of the proof of Proposition 7.33 . . . . 7.5 Control in (ext) M, Part II . . . . . . . . . . . . . . . . . . 7.5.1 Estimate for η . . . . . . . . . . . . . . . . . . . . 7.5.2 Crucial lemmas . . . . . . . . . . . . . . . . . . . . 7.5.3 Proof of Proposition 7.35, Part I . . . . . . . . . . 7.5.4 Proof of Proposition 7.35, Part II . . . . . . . . . . 7.6 Conclusion of the proof of Theorem M4 . . . . . . . . . . 7.7 Proof of Theorem M5 . . . . . . . . . . . . . . . . . . . .

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295 295 295 296 299 301 302 305 308 311 311 314 315 322 332 336 336 338 339 340 343 346 347 347 355 359 362 366

8 Initialization and Extension (Theorems M6, M7, M8) 8.1 Proof of Theorem M6 . . . . . . . . . . . . . . . . . . . 8.2 Proof of Theorem M7 . . . . . . . . . . . . . . . . . . . 8.3 Proof of Theorem M8 . . . . . . . . . . . . . . . . . . . 8.3.1 Main norms . . . . . . . . . . . . . . . . . . . . . 8.3.2 Control of the global frame . . . . . . . . . . . . 8.3.3 Iterative procedure . . . . . . . . . . . . . . . . . 8.3.4 End of the proof of Theorem M8 . . . . . . . . . 8.4 Proof of Proposition 8.7 . . . . . . . . . . . . . . . . . . 8.4.1 A wave equation for ρ˜ . . . . . . . . . . . . . . . 8.4.2 Control of g (r) . . . . . . . . . . . . . . . . . . 8.4.3 End of the proof of Proposition 8.7 . . . . . . . . 8.5 Proof of Proposition 8.8 . . . . . . . . . . . . . . . . . . 8.5.1 A wave equation for α + Υ2 α . . . . . . . . . . . 8.5.2 End of the proof of Proposition 8.8 . . . . . . . . 8.6 Proof of Proposition 8.9 . . . . . . . . . . . . . . . . . . 8.6.1 Control of α and Υ2 α . . . . . . . . . . . . . . . 8.6.2 Control of α . . . . . . . . . . . . . . . . . . . . . 8.6.3 End of the proof of Proposition 8.9 . . . . . . . . 8.7 Proof of Proposition 8.10 . . . . . . . . . . . . . . . . . 8.7.1 r-weighted divergence identities for Bianchi pairs

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372 372 376 387 389 391 393 396 399 399 400 405 408 408 417 418 418 420 424 424 425

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8.7.2 8.7.3 8.8 Proof 8.8.1 8.8.2 8.8.3 8.8.4 8.8.5 8.9 Proof 8.9.1 8.9.2 8.10 Proof

End of the proof of Proposition 8.10 . . . . . . . . . . . . . Proof of (8.3.12) . . . . . . . . . . . . . . . . . . . . . . . . of Proposition 8.11 . . . . . . . . . . . . . . . . . . . . . . . Proof of Proposition 8.31 . . . . . . . . . . . . . . . . . . . Weighted estimates for transport equations along e4 in (ext) M Several identities . . . . . . . . . . . . . . . . . . . . . . . . Proof of Proposition 8.32 . . . . . . . . . . . . . . . . . . . Proof of Proposition 8.33 . . . . . . . . . . . . . . . . . . . of Proposition 8.12 . . . . . . . . . . . . . . . . . . . . . . . Weighted estimates for transport equations along e3 in (int) M Proof of Proposition 8.42 . . . . . . . . . . . . . . . . . . . of Proposition 8.13 . . . . . . . . . . . . . . . . . . . . . . .

435 440 442 444 454 460 464 471 479 480 482 485

9 GCM Procedure 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Main assumptions . . . . . . . . . . . . . . . 9.1.2 Elliptic Hodge lemma . . . . . . . . . . . . . 9.2 Deformations of S surfaces . . . . . . . . . . . . . . 9.2.1 Deformations . . . . . . . . . . . . . . . . . . 9.2.2 Pullback map . . . . . . . . . . . . . . . . . . 9.2.3 Comparison of norms between deformations . 9.2.4 Adapted frame transformations . . . . . . . . 9.3 Frame transformations . . . . . . . . . . . . . . . . . 9.3.1 Main GCM equations . . . . . . . . . . . . . 9.3.2 Equation for the average of a . . . . . . . . . 9.3.3 Transversality conditions . . . . . . . . . . . 9.4 Existence of GCM spheres . . . . . . . . . . . . . . 9.4.1 The linearized GCM system . . . . . . . . . . 9.4.2 Comparison of the Hawking mass . . . . . . . 9.4.3 Iteration procedure for Theorem 9.32 . . . . . 9.4.4 Existence and boundedness of the iterates . . 9.4.5 Convergence of the iterates . . . . . . . . . . 9.5 Proof of Proposition 9.37 and of Corollary 9.38 . . . 9.5.1 Proof of Proposition 9.37 . . . . . . . . . . . 9.5.2 Proof of Corollary 9.38 . . . . . . . . . . . . . 9.6 Proof of Proposition 9.43 . . . . . . . . . . . . . . . 9.6.1 Pullback of the main equations . . . . . . . . 9.6.2 Basic lemmas . . . . . . . . . . . . . . . . . . 9.6.3 Proof of the estimates (9.6.5), (9.6.6), (9.6.7) 9.7 A corollary to Theorem 9.32 . . . . . . . . . . . . . . 9.8 Construction of GCM hypersurfaces . . . . . . . . . 9.8.1 Definition of Σ0 . . . . . . . . . . . . . . . . 9.8.2 Extrinsic properties of Σ0 . . . . . . . . . . . 9.8.3 Construction of Σ0 . . . . . . . . . . . . . . .

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486 486 488 489 489 489 490 492 496 504 513 518 519 520 524 526 527 530 535 538 538 542 545 545 548 556 559 566 569 570 583

10 Regge-Wheeler Type Equations 10.1 Basic Morawetz estimates . . . . . . . . . . . . 10.1.1 Structure of the proof of Theorem 10.1 . 10.1.2 A simplified set of assumptions . . . . . 10.1.3 Functions depending on m and r . . . .

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603 607 609 613 618 625 632 636 642 644 651 653 656 659 659 668 675 675 676 682 682 682 683 696 701 706 711

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B Appendix to Chapter 8 B.1 Proof of Proposition 8.14 . . . . . . . . . . . . . . . . . . . . . . .

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10.4

10.5

10.1.4 Deformation tensors of the vectorfields R, T, X 10.1.5 Basic integral identities . . . . . . . . . . . . . 10.1.6 Main Morawetz identity . . . . . . . . . . . . . 10.1.7 A first estimate . . . . . . . . . . . . . . . . . . 10.1.8 Improved lower bound in (ext) M . . . . . . . . 10.1.9 Cut-off correction in (int) M . . . . . . . . . . . 10.1.10 The redshift vectorfield . . . . . . . . . . . . . 10.1.11 Combined estimate . . . . . . . . . . . . . . . . 10.1.12 Lower bounds for Q . . . . . . . . . . . . . . . 10.1.13 First Morawetz estimate . . . . . . . . . . . . . 10.1.14 Analysis of the error term E . . . . . . . . . . 10.1.15 Proof of Theorem 10.1 . . . . . . . . . . . . . . Dafermos-Rodnianski rp -weighted estimates . . . . . . 10.2.1 Vectorfield X = f (r)e4 . . . . . . . . . . . . . . 10.2.2 Energy densities for X = f (r)e4 . . . . . . . . 10.2.3 Proof of Theorem 10.37 . . . . . . . . . . . . . Higher weighted estimates . . . . . . . . . . . . . . . . 10.3.1 Wave equation for ψˇ . . . . . . . . . . . . . . . 10.3.2 The rp -weighted estimates for ψˇ . . . . . . . . Higher derivative estimates . . . . . . . . . . . . . . . 10.4.1 Basic assumptions . . . . . . . . . . . . . . . . 10.4.2 Strategy for recovering higher order derivatives 10.4.3 Commutation formulas with the wave equation 10.4.4 Some weighted estimates for wave equations . . 10.4.5 Proof of Theorem 5.17 . . . . . . . . . . . . . . 10.4.6 Proof of Theorem 5.18 . . . . . . . . . . . . . . More weighted estimates for wave equations . . . . . .

A Appendix to Chapter 2 A.1 Proof of Proposition 2.64 . . . . . . . . . . A.2 Proof of Proposition 2.71 . . . . . . . . . . A.3 Proof of Lemma 2.72 . . . . . . . . . . . . . A.4 Proof of Proposition 2.73 . . . . . . . . . . A.5 Proof of Proposition 2.74 . . . . . . . . . . A.6 Proof of Proposition 2.90 . . . . . . . . . . A.7 Proof of Lemma 2.92 . . . . . . . . . . . . . A.8 Proof of Corollary 2.93 . . . . . . . . . . . . A.9 Proof of Lemma 2.91 . . . . . . . . . . . . . A.10 Proof of Proposition 2.99 . . . . . . . . . . A.11 Proof of Proposition 2.100 . . . . . . . . . . A.12 Proof of the Teukolsky-Starobinsky identity A.13 Proof of Proposition 2.107 . . . . . . . . . . A.14 Proof of Theorem 2.108 . . . . . . . . . . . A.14.1 The Teukolsky equation for α . . . . A.14.2 Commutation lemmas . . . . . . . . A.14.3 Main commutation . . . . . . . . . . A.14.4 Proof of Theorem 2.108 . . . . . . .

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xi

CONTENTS

C Appendix to Chapter 9 C.1 Proof of Lemma 9.11 . . . . . . . . . . . . . . . . . . . . . . . . . .

806 806

D Appendix to Chapter 10 D.1 Horizontal S-tensors . . . . . . . . . . D.1.1 Mixed tensors . . . . . . . . . . D.1.2 Invariant Lagrangian . . . . . . D.1.3 Comparison of the Lagrangians D.1.4 Energy-momentum tensor . . . D.2 Standard calculation . . . . . . . . . . D.3 Vectorfield Xf . . . . . . . . . . . . . D.4 Proof of Proposition 10.47 . . . . . . .

819 819 820 820 821 822 823 824 827

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836

List of Figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 3.1 3.2 3.3

The initial value problem for Einstein vacuum equations . . Minkowski in standard coordinates . . . . . . . . . . . . . . Penrose diagram of Minkowski . . . . . . . . . . . . . . . . Kruskal’s maximal extension of Schwarzschild . . . . . . . . Complete Penrose diagram of Schwarzschild . . . . . . . . . Exterior region of Schwarzschild . . . . . . . . . . . . . . . Exterior region of Kerr . . . . . . . . . . . . . . . . . . . . . Kerr solution on a fixed time slice . . . . . . . . . . . . . . Behavior of null geodesics outside and inside the black hole The GCM admissible spacetime M . . . . . . . . . . . . . . The initial data layer L0 . . . . . . . . . . . . . . . . . . . . The GCM admissible spacetime M . . . . . . . . . . . . . . The Penrose diagram of the spacetime M . . . . . . . . . .

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3 4 5 6 7 7 8 9 12 20 91 95 118

Acknowledgments

This work would be inconceivable without the remarkable advances made in the last sixty years on black holes. The works of Regge-Weeler, Israel, Carter, Teukolsky, Chandrasekhar, Wald, etc., made during the so-called golden age of black hole physics in the sixties and seventies, have greatly influenced our understanding of invariant quantities and the wave equations they satisfy. The advances made in the last fifteen years, which have led to the development of new mathematical methods to derive the decay of waves on black hole spacetimes, are even more immediately relevant to our work. In particular we would like to single out the direct influence of Dafermos-Holzegel-Rodnianski [26] in the gestation of our own ideas in this book. Finally, the work on the nonlinear stability of the Minkowski space in [20], a milestone in the mathematical GR, has significantly instructed our work here. We would like to thank E. Giorgi for her careful proofreading of various sections of the manuscript. Various discussions we had with S. Aksteiner were very useful. Finally we thank our wives Anca and Emilie for their incredible patience, understanding and support during our many years of work on this project. The first author has been supported by the NSF grant DMS 1362872. He would like to thank the mathematics departments of Paris 6, Cergy-Pontoise and IHES for their hospitality during his many visits in the last six years. The second author is supported by the ERC grant ERC-2016 CoG 725589 EPGR.

Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Chapter One Introduction 1.1

BASIC NOTIONS IN GENERAL RELATIVITY

We provide a quick review of the basic concepts of general relativity relevant to this work. For a proper introduction to the subject we refer to the books by R. Wald [66] and S. Caroll [15]. 1.1.1

Spacetime and causality

The main object of Einstein’s general relativity is the spacetime. To define a spacetime, consider a four dimensional Lorentzian manifold (M, g), with g denoting a Lorentzian metric of signature (−, +, +, +). Two Lorentzian manifolds (M, g), (M0 , g0 ) are equivalent if there exists a diffeomorphism Φ : M → M0 such that g = Φ# (g0 ). A spacetime is simply a class of equivalence of such Lorentzian manifolds. A Lorentzian metric divides vectors X in a tangent space Tp (M) into timelike, null and spacelike according to whether g(X, X) is, respectively, negative, zero or positive. A curve γ(t) is said to be timelike, respectively null, if its tangent vector γ(t) ˙ is timelike or null. It is called causal if it is either timelike or null. Remark 1.1. Observers in general relativity are identified to timelike curves, and freely moving observers correspond to timelike geodesics. Points of M are referred to as events and the proper time of an observer γ(t) between the events γ(t1 ), γ(t2 ) is the integral, Z t2 q
−g γ(t), ˙ γ(t) ˙ dt. t1

Massless particles, on the other hand, follow null geodesics. The proper time of such a particle, i.e., the proper time of the corresponding null geodesic, is the affine parameter of the geodesic vectorfield associated to the curve. Given a set S ⊂ M, we denote by I + (S) the set of all points in M which can be reached by future directed timelike curves1 originating at S, called the future set of S. The set J + (S), consisting of points which can be reached by future directed causal curves from S, is called the causal future of S. One defines in the same manner the past and causal pasts I − (S) and J − (S). A hypersurface Σ is called spacelike or null, if the direction normal to it is timelike, respectively null. Typical spacelike hypersurfaces are given by the level surfaces of time functions t, i.e., non-degenerate functions on M ( dt 6= 0) such that 1 We

assume the spacetime to be time oriented, i.e., there exists a globally defined nondegenerate timelike vectorfield T . In particular, a causal vectorfield X is future oriented if g(T, X) < 0.

2

CHAPTER 1

its gradient −gµν ∂µ t∂ν is timelike. Typical null hypersurfaces are given by level surfaces of optical functions u, i.e., non-degenerate functions u : M → R verifying gµν ∂µ u∂ν u = 0,

du 6= 0.

(1.1.1)

In that case the gradient L := −gµν ∂µ u∂ν is both null and geodesic, i.e., g(L, L) = 0 and DL L = 0. A spacelike hypersurface Σ is said to be a Cauchy hypersurface in M if any in-extendible causal curve intersects Σ at precisely one point. Spacetimes which admit such hypersurfaces rule out causal pathologies such as the presence of closed timelike curves. A spacetime is called globally hyperbolic if it possesses such a hypersurface and, in addition, all sets of the form J + (p) ∩ J − (q) are compact. 1.1.2

The initial value formulation for Einstein equations

Let (M, g) a spacetime. Einstein equations are given by 1 Rαβ − gαβ R = Tαβ 2

(1.1.2)

with Rαβ the Ricci curvature of g, R the scalar curvature of g, and Tαβ the energy-momentum tensor of some matterfield defined on (M, g). An initial data set consists of a 3 dimensional manifold Σ(0) , a complete Riemannian metric g(0) , a symmetric 2-tensor k(0) , and a well specified set of initial conditions corresponding to the matterfields under consideration. These have to verify a well known set of constraint equations. We restrict the discussion to asymptotically flat initial data sets, i.e., outside a sufficiently large compact set K, Σ(0) \ K is diffeomorphic to the complement of the unit ball in R3 and admits a system of coordinates in which g(0) is asymptotically euclidean, and k(0) vanishes asymptotically at appropriate order. A Cauchy development of an initial data set is a globally hyperbolic spacetime (M, g), verifying the Einstein equations (1.1.2) in the presence of a matterfield with energy-momentum T and an embedding i : Σ → M such that i∗ (g(0) ), i∗ (k(0) ) are the first and second fundamental forms of i(Σ(0) ) in M. We restrict our attention to the Einstein vacuum equations (EVE), i.e., the case when the energy-momentum tensor vanishes identically and the equations take the purely geometric form, Rαβ = 0.

(1.1.3)

In that case, the constraint equations mentioned above take the form div k(0) − ∇ trk(0) = 0,

R(0) − |k(0) |2 + (trk(0) )2 = 0.

(1.1.4)

Here ∇ denotes the covariant derivative on Σ(0) , div the usual divergence of a symmetric 2-tensor, defined with respect to ∇, and R(0) the scalar curvature of the metric g(0) . Moreover |k(0) | and trk(0) are the Riemannian norm and trace of k(0) with respect to g(0) . The most basic question concerning the initial value problem, solved in a satisfactory way for very large classes of evolution equations, is that of local existence and uniqueness of solutions. For the Einstein equations, this type of result was

3

INTRODUCTION

first established by Y. Choquet-Bruhat [13] with the help of wave coordinates.2 According to this result any smooth initial data set admits a smooth, unique (up to an isometry) globally hyperbolic Cauchy development.3 In the case of nonlinear systems of partial differential equations, the local existence and uniqueness result leads, through a straightforward extension argument, to a result concerning the maximal time interval of existence. The formulation of the same type of result for the Einstein equations is a little more subtle; something similar was achieved in [14], see also [60] for a modern version of the result. Theorem 1.2 (Bruhat-Geroch). For each smooth initial data set there exists a unique, smooth, maximal future globally hyperbolic development (MFGHD).

Figure 1.1: The initial value problem for Einstein vacuum equations

1.1.3 1.1.3.1

Special solutions Minkowski space

The Minkowski space consists of the manifold R1+3 together with a Lorentzian metric m and a distinguished system of coordinates xα , α = 0, 1, 2, 3, called inertial, relative to which the metric has the diagonal form mαβ = diag(−1, 1, 1, 1). We write, splitting the spacetime coordinates xα into the time component x0 = t and space components x = x1 , x2 , x3 , m = −dt2 + (dx1 )2 + (dx2 )2 + (dx3 )2 . In polar coordinates (t, r, θ, ϕ), m = −dt2 + dr2 + r2 dσS2 ,

dσS2 := dθ2 + sin2 θdϕ2 .

The standard optical functions in R1+3 are given by u = t − r, u = t + r, often called retarded and advanced time coordinates. One can compactify the Minkowski space by constructing a map P : (u, u, ω) → (U, U , ω), ω ∈ S2 , where u = tan U,

u = tan U ,

−

π π 2m and two regions r < 2m of the Schwarzschild metric to obtain a metric which is smooth along H = {r = 2m}, see [66] for details, called the Schwarzschild horizon. The portion of r < 2m to the future of the hypersurface t = 0 is a black hole whose future boundary r = 0 is singular. The similar region to the past of t = 0 is called a white hole. The region r > 2m, called the domain of outer communication (DOC), is free of singularities.

6

CHAPTER 1

Figure 1.4: Kruskal’s maximal extension of Schwarzschild

To explicitly extend the metric, introduce the tortoise coordinate r∗ and the optical functions u and u by  r  r∗ := r + 2m ln −1 , u := t − r∗ , u := t + r∗ , 2m and Kruskal renormalized null coordinates, u

u

u0 := −e− 4m ,

u0 := e 4m ,

relative to which the metric takes the form r

ds2 = −

16m2 e− 2m 0 0 du du + r2 dσS2 . r

Observe now that r = 2m corresponds precisely to u0 u0 = 0. Indeed r is an implicit function of u0 u0 through the relation  r  r − 1 e 2m = −u0 u0 . 2m In the new coordinates, the Schwarzschild metric thus extends past r = 2m as illustrated in figure 1.4. We can also conformally compactify the Schwarzschild space by proceeding with the transformation U := arctan(u0 ),

U := arctan(u0 ).

The completed, conformally compactified spacetime is provided by figure 1.5. Here, as for Minkowski space, the boundaries I + and I − , called future and past null infinities, are idealized boundaries of the spacetime corresponding to end points, of future directed, respectively past directed, null geodesics. The points i+ and i− correspond to end points of future and past timelike geodesics, while i0 corresponds to spacelike infinity. Note that the black hole region can be identified as
the complement of the past of future null infinity, i.e., the complement of J − I + .

7

INTRODUCTION

Figure 1.5: Complete Penrose diagram of Schwarzschild

Figure 1.6: Exterior region of Schwarzschild

Similarly
the white hole region is the complement  of the future of past null infinity J + I − . The null hypersurface H = r = 2m , called the event horizon, is the boundary of the black hole and of the white hole. In figure 1.6, representing one connected component of DOC, we note the presence of the timelike hypersurface r = 3m on which null geodesics can be trapped. 1.1.3.3

Kerr space

The Schwarzschild family is included in a larger two parameter family of solutions K(a, m) discovered by Kerr. A given Kerr spacetime, with 0 ≤ |a| ≤ m, has a

8

CHAPTER 1

well defined domain of outer communication r > r+ := m + (m2 − a2 )1/2 . In Boyer-Lindquist coordinates, well adapted to r > r+ , the Kerr metric has the form
∆ − a2 sin2 θ 2 4amr q2 2 Σ2 2 2 2 gK = − dt − sin θdtdϕ + dr + q dθ + sin2 θdϕ2 q2 q2 ∆ q2 with q 2 = r2 + a2 cos2 θ, ∆ = r2 + a2 − 2mr, Σ2 = (r2 + a2 )2 − a2 (sin θ)2 ∆. Note that ∆(r+ ) = 0. As in the Schwarzschild case, the exterior Kerr metric extends smoothly across the hypersurface r = r+ . The future and past sets of any point in the domain of outer communication intersect any timelike curve, passing through points of arbitrary large values of r, in finite time as measured relative to proper time along the curve. This fact is violated by points in the region r ≤ r+ , which consists of the union between a black hole region, extended towards the future, and a white hole region to the past. Thus physical signals (i.e., future timelike or null geodesics) which initiate at points in r ≤ r+ cannot be registered by far away observers.5 The domain of outer communication {r > r+ } is real analytic. The boundary of the domain of outer communication {r = r+ } is called the event horizon. In the non-degenerate case, |a| < m, the event horizon consists of two null hypersurfaces intersecting transversally on a compact 2-sphere. The Kerr solution can also be conformally compactified in the same manner as Minkowski and Schwarzschild. We can thus talk about the future and past null infinities I + , I − as well as i0 , i+ , i− . As before, I + is a complete null hypersurface, smooth away from i0 .

Figure 1.7: Exterior region of Kerr The exterior Kerr metrics are stationary, which means, roughly, that the coefficients of the metric are independent of the time variable t. One can reformulate 5 They must end in the singularity at r = 0, in Schwarzschild spacetime. Their behavior in Kerr is more complicated due to the presence of a Cauchy horizon at r = r− along which the spacetime remains smooth.

9

INTRODUCTION

this by saying that the vectorfield T = ∂t is Killing6 (everywhere in the domain of outer communication) and timelike at points with r large, i.e., the so-called asymptotic region (where the spacetime is close to flat). One can also easily check that T is tangent to the horizon H = N ∪ N , which is itself a null hypersurface, i.e., the restriction of the metric to the tangent space to H is degenerate (see figure 1.7). In addition to being stationary, the coefficients of the Kerr metric are independent of the coordinate ϕ. Thus Kerr is stationary and axially symmetric. It has been conjectured that all asymptotically flat stationary solutions of the Einstein vacuum equations must be Kerr solutions. The conjecture has been verified only if additional assumptions are made, see [35] for a recent survey of known results. The Schwarzschild metrics, corresponding to a = 0, are not just axially symmetric but spherically symmetric, which means that the metric is left invariant by the whole rotation group of the standard sphere S2 . A well known theorem of Birkhoff shows that they are the only such solutions of the Einstein vacuum equations. Another peculiarity of a Schwarzschild metric, not true in the case of Kerr, is that the stationary Killing vectorfield T = ∂t is orthogonal to the hypersurface t = 0. A stationary spacetime which has this property is called static. This is also equivalent to the fact that the Schwarzschild metric is invariant with respect to the reflection t → −t. Moreover, T is timelike for all r > 2m and null along the Schwarzschild horizon H = {r = 2m}. This is not the case for Kerr solutions in which case T = ∂t is only timelike for r > m + (m2 − a2 cos2 θ)1/2 , null for r = m + (m2 − a2 cos2 θ)1/2 and spacelike in the region between r+ and r = m + (m2 − a2 cos2 θ)1/2 , called the ergosphere. Finally we remark that the Kerr family is not physically relevant for |a| > m, hence the restriction to |a| ≤ m.

Figure 1.8: Kerr solution on a fixed time slice To summarize: 1. The Kerr family K(a, m), 0 ≤ |a| ≤ m, provides a two parameter family of asymp6 A vectorfield X is said to be Killing if its associated 1-parameter flow consists of isometries of g, i.e., the Lie derivative of the metric g with respect to X vanishes, LX g = 0.

10

2.

3.

4.

5.

CHAPTER 1

totically flat solutions of the Einstein vacuum equations exhibiting a smooth domain of outer communication and its complement, separated by the event horizon {r = r+ }. For |a| < m, the event horizon consists of two null hypersurfaces intersecting transversally on a compact 2-sphere. All Kerr solutions are stationary, i.e., they admit a Killing vectorfield T which is timelike in the asymptotic region. The Schwarzschild spacetime (i.e., a = 0) is also static. Moreover the Kerr family is axially symmetric, i.e., it admits another Killing vectorfield Z which vanishes on the axis of symmetry. The Schwarzschild spacetime is spherically symmetric. The stationary vectorfield T is tangent along the horizon and spacelike for all 0 < |a| ≤ m. It remains spacelike in a small region of DOC called ergoregion. In the particular case a = 0, T is null along the horizon and timelike everywhere in DOC. In all cases 0 ≤ |a| ≤ m, DOC contains trapped null geodesics, i.e., null geodesics which are entirely contained in a region of DOC with a bounded value of r. In the case a = 0, all trapped null geodesics are either tangent to the timelike surface {r = 3m} or asymptotic to it. All physically acceptable Kerr solutions, i.e., |a| ≤ m, have complete future and past null infinities corresponding to r = ∞. Here are some other important properties of the Kerr family.

• The Kerr solution has a remarkable algebraic feature, encoded in the so-called Petrov type D property, according to which it admits, at every point a pair of null vectors (l, l), normalized by the condition g(l, l) = −2, called principal null vectors, such that all components of the Riemann curvature tensor vanish identically except for the two independent components R(l, l, l, l),

?

R(l, l, l, l),

with ?R the Hodge dual of R. • In addition to the symmetries provided by the Killing vectorfields T and Z, the Kerr solution possesses a nontrivial Killing tensor, i.e., a symmetric 2-covariant tensor C (the Carter tensor) verifying D(α Cβγ) = 0. • The Kerr family is distinguished among all stationary solutions of EVE by the vanishing of a four tensor called the Mars-Simon tensor, see [51]. 1.1.4

Stability of Minkowski space

The Minkowski space (R1+3 , m) is the simplest solution of the Einstein vacuum equations. Note that it belongs to the Kerr family and corresponds to the particular case a = m = 0. Among all Kerr solutions, the Minkowski space is the only one free of pathologies such as singular boundaries, or the presence of Cauchy horizons. In particular, it is geodesically complete, i.e., any freely moving observer in M can be extended indefinitely, as measured relative to its proper time. Such a spacetime is said to have a regular MFGHD. Does this property persist under small perturbations? The result stated below is a rough version of the global stability of Minkowski.

INTRODUCTION

11

The complete result also provides very precise information about the decay of the curvature tensor along null and timelike directions as well as much other geometric information concerning the causal structure of the corresponding spacetime, see [20], as well as [42], [49] and [7]. Of particular interest are peeling properties, i.e., the precise decay rates of various components of the curvature tensor along future null geodesics. Theorem 1.4 (Global stability of Minkowski). The maximal future development of an asymptotically flat initial data set, sufficiently close to that of Minkowski space, in an appropriate topology, is geodesically complete and converges to the Minkowski space. Here are, very schematically, some of the main ideas in the proof of the stability of Minkowski space. 1. Perturbations radiate and decay sufficiently fast (just fast enough!) to insure convergence. 2. Interpret the Bianchi identities as a Maxwell like system. This is an effective, invariant, way to treat the hyperbolic character of the equations. 3. Rely on four important PDE advances of late last century: a) Vectorfield approach to get decay based on approximate Killing and conformal Killing symmetries of the equations, see [39], [40], [41], [19]. b) Generalized energy estimates using both the Bianchi identities and the approximate Killing and conformal Killing vectorfields. c) The null condition identifies the deep mechanism for nonlinear stability, i.e., the specific structure of the nonlinear terms enables stability despite the slow decay rate of the perturbations, see [38], [40], [18]. d) Involved bootstrap argument according to which one makes educated assumptions about the behavior of the spacetime and then proceeds to show that they are in fact satisfied. This amounts to a conceptual linearization, i.e., a method by which the equations become, essentially, linear7 without actually linearizing them. 1.1.5

Cosmic censorship

Unlike the situation described in Theorem 1.4, we expect maximal developments of typical, nonsmall, initial data sets to be incomplete, with singular boundaries. As shown by D. Christodoulou [21], trapped surfaces can form in evolution starting with regular initial conditions.8 Together with the well known singularity theorem of R. Penrose, these results show that there exists a large class of regular initial data whose MFGHD is incomplete. The unavoidable presence of singularities, for sufficiently large initial data sets, as well as the analysis of explicit examples (such as Schwarzschild and Kerr) have led Penrose to formulate two fundamental conjectures, concerning the character of general solutions to the Einstein equations. Here we restrict our discussion only to the so-called weak cosmic censorship conjecture (WCC), which is the only one relevant to the problem of stability. To understand the statement of WCC, consider 7 With 8 That

quadratic and higher order terms satisfying the null condition on the right-hand side. is, free of trapped surfaces. See also more recent results in [45], [44] and [2].

12

CHAPTER 1

the different behavior of null rays in Schwarzschild and Minkowski spacetimes. In Minkowski space, light originating at any point p = (t0 , x0 ) propagates, towards future, along the null rays of the null cone t − t0 = |x − x0 |. Any free observer in R1+3 , following a straight timelike line, will necessarily meet this light cone in finite time, thus experiencing the event p. On the other hand, any point p in the trapped region r < 2m of the Schwarzschild space is such that all null rays initiating at p remain trapped in the region r < 2m. In particular events causally connected to the singularity at r = 0 cannot influence events in the domain of outer communication r > 2m, which is thus entirely free of singularities. The same holds true in any Kerr solution with 0 ≤ |a| ≤ m.

Figure 1.9: Behavior of null geodesics outside and inside the black hole WCC is an optimistic extension of this fact to future developments of general, asymptotically flat initial data sets. The desired conclusion of the conjecture is that any such development, with the possible exception of a non-generic set of initial conditions, has the property that any sufficiently distant observer will not encounter singularities. To make this more precise, one needs to define what a sufficiently distant observer means. This is typically done by introducing the notion of future null infinity I + which provides end points for the null geodesics that propagate to asymptotically large distances. As in the cases analyzed above, future null infinity is constructed by conformally embedding the physical spacetime (M, g) to a larger f g e) such that g e = Ω2 g in M, with a null boundary I + (where spacetime (M, Ω = 0, dΩ 6= 0). Definition 1.5. The future null infinity I + is said to be complete9 if any future null geodesic along it can be indefinitely extended relative to an affine parameter.10 Conjecture (Weak Cosmic Censorship conjecture). Generic asymptotically flat initial data sets have maximal future developments possessing a complete future null infinity. Once the completeness of future null infinity has been established, one can then define the black hole region B to be the complement of the causal past of null 9 A more precise definition of complete future null infinity, which avoids the technical and murky issue of the precise degree of smoothness of the conformal compactification, was proposed by Christodoulou in [17]. 10 This can be informally reformulated, for MFGHD spaces, by stating that there exists a sequence of relatively compact sets Kn exhausting the initial hypersurface Σ(0) such that the proper future time of observers starting in Kn+1 \ Kn tends to infinity as n → ∞.

13

INTRODUCTION

infinity B := M \ J − (I + ).

(1.1.7)

The boundary H+ of B is called the event horizon of the black hole. 1.2

STABILITY OF KERR CONJECTURE

The nonlinear stability of the Kerr family is one of the most pressing issues in mathematical GR today. Roughly, the problem is to show that all spacetime developments of initial data sets, sufficiently close to the initial data set of a Kerr spacetime, behave in the large like a (typically another) Kerr solution. This is not only a deep mathematical question but one with serious astrophysical implications. Indeed, if the Kerr family would be unstable under perturbations, black holes would be nothing more than mathematical artifacts. Here is a more precise formulation of the conjecture. Conjecture (Stability of Kerr conjecture). Vacuum initial data sets, sufficiently close to Kerr initial data, have a maximal development with complete future null infinity and with domain of outer communication11 which approaches (globally) a nearby Kerr solution. There are four, related, major obstacles in passing from the stability of Minkowski to that of the Kerr family. 1. The first can be understood in the general framework of nonlinear hyperbolic or dispersive equations. Given a nonlinear equation N [φ] = 0 and a stationary solution φ0 we have two notions of stability, orbital stability, according to which small perturbations of φ0 lead to solutions φ which remain close, in some norm (typically L2 based ) for all time, and asymptotical stability, according to which the perturbed solutions converge, as t → ∞, to a nearby stationary solution. Note that the second notion is far stronger, and much more precise, than the first and that orbital stability can only be established (without appealing to the stronger version) only for equations with very weak nonlinearities. For quasilinear equations, such as the Einstein field equations, a proof of stability requires, necessarily, a proof of asymptotic stability. This must then be based on a detailed understanding of the decay properties of the linearized12 equations. One is thus led to study the linearized equations N 0 [φ0 ]ψ = 0, with N 0 [φ0 ] the Fr´echet derivative of N at φ0 , which, in many important cases, are hyperbolic13 systems with variable coefficients that typically present instabilities. In the exceptional situation, when nonlinear stability can ultimately be established, one can tie all the instability modes of the linearized system to two properties of the 11 This presupposes the existence of an event horizon. Note that the existence of such an event horizon can only be established upon the completion of the proof of the conjecture. 12 It is irrelevant whether a specific linearization procedure needs to be implemented; what is important here is to identify the linear mechanism for decay, such as the Maxwell system in the case of the stability of Minkowski space mentioned above. 13 In the case of EVE the linearized equations are linear hyperbolic only after we mod out the linearized version of general coordinate transformations.

14

CHAPTER 1

nonlinear equation: a) The presence of a continuous14 family of other stationary solutions of N [φ] = 0 near φ0 . b) The presence of a continuous family of diffeomorphisms15 of the background manifold which map, by pullback, solutions to solutions. For a typical stationary solution φ0 , both properties exist and generate nontrivial solutions of the linearized equation N 0 [φ0 ]ψ = 0. In the case of relatively simple scalar nonlinear equations, where the symmetry group of the equation is small, an effective strategy of dealing with this problem (known under the name of modulation theory) has been developed, see for example [53], [55]. In the case of the Einstein equations this problem is compounded by the large invariance group of the equations, i.e., all diffeomorphisms of the spacetime manifold. To deal with both problems and establish stability one has to: • Track the parameters (af , mf ) of the final Kerr spacetime. • Track the coordinate system (gauge condition) relative to which we have decay for all linearized quantities. Such a coordinate system cannot be imposed a priori, it has to emerge dynamically in the construction of the spacetime. 2. As described earlier, the fundamental insight in the stability of the Minkowski space was that we can treat the Bianchi identities as a Maxwell system in a slightly perturbed Minkowski space by using the vectorfield method. This cannot work for perturbations of Kerr due to the fact that some of the null components of the curvature tensor16 are nontrivial in Kerr. 3. Even if we can establish a useful version of linearization (i.e., one which addresses the above mentioned problems), there are still major obstacles in understanding their decay properties. Indeed, when one considers the simplest, relevant, linear equation on a fixed Kerr background, i.e., the wave equation g ψ = 0 (often referred to as the poor man’s linearization of EVE), one encounters serious difficulties even to prove the boundedness of solutions for the most reasonable, smooth, compactly supported data. Below is a very short description of these. • The problem of trapped null geodesics. This concerns the existence of null geodesics17 neither crossing the event horizon nor escaping to null infinity, along which solutions can concentrate for arbitrary long times. This leads to degenerate energy estimates which require a very delicate analysis. • The trapping properties of the horizon. The horizon itself is ruled by null geodesics, which do not communicate with null infinity and can thus concentrate energy. This problem was solved by understanding the so-called redshift effect associated to the event horizon, which more than counteracts this type of trapping. • The problem of superradiance. This is essentially the failure of the stationary Killing field T = ∂t to be everywhere timelike in the domain of outer communications and, thus, the failure of the associated conserved energy to be positive. Note that this problem is absent in Schwarzschild and, in general, 14 This is responsible for the fact that a small perturbation of the fixed stationary solution φ 0 may not converge to φ0 but to another nearby stationary solution. In the particular case of the stability of Kerr we have a two parameter family of solutions K(a, m). 15 In the case of EVE, any diffeomorphism has that property. 16 With respect to the so-called principal null directions. 17 In the Schwarzschild case, these geodesics are located on the so-called photon sphere r = 3m.

15

INTRODUCTION

for axially symmetric solutions. • Superposition problem. This is the problem of combining the estimates in the near region, close to the horizon (including the ergoregion and trapping), with estimates in the asymptotic region, where the spacetime looks Minkowskian. 4. The full linearized system of EVE around Kerr, usually referred to as the linearized gravity system (LGS), whatever its formulation, presents far more difficulties beyond those mentioned above concerning the poor man’s linear scalar wave equation on Kerr, see the discussion below. Historically, two versions of LGS have been considered. a) At the level of the metric itself, i.e., if G denotes the Einstein tensor, Gαβ = Rαβ − 12 Rgαβ , G0 (g0 ) δg = 0.

(1.2.1)

b) Via the Newman-Penrose (NP) formalism, based on null frames. In what follows we review the main known results concerning solutions to the linearized equations on a Kerr background. 1.2.1

Formal mode analysis

The first important results concerning both items 3 and 4 above were obtained by physicists based on the classical method of separation of variables and formal mode analysis. In the particular case where g0 is the Schwarzschild metric, the linearized equations (1.2.1) can be formally decomposed into modes, by associating t-derivatives with multiplication by iω and angular derivatives with multiplication by l, i.e., the eigenvalues of the spherical Laplacian. A similar decomposition, using oblate spheroidal harmonics, can be done in Kerr. The formal study of fixed modes from the point of view of metric perturbations as in (1.2.1) was initiated by ReggeWheeler [58] who discovered the master Regge-Wheeler equation for odd-parity perturbations. This study was completed by Vishveshwara [65] and Zerilli [69]. A gauge-invariant formulation of metric perturbations was then given by Moncrief [56]. An alternative approach via the Newman-Penrose (NP) formalism was first undertaken by Bardeen-Press [6]. This latter type of analysis was later extended to the Kerr family by Teukolsky [64] who made the important discovery that the extreme curvature components, relative to a principal null frame, satisfy decoupled, separable, wave equations. These extreme curvature components also turn out to be gauge invariant in the sense that small perturbations of the frame lead to quadratic errors in their expression. The full extent of what could be done by mode analysis, in both approaches, can be found in Chandrasekhar’s book [11]. Chandrasekhar also introduced (see [12]) a transformation theory relating the two approaches. More precisely, he exhibits a transformation which connects the Teukolsky equations to the Regge-Wheeler one. This transformation was further elucidated and extended by R. Wald [67] and recently by Aksteiner et al. [1]. The full mode stability, i.e., lack of exponentially growing modes, for the Teukolsky equation on Kerr is due to Whiting [68] (see also [61] for a stronger quantitive version).

16

CHAPTER 1

1.2.2

Vectorfield method

Note that mode stability is far from establishing even boundedness of solutions to the linearized equations. To achieve that and, in addition, to derive realistic decay estimates one needs an entirely different approach based on a far-reaching extension of the classical vectorfield method18 used in the proof of the nonlinear stability of Minkowski [20]. The new vectorfield method compensates for the lack of enough Killing and conformal Killing vectorfields on a Schwarzschild or Kerr background by introducing new vectorfields whose deformation tensors have coercive properties in different regions of spacetime, not necessarily causal. The new method has emerged in the last fifteen years in connection to the study of boundedness and decay for the scalar wave equation in the Kerr space K(a, m), ga,m ψ = 0.

(1.2.2)

The starting and most demanding part of the new method is the derivation of a global, simultaneous, Energy-Morawetz estimate which degenerates in the trapping region. This task is somewhat easier in Schwarzschild, or for axially symmetric solutions in Kerr, where the trapping region is restricted to a smooth hypersurface. The first such estimates, in Schwarzschild, were proved by Blue and Soffer in [8], [9], followed by a long sequence of further improvements in [10], [22], [54], etc. See also [36] and [62] for a vectorfield method treatment of the axially symmetric case in Kerr with applications to nonlinear equations. In the absence of axial symmetry the derivation of an Energy-Morawetz estimate in Kerr(a, m), |a/m|  1 requires a more refined analysis involving either Fourier decompositions, see [24], [63], or a systematic use of the second order Carter operator, see [3]. The derivation of such an estimate in the full sub-extremal case |a| < m is even more subtle and was recently achieved by Dafermos, Rodnianski and Shlapentokh-Rothman [28] by combining mode decomposition with the vectorfield method. Once an Energy-Morawetz estimate is established one can commute with the time translation vectorfield and the so-called redshift vectorfield,19 first introduced in [22], to derive uniform bounds for solutions. The most efficient way to also get decay, and solve the superposition problem, is due to Dafermos and Rodnianski, see [23], based on the presence of a family of rp -weighted, quasi-conformal vectorfields defined in the far r region of spacetime.20 18 Method

based on the symmetries of Minkowski space to derive uniform, robust decay for nonlinear wave equations, see [39], [40], [41], [19]. 19 Note that the redshift vectorfield is also used as a multiplier in the derivation of the EnergyMorawetz estimate. 20 These replace the scaling and inverted time translation vectorfields used in [39] or their corresponding deformations used in [20]. A recent improvement of the method, relevant to our work here, allowing one to derive higher order decay can be found in [5]. See also [57] for further extensions of this method.

INTRODUCTION

1.3

1.3.1

17

NONLINEAR STABILITY OF SCHWARZSCHILD UNDER POLARIZED PERTURBATIONS Bare-bones version of our theorem

The goal of the book is to prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, i.e., solutions of the Einstein vacuum equations (1.1.3) for asymptotically flat 1 + 3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield Z with closed orbits. This class of perturbations allows us to restrict our analysis to the case when the final state of evolution is itself a Schwarzschild spacetime. This is not the case in general, as a typical perturbation of Schwarzschild may approach a member of the Kerr family with small angular momentum. The simplest version of our main theorem can be stated as follows. Theorem 1.6 (Main Theorem (first version)). The future globally hyperbolic development of an axially symmetric, polarized,21 asymptotically flat initial data set, sufficiently close (in a specified topology) to a Schwarzschild initial data set of mass m0 > 0, has a complete future null infinity I + and converges in its causal past J −1 (I + ) to another nearby Schwarzschild solution of mass m∞ close to m0 . Our theorem is an important step in the long-standing effort to prove the full nonlinear stability of Kerr spacetimes K(a, m), in the sub-extremal regime |a| < m. We give a succinct review below of some of the most important results which have been obtained so far in this direction. 1.3.2

Linear stability of the Schwarzschild spacetime

A first quantitative (i.e., which provides precise decay estimates) proof of the linear stability of Schwarzschild spacetime has recently been established22 by Dafermos, Holzegel and Rodnianksi in [26], via the NP formalism (expressed in a double null foliation23 ). It is important to note that while the Teukolsky equation (in the NP formalism) is separable, and thus amenable to mode analysis, it is not Lagrangian and thus cannot be treated by direct energy type estimates. To overcome this difficulty [26] relies on a new physical space version of the Chandrasekhar transformation [12], which takes solutions of the Teukolsky equations to solutions of Regge-Wheeler, which is manifestly both Lagrangian and coercive. After quantitative decay has been established for this latter equation, based on the new vectorfield method, the physical space form of the transformation allows one to derive quantitative decay for solutions of the original Teukolsky equation. Once decay estimates for the Teukolsky equation have been established, the remaining work in [26] is to bound all other curvature and Ricci coefficients associated to the double null foliation. This last step requires carefully chosen gauge conditions along the 21 See section 2.1.1 for a precise definition of axial symmetry and polarization. This property is preserved by the Einstein equations, i.e., if the data is axially symmetric, polarized, so is its development. 22 A somewhat weaker version of linear stability of Schwarzschild was subsequently proved in [34] by using the original, direct, Regge-Wheeler, Zerilli approach combined with the vectorfield method and adapted gauge choices. See also [37] for an alternate proof of linear stability of Schwarzschild using wave coordinates. 23 This is possible in Schwarzschild where the principal null directions are integrable.

18

CHAPTER 1

event horizon of the fixed Schwarzschild background. This final gauge is itself then quantitatively bounded in terms of the initial data, thus giving a comprehensive statement of linear stability. 1.3.3

Main ideas in the proof of Theorem 1.6

In the passage from linear to nonlinear stability of Schwarzschild one has to overcome major new difficulties. Some are similar to those encountered in the stability of Minkowski [20], such as: 1. Need of an appropriate geometric setting which takes into account the decay and peeling properties of the curvature. In [20] this was achieved with the help of the foliation of the perturbed spacetime given by two optical functions (int) u and (ext) u and a maximal time function t. The exterior optical function (ext) u, which was initialized at infinity, was essential to derive the decay and peeling properties along null directions while (int) u, initialized on a timelike axis, was responsible for covering the interior, non-radiative, back scattering decay. 2. The peeling and decay estimates have to be derived by some version of the geometric vectorfield method which relates decay to generalized energy type estimates. 3. The peeling and decay estimates mentioned above should be sufficiently strong to be able to deal with the error terms generated by the vectorfield method. For this to happen, the error terms need to exhibit an appropriate null structure. The new main difficulties are as follows: 1. One needs a procedure which allows one to take into account the change of mass and detect its final value. Note also that we need to restrict the nature of the perturbations to insure that the final state of a perturbation of Schwarzschild is still Schwarzschild. 2. While in the stability of Minkowski space all components of the curvature tensor were expected to approach zero, this is no longer true. Indeed, the middle curvature component (relative to an adapted null frame) ought to converge to its respective value in the final Schwarzschild spacetime. This statement is unfortunately hard to quantify since that value depends both on the final mass and on the corresponding Schwarzschild coordinates. Moreover, some of the other curvature components, which are expected to converge to zero, are also ill defined since a small change of the null frame can produce small linear distortion to the basic equation which these curvature components verify. Note that this difficulty was absent in the stability of the Minkowski space where small changes in the frame produce only quadratic errors. 3. The classical vectorfield method used in the nonlinear stability of Minkowski space was based on the construction, together with the spacetime, of an adequate family of approximate Killing and conformal Killing vectorfields which mimic the role played by the corresponding vectorfields in Minkowski space in establishing uniform decay estimates. The Schwarzschild space, however, has a much more limited set of Killing vectorfields and no useful conformal Killing ones. As mentioned above, this problem appears already in the analysis of the standard scalar linear wave equation in Schwarzschild. 4. As in the stability of the Minkowski space, one needs to make gauge conditions to insure that we are measuring decay relative to an appropriate center of mass

INTRODUCTION

19

frame. Yet, as we saw above, it is no longer true that small perturbations of the null frame produce only quadratic errors for the curvature, as was the case in the stability of Minkowski space. In fact, the center of mass frame of the perturbed black hole continuously changes in response to incoming radiation. This, the so-called recoil problem, does not occur in linear theory. Here is a very short summary of how we solve these new challenges in our work. 1. We resolve the first difficulty by restricting our analysis to axially symmetric, polarized perturbations and by tracking the mass using a quantity, called the quasi-local Hawking mass, for which we derive simple propagation equations which establish monotonicity of the mass up to errors which are quadratic with respect to the perturbations. 2. We resolve the second difficulty by making use of the fact that the extreme components of the curvature are, up to quadratic terms, invariant under null frame transformations. As in [26], we also make use of a transformation, similar to that of Chandrasekhar mentioned above, which maps the extreme components of the curvature to a new quantity q, defined up to quadratic errors, that verifies a Regge-Wheeler type equation. Once we manage to control q, i.e., to derive quantitative decay estimates for it, we can also control, in principle,24 the two extreme curvature invariants α and α, the first by inverting the Chandrasekhar transformation and the second by using a variant of the Teukolsky-Starobinsky identities. One is then left with the arduous task of recovering25 all other null components of the curvature tensor and all connection coefficients. 3. The third difficulty manifests itself in the most sensitive part of the entire argument, i.e., in the task of deriving quantitative decay estimates for q by making use of the Regge-Wheeler type equation it verifies. To do this we rely on the new vectorfield method as outlined in section 1.2.2 above. The main new difficulties are: a) The vectorfield method introduces new error terms, not present in linear theory. To estimate these terms we need precise decay information, off the final Schwarzschild space, for all connection coefficients and curvature of the perturbation. b) The most difficult terms are those due to the quadratic errors made in the derivation of the Regge-Wheeler equation for q. As in the proof of the stability of the Minkowski space, the precise rates of decay for various curvature and connection coefficients, i.e., the peeling properties of the perturbation, and the precise structure of these error terms is of fundamental importance. 4. We solve the fourth and most important new difficulty by a procedure we call General Covariant Modulation (GCM). This procedure, which takes advantage of the full covariance of the Einstein equations, allows us to construct the perturbed spacetime by a continuity argument involving finite GCM admissible spacetimes M as represented in figure 1.10. The past boundaries C 1 ∪ C1 are incoming and outgoing null hypersurfaces on which the initial perturbation is prescribed. The future boundaries consist of the union A ∪ C ∗ ∪ C∗ ∪ Σ∗ where A and Σ∗ are spacelike, C ∗ is incoming null, C∗ outgoing null. The boundary A is chosen 24 Provided 25 In

that one can deal with the nonlinear terms. the linear setting this was partially achieved in [25].

HH

(int)

1

Figure 1.10: The GCM admissible spacetime M

so that, in the limit when M converges to the final state, it is included in the perturbed black hole. The spacelike boundary Σ∗ plays a fundamental role in our construction as seen below. The spacetime M also contains a timelike hypersurface T which divides M into an exterior region we call (ext) M and an interior one (int) M. We say that M is a GCM admissible spacetime if it verifies the following properties. a) The far region (ext) M is foliated by a geodesic foliation induced by an outgoing optical function u initialized on Σ∗ b) The near region (int) M is foliated by a geodesic foliation induced by an incoming optical function u initialized at T such that its level sets on T coincide with those of u. c) The foliation induced on Σ∗ is such that specific geometric quantities take Schwarzschildian values. We refer to these as GCM conditions. These conditions are dynamically reset in the continuation process on which our proof is based. d) The area radius r(u) of the spheres of constant u along Σ∗ is far greater than the corresponding value of u. This condition allows us to simplify somewhat the null structure and Bianchi equations induced on Σ∗ and corresponds to the expectation that the spacelike hypersurfaces Σ∗ converges to the null infinity of the final state of the perturbation. 5. The GCM conditions together with the control derived on q, α and α mentioned earlier allow us to control all null connection and curvature coefficients along on Σ∗ , i.e., to derive appropriated decay estimates for them. These estimates can then be transported to (ext) M using the full scope of the null structure and null Bianchi identities associated to the outgoing geodesic foliation.

T

T

T

(i

nt ( ) in

Mt) M

) MM

+

xt(e ) xt

t) M

⌃⌃

T

⇤

CC 1

1

C1 C

M

AA T

M

I+ I

⇤ ⇤

CC

M

M

⇤

++

⇤

C⇤C

(ext)

(e

(i n

(int)

t)

T

t) (e x M

⌃⇤

M

(ext)

M

T I+

C1

I+

I+

+

(i n

(e x t)

⌃A⇤ ⇤

⇤

H A

CC (int) 1⇤ M

I+

⌃⇤

C1

C1

I+C 1

Cxt) 1 M

(ext) C M 1⇤

C1

C

A

A

⌃⇤

C⇤

C1

C⇤

C1

C1

C⇤

+

H

⌃⇤

H+

C1

C⇤

C1

C⇤

C1

M

M

M

A

CHAPTER 1

M

)

A

)

T

nt

A

nt

(i

xt )

C⇤

(i

+

(e

C⇤

I+

C1

⌃nt) ⇤M

C⇤

T

TI

C⇤

⌃⇤

A

(i

C⇤

C1

C⇤

(e

C⇤

) xt

C⇤

H+

I+

H+

H+

(e

20

H+

INTRODUCTION

21

6. The decay estimates in (ext) M can then be used as initial conditions along the timelike hypersurface T for the incoming foliation of (int) M. These allow us to also derive appropriate decay estimates for all null connection and curvature coefficients of the foliation induced by u. 7. The precise decay estimates derived in 5 are sufficiently strong to allow us to control all error terms generated in the process of estimating q, as mentioned in 3. Note that in figure 1.10, one starts with initial conditions on the union of null hypersurfaces C1 ∪ C 1 rather than an initial spacelike hypersurface Σ(0) . One can justify this simplification based on the results of [42], [43], see Remark 3.12. The full red line H+ represents the future event horizon of the perturbed Schwarzschild. The line T represents the timelike hypersurface separating (int) M from (ext) M. In deriving decay estimates the precise choice of T is irrelevant. A choice, however, needs to be made in order to avoid a derivative loss for our top energy estimates.26 The spacetime is constructed by a continuity argument, i.e., we assume that the spacetime terminating at C∗ ∪ C ∗ saturates a given bootstrap assumption (BA) and show, by a long sequence of a priori estimates which take advantage of the smallness of the initial perturbation, that (BA) can be improved and the spacetime extended past C∗ ∪ C ∗ ∪ Σ∗ . Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations, i.e., perturbations for which new ideas, such as our GCM procedure are needed. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space. We are thus confident that our procedure may apply in a more general setting. We would like to single out two other recent important contributions to nonlinear stability of black holes. In the context of asymptotically flat Einstein vacuum equations the result of Dafermos-Holzegel-Rodnianski [25] constructs a class of Kerr black hole solutions starting from future infinity while Hintz-Vasy [31]27 prove the nonlinear stability of Kerr-de Sitter, for small angular momentum, in the context of the Einstein vacuum equations with a nontrivial positive cosmological constant. Though the two results are very different they share in common the fact that the perturbations they treat decay exponentially. This makes the analysis significantly easier than in our case when the decay is barely enough to control the nonlinear terms. 1.3.4

Beyond polarization

While we believe that the general strategy outlined in this work can be extended to the general case of perturbations of Kerr, there are a few major conceptual roadblocks, all connected to our symmetry assumption, which have to be overcome.28 We describe the main effects played by polarization in our work as follows: 1. Polarization plays a crucial role in our nonlinear version of the Chandrasekhar transformation allowing us to pass from the Teukolsky equation for α to the 26 See

[20] for a similar situation. also [32] for the stability of Kerr-Newman de Sitter. 28 Note nevertheless that many steps in the proof of Theorem 1.6 do not depend at all on polarization. 27 See

22

CHAPTER 1

Regge-Wheeler equation for q mentioned above in section 1.3.3. 2. Polarization is also used in the derivation of the Morawetz type estimates for q in Chapter 10, where we take advantage of the simpler nature of the trapped set for Schwarzschild metrics. 3. The GCM construction, which plays a fundamental role in this book, also makes essential use of polarization. There are fundamental conceptual difficulties to pass from the polarized case to the general case which will require new ideas. 4. A full stability result has to identify not only the final value of the mass m, but also the final value of the angular momentum. In our work, the final value of m is tracked with the help of the Hawking mass. Though there exists in the literature several interesting proposals29 for a quasi-local definition of angular-momentum, in the spirit of the Hawking mass, none of them seem suitable to the dynamical approach developed here. 1.3.5

Note added in proof

We would like to report on additional progress made on the Kerr stability problem since we submitted our work, and dealing with the issues raised in section 1.3.4.30 • A notable development is the extension of the Chandrasekhar transformation to the case of linear Spin-2 equations in Kerr, as well as a methodology to derive, based on it, boundedness and decay estimates for the Teukolsky variable α. This was done independently by Ma [50] and Dafermos-Holzegel-Rodnianski [27]. In collaboration with E. Giorgi, we have extended, see [29], the derivation of the generalized Regge-Wheeler equation to realistic perturbations of Kerr, i.e., perturbations consistent with the decay and boundedness properties established in this work. Our derivation is based on a new geometric formalism which takes into account the lack of integrability of the principal null frames of Kerr. • The most important advance, achieved in [47], [48], concerns the removal of any symmetry assumption in the construction of GCM spheres. • The construction of intrinsic GCM spheres in [48] also leads to a canonical definition of the angular momentum on such spheres.

1.4

ORGANIZATION

The book is organized as follows. In Chapter 2 we introduce the main quantities, equations and basic tools needed later. It is our main reference kit providing all main null structure and null Bianchi equations, in general null frames, in the context of axially symmetric polarized spacetimes. Though we work with the reduced equations, i.e., the equations reduced by the symmetries, most of the work in the book does not really depend of the reduction. Besides insuring that the final state is a Schwarzschild space the reduction only plays a significant role in the GCM construction. Chapter 3, the heart of the book, contains the precise version of our main theorem, its main conclusions as well as a full strategy of its proof, divided in nine 29 See,

for example, [59] or [16]. not directly connected to our strategy, we would like also to refererence the work of Andersson-B¨ ackdahl-Blue-Ma [4] and H¨ afner-Hintz-Vasy [30] on the linear stability of Kerr. 30 Though

INTRODUCTION

23

supporting intermediate results, Theorems M0–M8. We also give a short description of the proof of each theorem. In the other chapters of this book we give complete proofs of Theorems M0–M8 and a full description of our GCM procedure. The reader versed in the formalism of null structure and Bianchi equations, as discussed in [20], is encouraged to glance fast over Chapter 2, to get familiarized with the notation, and then move directly to Chapter 3.

Chapter Two Preliminaries 2.1

AXIALLY SYMMETRIC POLARIZED SPACETIMES

2.1.1

Axial symmetry

We consider vacuum, four dimensional, simply connected, axially symmetric spacetimes (M, g, Z) with g Lorentzian and Z an axial Killing vectorfield on M. We denote by A the axis of symmetry, i.e., the points on M for which X := g(Z, Z) = 0. In the case of interest for us we assume dX 6= 0 and that A is a smooth manifold of codimension 2. The Ernst potential of the spacetime is given by σµ

:= Dµ (−Zα Zα ) − i ∈µβγδ Zβ Dγ Zδ .

The 1-form σµ dxµ is closed and thus there exists a function σ : M → C, called the Z-Ernst potential, such that σµ = Dµ σ. Note also that Dµ g(Z, Z) = 2Gµλ Zλ = − 0) is given by e3 = L = Υ−1 ∂t − ∂r ,

e4 = ΥL = ∂t + Υ∂r ,

Υ=1−

All Ricci coefficients vanish except χ=

Υ , r

1 χ=− , r

ω=−

m , r2

ω = 0.

2m . (2.1.51) r

51

PRELIMINARIES

2. The null frame (e3 , e4 ) for which e4 is geodesic is given by e4 = L = Υ−1 ∂t + ∂r ,

e3 = ΥL = ∂t − Υ∂r .

All Ricci coefficients vanish except χ=

1 , r

χ=−

Υ , r

ω = 0,

ω=

m . r2

Note that the null pair (2.1.51) is regular along the future event horizon as can be easily seen by studying the behavior15 of future directed ingoing null geodesics near r = 2m.

2.2

MAIN EQUATIONS

In this section we translate the null structure and null Bianchi identities associated to an S-foliation in the reduced picture. We start with general, Z-invariant, Sfoliation. We then consider the special case of geodesic foliations. 2.2.1

Main equations for general S-foliations

We consider a fixed Z-invariant S-foliation with a fixed Z-invariant null frame e3 , e4 . 2.2.1.1

Null structure equations

We simply translate the well known spacetime null structure equations (see16 proposition 7.4.1 in [20]) in the reduced picture. Thus the spacetime equation17 ∇ / 3χ b + trχ χ b =

b − 2ωb b −α ∇ / ⊗ξ χ + (η + η − 2ζ)⊗ξ

becomes18 e3 (ϑ) + κ ϑ = 2(eθ (ξ) − eθ (Φ)ξ) − 2ω ϑ + 2(η + η − 2ζ) ξ − 2α.

(2.2.1)

The spacetime equation 1 e3 (trχ) + trχ2 = 2div / ξ − 2ωtrχ + 2ξ · (η + η − 2ζ) − χ b·χ b 2 becomes 1 e3 (κ) + κ2 + 2ω κ = 2

1 2(eθ ξ + eθ (Φ)ξ) + 2(η + η − 2ζ)ξ − ϑ ϑ. (2.2.2) 2

15 That is, the null geodesics in the direction of L reach the horizon in finite proper time. Note that, on the other hand, the past null geodesics in the direction of L still meet the horizon in infinite proper time. 16 Note however that the notations in [20] are different, see section 7.3 for the definitions. 17 For convenience we drop the (1+3) labels in what follows. 18 Recall that (1+3)χ bθθ = 12 ϑ.

52

CHAPTER 2

The spacetime equation 1 ∇ / 4χ b + trχ χ b = 2

1 b + 2ωb b + η ⊗η b ∇ / ⊗η χ − trχb χ + ξ ⊗ξ 2

becomes 1 e4 ϑ + κ ϑ − 2ωϑ = 2

1 2(eθ η − eθ (Φ)η) − trχ ϑ + 2(ξ ξ + η 2 ). 2

The spacetime equation 1 ∇ / 4 trχ + trχ trχ = 2div / η + 2ρ + 2ω trχ − χ b·χ b + 2(ξ · ξ + η · η) 2 becomes 1 e4 (κ) + κ κ − 2ωκ = 2

1 2(eθ η + eθ (Φ)η) + 2ρ − ϑ ϑ + 2(ξ ξ + η η). 2

The spacetime equation 1 1 = −β − 2∇ /ω − χ b · (ζ + η) − trχ(ζ + η) + 2ω(ζ − η) + (b χ + trχ)ξ + 2ωξ 2 2

∇ / 3ζ

becomes (note that

(1+3)

β = −β !)

1 e3 ζ + κ(ζ + η) − 2ω(ζ − η) 2

=

1 1 1 β − 2eθ (ω) + 2ωξ + κ ξ − ϑ(ζ + η) + ϑ ξ. 2 2 2

The spacetime equation 1 ∇ / 4ξ − ∇ / 3 η = −β + 4ωξ + χ b · (η − η) + trχ(η − η) 2 becomes19 1 1 e4 (ξ) − e3 (η) = β + 4ωξ + κ(η − η) + ϑ(η − η). 2 2 The spacetime equation ∇ / 4ω + ∇ / 3ω

=

ρ + 4ωω + ξ · ξ + ζ · (η − η) − η · η

becomes e4 ω + e3 ω

= ρ + 4ωω + ξ ξ + ζ(η − η) − η η.

The spacetime Codazzi equation (1+3)

div / (1+3)χ b

19 Note

that

=

(1+3)β

1 β + ( (1+3)∇ / (1+3)trχ − 2

(1+3)

= −β and / d?1 f = −eθ (f ).

(1+3)

trχ (1+3)ζ) +

(1+3)

χ b·

(1+3)

ζ

53

PRELIMINARIES

becomes20 1 (eθ (ϑ) + 2eθ (Φ)ϑ) 2

=

1 1 −β + (eθ (κ) − κζ) + ϑζ. 2 2

The Gauss equation K

= −

1 (1+3) 1 trχ (1+3)trχ + (1+3)χ b (1+3)χ b− 4 2

(1+3)

ρ

becomes K

1 1 = − κκ + ϑϑ − ρ. 4 4

We summarize the results in the following proposition. Proposition 2.56. e3 (ϑ) + κ ϑ + 2ω ϑ = −2α − 2 /d?2 ξ + 2(η + η − 2ζ) ξ, 1 1 e3 (κ) + κ2 + 2ω κ = 2 /d1 ξ + 2(η + η − 2ζ)ξ − ϑ2 , 2 2 1 1 ? e4 ϑ + κ ϑ − 2ωϑ = −2 /d2 η − κ ϑ + 2(ξ ξ + η 2 ), 2 2 1 1 e4 (κ) + κ κ − 2ωκ = 2 /d1 η + 2ρ − ϑ ϑ + 2(ξ ξ + η η), 2 2 1 1 ? e3 ζ + κ(ζ + η) − 2ω(ζ − η) = β + 2 /d1 ω + 2ωξ + κ ξ 2 2 1 1 − ϑ(ζ + η) + ϑ ξ, 2 2 1 1 e4 (ξ) − 4ωξ − e3 (η) = β + κ(η − η) + ϑ(η − η), 2 2 e4 ω + e3 ω = ρ + 4ωω + ξ ξ + ζ(η − η) − η η.

20 Note

that

(1+3)β

θ

= −β,

(1+3)χ b

=

1 ϑ. 2

(2.2.3)

54

CHAPTER 2

In view of the symmetry e3 − e4 , we also derive e4 (ϑ) + κ ϑ + 2ωϑ = −2α − 2 /d?2 ξ + 2(η + η + 2ζ)ξ, 1 1 e4 (κ) + κ2 + 2ω κ = 2 /d1 ξ + 2(η + η + 2ζ)ξ − ϑ2 , 2 2 1 1 ? e3 ϑ + κ ϑ − 2ωϑ = −2 /d2 η − κ ϑ + 2(ξ ξ + η 2 ), 2 2 1 1 e3 (κ) + κ κ − 2ωκ = 2 /d1 η + 2ρ − ϑ ϑ + 2(ξ ξ + η η), 2 2 (2.2.4) 1 1 ? −e4 ζ + κ(−ζ + η) + 2ω(ζ + η) = β + 2 /d1 ω + 2ωξ + κ ξ 2 2 1 1 − ϑ(−ζ + η) + ϑ ξ, 2 2 1 1 e3 (ξ) − e4 (η) = β + 4ωξ + κ(η − η) + ϑ(η − η), 2 2 e4 ω + e3 ω = ρ + 4ωω + ξ ξ + ζ(η − η) − η η. We also have the Codazzi equations /d2 ϑ /d2 ϑ

= −2β − /d?1 κ − ζκ + ϑ ζ,

= −2β − /d?1 κ + ζκ − ϑ ζ,

and the Gauss equation 1 1 K = −ρ − κ κ + ϑ ϑ. 4 4 2.2.2

Null Bianchi identities

We now translate the spacetime null Bianchi identities of [20] (see proposition 7.3.2) in the reduced picture. The spacetime equation (note that D /2?β := − 12 (1+3)∇ / ⊗ β) 1 ∇ / 3 α + trχ α 2

= −2 D /2? β + 4ωα − 3(b χρ + ? χ b ?ρ) + (ζ + 4η) ⊗ β

becomes (note that ?ρ = 0) 1 e3 α + κα 2

3 (eθ (β) − (eθ Φ)β) + 4ωα − ϑρ + (ζ + 4η)β. 2

=

(2.2.5)

The spacetime equation ∇ / 4 β + 2trχβ

=

div / α − 2ωβ + (2ζ + η) · α + 3(ξρ + ?ξ ?ρ)

becomes e4 β + 2κβ

=

(eθ α + 2(eθ Φ)α) − 2ωβ + (2ζ + η)α + 3ξρ.

The spacetime equation ∇ / 3 β + trχβ

=

D /1?(−ρ, ?ρ) + 2b χ · β + 2ω β + ξ · α + 3(ηρ + ?η ?ρ)

(2.2.6)

55

PRELIMINARIES

becomes (recall

(1+3)

β θ = −β)

e3 β + κβ

=

eθ (ρ) + 2ωβ + 3ηρ − ϑβ + ξα.

(2.2.7)

The spacetime equation 3 e4 ρ + trχρ 2

1 div / β− χ b · α + ζ · β + 2(η · β − ξ · β) 2

=

becomes 3 e4 ρ + κρ = 2

1 (eθ (β) + (eθ Φ)β) − ϑ α + ζ β + 2(η β + ξ β). 2

(2.2.8)

Indeed note that (1+3)

χ b·

(1+3)

α = 2 (1+3)χ b θθ (1+3)αθθ = ϑα.

All other equations in the proposition below are derived using the e3 − e4 symmetry. We summarize the results in the following proposition. Proposition 2.57. 1 3 e3 α + κα = − /d?2 β + 4ωα − ϑρ + (ζ + 4η)β, 2 2 e4 β + 2κβ = /d2 α − 2ωβ + (2ζ + η)α + 3ξρ, e3 β + κβ = − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα, 3 1 e4 ρ + κρ = /d1 β − ϑ α + ζ β + 2(η β + ξ β), 2 2

(2.2.9) 3 1 e3 ρ + κρ = /d1 β − ϑ α − ζ β + 2(η β + ξ β), 2 2 e4 β + κβ = − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα,

e3 β + 2κ β = /d2 α − 2ω β + (−2ζ + η)α + 3ξρ, 1 3 e4 α + κ α = − /d?2 β + 4ωα − ϑρ + (−ζ + 4η)β. 2 2 2.2.2.1

Mass aspect functions

We define the mass aspect functions 1 µ : = − /d1 ζ − ρ + ϑϑ, 4 1 µ : = /d1 ζ − ρ + ϑϑ. 4

(2.2.10)

One can derive useful propagation equations, in the e4 direction for µ and in the e3 direction for µ, by using the null structure and null Bianchi equations, see [20] and [42]. In the next section we will do this in the context of null geodesic foliations.

56

CHAPTER 2

2.2.3

Hawking mass

Definition 2.58. The Hawking mass m = m(S) of S is defined by the formula Z 2m 1 =1+ κκ. (2.2.11) r 16π S Proposition 2.59. The following identities hold true. 1. The average of ρ is given by the formulas ρ

= −

2m 1 + r3 16πr2

Z ϑϑ.

(2.2.12)

S

2. The average of the mass aspect function is 2m . r3

(2.2.13)

4Υ −κ ˇκ ˇ r2

(2.2.14)

µ = µ= 3. The average of κ and κ are related by κκ = − where Υ = 1 −

2m r .

Proof. We have from the Gauss equation K

1 1 = − κκ + ϑϑ − ρ. 4 4

Integrating on S and using the Gauss Bonnet formula, we infer Z Z Z 1 1 4π = − κκ + ϑϑ − ρ. 4 S 4 S S Together with the definition of the Hawking mass, we infer   Z Z Z 1 1 ρ = −4π 1 + κκ + ϑϑ 16π S 4 S S Z 8πm 1 = − + ϑϑ r 4 S and hence ρ =

−

2m 1 + r3 16πr2

Z ϑϑ S

which proves our first identity. The second identity follows easily from the definition of µ, µ and the first formula. Thus, for example, µ=

1 |S|

Z µ = S

1 |S|

 Z  Z 1 1 2m − /d1 ζ − ρ + ϑϑ = −ρ + ϑϑ = 3 . 4 4|S| r S S

To prove the last identity we remark that, in view of the definition of the Hawking

57

PRELIMINARIES

mass, −Υ =

2m −1 r

=

1 16π

Z κκ = S



1 16π

|S|κ κ +

Z

 κ ˇκ ˇ

S

and hence 16πΥ 1 − |S| |S| 4Υ = − 2 −κ ˇκ ˇ. r

κκ = −

Z κ ˇκ ˇ S

This concludes the proof of the proposition. 2.2.4

Outgoing geodesic foliations

We restrict our attention to geodesic foliations, i.e., geodesic foliations by Z-invariant optical functions. 2.2.4.1

Basic definitions

Assume given an outgoing optical function u, i.e., Z-invariant solution of the equation gαβ ∂α u∂β u = g ab ∂a u∂b u = 0 and L = −g ab ∂b u∂a its null geodesic generator. We choose e4 such that e4 = ςL,

L(ς) = 0.

(2.2.15)

Remark 2.60. In our definition of a GCM admissible spacetime, see section 3.1, we initialize ς on the spacelike hypersurface Σ∗ . We then choose s such that e4 (s) = 1.

(2.2.16)

The functions u, s generate what is called an outgoing geodesic foliation. Let Su,s be the 2-surfaces of intersection between the level surfaces of u and s. We choose e3 the unique Z-invariant null vectorfield orthogonal to Su,s and such that g(e3 , e4 ) = −2. We then let eθ be unit tangent to Su,s , Z-invariant and orthogonal to Z. We also introduce Ω := e3 (s).

(2.2.17)

Lemma 2.61. We have ω = ξ = 0,

η = −ζ,

(2.2.18)

58

CHAPTER 2

2 , e3 (u) e4 (ς) = 0, ς=

(2.2.19)

eθ (log ς) = η − ζ,

eθ (Ω) = −ξ − (η − ζ)Ω, e4 (Ω) = −2ω.

Proof. Recall that L is geodesic, e4 = ςL and L(ς) = 0. This immediately implies that e4 is geodesic, and hence we have ω = ξ = 0. Applying the vectorfield [e4 , eθ ] = (η + ζ)e4 + ξe3 − χeθ to s, and since e4 (s) = 1 and eθ (s) = 0, we derive η + ζ = 0. Next, note that e3 (u) = g(e3 , −L) = −ς −1 g(e3 , e4 ) =

2 ς

and hence ς

=

2 . e3 (u)

Applying the vectorfield [e3 , eθ ] = ξe4 + (η − ζ)e3 − χeθ to u and making use of the relation e4 (u) = eθ (u) = 0 we deduce (η − ζ)e3 (u) = e3 (eθ u) − eθ e3 (u) = −eθ e3 (u) which together with the identity ς = 2/e3 (u) yields   2 η − ζ = −eθ log(e3 u) = −eθ log = eθ (log ς) ς and hence eθ (log ς)

= η − ζ.

Applying the vectorfield [e3 , eθ ] = ξe4 + (η − ζ)e3 − χeθ

59

PRELIMINARIES

to s we deduce, since e4 (s) = 1, eθ (s) = 0 and e3 (s) = Ω, eθ (Ω)

= −ξ − (η − ζ)Ω.

Finally, applying [e4 , e3 ] = −2ωe4 − 2(η − η)eθ + 2ωe3 to s, and using e4 (s) = 1 and eθ (s) = 0, we infer e4 (e3 (s)) = −2ω, i.e., e4 (Ω) = −2ω as desired. Remark 2.62. In the particular case when ς is constant we have η = ζ = −η. In Schwarzschild, relative to the standard outgoing geodesic frame, we have   2m ς = 1, Ω = −Υ = − 1 − . r 2.2.4.2

Basic equations

Proposition 2.63. Relative to an outgoing geodesic foliation we have: 1. The reduced null structure equations take the form e4 (ϑ) + κ ϑ = −2α, 1 1 e4 (κ) + κ2 = − ϑ2 , 2 2 e4 ζ + κζ = −β − ϑζ, 1 1 e4 (η − ζ) + κ(η − ζ) = − ϑ(η − ζ), 2 2 1 1 e4 ϑ + κ ϑ = 2 /d?2 ζ − κ ϑ + 2ζ 2 , 2 2 1 1 e4 (κ) + κ κ = −2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 , 2 2 e4 ω = ρ + ζ(2η + ζ), 1 1 e4 (ξ) = −e3 (ζ) + β − κ(ζ + η) − ϑ(ζ + η), 2 2 and e3 (ϑ) + κ ϑ + 2ω ϑ = −2α − 2 /d?2 ξ + 2(η − 3ζ) ξ, 1 1 e3 (κ) + κ2 + 2ω κ = 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 , 2 2 1 1 1 1 ? e3 ζ + κ(ζ + η) − 2ω(ζ − η) = β + 2 /d1 ω + κ ξ − ϑ(ζ + η) + ϑ ξ, 2 2 2 2 1 1 ? 2 e3 ϑ + κ ϑ − 2ωϑ = −2 /d2 η − κ ϑ + 2η , 2 2 1 1 e3 (κ) + κ κ − 2ωκ = 2 /d1 η + 2ρ − ϑ ϑ + 2η 2 , 2 2

60

CHAPTER 2

and /d2 ϑ /d2 ϑ K

= −2β − /d?1 κ − ζκ + ϑ ζ, = −2β − /d?1 κ + ζκ − ϑ ζ, 1 1 = −ρ − κ κ + ϑ ϑ. 4 4

2. The null Bianchi identities are given in this case by 1 3 e3 α + κα = − /d?2 β + 4ωα − ϑρ + (ζ + 4η)β, 2 2 e4 β + 2κβ = /d2 α + ζα, e3 β + κβ = − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα, 3 1 e4 ρ + κρ = /d1 β − ϑ α − ζβ, 2 2 3 1 e3 ρ + κρ = /d1 β − ϑ α − ζ β + 2(η β + ξ β), 2 2 e4 β + κβ = − /d?1 ρ − 3ζρ − ϑβ,

e3 β + 2κ β = /d2 α − 2ω β + (−2ζ + η)α + 3ξρ, 1 3 e4 α + κ α = − /d?2 β − ϑρ − 5ζβ. 2 2

3. The mass aspect function µ = − /d1 ζ − ρ + 14 ϑϑ, defined in (2.2.10), verifies the transport equation 3 e4 (µ) + κµ = 2 Err[e4 µ] :

=

Err[e4 µ], 1 2 1 κζ + eθ (κ)ζ + /d1 (ϑζ) − κϑ2 . 2 8

Proof. Concerning the null structure equations we only need to derive the equation for η − ζ. According to Proposition 2.56 we have 1 1 = β + 4ωξ + κ(η − η) + ϑ(η − η) 2 2

e3 (ξ) − e4 (η) which becomes e4 η

=

1 1 −β − κ(η − η) − ϑ(η − η) 2 2

and 1 −e4 ζ + κ(−ζ + η) + 2ω(ζ + η) 2

=

1 1 1 β + 2 /d?1 ω + 2ωξ + κ ξ − ϑ(−ζ + η) − ϑ ξ 2 2 2

which becomes e4 ζ

= −κζ − β − ϑζ.

61

PRELIMINARIES

Hence, e4 (ζ − η)

= =

1 1 −κζ − ϑζ + κ(η − η) + ϑ(η − η) 2   2  1 1 κ −ζ + (η − η) + ϑ −ζ + (η − η) . 2 2

Since ζ = −η we deduce −ζ + 12 (η − η) = 12 (−ζ + η) and thus, e4 (ζ − η)

=

−κ(ζ − η) − ϑ(ζ − η)

as desired. The Bianchi equations follow immediately from the general equations derived in the previous section. It only remains to check the equation verified by the mass aspect function µ. We have 1 = −[e4 , /d1 ]ζ − /d1 e4 (ζ) − e4 (ρ) + e4 (ϑϑ) 4 1 1 ? 2 = κ /d1 ζ − ϑ /d2 ζ + e4 (Φ)ζ − βζ − /d1 (−κζ − β − ϑζ) 2 2  3 1 − − κρ + /d1 β − ϑ α − ζβ 2 2   1 1 1 1 + ϑ − κ ϑ + 2 /d?2 ζ − κ ϑ + 2ζ 2 + ϑ (−κ ϑ − 2α) 4 2 2 4   3 1 1 ? 1 = κ /d1 ζ + ρ − ϑϑ − ϑ /d2 ζ + (κ − ϑ)ζ 2 + eθ (κ)ζ + /d1 (ϑζ) 2 4 2 2   1 1 + ϑ 2 /d?2 ζ − κϑ + 2ζ 2 4 2

e4 (µ)

and hence 3 e4 (µ) + κµ = 2

1 2 1 κζ + eθ (κ)ζ + /d1 (ϑζ) − κϑ2 2 8

as desired. This concludes the proof of the proposition. 2.2.4.3

Transport equations for S-averages

Proposition 2.64. For any scalar function f , we have Z  Z e4 f = (e4 (f ) + κf ), S Z  ZS  Z  e3 f = (e3 (f ) + κf ) + Err e3 f , S

S

S

(2.2.20)

62

CHAPTER 2

where the error term is given by the formula  Z  Z Z Err e3 f : = −ς −1 ςˇ (e3 (f ) + κf ) + ς −1 ςˇ(e3 (f ) + κf ) S S S Z Z
ˇ + ς −1 Ωˇ + Ω ς (e4 f + κf ) − ς −1 Ω ςˇ(e4 f + κf ) S S Z ˇ ς(e4 f + κf ). − ς −1 Ω S

In particular, we have e4 (r) =

r κ, 2

e3 (r) =

r (κ + A) 2

(2.2.21)

where
ˇ + ς −1 Ωˇ ˇ = −ς −1 κˇ ς +κ Ω ς + ς −1 ςˇκ ˇ − ς −1 Ω ςˇκ ˇ − ς −1 Ωςκ. (2.2.22)

A:

Proof. See section A.1. Corollary 2.65. For a reduced scalar f , we have Z     Z  3 1 e4 f eΦ = e4 (f ) + κ − ϑ f eΦ 2 2 S S and Z e3

Φ

fe S



Z  =

 e3 (f ) +

S

   Z  3 1 Φ Φ κ − ϑ f e + Err e3 fe . 2 2 S

Proof. In view of Proposition 2.64, we have Z  Z   e4 f eΦ = e4 (f eΦ ) + κf eΦ S ZS   = e4 (f ) + (κ + e4 Φ)f eΦ    ZS  3 1 = e4 (f ) + κ − ϑ f eΦ 2 2 S as desired. Also, using again Proposition 2.64, we have Z   Z  Z   e3 f eΦ = e3 (f eΦ ) + κf eΦ + Err e3 f eΦ S S S Z  Z   Φ = e3 (f ) + (κ + e3 Φ)f e + Err e3 f eΦ S     SZ  Z  3 1 Φ = e3 (f ) + κ − ϑ f e + Err e3 f eΦ 2 2 S S as desired.

63

PRELIMINARIES

Corollary 2.66. Given a scalar function f , we have e4 (f ) = e4 (f ) + κˇ fˇ, e4 (fˇ) = e4 (f ) − e4 (f ) − κˇ fˇ,

(2.2.23)

and
e3 f = e3 (f ) + Err[e3 f ], e3 (fˇ) = e3 (f ) − e3 (f ) − Err[e3 (f )],

(2.2.24)

where  
Err[e3 (f )] = −ς −1 ςˇ e3 f + κf − κf + ς −1 ςˇ(e3 f + κf ) − ςˇκ ˇf

ˇ + ς −1 Ωˇ (2.2.25) + Ω ς e4 f + κf − κ f     ˇ 4 f + κf ) − Ως ˇ κf + κ − ς −1 Ω ςˇ(e4 f + κf ) − ςˇκ ˇ f − ς −1 Ως(e ˇ fˇ. Proof. We have, recalling Lemma 2.45 and |S| = 4πr2 , R  Z f 1 e4 (|S|) e4 r S e4 (f ) = e4 = (e4 (f ) + κf ) − f = e4 (f ) + κf − 2 f |S| |S| S |S| r = e4 (f ) + κ f − κ f = e4 (f ) + κˇ fˇ. This also yields e4 (fˇ)

=

e4 (f ) − e4 (f ) = e4 (f ) − e4 (f ) − κ ˇ fˇ

as desired. Similarly, R

e3 (f )

f |S|



1 = e3 |S|

Z



2e3 (r) − f r S  Z  Z 1 1 = (e3 f + κf ) + Err e3 f − (κ + A)f |S| S |S|  Z S  1 = e3 (f ) + κf − κf + Err e3 f − Af |S|  Z S 1 = e3 (f ) + κ ˇ fˇ + Err e3 f − Af . |S| S = e3

S

f

We deduce e3 (f )

=

e3 (f ) + Err[e3 (f )]

64

CHAPTER 2

 R
 where, recalling the definitions of the error terms Err e3 S f and A,  Z  1 ˇ Err[e3 (f )] = κ ˇf + Err e3 f − Af |S| S
ˇ + ς −1 Ωˇ = κ ˇ fˇ − ς −1 ςˇ e3 f + κf + ς −1 ςˇ(e3 f + κf ) + Ω ς e4 f + κf ˇ 4 f + κf ) ς −1 Ω ςˇ(e4 f + κf ) − ς −1 Ως(e  
ˇ + ς −1 Ωˇ ˇ f −ς −1 κˇ ς + ς −1 ςˇκ ˇ+κ Ω ς − ς −1 Ω ςˇκ ˇ − ς −1 Ωςκ ,

− − i.e., Err[e3 (f )]

 
= κ ˇ fˇ − ς −1 ςˇ e3 f + κf − κf + ς −1 ςˇ(e3 f + κf ) − ςˇκ ˇf  

ˇ + ς −1 Ωˇ + Ω ς e4 f + κf − κ f − ς −1 Ω ςˇ(e4 f + κf ) − ςˇκ ˇf   ˇ 4 f + κf ) − Ωςκf ˇ − ς −1 Ως(e

as stated. Finally e3 (fˇ)

= e3 f − e3 (f ) = e3 f − e3 (f ) − Err[e3 f ]

which ends the proof of the corollary. The following is also an immediate application of Proposition 2.64. Corollary 2.67. If f verifies the scalar equation p e4 (f ) + κf = F, 2 then e4 (rp f ) 2.2.4.4

= rp F.

Commutation identities revisited

We revisit the general commutation identities of Lemma 2.54 in an outgoing geodesic foliation. Lemma 2.68. The following commutation formulae holds true: 1. If f ∈ sk ,   1 [r /dk , e4 ]f = r Comk (f ) + κ ˇ /dk f , 2   1 [r /dk , e3 ]f = r Comk (f ) + (−A + κ ˇ ) /dk f . 2 2. If f ∈ sk−1 ,

(2.2.26)

65

PRELIMINARIES

 1 ? =r + κ ˇ /dk f , 2   1 ∗ ? ? [r /dk , e3 ]f = r Comk (f ) + (−A + κ ˇ ) /dk f . 2 [r /d?k , e4 ]f



Com∗k (f )

(2.2.27)

Also, we have 1 = − ϑ /d?k+1 f 2 1 Comk (f ) = − ϑ /d?k+1 f 2 1 ∗ Comk (f ) = − ϑ /dk−1 f 2 − (k − 1)βf, 1 Com∗k (f ) = − ϑ /dk−1 f 2 Comk (f )

+ (ζ − η)e3 f − kηe3 Φf − ξ(e4 f + ke4 (Φ)f ) − kβf, + kζe4 Φf − kβf, − (ζ − η)e3 f − (k − 1)ηe3 Φf + ξ(e4 f − (k − 1)e4 (Φ)f ) + (k − 1)ζe4 Φf − (k − 1)βf.

Proof. We make use of the commutation Lemma 2.54 and the definition of A, see Proposition 2.64, to write, for f ∈ sk , [r /dk , e4 ]f

[r /dk , e3 ]f

= r[ /dk , e4 ]f − e4 (r) /dk f 1 r = rκ /dk f + rComk (f ) − κ /dk f 2 2  1 = r Comk (f ) + κ ˇ /dk f 2 = r[ /dk , e4 ]f − e3 (r) /dk f 1 r = rκ /dk f + rComk (f ) − (A + κ) /dk f 2 2  1 = r Comk (f ) + (−A + κ ˇ ) /dk f . 2

The remaining formulae are proved in the same manner. Also, the form of Comk (f ), Comk (f ), Com∗k (f ) and Com∗k (f ) follows from Lemma 2.54 and the fact that we have ξ = η + ζ = 0 in an outgoing geodesic foliation. We also record here for future use the following lemma. Lemma 2.69. Let T = 12 (e3 + Υe4 ), with Υ = 1 − 2m r . We have,     m m 2 e4 (m) [T, e4 ] = ω− 2 − κ− + e4 + (η + ζ)eθ , r 2r r r     (2.2.28)  m m 2Υ m e3 (m) [T, e3 ] = −Υ ω − 2 − κ+ − A+ e4 − (η + ζ)Υeθ . r 2r r 2r r

66

CHAPTER 2

Proof. Recall that [e3 , e4 ] = 2ωe4 + 2(η + ζ)eθ . Thus,     1 1 2m [T, e4 ] = [e3 + Υe4 , e4 ] = 2ωe4 + 2(η + ζ)eθ − e4 1 − e4 2 2 r   m e4 (m) = ω − 2 e4 (r) + e4 + (η + ζ)eθ r r   m m e4 (m) = ω − 2 − 2 (e4 (r) − 1) + e4 + (η + ζ)eθ r r r   e (m)  m m r 4 = ω− 2 − 2 κ−1 + e4 + (η + ζ)eθ r r 2 r     m m 2 e4 (m) = ω− 2 − κ− + e4 + (η + ζ)eθ r 2r r r and, [T, e3 ]

= = = =

    1 1 2m [e3 + Υe4 , e3 ] = Υ (−2ωe4 − 2(η + ζ)eθ ) − e3 1 − e4 2 2 r   m e3 (m) −Υω − 2 e3 (r) + e4 − Υ(η + ζ)eθ r r    m m mr e3 (m) −Υ ω − 2 − Υ 2 − 2 (κ + A) + e4 − Υ(η + ζ)eθ r r r 2 r      m m 2Υ m e3 (m) −Υ ω − 2 − κ+ − A+ e4 − Υ(η + ζ)eθ r 2r r 2r r

which concludes the proof of the lemma. Remark 2.70. When applying the formulas of Lemma 2.69 to a k-reduced scalar f ∈ sk , the term (η+ζ)eθ (f ) should correspond to a reduced scalar. In fact, recalling Remark 2.26, we can write ζeθ (f )

=

1 ζ ( /dk f − /d?k+1 f ) 2

which can indeed be shown to be a k-reduced scalar in sk . 2.2.4.5

Derivatives of the Hawking mass

Proposition 2.71 (Derivatives of the Hawking mass). We have the following identities for the Hawking mass, Z r e4 (m) = Err1 , (2.2.29) 32π S

67

PRELIMINARIES

and e3 (m)

Z Z
r
r −1 ˇ = 1 − ς ςˇ Err1 + Ω + ς Ωˇ ς Err1 32π S 32π S Z
r +ς −1 ςˇ 2ρˇ κ + 2ˇ ρκ + 2κ /d1 η + 2κ /d1 ξ + Err2 32π S Z −1 r ˇ (2ρˇ −ς (Ωˇ ς + Ως) κ + 2ˇ ρκ − 2κ /d1 ζ + Err2 ) 32π S h i m ˇ − ς −1 −ˇ ςκ ˇ + Ω ςˇκ ˇ + Ωςκ , (2.2.30) r −1

where Err1

:=

Err1

:=

Err2

:=

Err2

:=

1 1 2ˇ κρˇ + 2eθ (κ)ζ − κϑ2 − κ ˇ ϑϑ + 2κζ 2 , 2 2
1 1 2ˇ ρκ ˇ − 2eθ (κ)η − 2eθ (κ)ξ − κ ˇ ϑϑ + 2κη 2 + 2κ η − 3ζ ξ − κϑ2 , 2 2 1 2 1 2 2ˇ ρκ ˇ − κϑ − κϑϑ + 2κζ , 2 2 
1 1 2 2ˇ ρκ ˇ + κ 2η − ϑϑ + 2κ η − 3ζ ξ − κϑ2 . 2 2

Proof. The proof relies R on the definition of the Hawking mass m given by the 1 formula 2m = 1 + r 16π S κκ, Proposition 2.64, and the null structure equations for e4 (κ), e4 (κ), e3 (κ) and e3 (κ) provided by Proposition 2.63. We refer to section A.2 for the details. 2.2.4.6

Transport equations for main averaged quantities

Lemma 2.72. The following equations hold true:     2 1 2 1 1 2 e4 κ − + κ κ− = − ϑ2 + κ ˇ , r 2 r 4 2   (2.2.31)  m 2m m 2 e4 (m) e4 ω − 2 = ρ + 3 + 2 κ − − + 3ζ(2η + ζ) + κ ˇ ω ˇ, r r r r r2 and     2 1 2 e3 κ − + κ κ− (2.2.32) r 2 r       2 4 m 2m 1 = 2ω κ − + ω − 2 + 2 ρ + 3 − ς −1 − κ κ + 2ω κ + 2ρ ςˇ r r r r 2   

1 1 1 ˇ 2 ˇ + ς −1 Ωˇ − κ2 Ω ς − ς −1 κˇ ς+ κ Ω + ς −1 Ωˇ ς + Err e3 κ − , 2 r r r

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CHAPTER 2

where    2 Err e3 κ − r

:=

1 1 1 1 ˇ 2η 2 + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + ς −1 ςˇκ ˇ − ς −1 Ω ςˇκ 2 2 r r   1 1 ˇ − ς −1 ςˇ 1 κ − ς −1 Ωςκ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 r 2 2 "   1 1 −1 2 +ς ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η 2 2 #  
1 1 ˇ + ς −1 Ωˇ −ˇ ςκ ˇκ + Ω ς κ ˇ 2 − ϑ2 2 4 !   1 1 −ς −1 Ω ςˇ κ2 − ϑ2 − ςˇκ ˇκ 2 4 !   1 2 1 2 −1 ˇ ˇ −ς Ως κ − ϑ − Ως κ κ + κ ˇκ ˇ. (2.2.33) 2 4

Proof. The proof relies on Corollary 2.66 and the null structure equations for e4 (κ) and e3 (κ) provided by Proposition 2.63. We refer to section A.3 for the details. 2.2.4.7

Transport equations for main checked quantities

Proposition 2.73 (Transport equations for checked quantities). We have the following transport equations in the e4 direction: e4 κ ˇ + κκ ˇ = Err[e4 κ ˇ ], 1 2 1 2 1 2 Err[e4 κ ˇ] : = − κ ˇ − κ ˇ − (ϑ − ϑ2 ), 2 2 2 1 1 e4 κ ˇ + κˇ κ+ κ ˇ κ = −2 /d1 ζ + 2ˇ ρ + Err[e4 κ ˇ ], 2 2     (2.2.34) 1 1 1 1 2 2 Err[e4 κ ˇ] : = − κ ˇκ ˇ− κ ˇκ ˇ + − ϑϑ + 2ζ − − ϑϑ + 2ζ , 2 2 2 2 e4 ω ˇ = ρˇ + Err[e4 ω ˇ ], Err[e4 ω ˇ ] : = −ˇ κω ˇ + (ζ(2η + ζ) − ζ(2η + ζ)), 3 3 e4 ρˇ + κˇ ρ + ρˇ κ = /d1 β + Err[e4 ρˇ], 2 2     3 1 1 1 Err[e4 ρˇ] : = − κ ˇ ρˇ + κ ˇ ρˇ − ϑα + ζβ + ϑα + ζβ , 2 2 2 2 3 3 e4 µ ˇ + κˇ µ + µˇ κ = Err[e4 µ ˇ], 2 2 3 1 Err[e4 µ ˇ] : = − κ ˇµ ˇ+ κ ˇµ ˇ + Err[e4 µ] − Err[e4 µ], 2 2 ˇ = −2ˇ ˇ e4 (Ω) ω+κ ˇ Ω.

(2.2.35)

PRELIMINARIES

69

Also in the e3 direction, 1 e3 (ˇ κ) = 2 /d1 η + 2ˇ ρ − (κˇ κ + κˇ κ) + 2 (ωˇ κ + κˇ ω) 2  
1 1 ˇ + ς −1 Ωˇ + ς −1 − κ κ + 2ω κ + 2ρ ςˇ + κ2 Ω ς + Err[e3 κ ˇ ], 2 2   1 e3 (ˇ κ) + κ κ ˇ = 2 /d1 ξ − 2 (ˇ ωκ+ωκ ˇ ) + ς −1 ςˇ − κ2 − 2ω κ (2.2.36) 2  
1 ˇ + ς −1 Ωˇ − Ω ς − κ κ + 2ρ + Err[e3 (ˇ κ)], 2
3 3 3 3 ˇ + ς −1 Ωˇ e3 ρˇ + κˇ ρ = − ρˇ κ + /d1 β − κ ρς −1 ςˇ + κ ρ Ω ς + Err[e3 ρˇ], 2 2 2 2 with error terms given by   1
1 Err[e3 κ ˇ ] := 2 η 2 − η 2 − κ ˇκ ˇ + 2ˇ ωκ ˇ− ϑϑ − ϑϑ 2 2   1 1 + ς −1 ςˇ κ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 2 2 !   1 1 −1 2 −ς ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η − ςˇκ ˇκ 2 2 (2.2.37) !    
1 1 1 2 1 2 ˇ + ς −1 Ωˇ − Ω ς κ ˇ 2 − ϑ2 + ς −1 Ω ςˇ κ − ϑ − ςˇκ ˇκ 2 4 2 4 !   1 1 ˇ ˇ κκ − κ + ς −1 Ως κ2 − ϑ2 − Ως ˇκ ˇ, 2 4  1 2 1 2 Err[e3 (ˇ κ)] := − κ ˇ − 2ˇ ωκ ˇ + 2(η − 3ζ)ξ − 2(η − 3ζ)ξ − ϑ − ϑ2 2 2 !   1 2 1 2 −1 −ς ςˇ κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ − ςˇκ ˇκ 2 2 !   (2.2.38) 1 1 −1 2 + ς Ω ςˇ κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ − ςˇκ ˇκ 2 2 !   1 1 −1 ˇ ˇ κκ − κ +ς Ως κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 − Ως ˇ2 , 2 2

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CHAPTER 2

and 

   1 1 ϑα + ζβ − 2ηβ − 2ξβ + ϑα + ζβ − 2ηβ − 2ξβ − 2 2   1 1 + ς −1 ςˇ − κ ˇ ρˇ − ϑ α − ζ β + 2(η β + ξ β) 2 2 !   1 1 −1 −ς ςˇ − κρ + /d1 β − ϑ α − ζ β + 2(η β + ξ β) − ςˇκ ˇρ 2 2  
1 1 ˇ + ς −1 Ωˇ − Ω ς − κ ˇ ρˇ − ϑ α − ζβ 2 2 !   1 1 −1 + ς Ω ςˇ − κρ + /d1 β − ϑ α − ζβ − ςˇκ ˇρ 2 2 !   1 1 −1 ˇ − κρ + /d1 β − ϑ α − ζβ − Ως ˇ κ −κ +ς Ως ˇ ρˇ. 2 2

Err[e3 ρˇ] := −

3 κ ˇ ρˇ 2

(2.2.39)

Proof. The proof relies on Corollary 2.66 and the null structure equations of Proposition 2.63. We refer to section A.4 for the details. 2.2.5

Additional equations

We derive below additional equations for ω, η, ξ. Proposition 2.74. The following identities hold true for a general forward geodesic foliation. • The scalar ω verifies 2 /d?1 ω

  1 1 1 = − κξ + κ + 2ω + ϑ η + e3 (ζ) − β 2 2 2 1 1 1 + κζ − 2ωζ + ϑζ − ϑξ. 2 2 2

• The reduced 1-form η verifies 2 /d2 /d?2 η

= −

Err[ /d2 /d?2 η]

= −


κ −e3 (ζ) + β − e3 (eθ (κ)) − κ



1 κζ − 2ωζ 2

 + 6ρη − κeθ κ

1 κeθ (κ) + 2ωeθ (κ) + 2eθ (ρ) + Err[ /d2 /d?2 η], 2     1 1 1 1 2 /d1 η − κϑ + 2η 2 η + 2eθ (η 2 ) − κ ϑζ − ϑξ − ϑeθ (κ) 2 2 2 2   1 1 1 3 2 /d1 η − ϑϑ + 2η 2 ζ − eθ (ϑ ϑ) − ϑ2 ξ − ϑϑη. 2 2 2 2

71

PRELIMINARIES

• The reduced 1-form ξ verifies
3 = −e3 (eθ (κ)) + κ e3 (ζ) − β + κ2 ζ − κeθ κ + 6ρξ − 2ωeθ (κ) 2 + Err[ /d2 /d?2 ξ],   1 1 2 1 ? Err[ /d2 /d2 ξ] = 2 /d1 ξ + κ ϑ + 2ηξ − ϑ η + 2eθ (ηξ) − eθ (ϑ2 ) 2 2 2   1 1 1 1 + κ ϑζ − ϑξ − ϑeθ κ − ϑϑξ 2 2 2 2     1 − ζ 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 + ξ − ϑϑ − 2 /d1 ζ + 2ζ 2 2 − 6ηζξ − 6eθ (ζξ). 2 /d2 /d?2 ξ

Proof. The proof relies on the null structure equations of Proposition 2.63, in particular the ones for e3 (ζ), e3 (κ) and e3 (κ). We refer to section A.5 for the details. 2.2.6

Ingoing geodesic foliation

All the equations of section 2.2.4 for outgoing geodesic foliations have a counterpart for ingoing geodesic foliations. The corresponding equations can be easily deduced from the ones in section 2.2.4 by performing the following substitutions: u → u,

s → s,

α → α,

β →, β,

e4 → e3 ,

Cu → Cu ,

e3 → e4 ,

Su,s → Su,s ,

eθ → eθ ,

ρ → ρ,

ξ → ξ,

ω → ω,

κ → κ,

ϑ → ϑ,

ω → ω,

ξ → ξ,

r → r,

m → m,

e4 (s) = 1 → e3 (s) = −1,

µ → µ,

ϑ → ϑ,

β → β,

η → η,

α → α,

η → η,

ζ → −ζ, κ → κ, 2 2 Ω = e3 (s) → Ω = e4 (s), ς = →ς= , e3 (u) e4 (u) 2Υ m m →κ− , ω − 2 → ω + 2, r r r 2m 2m → µ − 3 , Ω + Υ → Ω − Υ, ς − 1 → ς − 1, r3 r

2 2 2Υ →κ+ , κ+ r r r 2m 2m ρ+ 3 →ρ+ 3 , µ− r r 2 2 A = e3 (r) − κ → A = e4 (r) − κ. r r κ−

2.2.7

Adapted coordinates systems

2.2.7.1

(u, s, θ, ϕ) coordinates

Proposition 2.75. Consider, in addition to the functions u, s, ϕ, an additional Z-invariant function θ. Then, relative to the coordinates system (u, s, θ, ϕ), the following hold true: 1. The spacetime metric takes the form  2 1 g = −2ςduds + ς 2 Ωdu2 + γ dθ − ς(b − Ωb)du − bds + e2Φ (dϕ)2 (2.2.40) 2

72

CHAPTER 2

where Ω = e3 (s),

b = e4 (θ),

b = e3 (θ),

γ −1 = eθ (θ)2 .

2. In these coordinates the reduced frame takes the form   1 1 1√ √ ∂s = e4 − b γeθ , ∂u = ς e3 − Ωe4 − γ(b − bΩ)eθ , 2 2 2 √ ∂θ = γeθ .

(2.2.41)

(2.2.42)

3. In the particular case when b = e4 (θ) = 0 we have e4 (γ) = 2χγ,

e4 (b) = −2(ζ + η)γ −1/2 .

(2.2.43)

Proof. First, from the fact that (e3 , e4 , eθ ) forms a null frame, we easily verify that (2.2.42) holds. Then, (2.2.40) immediately follows from (2.2.42) and the fact that (e3 , e4 , eθ ) forms a null frame. To prove the last statement, when b = e4 (θ) = 0, we start with [e4 , e3 ] = 2ωe3 − 2ωe4 + 2(η − η)eθ = −2(ζ + η)eθ − 2ωe4 . Applying this to θ we derive [e4 , e3 ](θ) = (−2(ζ + η)eθ − 2ωe4 )(θ) = −2(ζ + η)eθ (θ) = −2(ζ + η)γ −1/2 . We deduce e4 (b) = e4 (e3 (θ)) = −2(ζ + η)γ −1/2 . To prove the equation for γ we make use of [e4 , eθ ] = (η + ζ)e4 + ξe3 − χeθ = −χeθ so that e4 eθ (θ) = [e4 , eθ ](θ)

=

−χeθ (θ) = −χγ −1/2 .

Thus e4 (γ −1/2 ) = −χγ −1/2 from which e4 (γ) = 2χγ. This concludes the proof of the lemma. Remark 2.76. In Schwarzschild, relative to the above coordinate system, we have ς = 1,

Ω = −Υ,

b = b = 0,

γ = r2 ,

eΦ = r sin θ,

so that we obtain outgoing Eddington-Finkelstein coordinates. Remark 2.77. The (u, s, θ, ϕ) coordinates system, with the choice b = 0 (i.e., θ is

73

PRELIMINARIES

transported by e4 (θ) = 0), will be used in section 3.7 and Chapter 9 in connection with our GCM procedure. 2.2.7.2

(u, r, θ, ϕ) coordinates

Proposition 2.78. Consider, in addition to the functions u, r, ϕ, an additional Z-invariant function θ. Relative to the coordinates (u, r, θ, ϕ) the following hold true: 1. The spacetime metric takes the form  2 4ς ς 2 (κ + A) 2 1 b g = − dudr + du + γ dθ − ςbdu − Θ rκ κ 2 2

(2.2.44)

where b = e4 (θ),

b = e3 (θ),

γ=

1 (eθ (θ))2

(2.2.45)

and 4 Θ := dr − ς rκ



κ+A κ

 du.

2. The reduced coordinates derivatives take the form √ 2 γ 2 ∂r = e4 − beθ , rκ rκ √ ∂θ = γeθ ,       1 1κ+A 1√ κ+A ∂u = ς e3 − e4 − γ b− b eθ . 2 2 κ 2 κ 3. To control eΦ , we will rely on the following transport equation  Φ  e eΦ −1 = (ˇ κ − ϑ) . e4 r sin θ 2r sin θ

(2.2.46)

(2.2.47)

Proof. First, from the fact that (e3 , e4 , eθ ) forms a null frame, we easily verify that (2.2.46) holds. Then, (2.2.44) immediately follows from (2.2.46) and the fact that (e3 , e4 , eθ ) forms a null frame. It remains to prove (2.2.47). It follows from  Φ      e eΦ e4 (r) eΦ 1 κ e4 −1 = e4 (Φ) − = (κ − ϑ) − r sin θ r sin θ r r sin θ 2 2 =

eΦ (ˇ κ − ϑ) 2r sin θ

which concludes the proof of the lemma. Remark 2.79. In Schwarzschild, relative to the above coordinate system, we have κ=

2 , r

κ=−

2Υ , r

ς = 1,

A = 0,

b = b = 0,

γ = r2 ,

eΦ = r sin θ,

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CHAPTER 2

so that we obtain outgoing Eddington-Finkelstein coordinates. Remark 2.80. The (u, r, θ, ϕ) coordinates system, with the choice (2.2.52) for θ introduced below, will be used in Proposition 3.17 to prove the convergence to the outgoing Eddington-Finkelstein coordinates of Schwarzschild. 2.2.7.3

(u, r, θ, ϕ) coordinates

We easily deduce an analog statement relative to (u, r, θ, ϕ) coordinates. Proposition 2.81. Consider, in addition to the functions u, r, ϕ, an additional Z-invariant function θ. Relative to the coordinates (u, r, θ, ϕ) the following hold true: 1. The spacetime metric takes the form  2 4ς ς 2 (κ + A) 2 1 b g = − dudr + du + γ dθ − ςbdu − Θ rκ κ 2 2

(2.2.48)

where b = e4 (θ),

b = e3 (θ),

γ=

1 (eθ (θ))2

(2.2.49)

and 4 Θ := dr − ς rκ



κ+A κ

 du.

2. The reduced coordinates derivatives take the form √ 2 γ 2 ∂r = e3 − beθ , rκ rκ √ ∂θ = γeθ ,       1 1κ+A 1√ κ+A ∂u = ς e4 − e3 − γ b− b eθ . 2 2 κ 2 κ 3. To control eΦ , we will rely on the following transport equation  Φ  e eΦ e3 −1 = (ˇ κ − ϑ) . r sin θ 2r sin θ

(2.2.50)

(2.2.51)

Remark 2.82. In Schwarzschild, relative to the above coordinate system, we have κ=

2 , r

κ=−

2Υ , r

ς = 1,

A = 0,

b = b = 0,

γ = r2 ,

eΦ = r sin θ,

so that we obtain ingoing Eddington-Finkelstein coordinates. Remark 2.83. The (u, r, θ, ϕ) coordinates system, with the choice (2.2.52) for θ introduced below, will be used in Proposition 3.18 to prove the convergence to the ingoing Eddington-Finkelstein coordinates of Schwarzschild.

75

PRELIMINARIES

2.2.7.4

Initialization of θ

We now introduce the coordinate function θ that will be used for the (u, r, θ, ϕ) coordinates system and for the (u, r, θ, ϕ) coordinates system, see Remarks 2.80 and 2.83. Lemma 2.84. Let θ ∈ [0, π] be the Z-invariant scalar on M defined by θ := cot−1 (reθ (Φ)) .

(2.2.52)

Then, eΦ r sin θ

=

√

1+a

(2.2.53)

e2Φ + (eθ (eΦ ))2 − 1. r2

(2.2.54)

where a :=

Moreover, we have in an outgoing geodesic foliation reθ (θ)

=

e3 (θ)

=

e4 (θ)

=

r2 (K − r12 ) , 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + A + ϑ) eθ (Φ) + rξe4 (Φ) + rηe3 (Φ) − , 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + ϑ) eθ (Φ) − rζe3 (Φ) − , 1 + (reθ (Φ))2

1+

and analog identities hold for an ingoing geodesic foliation. Proof. In view of the definition of θ, we have θ ∈ [0, π], sin θ ≥ 0 and sin θ

=

√

1 1 eΦ =p =p 1 + cot θ2 1 + (reθ (Φ))2 e2Φ + (reθ (eΦ ))2 eΦ

= r

q

e2Φ r2

eΦ = √ . r 1+a + (eθ (eΦ ))2

Hence eΦ r sin θ

r =

√ e2Φ + (eθ (eΦ ))2 = 1 + a. 2 r

Also, we compute reθ (θ)

=

−

r2 eθ eθ (Φ) . 1 + (reθ (Φ))2

Next, recall that we have eθ eθ (Φ)

= −K − (eθ (Φ))2 .

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CHAPTER 2

We infer reθ (θ)

r2 (K − r12 ) r2 (K + (eθ (Φ))2 ) = 1 + 1 + (reθ (Φ))2 1 + (reθ (Φ))2

=

as desired. Also, we have in an outgoing geodesic foliation e4 (θ)

= =

= =

re4 eθ (Φ) + e4 (r)eθ (Φ) 1 + (reθ (Φ))2 r(D4 Dθ Φ + DD4 eθ Φ) + e4 (r)eθ (Φ) − 1 + (reθ (Φ))2   rβ + r e4r(r) − e4 (Φ) eθ (Φ) − rζe4 (Φ) − 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + ϑ) eθ (Φ) − rζe4 (Φ) − . 1 + (reθ (Φ))2 −

Finally, we compute in an outgoing geodesic foliation e3 (θ)

re3 eθ (Φ) + e3 (r)eθ (Φ) 1 + (reθ (Φ))2 r(D3 Dθ Φ + DD3 eθ Φ) + e3 (r)eθ (Φ) = − 1 + (reθ (Φ))2   rβ + r e3r(r) − e3 (Φ) eθ (Φ) + rξe4 (Φ) + rηe3 (Φ) = − 1 + (reθ (Φ))2 r rβ + 2 (−ˇ κ + A + ϑ) eθ (Φ) + rξe4 (Φ) + rηe3 (Φ) = − . 1 + (reθ (Φ))2 = −

This concludes the proof of the lemma. In view of (2.2.53), we will need to control the quantity a defined in (2.2.54). To this end, we will need the following lemma. Lemma 2.85. The quantity a defined in (2.2.54) vanishes on the axis of symmetry and verifies the following identities in an outgoing geodesic foliation: e4 (a)

=

eθ (a)

=

e3 (a)

=

  (ˇ κ − ϑ)e2Φ + 2eθ (eΦ ) β − e4 (Φ)ζ eΦ , 2 r      2m 1 4Υ 1 2Φ κκ + 2 − ϑϑ , 2eθ (Φ)e ρ+ 3 + r 4 r 4   2Φ   κ ˇ−A−ϑ e Φ + 2e (e ) β + e (Φ)η + ξe (Φ) eΦ , θ 3 4 r2

and analog identities hold in an ingoing geodesic foliation. Proof. The vanishing on the axis follow easily from the fact that both e2Φ and (eθ (eΦ ))2 − 1 vanish on the axis (see (2.1.13)). To prove the second part of the

77

PRELIMINARIES

lemma we recall that, with respect to the reduced metric (see equation (2.1.4)), Rab = Da Db Φ + Da ΦDb Φ, and (see Definition 2.50) R3θ = β,

R4θ = β,

Starting with the definition a = geodesic foliation e4 (a)

= = =

e2Φ r2

Rθθ = R34 = ρ,

R34 = ρ.

+ (eθ (eΦ ))2 − 1, we compute in an outgoing

2e4 (Φ)e2Φ 2e4 (r)e2Φ − + 2eθ (eΦ )e4 (eθ (eΦ )) 2 r r3   (κ − ϑ)e2Φ κe2Φ Φ − + 2e (e ) e (e (Φ)) + e (Φ)e (Φ) eΦ θ 4 θ θ 4 r2 r2   (ˇ κ − ϑ)e2Φ Φ + 2e (e ) β − e (Φ)ζ eΦ . θ 4 r2

Also eθ (a)

= = = = = =

2eθ (Φ)e2Φ + 2eθ (eΦ )eθ (eθ (eΦ )) r2   2eθ (Φ)e2Φ + 2eθ (eΦ ) eθ (eθ (Φ)) + eθ (Φ)2 eΦ 2 r   2eθ (Φ)e2Φ Φ + 2e (e ) ρ + D Φ eΦ θ D e θ θ r2   2eθ (Φ)e2Φ 1 1 + 2eθ (eΦ ) ρ + χe3 Φ + χe4 Φ eΦ 2 r 2 2   2eθ (Φ)e2Φ 1 1 Φ + 2e (e ) ρ + κκ − ϑϑ eΦ θ r2 4 4      2m 1 4Υ 1 2eθ (Φ)e2Φ ρ+ 3 + κκ + 2 − ϑϑ . r 4 r 4

Finally, we have in an outgoing geodesic foliation e3 (a)

= = =

2e3 (Φ)e2Φ 2e3 (r)e2Φ − + 2eθ (eΦ )e3 (eθ (eΦ )) 3 r2 r     κ + A e2Φ (κ − ϑ)e2Φ Φ − + 2e (e ) e (e (Φ)) + e (Φ)e (Φ) eΦ θ 3 θ θ 3 2 2 r r     κ ˇ − A − ϑ e2Φ Φ + 2e (e ) β + e (Φ)η + ξe (Φ) eΦ . θ 3 4 r2

This concludes the proof of the lemma. Remark 2.86. The function θ defined by (2.2.52) defines • together with the functions (u, r, ϕ), a regular coordinates system with the axis of symmetry corresponding to θ = 0, π, • together with the functions (u, r, ϕ), a regular coordinates system with the axis of symmetry corresponding to θ = 0, π.

78 2.3

CHAPTER 2

PERTURBATIONS OF SCHWARZSCHILD AND INVARIANT QUANTITIES

Recall that in Schwarzschild all Ricci coefficients ξ, ξ, ϑ, ϑ, η, η, ζ and curvature components α, α, β, β vanish identically. In addition the check quantities κ ˇ, κ ˇ, ω ˇ, ω ˇ and ρˇ also vanish. Thus, roughly, we expect that in perturbations of Schwarzschild these quantities stay small, i.e., of order O() for a sufficiently small . More precisely we say that a smooth, vacuum, Z-invariant, polarized spacetime is an O()-perturbation of Schwarzschild, or simply O()-Schwarzschild, if the following are true relative to a Z-invariant null frame e3 , e4 , eθ : ξ, ξ, ϑ, ϑ, η, η, ζ, κ ˇ, κ ˇ, ω ˇ, ω ˇ

α, α, β, β , ρˇ = O()

(2.3.1)

Moreover, r e3 (r) − κ = O(), 2

r e4 (r) − κ = O(), 2

(2.3.2)

where r is the area radius of the 2-spheres generated by eθ , eϕ , see (2.1.12). In reality, of course, we expect that small perturbations of Schwarzschild remain not only close to the original Schwarzschild but also converge to a nearby Schwarzschild solution but for the discussion below this will suffice. 2.3.1

Null frame transformations

Our definition of O()-Schwarzschild perturbations does not specify a particular frame. In what follows we investigate how the main Ricci and curvature quantities change relative to frame transformations, i.e., linear transformations which take null frames into null frames. Lemma 2.87. A general null transformation can be written in the form   1 2 0 e4 = λ e4 + f eθ + f e3 , 4     1 1 1 1 0 eθ = 1 + f f eθ + f e4 + f 1 + f f e3 , 2 2 2 4      1 1 2 2 1 1 2 0 −1 e3 = λ 1 + f f + f f e3 + f 1 + f f eθ + f e4 . 2 16 4 4

(2.3.3)

Proof. It is straightforward to check that the transformation (2.3.3) takes null frames into null frames. One can also check that it can be written in the form type(3) ◦ type(1) ◦ type(2) where the type 1 transformations fix e3 , i.e., (λ = 1, f = 0), type 2 transformations fix e4 , i.e., (λ = 1, f = 0) and type 3 transformations keep the directions of e3 , e4 , i.e., (f = f = 0). Remark 2.88. Note that f, f are reduced from spacetime 1 forms while λ is reduced from a scalar. Remark 2.89. A transformation consistent with O()-Schwarzschild spacetimes must have f, f = O() and a := log λ = O().

PRELIMINARIES

79

Proposition 2.90 (Transformation formulas). Under a general transformation of type (2.3.3), the Ricci coefficients and curvature components transform as follows:   1 1 ξ 0 = λ2 ξ + λ−1 e04 (f ) + ωf + f κ + λ2 Err(ξ, ξ 0 ), 2 4 1 Err(ξ, ξ 0 ) = f ϑ + l.o.t., 4   (2.3.4) 1 1 0 ξ = λ−2 ξ + λe03 (f ) + ω f + f κ + λ−2 Err(ξ, ξ 0 ), 2 4 1 1 Err(ξ, ξ 0 ) = − λf 2 e03 (f ) + f ϑ + l.o.t., 8 4 1 ζ 0 = ζ − e0θ (log(λ)) + (−f κ + f κ) + f ω − f ω + Err(ζ, ζ 0 ), 4 1 0 1 0 Err(ζ, ζ ) = f eθ (f ) + (−f ϑ + f ϑ) + l.o.t., 2 4 1 0 1 0 η = η + λe3 (f ) + κf − f ω + Err(η, η 0 ), 2 4 1 0 Err(η, η ) = f ϑ + l.o.t., 4 1 1 0 η = η + λ−1 e04 (f ) + κf − f ω + Err(η, η 0 ), 2 4 1 2 −1 0 1 0 Err(η, η ) = − f λ e4 (f ) + f ϑ + l.o.t., 8 4

(2.3.5)

κ0 = λ (κ + /d1 0 (f )) + λErr(κ, κ0 ), 1 Err(κ, κ0 ) = f (ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω + l.o.t.,
4 (2.3.6) κ0 = λ−1 κ + /d1 0 (f ) + λ−1 Err(κ, κ0 ), 1 1 Err(κ, κ0 ) = − f 2 e0θ (f ) + f (−ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω + l.o.t., 4 4 ϑ0 = λ (ϑ − /d?2 0 (f )) + λErr(ϑ, ϑ0 ), 1 Err(ϑ, ϑ0 ) = f (ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t.
4 −1 (2.3.7) 0 −1 ?0 ϑ =λ ϑ − /d2 (f ) + λ Err(ϑ, ϑ0 ), 1 1 Err(ϑ, ϑ0 ) = − f 2 e0θ (f ) + f (−ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t., 4 4

80

CHAPTER 2

 1 −1 0 ω = λ ω − λ e4 (log(λ)) + λErr(ω, ω 0 ), 2 1 1 1 1 1 1 Err(ω, ω 0 ) = f e04 (f ) + ωf f − f η + f ξ + f ζ − κf 2 + 4 2 2 2 2 8 1 2 − ωf + l.o.t., 4   1 0 0 −1 ω =λ ω + λe3 (log(λ)) + λ−1 Err(ω, ω 0 ), 2 1 1 1 1 1 Err(ω, ω 0 ) = − f e03 (f ) + ωf f − f η + f ξ − f ζ − κf 2 + 4 2 2 2 8 1 2 − ωf + l.o.t. 4 0



1 ffκ 8 (2.3.8)

1 ffκ 8

The lower order terms we denote by l.o.t. are linear with respect to the Ricci coefficients Γ = {ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ} and quadratic or higher order in f, f , and do not contain derivatives of these latter. Also, α0 = λ2 α + λ2 Err(α, α0 ), 3 Err(α, α0 ) = 2f β + f 2 ρ + l.o.t., 2   3 0 β = λ β + ρf + λErr(β, β 0 ), 2 1 Err(β, β 0 ) = f α + l.o.t., 2 ρ0 = ρ + Err(ρ, ρ0 ), 3 Err(ρ, ρ0 ) = ρf f + f β + f β + l.o.t., 2   3 0 β = λ−1 β + ρf + λ−1 Err(β, β 0 ), 2 1 Err(β, β 0 ) = f α + l.o.t., 2 0 α = λ−2 α + λ−2 Err(α, α0 ), 3 Err(α, α0 ) = 2f β + f 2 ρ + l.o.t. 2

(2.3.9)

The lower order terms we denote by l.o.t. are linear with respect to the curvature quantities α, β, ρ, β, α and quadratic or higher order in f, f , and do not contain derivatives of these latter. Proof. See section A.6. Lemma 2.91. In the particular case when λ = 1, f = 0, we have 1 e04 = e4 + f eθ + f 2 e3 , 4 1 0 eθ = eθ + f e3 , 2 e03 = e3 ,

81

PRELIMINARIES

and ξ0

ω0 ζ0 η0

1 1 1 1 1 1 1 = ξ + e04 f + κf + f ω + f ϑ + f 2 η − f 2 η + f 2 ζ − f 3 κ 2 4 4 4 4 2 16 1 1 1 − f 3 ω − f 3 ϑ − f 4 ξ, 4 16 16 1 1 1 1 1 1 = ω + f ζ − ηf − f 2 ω − f 2 κ − f 2 ϑ − f 3 ξ, 2 2 4 8 8 8     1 1 1 = ζ− κ+ω f −f ϑ + fξ , 4 4 2 1 1 = η + e03 (f ) − f ω − f 2 ξ. 2 4

Proof. The proof follows from Proposition 2.90 by setting λ = 1, f = 0. Since we need precise formulas for the error terms, we provide a proof in section A.9. Lemma 2.92 (Transport equations for (f, f , λ)). Assume that we have in the new null frame (e03 , e04 , e0θ ) of type (2.3.3) ξ 0 = 0,

ω 0 = 0,

ζ 0 + η 0 = 0.

Then, (f , f, log(λ)) satisfy the following transport equations λ−1 e04 (f ) +

κ

 + 2ω f

2 λ−1 e04 (log(λ)) κ λ−1 e04 (f ) + f 2

= −2ξ + E1 (f, Γ), =

2ω + E2 (f, Γ),

= −2(ζ + η) + 2e0θ (log(λ)) + 2f ω + E3 (f, f , Γ),

where E1 , E2 and E3 are given by E1 (f, Γ) E2 (f, Γ) E3 (f, f , Γ)

1 = − ϑf + l.o.t., 2 1 1 = f ζ − f 2 ω − ηf − f 2 κ + l.o.t., 2 4 1 0 = −f eθ (f ) − f ϑ + l.o.t. 2

Here, l.o.t. denote terms which are cubic or higher order in f, f (or in f only in the ˇ and do not contain derivatives of these quantities, where case of E1 and E2 ) and Γ ˇ Γ and Γ denotes the Ricci coefficients and renormalized Ricci coefficients w.r.t. the original null frame (e3 , e4 , eθ ). Proof. See section A.7. To avoid a potential log loss for the third equation in Lemma 2.92, i.e., the transport equation for f , we state the following renormalized version of the lemma. Corollary 2.93. Assume given a null frame (e3 , e4 , eθ ) associated to an outgoing geodesic foliation as in section 2.2.4, and let r denote the corresponding area radius. Assume that we have in the new null frame (e03 , e04 , e0θ ) of type (2.3.3) ξ 0 = 0,

ω 0 = 0,

ζ 0 + η 0 = 0.

82

CHAPTER 2

Then, (f , f, log(λ)) satisfy the following transport equations λ−1 e04 (rf )

= E10 (f, Γ),

λ−1 e04 (log(λ)) = E20 (f, Γ),   λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω = E30 (f, f , λ, Γ), where E10 (f, Γ) E20 (f, Γ) E30 (f, f , λ, Γ)

r r = − κ ˇ f − ϑf + l.o.t., 2 2 1 2 1 = f ζ − f ω − ηf − f 2 κ + l.o.t., 2  4  r 2 2 = − κ ˇf + r κ ˇ− κ− e0θ (log(λ)) 2 r   r +r2 /d01 (f ) + λ−1 ϑ0 e0θ (log(λ)) − κ ˇ Ωf + rE3 (f, f , Γ) 2 2 0 −2r eθ (E2 (f, Γ)) + rΩE1 (f, Γ),

and where E1 , E2 and E3 are given in Lemma 2.92. Proof. See section A.8. 2.3.2

Schematic notation Γg and Γb

Many of the identities which we present below contain a huge number of O(2 ) terms. In what follows we introduce schematic notation meant to keep track of the most important error terms. Note that the decomposition below between the terms Γg and Γb is consistent with our main bootstrap assumptions BA-E on energy and BA-D on decay, see section 3.4.1. Definition 2.94. We divide the small connection coefficient terms (relative to an arbitrary null frame) into21     2 1 2 (0) (0) Γg = rξ, ϑ, ζ, η, e4 (r) − κ, eθ (r) , Γb = η, ϑ, ξ, e3 (r) − κ . r r r For higher derivatives we introduce n o (0) 2 Γ(1) g = dΓg , r eθ (ω), reθ (κ), reθ (κ) ,

n o (1) (0) Γb = dΓb , reθ (ω) ,

and for s ≥ 2, ≤s Γ(s) Γg , g =d

(s)

Γb = d≤s Γb ,

where we have introduced the notations d = {e3 , re4 , d/}, 21 In the frames we are using, we have in fact ξ = 0 for r ≥ 4m so that it behaves in fact better 0 (0) than the other components of Γg .

83

PRELIMINARIES

with angular derivatives d/ of reduced scalars in sk defined by (2.1.36). Remark 2.95. According to the main bootstrap assumptions BA-E, BA-D (see section 3.4.1), the terms Γb behave worse in powers of r than the terms in Γg . (s) (s) (s) Thus, in the calculations below, we replace the terms of the form Γg + Γb by Γb . (s) (s) Given the form of the bootstrap assumptions, we may also replace r−1 Γb by Γg . ˇ R. ˇ We also include in We will denote l.o.t. the cubic and higher error terms in Γ, l.o.t. terms which decay faster in powers of r than the main quadratic terms. 2.3.3

The invariant quantity q

Note from the transformation formulas of Proposition 2.90 that the only quantities which remain invariant up to quadratic or higher order error terms are α, α and ρ. Among these only α, α vanish in Schwarzschild. We call such quantities O(2 ) invariant. In what follows we show that, in addition to these two invariants, there exist other important invariants. Lemma 2.96. The expression   1 e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω 2 − 8ω κ + κ2 α 2 is an O(2 ) invariant. It is also a conformal invariant, i.e., invariant under transformations (2.3.3) with f = f = 0. Proof. Clearly the quantity vanishes in Schwarzschild and is an O(2 ) invariant. For a conformal transformation, the result follows by a straightforward application of the transformation properties of Proposition 2.90 in the particular case where f = f = 0. Remark 2.97. Alternatively one can also define the corresponding quantity obtained by interchanging e3 , e4 , i.e.,   1 e4 (e4 (α)) + (2κ − 6ω)e4 (α) + −4e4 (ω) + 8ω 2 − 8ωκ + κ2 α. 2 Note that it differs by O(2 ) from the previous one. Definition 2.98. Given a general null frame (e4 , e3 , eθ ), and given a scalar function r satisfying the assumptions for section 2.3.2, i.e., 2 e4 (r) − κ ∈ Γg , r

1 eθ (r) ∈ Γg , r

2 e3 (r) − κ ∈ Γb , r

we define our main quantity q as " q :=r4 e3 (e3 (α)) + (2κ − 6ω)e3 (α)  # 1 2 + −4e3 (ω) + 8ω − 8ω κ + κ α . 2 

2

(2.3.10)

84 2.3.4

CHAPTER 2

Several identities for q

In this section, we state three identities involving the quantity q defined by (2.3.10). All calculations are made in a general frame. Proposition 2.99. We have   3 3 4 ? ? q = r /d2 /d1 ρ + κρϑ + κρϑ + Err[q] 4 4

(2.3.11)

with error term written schematically in the form Err[q]

= r4 e3 η · β + r2 d≤1 Γb · Γg ).

(2.3.12)

Proof. See section A.10 The following consequence of Proposition 2.99 will prove to be very useful in the sequel. Proposition 2.100. We have   3 3 3 3 e3 (rq) = r5 /d?2 /d?1 /d1 β − ρ /d?2 /d?1 κ − κρ /d?2 ζ − κρα + (2ρ2 − κκρ)ϑ 2 2 2 4 +Err[e3 (rq)], (2.3.13) where the error term Err[e3 (rq)] is given schematically by

Err[e3 (rq)] = rΓb q + r5 d≤1 e3 η · β + r3 d≤2 Γb · Γg .

(2.3.14)

Proof. See section A.11. We deduce from Proposition 2.100 the following nonlinear version of the TeukolskyStarobinsky identity. Proposition 2.101. The following identity holds true in (int) M:    3  2 2 7 ? ? e3 (r e3 (rq)) + 2ωr e3 (rq) = r /d2 /d1 /d1 /d2 α + ρ κe4 − κe3 α + Err[T S], 2 (2.3.15) where the error term Err[T S] is given schematically by  Err[T S] = r4 d/Γb + rΓb · Γb ) · α + r2 Γb e3 (rq) + (d≤1 Γb )rq

+ r7 d≤2 e3 η · β + r5 d≤3 Γb · Γg . Proof. See section A.12.

2.4

INVARIANT WAVE EQUATIONS

In this section, we write wave equations for the invariant quantities α, α and q.

85

PRELIMINARIES

2.4.1

Preliminaries

Lemma 2.102. With respect to a general S-foliation we have, for a reduced scalar ψ ∈ s0 ,     1 1 1 g ψ = − (e3 e4 + e4 e3 ) ψ + 4 / ψ + ω − κ e4 ψ + ω − κ e3 ψ 2 2 2 (2.4.1) + (η + η)eθ ψ. Also, g ψ

=

g ψ

=

 −e3 e4 ψ + 4 / ψ + 2ω −  −e4 e3 ψ + 4 / ψ + 2ω −

 1 κ e4 ψ − 2  1 κ e3 ψ − 2

1 κe3 ψ + 2ηeθ ψ, 2 1 κe4 ψ + 2ηeθ ψ. 2

Proof. We calculate, in spacetime, g ψ

= g34 D3 D4 ψ + g43 D4 D3 ψ + δ AB DA DB ψ 1 = − (D3 D4 + D4 D3 )ψ + gAB DA DB ψ. 2

Now, δ AB DA DB ψ D3 D4 ψ D4 D3 ψ

1 (1+3) 1 trχe3 ψ − (1+3)trχe4 ψ, 2 2 = e3 e4 ψ − 2ωe4 ψ − 2ηeθ ψ, = 4 /ψ −

= e4 e3 ψ − 2ωe3 ψ − 2ηeθ ψ.

Hence, g ψ

1 1 1 = − (e3 e4 + e4 e3 )ψ + 4 / ψ − (1+3)trχe3 ψ − (1+3)trχe4 ψ 2 2 2 + ωe4 ψ + ηeθ ψ + ωe3 ψ + ηeθ ψ   1 1 (1+3) = − (e3 e4 + e4 e3 )ψ + 4 /ψ + ω − trχ e4 ψ 2 2   1 + ω − (1+3)trχ e3 ψ + (η + η)eθ ψ. 2

Since 1 1 e4 e3 ψ = e3 e4 ψ + ωe3 ψ − ωe4 ψ + (η − η)eθ ψ 2 2 we also have 

g ψ Since κ =

=

 1 (1+3) 1 −e3 e4 ψ + 4 / ψ + 2ω − trχ e4 ψ − (1+3)trχe3 ψ + 2ηeθ ψ. 2 2

(1+3)

trχ, κ =

(1+3)

trχ, this concludes the proof of the lemma.

86

CHAPTER 2

Definition 2.103. Given a reduced k-scalar ψ ∈ sk we define 1 1 1 k ψ = − (e3 e4 + e4 e3 ) ψ + 4 / k ψ + (ω − trχ)e4 ψ + (ω − trχ)e3 ψ 2 2 2 + (η + η)eθ ψ.

(2.4.2)

Equivalently, we have k ψ k ψ

 = −e3 e4 ψ + 4 / k ψ + 2ω −  = −e4 e3 ψ + 4 / k ψ + 2ω −

 1 κ e4 ψ − 2  1 κ e3 ψ − 2

1 κe3 ψ + 2ηeθ ψ, 2 1 κe4 ψ + 2ηeθ ψ. 2

Remark 2.104. Not that the terms ηeθ ψ, ηeθ ψ have to be interpreted as in Remark 2.26, i.e., ηeθ ψ

=

1 η ( /dk ψ − /d?k+1 ψ) . 2

The term η /dk ψ is the reduced form of a tensor product of (1+3)η with D /k (1+3) ψ ? (1+3) while η /dk+1 ψ is the reduced form of a contraction between the 1-form η and ? (1+3) k + 1 tensor D /k+1 ψ. Remark 2.105. Recall that (see Definition 2.23), 4 / kf


2 := eθ (eθ f ) + eθ (Φ)eθ f − k 2 eθ (Φ) f.

Thus, for a ψ ∈ sk , we have,
2 4 / kψ = 4 / ψ − k 2 eθ (Φ) ψ. 2.4.1.1

Spacetime interpretation of Definition 2.103

The linearized equation verified by our main quantity q, which will be derived in the next section, has the form 2 ψ = V ψ

(2.4.3)

with V a scalar potential. In what follows we give simple spacetime interpretation of the equation (see Appendix D for more details). Given a mixed spacetime tensor in Tk M ⊗ TlS M of the form Uµ1 ...µk ,A1 ...AL where eµ is an orthonormal frame on M with (eA )A=1,2 tangent to S. We define ˙ µ Uν ...ν ,A ...A D 1 1 L k

= −

eµ Uν1 ...νk ,A1 ...Al − UDµ ν1 ...νk ,A1 ...Al − . . . − Uν1 ...Dµ νk ,A1 ...Al Uν1 ...νk ,D˙ µ A1 ...Al − Uν1 ...νk ,A1 ...D˙ µ Al

˙ µ A denoting the projection of De eA on S. One can easily check the comwith D µ mutator formulae ˙ µD ˙ ν −D ˙ νD ˙ µ )ΨA (D ˙ µD ˙ ν −D ˙ νD ˙ µ )ΨλA (D

=

RA B µν ΨB ,

=

Rλ σ µν ΨσA + RA B µν ΨλB .

87

PRELIMINARIES

Define ˙ µD ˙ ν Ψ. ˙ g Ψ := gµν D  Consider the following Lagrangian for Ψ = ΨAB ∈ S2 :   ˙ µ ΨA A D ˙ µ ΨB B + V ΨA A ΨB B . L[Ψ] = g/ A1 B1 g/ A2 B2 gµν D 1 2 1 2 1 2 1 2 Proposition 2.106. The Euler-Lagrange equations for the Lagrangian L[Ψ] above are given by ˙ =VΨ Ψ

(2.4.4)

and its reduced form ψ = Ψθθ is precisely (2.4.3). Proof. Straightforward verification. 2.4.2

Wave equations for α, α, and q

We start with the wave equations for α and α, which are derived in a general null frame. Proposition 2.107. The following identities hold true. 1. The invariant quantity α ∈ s2 verifies the Teukolsky wave equation, 2 α = −4ωe4 (α) + (4ω + 2κ)e3 (α) + V α + Err[g α], 1 V = −4ρ − 4e4 (ω) − 8ωω + 2ω κ − 10κ ω + κ κ, 2

(2.4.5)

where Err[g α] is given schematically by   Err(g α) = Γg e3 (α) + r−1 d≤1 (η, Γg )(α, β) + ξ(e3 (β), r−1 dˇ ρ) + l.o.t. where l.o.t. denote terms which are quadratic and enjoy better decay properties or are higher order and decay at least as good. 2. The invariant quantity α ∈ s2 verifies the Teukolsky wave equation, 2 α = −4ωe3 (α) + (4ω + 2κ)e4 (α) + V α + Err[g α], 1 V = −4ρ − 4e3 (ω) − 8ωω + 2ωκ − 10κ ω + κ κ, 2

(2.4.6)

where Err(g α)

= r−1 d(Γb α) + d(Γb β) + l.o.t.

Proof. See section A.13. We may now state the wave equation satisfied by q. Theorem 2.108. The invariant scalar quantity q defined in (2.3.10) verifies the

88

CHAPTER 2

equation 2 q + κκ q = Err[2 q]

(2.4.7)

where Err[2 q] is O(2 ). If q is defined relative to a null frame satisfying, in addition to the assumptions of section 2.3.2, that η ∈ Γg and ξ = 0 for r ≥ 4m0 , the error term is then given schematically by   Err[2 q] = r2 d≤2 (Γg · (α, β)) + e3 r3 d≤2 (Γg · (α, β)) +d≤1 (Γg · q) + l.o.t.

(2.4.8)

Proof. See section A.14. Remark 2.109. Note that the main frame used in this book is an outgoing geodesic null frame in r ≥ 4m0 so that ξ = 0, but unfortunately, as it turns out, η ∈ Γb . This would not allow us to control the error term appearing in (2.4.7). To overcome this problem, we are forced to define q relative to a different frame where ξ = 0 still holds for r ≥ 4m0 and for which we have in addition η ∈ Γg , see Proposition 3.26 for the existence of such a frame. See also the discussion at the beginning of section 3.4.6. The remark above leads us to the following. Remark 2.110. The quantity q we will be working with for the rest of the book is defined, according to equation (2.3.10), relative to the global frame of Proposition 3.26 for which η ∈ Γg . It is only in such a frame that q verifies the correct decay estimates.

Chapter Three Main Theorem 3.1

GENERAL COVARIANT MODULATED ADMISSIBLE SPACETIMES

Note that all definitions below are consistent with the framework of Z-invariant polarized spacetimes. 3.1.1

Initial data layer

Recall that m0 > 0 is given as the mass of the Schwarzschild solution to which the initial data is 0 close. Let δH > 0 be a sufficiently small constant which will be specified later. Definition 3.1 (Initial data layer). We consider a spacetime region (L0 , g), sketched below in figure 3.1, where • The metric g is a reduced metric from a Lorentzian spacetime metric g close to Schwarzschild in a suitable topology.1 • L0 = (ext) L0 ∪ (int) L0 . • The intersection (ext) L0 ∩ (int) L0 is nontrivial. Furthermore, our initial data layer (L0 , g) satisfies the following: 1. Boundaries. The future and past boundaries of L0 are given by ∂ + L0

∂ − L0

= A0 ∪ C (2,L0 ) ∪ C(2,L0 ) , = C(0,L0 ) ∪ C (0,L0 ) ,

where a) The past outgoing null boundary of the far region (ext) L0 is denoted by C(0,L0 ) . b) The past incoming null boundary of the near region (int) L0 is denoted by C (0,L0 ) . c) (ext) L0 is unbounded in the future outgoing null directions. d) The future outgoing null boundary of the far region (ext) L0 is denoted by C(2,L0 ) . e) The future incoming null boundary of the near region (int) L0 is denoted by C (2,L0 ) . f ) The future spacelike boundary of the near region (int) L0 is denoted by A0 .

2. Foliations of L0 and adapted null frames. The spacetime L0 =

(ext)

L0 ∪

1 This topology will be specified in our initial data layer assumptions, see (3.3.5) as well as section 3.2.4.

90

CHAPTER 3

(int)

L0 is foliated as follows:

a) The far region (ext) L0 is foliated by two functions (uL0 , (ext) sL0 ) such that • uL0 is an outgoing optical function on (ext) L0 whose leaves are denoted by C(uL0 ,L0 ) . • (ext) sL0 is an affine parameter along the level hypersurfaces of uL0 , i.e., (ext)

L0 ( (ext) sL0 ) = 1 where

(ext)

L0 := −g ab ∂b (uL0 )∂a .

• We denote by ( (ext) (e0 )3 , (ext) (e0 )4 , (ext) (e0 )θ ) the null frame adapted to the outgoing geodesic foliation (uL0 , (ext) sL0 ) on (ext) L0 . • Let (ext) rL0 denote the area radius of the 2-spheres S(uL0 , (ext) sL0 ) of this foliation. • The outgoing future null boundary C(2,L0 ) corresponds precisely to uL0 = 2 and the outgoing past null boundary C(0,L0 ) corresponds to uL0 = 0. • The foliation by uL0 of (ext) L0 terminates at the timelike boundary    δH (ext) . rL0 = 2m0 1 + 4 b) The near region

(int)

L0 is foliated by two functions (uL0 ,

• uL0 is an ingoing optical function on C(uL ,L0 ) .

(int)

(int)

sL0 ) such that

L0 whose leaves are denoted by

0

•

(int)

sL0 is an affine parameter along the level hypersurfaces of uL0 , i.e., (int)

L0 ( (int) sL0 ) = −1 where

(ext)

L0 := −g ab ∂b (uL0 )∂a .

• We denote by ( (int) (e0 )3 , (int) (e0 )4 , (int) (e0 )θ ) the null frame adapted to the outgoing geodesic foliation (uL0 , (int) sL0 ) on (int) L0 . • Let (int) rL0 denote the area radius of the 2-spheres S(uL0 , (int) sL0 ) of this foliation. • The (uL0 , (int) s) foliation is initialized on (ext) rL0 = 2m0 (1 + δ2H ) as it will be made precise below. • The foliation by uL0 of (int) L0 terminates at the spacelike boundary n o A0 = (int) rL0 = 2m0 (1 − 2δH ) where m0 and δH have been defined above. • The ingoing future null boundary C (2,L0 ) corresponds precisely to uL0 = 2 and the ingoing past null boundary C (0,L0 ) corresponds to uL0 = 0. • The foliation by uL0 of (int) L0 terminates at the timelike boundary n o (int) rL0 = 2m0 (1 + 2δH ) . 3. Initializations of the (uL0 , (int) sL0 ) foliation. The (uL0 , is initialized on (ext) rL0 = 2m0 (1 + δ2H ) by setting uL0

=

uL0 ,

(int)

sL0 =

(ext)

sL0

(int)

sL0 ) foliation

91

MAIN THEOREM

and, with λ0 = (ext) λ0 = 1 − (int)

(e0 )4 = λ0 (ext) (e0 )4 ,

2m0 (ext) r L0

(int)

,

(ext) (e0 )3 = λ−1 (e0 )3 , 0

(int)

(e0 )θ =

(ext)

(e0 )θ .

4. Coordinates system on (ext) L0 ( (ext) rL0 ≥ 4m0 ). In (ext) L0 ( (ext) rL0 ≥ 4m0 ), there exists adapted coordinates (uL0 , (ext) sL0 , θL0 , ϕ) with b = 0, see Proposition 2.75, such that the spacetime metric g takes the form g = −2duL0 d



(ext)



2

sL0 + ΩL0 (duL0 ) + γL0

 dθL0

1 − bL0 duL0 2

2

+ e2Φ dϕ2 . (3.1.1)

Figure 3.1: The initial data layer L0

3.1.2

Main definition

Recall that m0 > 0 is given as the mass of the Schwarzschild solution to which the initial data is 0 close, and that δH > 0 is a sufficiently small constant which will be specified later. Definition 3.2 (GCM admissible spacetime). We consider a spacetime (M, g), sketched in figure 3.2, where • The metric g is a reduced metric from a Lorentzian spacetime metric g close to

92

CHAPTER 3

Schwarzschild in a suitable topology.2 • M = (ext) M ∪ (int) M. • T = (ext) M ∩ (int) M is a timelike hypersurface. (M, g) is called a general covariant modulated admissible (or shortly GCM admissible) spacetime if it is defined as follows. 1. Boundaries. The future and past boundaries of M are given by ∂+M = −

∂ M =

A ∪ C ∗ ∪ C ∗ ∪ Σ∗ , C1 ∪ C 1 ,

where a) The past boundary C1 ∪ C 1 is included in the initial data layer L0 , defined in section 3.1.1, in which the metric on M is specified to be a small perturbation of the Schwarzschild data. b) The future spacelike boundary of the far region (ext) M is denoted by Σ∗ . c) The future outgoing null boundary of the far region (ext) M is denoted by C∗ . d) The future incoming null boundary of the near region (int) M is denoted by C∗. e) The future spacelike boundary of the near region (int) M is denoted by A. f ) The timelike boundary T , separating (ext) M from (int) M, starts at C 1 ∩ C1 and terminates at C ∗ ∩ C∗ .

2. Foliations of M and adapted null frames. The spacetime M = (int) M is foliated as follows:

(ext)

M∪

a) The far region (ext) M is foliated by two functions (u, (ext) s) such that • u is an outgoing optical function on (ext) M, initialized on Σ∗ , whose leaves are denoted by C(u). • (ext) s is an affine parameter along the level hypersurfaces of u, i.e., L( (ext) s) = 1 where L := −g ab ∂b u∂a . • The (u, (ext) s) foliation is initialized on Σ∗ as it will be made precise below. • We denote by ( (ext) e3 , (ext) e4 , (ext) eθ ) the null frame adapted to the outgoing geodesic foliation (u, (ext) s) on (ext) M where (ext) e4 = L. • Let (ext) r and (ext) m respectively the area radius and the Hawking mass of the 2-spheres S(u, (ext) s) of this foliation. • The outgoing future null boundary C∗ corresponds precisely to u = u∗ and the outgoing past null boundary C1 corresponds to u = 1. • The foliation by u of (ext) M terminates at the timelike boundary n o T = (ext) r = rT 2 This

3.2.

topology will be specified in our bootstrap assumptions, see (3.3.6) as well as section

93

MAIN THEOREM

where rT satisfies3     δH 3δH 2m0 1 + ≤ rT ≤ 2m0 1 + . 2 2 b) The near region (int) M is foliated by two functions (u, (int) s) such that • u is an ingoing optical function on (int) M, initialized on T , whose leaves are denoted by C(u). • (int) s is an affine parameter along the level hypersurfaces of u, i.e., L( (int) s) = −1 where L := −g ab ∂b u∂a . • The (u, (int) s) foliation is initialized on T as it will be made precise below. • We denote by ( (int) e3 , (int) e4 , (int) eθ ) the null frame adapted to the outgoing geodesic foliation (u, (int) s) on (int) M where (int) e3 = L. • Let (int) r and (int) m respectively the area radius and the Hawking mass of the 2-spheres S(u, (int) s) of this foliation. • The foliation by u of (int) M terminates at the spacelike boundary n o A = (int) r = 2m0 (1 − δH ) where m0 and δH have been defined above. • The ingoing future null boundary C ∗ corresponds precisely to u = u∗ and the ingoing past null boundary C 1 corresponds to u = 1.

3. GCM foliation of Σ∗ . The (u, (ext) s) foliation of spacelike hypersurface Σ∗ has the following properties:

(ext)

M restricted to the

a) There exists a constant cΣ∗ such that Σ∗ := {u +

(ext)

r = cΣ∗ }.

b) We have4 r  u∗ on Σ∗ . c)

(ext)

(3.1.2)

s satisfies5 (ext)

s=

(ext)

r on Σ∗ .

d) We say that Σ∗ is a general covariant modulated hypersurface6 (or shortly GCM hypersurface) if relative to the above defined null frame of (ext) M, the 3 A specific choice of r will be made in section 3.8.9, see (3.8.8), in the context of a Lebesgue T point argument needed to recover the top order derivatives. 4 See (3.3.4) for the precise condition. 5 Recall that (ext) s satisfies on (ext) M the transport equation L( (ext) s) = 1 and thus needs to be initialized on a hypersurface transversal to L, chosen here to be Σ∗ . 6 More generally, a GCM hypersurface is one with the property that we can specify, using the full covariance of the Einstein equations, a number of vanishing conditions (equal to the number ˇ of degrees of freedom of the diffeomorphism group) for well-chosen components of Γ.

94

CHAPTER 3

following conditions hold7 along Σ∗ : 2 κ = , /d?2 /d?1 κ = 0, /d?2 /d?1 µ = 0, Z Z r 2m Φ ηe = 0, ξeΦ = 0, a SP = −1 − , r S S

(3.1.3)

where a is the unique scalar function such that ν = e3 + ae4 is tangent to Σ∗ , and SP denotes the south poles of the spheres on Σ∗ . Moreover we also assume Z Z Φ βe = 0, eθ (κ)eΦ = 0, with S∗ := Σ∗ ∩ C∗ . (3.1.4) S∗

S∗

Note that the role of the GCM foliation of Σ∗ is to initialize the (u, (ext) s) foliation of (ext) M. e) In view of the definition of ν and ς, we have ν(u) = e3 (u) + ae4 (u) = 2/ς. ν being tangent to Σ∗ , u is thus transported along Σ∗ , and hence defined up to a constant. To calibrate u on Σ∗ , we fix the value u = 1 as follows: S1 = Σ∗ ∩ {u = 1} is such that S1 ∩ C(1,L0 ) ∩ SP 6= ∅,

(3.1.5)

i.e., S1 is the unique sphere of Σ∗ such that its south pole intersects the south pole of one of the spheres of the outgoing null cone C(1,L0 ) of the initial data layer. 4. Initialization the (u, tialized on T such that

(int)

s) foliation on T . The (u, (int)

u = u,

s=

(ext)

(int)

s) foliation is ini-

s.

In particular, the 2-spheres S(u, (int) s) coincide on T with S(u, (ext) s) and (int) r = (ext) r. Moreover, the null frame ( (int) e3 , (int) e4 , (int) eθ ) is defined on T by the following renormalization, (int)

e4 = λ (ext) e4 ,

(int)

e3 = λ−1

(ext)

(int)

e3 ,

eθ =

(ext)

eθ on T

where λ=

(ext)

λ=1−

2 (ext) m . (ext) r

Remark 3.3. In Schwarzschild, u = t − r∗ , u = t + r∗ , with (ext) (int)

7 The

e4 e4

= =

Υ−1 ∂t + ∂r ,

(ext)

∂t + Υ∂r ,

(int)

dr∗ dr

= Υ−1 , and

e3 = ∂t − Υ∂r ,

e3 = Υ−1 ∂t − ∂r .

existence of such hypersurfaces is an essential part of our construction.

HH

M

(int)

M

⌃⌃

) MM

M

nt ( ) in (i

(e

t) M

Mt)

+

xt(e ) xt

T

⇤

CC 1

1

C1 C

1

AA T

T

M

I+ I

⇤ ⇤

CC

(ext)

⇤

++

⇤

I+

C⇤C

M

(int)

t)

t) (e x M (i n

⌃⇤

M

(ext)

M

T I+

C1

I+

+

(i n

(e x t)

⌃A⇤ ⇤

⇤

H A

CC (int) 1⇤ M

I+

⌃⇤

C1

C1

I+C 1

Cxt) 1 M

(ext) C M 1⇤

C1

C

A

A

⌃⇤

C⇤

C1

C⇤

C1

C1

C⇤

+

H

⌃⇤

H+

C1

C⇤

C1

C⇤

C1

M

M

T

T

T

M

A

95

M

3.1.3

)

A

)

T

nt

A

nt

(i

xt )

C⇤

(i

+

(e

C⇤

I+

C1

⌃nt) ⇤M

C⇤

T

TI

C⇤

⌃⇤

A

(i

C⇤

C1

C⇤

(e

C⇤

) xt

C⇤

H+

I+

H+

H+

(e

MAIN THEOREM

H+

Figure 3.2: The GCM admissible spacetime M

Renormalized curvature components and Ricci coefficients

For convenience, we introduce in this section a notation for renormalized curvature components and Ricci coefficients. Definition 3.4 (Renormalized curvature and Ricci coefficients in (ext) M). We introduce the following notations in (ext) M n o n o (ext) ˇ (ext) ˇ R = α, β, ρˇ, µ ˇ, β, α , Γ= κ ˇ , ϑ, ζ, η, κ ˇ , ϑ, ω ˇ, ξ , where, recall, ρˇ = ρ − ρ, µ ˇ = µ − µ, κ ˇ = κ − κ, κ ˇ = κ − κ, ω ˇ = ω − ω, and ξ = ω = 0, η = −ζ. Note that all the above quantities are defined with respect to the outgoing geodesic foliation of (ext) M (see section 2.2.4), and that the averages are taken with respect to the corresponding 2-spheres. Definition 3.5 (Renormalized curvature and Ricci coefficients in introduce the following notations in (int) M n o (int) ˇ R = α, β, ρˇ, µ ˇ , β, α , n o (int) ˇ Γ = ξ, ω ˇ, κ ˇ , ϑ, ζ, η, κ ˇ, ϑ ,

(int)

M). We

96

CHAPTER 3

where we have defined ρˇ = ρ − ρ, µ ˇ = µ − µ, κ ˇ = κ − κ, κ ˇ = κ − κ, ω ˇ = ω − ω, and we recall that

2m = 0. r3 Note that all the above quantities are defined with respect to the ingoing geodesic foliation of (int) M (see section 2.2.6), and that the averages are taken with respect to the corresponding 2-spheres. ξ = ω = 0, η = ζ, µ −

Remark 3.6. In Schwarzschild, we have (ext)

3.2

ˇ = 0, R

(int)

ˇ = 0, R

(ext) ˇ

Γ = 0,

(int) ˇ

Γ = 0.

MAIN NORMS

3.2.1

Main norms in

(ext)

M

All quantities appearing in this section are defined relative to the (ext) M frame adapted to the (u, (ext) s) foliation. In particular, recall that with respect to this frame, we have ξ = ω = 0, η = −ζ. Recall the definition (2.1.36) of higher order angular derivatives d/s of reduced scalars in sk . We introduce the notations d = {e3 , re4 , d/}. Definition 3.7. We introduce the vectorfield T defined on    1 2m T := 1− e4 + e3 . 2 r We also introduce the vectorfield N defined on (ext) M by    1 2m N := 1− e4 − e3 . 2 r Remark 3.8. In Schwarzschild, we have   2m0 T = ∂t , N = 1 − ∂r r in the standard (t, r, θ, ϕ) coordinates. We are ready to introduce our norms in

(ext)

M.

(ext)

M as (3.2.1)

(3.2.2)

97

MAIN THEOREM

L2 curvature norms in

3.2.1.1

(ext)

M

Let δB > 0 a small constant to be specified later. We introduce the weighted curvature norms, Z  2   (ext) ≥4m0 ˇ R0 [R] := sup r4+δB α2 + r4 β 2 1≤u≤u∗ Cu (r≥4m0 ) Z   + r4+δB (α2 + β 2 ) + r4 (ˇ ρ)2 + r2 β 2 + α2 ZΣ∗  + r3+δB (α2 + β 2 ) + r3−δB (ˇ ρ)2 + r1−δB β 2 (ext) M(r≥4m

0)

 +r−1−δB α2 ,



(ext)

0 ˇ R≤4m [R] 0

Z

2

:=

 2 3m ˇ 2, 1− |R| r (ext) M(r≤4m ) 0

and (ext)

ˇ := R0 [R]

(ext)

0 ˇ R≥4m [R] + 0

(ext)

0 ˇ R≤4m [R]. 0

For any nonzero integer k, we introduce the following higher derivatives norms 

(ext)

ˇ Rk [R]

2

:=



(ext)

ˇ R0 [d≤k R]

2

Z



+ (ext) M(r≤4m

0)

 ˇ 2 + |d≤k−1 R| ˇ2 . |d≤k−1 NR|

Remark 3.9. Note that the derivative in the N Rdirection, unlike all other first ˇ appears in the spacetime integral (ext) derivatives of R, with top number M(r≤4m0 ) of derivatives. This reflects the fact the N-derivatives do not degenerate at r = 3m in the Morawetz estimate. 3.2.1.2

L2 Ricci coefficients norms in

(ext)

M

For any k ≥ 2, we introduce the following norms Z "      2 (ext) ≥4m0 ˇ G Γ := r2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2 + (d≤k κ ˇ )2 k

Σ∗

# +

≤k

(d

≤k

2

ϑ) + (d

sup λ≥4m0 2−δB

≤k

η) + (d "

Z +

2

2

≤k

ω ˇ ) + (d

ξ)

2

  λ2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2

{r=λ} ≤k

(d

κ ˇ )2 + (d≤k ϑ)2 + (d≤k η)2 + (d≤k ω ˇ )2 #!

+

λ

+

λ−δB (d≤k ξ)2

,

98

CHAPTER 3



(ext)

  2 0 ˇ G≤4m Γ k

Z

≤k
2 ˇ , d Γ

:= (ext) M(≤4m

0)

and (ext)

3.2.1.3

  ˇ := Gk Γ

Decay norms in

(ext)

(ext)

  0 ˇ G≤4m Γ + k

(ext)

  0 ˇ G≥4m Γ . k

M

Let δdec > 0 a small constant to be specified later. We define   1 (ext) D0 [α] := sup r2 (2r + u)1+δdec + r3 (2r + u) 2 +δdec |α|, (ext) M   1 (ext) D0 [β] := sup r2 (2r + u)1+δdec + r3 (2r + u) 2 +δdec |β|, (ext) M   1 (ext) D0 [ˇ ρ] := sup r2 u1+δdec + r3 u 2 +δdec |ˇ ρ|, (ext) M

(ext)

D0 [ˇ µ]

sup r3 u1+δdec |ˇ µ|,

:=

(ext) M

(ext)

D0 [β]

sup r2 u1+δdec |β|,

:=

(ext) M

(ext)

D0 [α]

sup ru1+δdec |α|,

:=

(ext) M

and (ext)

ˇ D0 [R]

(ext)

:=

D0 [α] +

(ext)

+ (ext) D0 [β] +

D0 [β] +

(ext)

(ext)

D0 [ˇ ρ] +

(ext)

D0 [ˇ µ]

D0 [α].

Also, we introduce the following higher derivatives norms (ext)

ˇ D1 [R]

:= + +

ˇ + (ext) D0 [dR] ˇ D0 [R]   1 sup r3 (2r + u)1+δdec + r4 (2r + u) 2 +δdec |e3 (α)| (ext) M   1 sup r3 u1+δdec + r4 u 2 +δdec |e3 (β)| + sup r3 u1+δdec |e3 (ˇ ρ)|, (ext)

(ext) M

(ext) M

and for any integer k ≥ 2 (ext)

ˇ := Dk [R]

(ext)

ˇ D1 [d≤k−1 R].

99

MAIN THEOREM

Also, we define (ext)

D0 [ˇ κ]

sup r2 u1+δdec |ˇ κ|,   1 sup ru1+δdec + r2 u 2 +δdec |ϑ|, (ext) M   1 sup ru1+δdec + r2 u 2 +δdec |ζ|, (ext) M   1 sup ru1+δdec + r2 u 2 +δdec |ˇ κ|,

:=

(ext) M

(ext)

D0 [ϑ]

(ext)

(ext)

:=

D0 [ζ]

:=

D0 [ˇ κ]

:=

(ext) M

(ext)

D0 [ϑ]

sup ru1+δdec |ϑ|,

:=

(ext) M

(ext)

D0 [η]

sup ru1+δdec |η| +

:=

Z

(ext) M

(ext)

D0 [ˇ ω]

u2+2δdec η 2

 12 ,

Σ∗

sup ru1+δdec |ˇ ω |,

:=

(ext) M

(ext)

D0 [ξ]

sup ru1+δdec |ξ|,

:=

(ext) M

and (ext)

ˇ D0 [Γ]

:=

(ext)

+

D0 [ˇ κ] +

(ext)

(ext)

D0 [η] +

D0 [ϑ] +

(ext)

(ext)

D0 [ˇ ω] +

D0 [ζ] +

(ext)

(ext)

D0 [ˇ κ] +

(ext)

D0 [ϑ]

D0 [ξ].

Also, we introduce the following higher derivatives norms (ext)

ˇ := D1 [Γ]

(ext)

ˇ + sup r2 u1+δdec |e3 (ϑ, ζ, κ D0 [dΓ] ˇ )| (ext) M

and for any integer k ≥ 2 (ext)

ˇ := Dk [Γ]

(ext)

ˇ D1 [d≤k−1 Γ].

Remark 3.10. The integral bootstrap assumption on Σ∗ for η will only be needed in the proof of Proposition 3.20 and recovered in Proposition 7.22. In fact, other components satisfy an analog integral estimate on Σ∗ : this is the case of ϑ, ξ and rβ, see Proposition 7.22. But η is the only component for which we need to make this type of bootstrap assumption. 3.2.2

Main norms in

(int)

M

All quantities appearing in this section are defined relative to the adapted to the (u, (int) s) foliation. 3.2.2.1

L2 based norms in

(int)

M

We introduce the curvature norms  2 (int) ˇ R0 [R]

Z := (int) M

ˇ 2. |R|

(int)

M frame

100

CHAPTER 3

For any nonzero integer k, we introduce the following higher derivatives norms (int)

ˇ := Rk [R]

(int)

ˇ R0 [d≤k R].

For any k ≥ 0, we introduce the following norms Z  2 (int) ˇ ˇ 2. Gk [Γ] := |d≤k Γ| (int) M

3.2.2.2

Decay norms in

(int)

M

We define (int)

ˇ := sup u1+δdec |R|, ˇ D0 [R]

(int)

(int) M

ˇ := sup u1+δdec |Γ|. ˇ D0 [Γ] (int) M

Also, we introduce the following higher derivatives norms for any integer k ≥ 1 (int)

3.2.3

ˇ := Dk [R]

(int)

ˇ D0 [d≤k R],

(int)

ˇ := Dk [Γ]

(int)

ˇ D0 [d≤k Γ].

Combined norms

We define the following norms M by combining our above norms on (int) M (En)

Nk

(Dec) Nk

3.2.4

:= :=

ˇ + Rk [R] (ext) ˇ + Dk [R] (ext)

ˇ + Gk [Γ] (ext) ˇ + Dk [Γ]

(ext)

ˇ + Rk [R] (int) ˇ + Dk [R]

(int)

(int)

(ext)

M and

ˇ Gk [Γ], ˇ Dk [Γ].

(int)

Initial layer norm

Recall the notations of section 3.1.1 concerning the initial data layer L0 . Recall that the constant m0 > 0 is the mass of the initial Schwarzschild spacetime relative to which our initial perturbation is measured. We define the initial layer norm to be8 Ik 8 Recall

:=

(ext)

Ik +

(int)

Ik + I0k

that the initial data layer foliations satisfy η + ζ = 0, as well as ξ = ω = 0 on and η = ζ as well as ξ = ω = 0 on (int) L0 .

(ext) L

0,

101

MAIN THEOREM

where (ext)

 ρ + 2m0 + r2 |β| + r|α| r3 (ext) L 0
! 0 2 1 − 2m 2 2 r sup r |ϑ| + κ − + |ζ| + κ + r r (ext) L 0   m 0 sup r |ϑ| + ω − 2 + |ξ| r (ext) L 0 " γ sup r 2 − 1 + r|b| + |Ω + Υ| + |ς − 1| r (ext) L (ext) r ≥4m 0( 0 0) Φ # e +r − 1 , r sin θ  7 sup r 2 +δB (|α| + |β|) + r3

:=

I0

+ + +

(int)

:=

I0

+

I00

:=

 2m0 sup |α| + |β| + ρ + 3 + |β| + |α| r (int) L 0 "
2m0 2 1− r sup |ϑ| + κ − + |ζ| + κ + r (int) L 0 # m0 + ω + 2 + |ξ| , r 

sup (int) L

(ext) L 0∩ 0


|f | + |f | + | log(λ−1 0 λ)| ,

2 + |ϑ| r

λ0 = (ext) λ0 = 1 −

2m0 (ext) r L0

,

with Ik the corresponding higher derivatives norms obtained by replacing each component by d≤k of it. In the definition of I00 above, (f, f , λ) denote the transition functions of Lemma 2.87 from the frame of the outgoing part (ext) L0 of the initial data layer to the frame of the ingoing part (int) L0 of the initial data layer in the region (int) L0 ∩ (ext) L0 . Remark 3.11. Note that in the definition of (ext) Ik we allow a higher power of r in front of α, β and their derivatives than what is consistent with the results of [20] and [42]. The additional rδB power, for δB small, is consistent instead with the result of [43].

3.3 3.3.1

MAIN THEOREM Smallness constants

Before stating our main theorem, we first introduce the following constants that will be involved in its statement. • The constant m0 > 0 is the mass of the initial Schwarzschild spacetime relative to which our initial perturbation is measured. • The integer klarge corresponds to the maximum number of derivatives of the

102

CHAPTER 3

solution. • The size of the initial data layer norm is measured by 0 > 0. • The size of the bootstrap assumption norms are measured by  > 0. • δH > 0 measures the width of the region |r − 2m0 | ≤ 2m0 δH where the redshift estimate holds and which includes in particular the region (int) M. ˇ and R. ˇ • δdec is tied to decay estimates in u, u for Γ • δB is involved in the r-power of the rp -weighted estimates for curvature. In what follows, m0 is a fixed constant, δH , δB , and δdec are fixed, sufficiently small, universal constants, and klarge is a fixed, sufficiently large, universal constant, chosen such that 0 < δH , δdec , δB  min{m0 , 1},

δB > 2δdec ,

Then,  and 0 are chosen such that  0 ,   min δH , δdec , δB ,

1 klarge

klarge 

 , m0 , 1

1 . δdec

(3.3.1)

(3.3.2)

and 2

 = 03 .

(3.3.3)

Using the definition of 0 , we may now specify the behavior (3.1.2) of r on Σ∗ −2

dec inf r = 0 3 u1+δ . ∗

Σ∗

(3.3.4)

From now on, in the rest of the book, . means bounded by a constant depending only on geometric universal constants (such as Sobolev embeddings, elliptic estimates,...) as well as the constants m0 , δH , δdec , δB , klarge but not on  and 0 . 3.3.2

Statement of the main theorem

We are now ready to give the following precise version of our main theorem. Main Theorem (Main theorem, version 2). There exists a sufficiently large integer klarge and a sufficiently small constant 0 > 0 such that given an initial layer defined as in section 3.1.1 and satisfying the bound 5

Iklarge +5 ≤ 03 ,

(3.3.5)

there exists a globally hyperbolic development with a complete future null infinity I+ and a future horizon H+ together with foliations and adapted null frames verifying the admissibility conditions of section 3.1.2 such that the following bound is satisfied (En)

(Dec)

Nklarge + Nksmall ≤ C0

(3.3.6)

103

MAIN THEOREM

where C is a large enough universal constant and where ksmall is given by   1 ksmall = klarge + 1. (3.3.7) 2 In particular, • On

(ext)

M, we have 

|α|, |β| . |ˇ ρ| . |β| . |α| .

0

0 min , 2 1 +δ 3 dec r (u + 2r)1+δdec r (u + 2r) 2   0 0 min , , 1 r3 u 2 +δdec r2 u1+δdec 0 , r2 u1+δdec 0 , 1+δ ru dec

 ,

and 0 , dec r2 u1+δ   0 0 , |ϑ|, |ζ|, |ˇ κ| . min 1 r2 u 2 +δdec ru1+δdec 0 |η|, |ϑ|, |ˇ ω |, |ξ| . . 1+δ ru dec |ˇ κ| .

• On

(int)

ˇ = {ˇ ˇ = {α, β, ρˇ, β, α}, M we have, with Γ κ, ϑ, ζ, η, κ ˇ , ϑ, ω ˇ , ξ}, R ˇ R| ˇ . |Γ,

0 1+δ u dec

.

• The Bondi mass converges as u → +∞ along I+ to the final Bondi mass which we denote by m∞ . The final Bondi mass verifies the estimate m∞ m0 − 1 . 0 . In particular m∞ > 0. • The Hawking mass m satisfies 0 |m − m∞ | u1+δdec . 0  m0  1+δdec u   

on

(ext)

on

(int)

M,

M.

• The location of the future horizon H+ satisfies ! √ 0 r = 2m∞ + O on H+ . δdec u1+ 2

104 • On

• On

CHAPTER 3

(ext)

(int)

M, we have ρ + 2m∞ 3 r κ − 2 r
2 1 − 2mr∞ κ + r m ∞ ω − 2 r

(ext)

g

0

,

0



1 r3 u 2 +δdec r2 u1+δdec

,

0 , r2 u1+δdec   0 0 . min , 1 r2 u 2 +δdec ru1+δdec 0 . . 1+δ ru dec .

M, we have

ρ + 2m∞ , κ + 3 r • On by

 . min


2 1 − 2mr∞ 2 m∞ 0 , κ − , + ω . 1+δdec . 2 r r r u

M, the spacetime metric g is given in the (u, r, θ, ϕ) coordinates system =

gm∞ , (ext) M + O



0 u1+δdec

  (dr, du, rdθ)2 , r2 (sin θ)2 (dϕ)2

where gm∞ , (ext) M denotes the Schwarzschild metric of mass m∞ > 0 in outgoing Eddington-Finkelstein coordinates, i.e.,     2m∞ gm∞ , (ext) M := −2dudr − 1 − (du)2 + r2 (dθ)2 + (sin θ)2 (dϕ)2 . r • On by

(int)

M, the spacetime metric g is given in the (u, r, θ, ϕ) coordinates system 

g

= gm∞ , (int) M + O

0 u1+δdec



(dr, du, rdθ)2 , r2 (sin θ)2 (dϕ)2



where gm∞ , (ext) M denotes the Schwarzschild metric of mass m∞ > 0 in ingoing Eddington-Finkelstein coordinates, i.e.,     2m∞ gm∞ , (int) M := 2dudr − 1 − (du)2 + r2 (dθ)2 + (sin θ)2 (dϕ)2 . r Note that analog statements of the above estimates also hold for dk derivatives with k ≤ ksmall . Remark 3.12. In this book, we choose to specify the closeness to Schwarzschild of our initial data in the context of the Characteristic Cauchy problem. Note that the conclusions of our main theorem can be immediately extended to the case where the data are specified to be close to Schwarzschild on a spacelike hypersurface Σ. Indeed, one can reduce this latter case to our situation by invoking • The results in [42], [43] which allow us to control the causal region between Σ

105

MAIN THEOREM

and the outgoing part of the initial data layer.9 • A standard local existence result which controls the finite causal region between Σ and the ingoing part of the initial data layer. Remark 3.13. In the context of the previous remark, we note that the constant m0 > 0 appearing in the initial data layer norm of the assumption (3.3.5) of our main theorem does not necessarily coincide with the ADM mass of the corresponding initial data set on the spacelike hypersurface Σ. With respect to this ADM mass, we would recover the well known inequality stating that the final Bondi mass is smaller than the ADM mass. Remark 3.14. For most of the proof, it is sufficient to assume the following weaker analog of (3.3.5) for the initial data layer Iklarge +5 ≤ 0 . The only place where we need the stronger assumption (3.3.5) on the initial data layer is in section 8.1, see Remark 8.1.

3.4 3.4.1

BOOTSTRAP ASSUMPTIONS AND FIRST CONSEQUENCES Main bootstrap assumptions (En)

We assume that the combined norms Nk the following bounds:

(Dec)

and Nk

defined in section 3.2 verify

BA-E (Bootstrap Assumptions on energies and weighted energies) (En)

Nklarge ≤ ,

(3.4.1)

BA-D (Bootstrap Assumptions on decay) (Dec)

Nksmall ≤ .

(3.4.2)

In the remainder of section 3.4.1, we state several simple consequences of the bootstrap assumptions which will be proved in Chapter 4. 3.4.2

Control of the initial data

While the smallness constant involved in the bootstrap assumptions is  > 0, we need the smallness constant involved in the control of the initial data to be 0 > 0. This is achieved in the theorem below. Theorem M0. Assume that the initial data layer L0 , as defined in section 3.1.1, satisfies Iklarge +5 ≤ 0 . 9 Note

that the results of [43] are consistent with our initial data layer assumptions.

106

CHAPTER 3

Then under the bootstrap assumptions BA-D on decay, the following holds true on the initial data hypersurface C1 ∪ C 1 : ( h 7   i 9 max sup r 2 +δB |dk (ext) α| + |dk (ext) β| + r 2 +δB |dk−1 e3 ( (ext) α)| 0≤k≤klarge

C1



+ sup r3 C1

  ) k (ext) 2m 0 d ρ + 3 + r2 |dk (ext) β| + r|dk (ext) α| . 0 , r

"

  2m0 sup |dk (int) α| + |dk (int) β| + dk (int) ρ + 3 0≤k≤klarge C r 1 # max

+|dk (int) β| + |dk (int) α|

.

0 ,

and m sup − 1 . 0 . m 0 C ∪C 1

3.4.3

1

Control of averages and of the Hawking mass

The following two lemma are simple consequences of the bootstrap assumptions and will be proved in section 4.2. Lemma 3.15 (Control of averages). Assume given a GCM admissible spacetime M as defined in section 3.1.2 verifying the bootstrap assumption for some sufficiently small  > 0. Then, we have       2 2m 3 ≤ksmall sup u1+δdec r3 d≤ksmall κ − + r d ρ + . 0 , r r3 (ext) M      2Υ m  1+δdec 2 ≤ksmall 2 ≤ksmall sup u r d κ+ + r d ω− 2 . 0 , r r (ext) M       2 2m 3 ≤klarge 3 ≤klarge sup u r d κ− + r d ρ+ 3 . r r (ext) M      1 2Υ m  2 ≤klarge sup u 2 +δdec r2 d≤klarge κ + + r ω − . d r r2 (ext) M 1 2 +δdec

0 , 0 ,

      ≤k 2Υ ≤ksmall 2m small sup u κ− + d ρ+ 3 . 0 , d r r (int) M     2 ≤ksmall  m  sup u1+δdec d≤ksmall κ + + ω + . 0 , d r r2 (int) M 1+δdec

107

MAIN THEOREM

      ≤k 2Υ ≤klarge 2m large sup u κ− + d ρ+ 3 d r r (int) M      1 2 ≤klarge m  +δdec ≤klarge 2 sup u κ+ + d ω+ 2 d r r (int) M 1 2 +δdec

. 0 , . 0 .

Also, we have 

 1 sup u1+δdec r d≤ksmall Ω + Υ + u 2 +δdec r d≤klarge Ω + Υ . 0 , (ext) M

sup (int) M





 1 u1+δdec d≤ksmall Ω − Υ + u 2 +δdec d≤klarge Ω − Υ . 0 .

Finally, recall that µ and µ are given by the following formula µ=

2m r3

(ext)

on

M,

µ=

2m r3

on

(int)

M.

Lemma 3.16 (Control of the Hawking mass). Assume given a GCM admissible spacetime M as defined in section 3.1.2 verifying the bootstrap assumption for some sufficiently small  > 0. Then, we have   max sup u1+δdec |dk e3 (m)| + r|dk e4 (m)| . 0 , 0≤k≤klarge (ext) M   max sup u1+δdec |dk e3 (m)| + |dk e4 (m)| . 0 . 0≤k≤klarge

(int) M

The e4 derivatives behave better in powers of r, max

0≤k≤ksmall

max

0≤k≤klarge

sup r2 u1+δdec |dk e4 (m)| . 0 ,

(ext) M

1

sup r2 u 2 +δdec |dk e4 (m)| . 0 .

(ext) M

Moreover, m sup − 1 M m0 3.4.4

. 0 .

Control of coordinates system

The following two propositions on the existence of a suitable coordinates system both in (ext) M and in (int) M are also consequences of the bootstrap assumptions and will be proved in section 4.3. Proposition 3.17 (Control of a coordinates system on the Z-invariant scalar on M defined by (2.2.52), i.e., θ = cot−1 (reθ (Φ)) .

(ext)

M). Let θ ∈ [0, π] be (3.4.3)

Consider the (u, r, θ, ϕ) coordinates system introduced in Proposition 2.78. Then, relative to these (u, r, θ, ϕ) coordinates,

108

CHAPTER 3

1. The spacetime metric takes the form  2 4ς ς 2 (κ + A) 2 1 b g = − dudr + du + γ dθ − ςbdu − Θ rκ κ 2 2

(3.4.4)

where b = e4 (θ),

b = e3 (θ),

γ=

1 (eθ (θ))2

(3.4.5)

and 4 Θ= dr − ς rκ



κ+A κ

 du.

2. The reduced coordinates derivatives take the form √ 2 γ 2 ∂r = e4 − beθ , rκ rκ √ ∂θ = γeθ ,       1 1κ+A 1√ κ+A ∂u = ς e3 − e4 − γ b− b eθ . 2 2 κ 2 κ

(3.4.6)

3. The following estimates hold true:  1    γ   max sup ru 2 +δdec + u1+δdec dk 2 − 1 + r dk b . 0≤k≤ksmall (ext) M r
ˇ + dk (ς − 1) + r dk b . max sup u1+δdec dk Ω 0≤k≤ksmall

, .

(ext) M

Also, eΦ satisfies max

0≤k≤ksmall



sup

ru

1 2 +δdec

1+δdec

+u

(ext) M

   eΦ dk − 1 r sin θ

. .

Proposition 3.18 (Control of a coordinates system on (int) M). Let θ ∈ [0, π] be the Z-invariant scalar on M defined by (3.4.3). Consider the (u, r, θ, ϕ) coordinates system introduced in Proposition 2.81. Then, relative to these (u, r, θ, ϕ) coordinates, 1. The spacetime metric takes the form  2 4ς ς 2 (κ + A) 2 1 b g = − dudr + du + γ dθ − ςbdu − Θ rκ κ 2 2

(3.4.7)

where b = e4 (θ),

b = e3 (θ),

γ=

1 (eθ (θ))2

and 4 Θ := dr − ς rκ



κ+A κ

 du.

(3.4.8)

109

MAIN THEOREM

2. The reduced coordinates derivatives take the form √ 2 γ 2 ∂r = e3 − beθ , rκ rκ √ ∂θ = γeθ ,       1 1κ+A 1√ κ+A ∂u = ς e4 − e3 − γ b− b eθ . 2 2 κ 2 κ 3. The following estimates hold true: "   γ 1+δdec k ˇ max sup u d Ω + dk (ς − 1) + dk 2 − 1 0≤k≤ksmall (int) M r # k k + d b + d b

(3.4.9)

.

.

Also, eΦ satisfies max

0≤k≤ksmall

3.4.5

sup u

1+δdec

(int) M

 Φ  k e d − 1 . r sin θ

.

Pointwise bounds for higher order derivatives

We will need later to interpolate between the estimates provided by the bootstrap assumptions on decay and the bootstrap assumptions on energy. To this end, we will need the following consequence of the bootstrap assumptions on weighted energies. Proposition 3.19. The Ricci coefficients and curvature components satisfy the following pointwise estimates on M: n 7 δB

max sup r 2 + 2 |dk α| + |dk β| + r3 |dk µ| + |dk ρˇ| k≤klarge −5 M
+r2 |dk κ ˇ | + |dk ζ| + |dk ϑ| + |dk κ ˇ | + |dk β|
o +r |dk η| + |dk ϑ| + |dk ω ˇ | + |dk ξ| + |dk α| . . 3.4.6

Construction of a second frame in

(ext)

M

Recall that the quantity q satisfies the following wave equation, see (2.4.7), 2 q + κκq

=

Err[2 q]

where the nonlinear term Err[2 q] has the schematic structure exhibited in (2.4.8). Also, recall that according to our bootstrap assumption on decay and Proposition 3.19, η satisfies on (ext) M |d≤ksmall η| ≤

 ru1+δdec

,

 |d≤klarge −5 η| . . r

As discussed in Remark 2.109, this decay in r−1 is too weak to derive suitable decay for q. We thus need to provide another frame for (ext) M. This is the aim of the

110

CHAPTER 3

following proposition. Proposition 3.20. Let an integer kloss and a small constant δ0 > 0 satisfying10 16 ≤ kloss ≤

δdec (klarge − ksmall ), 3

δ0 :=

Let (e4 , e3 , eθ ) the outgoing geodesic null frame of frame (e04 , e03 , e0θ ) of (ext) M provided by

kloss . klarge − ksmall

(ext)

(3.4.10)

M. There exists another

1 e04 = e4 + f eθ + f 2 e3 , 4 1 0 eθ = eθ + f e3 , 2 e03 = e3 , such that the Ricci coefficients and curvature components with respect to that frame satisfy ξ 0 = 0,

max

0≤k≤ksmall +kloss

sup (ext) M

( 

 1 r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk Γ0g |

+ru1+δdec −2δ0 |dk Γ0b |   2 2Υ 0 0 0 0 +r2 u1+δdec −2δ0 dk−1 e03 κ0 − , κ0 + ,ϑ ,ζ ,η ,η r r  7 δB  1 + r 2 + 2 + r3 u 2 +δdec −2δ0 + r2 u1+δdec −2δ0 |dk (α0 , β 0 )|  9 δB  1 + r 2 + 2 + r3 u1+δdec + r4 u 2 +δdec −2δ0 |dk−1 e03 (α0 )|   1 + r3 u1+δdec + r4 u 2 +δdec −2δ0 |dk−1 e03 (β 0 )|   1 + r3 u 2 +δdec −2δ0 + r2 ru1+δdec −2δ0 |dk ρˇ0 | )   1+δdec −2δ0 2 k 0 k 0 +u r |d β | + r|d α | . , 10 Recall

from (3.3.1) and (3.3.7) that we have  0 < δdec  1,

δdec klarge  1,

ksmall =

 1 klarge + 1. 2

In particular, we have δdec (klarge − ksmall )  1 and hence there exists an integer kloss satisfying the required constraints.

111

MAIN THEOREM

where we have used the notation11   2 0 0 0 0 0 2Υ −1 0 0 0 0 −1 0 0 Γg = rω , κ − , ϑ , ζ , η , η , κ + , r (e4 (r) − 1), r eθ (r), e4 (m) , r r n o m Γ0b = ϑ0 , ω 0 − 2 , ξ 0 , r−1 (e03 (r) + Υ), r−1 e03 (m) . r Furthermore, f satisfies on |dk f | . |dk−1 e03 f | .

(ext)

M

 ru

1 2 +δdec −2δ0

+ u1+δdec −2δ0

 ru1+δdec −2δ0

,

for k ≤ ksmall + kloss + 2,

(3.4.11)

for k ≤ ksmall + kloss + 2.

Remark 3.21. The crucial point of Proposition 3.20 is that in the new null frame (e04 , e03 , e0θ ) of (ext) M, η 0 belongs to Γ0g and thus displays a better decay in r−1 than η corresponding to the outgoing geodesic frame (e4 , e3 , eθ ) of (ext) M. 3.5

GLOBAL NULL FRAMES

In this section, we construct 2 smooth global frames on M by matching the frame of (int) M on the one hand with a renormalization of the frame on (ext) M, and on the other hand, with a renormalization of the second frame of (ext) M given by Proposition 3.4.6. 3.5.1

Extension of frames

To construct the first global null frame, we need to extend the null frame of (int) M, i.e., ((int) e4 , (int) e3 , (int) eθ ), slightly into (ext) M, and the null frame of (ext) M, i.e., ((ext) e4 , (ext) e3 , (ext) eθ ), slightly into (int) M. We keep the same labels for the extended frame, i.e., ((int) e4 , (int) e3 , (int) eθ ) represents the extended frame of (int) M in (ext) M and vice versa. This convention also applies to the Ricci coefficients, curvature components, area radius and Hawking mass of the extended frames. Note that these extensions require, in addition to the initialization of the frames on T , to initialize 1. ((ext) e4 , (ext) e3 , (ext) eθ ) on C ∗ by ((ext) e4 , (ext) e3 , (ext) eθ ) = (((int) Υ)−1(int) e4 , (int) Υ(int) e3 , (int) eθ ). 2. ((int) e4 , (int) e3 , (int) eθ ) on C∗ by ((int) e4 , (int) e3 , (int) eθ ) = ((ext) Υ(ext) e4 , ((ext) Υ)−1(ext) e3 , (ext) eθ ). 11 Here, r and m denote respectively the area radius and the Hawking mass of the outgoing geodesic foliation of (ext) M, i.e., r = (ext) r and m = (ext) m. In particular, while eθ (r) = eθ (m) = 0, we have in general e0θ (r) 6= 0 and e0θ (m) 6= 0.

112

CHAPTER 3

3.5.2

Construction of the first global frame

We start with the definition of the region where the frame of renormalization of the frame of (ext) M will be matched.

(int)

M and a conformal

Definition 3.22. We define the matching region as the spacetime region     3 (ext) (int) Match := M∩ r ≤ 2m0 1 + δH 2     1 (int) (int) ∪ M∩ r ≥ 2m0 1 + δH , 2 where, as explained in the previous section, (int) r denotes the area radius of the ingoing geodesic foliation of (int) M and its extension to (ext) M. Here is our main proposition concerning our first global frame. Proposition 3.23. There exists a global null frame defined on and denoted by ((glo) e4 , (glo) e3 , (glo) eθ ) such that a) In

(ext)

(int)

M∪

(ext)

M

M \ Match, we have

((glo) e4 , (glo) e3 , (glo) eθ ) = b) In

(int)



(ext)

 Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ .

M \ Match, we have ((glo) e4 , (glo) e3 , (glo) eθ ) =



(int)

 e4 , (int) e3 , (int) eθ .

c) In the matching region, we have ˇ (glo) R) ˇ . u1+δdec dk ((glo) Γ, 0≤k≤ksmall −2 Match∩ (int) M ˇ (glo) R) ˇ . max sup u1+δdec dk ((glo) Γ, max

sup

0≤k≤ksmall −2 Match∩ (ext) M

Z max

0≤k≤klarge −1

where

(glo)

ˇ and R

Match

1 k (glo) ˇ (glo) ˇ 2 2 Γ, R) d (

.

, , ,

(glo) ˇ

Γ are given by   2m (glo) ˇ R = α, β, ρ + 3 , β, α , r   m 2Υ 2 (glo) ˇ Γ = ξ, ω + 2 , κ − , ϑ, ζ, η, η, κ + , ϑ, ω, ξ . r r r

d) Furthermore, we may also choose the global frame such that, in addition, one of the following two possibilities hold: i. We have on all

(ext)

M

((glo) e4 , (glo) e3 , (glo) eθ ) =



(ext)

 Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ .

113

MAIN THEOREM

ii. We have on all

(int)

M

((glo) e4 , (glo) e3 , (glo) eθ ) =



(int)

 e4 , (int) e3 , (int) eθ .

Remark 3.24. The global frame on M of Proposition 3.23 will be used to construct the second global frame in the next section, see Proposition 3.26. It will also be used to recover higher order derivatives in Theorem M8 (stated in section 3.6.2), see section 8.3.2. 3.5.3

Construction of the second global frame

We start with the definition of the region where the first global frame of M (i.e., the one of Proposition 3.23) and a conformal renormalization of the second frame of (ext) M (i.e., the one of Proposition 3.20) will be matched. Definition 3.25. We define the matching region as the spacetime region   7m0 Match0 := (ext) M ∩ ≤ (ext) r ≤ 4m0 , 2 where

(ext)

r denotes the area radius of the outgoing geodesic foliation of

(ext)

M.

Here is our main proposition concerning our second global frame. Proposition 3.26. Let an integer kloss and a small constant δ0 > 0 satisfy0 0 0 ing (3.4.10). There exists a global null frame ((glo ) e4 , (glo ) e3 , (glo ) eθ ) defined on (int) M ∪ (ext) M such that a) In

(ext)

M ∩ { (ext) r ≥ 4m0 }, we have   0 0 0 ((glo ) e4 , (glo ) e3 , (glo ) eθ ) = (ext) Υ (ext) e04 , (ext) Υ−1(ext) e03 , (ext) e0θ ,

where ((ext) e04 , (ext) e03 , (ext) e0θ ) denotes the second frame of of Proposition 3.20. 0 b) In (int) M ∪ ( (ext) M ∩ { (ext) r ≤ 7m 2 }), we have 0

0

(ext)

M, i.e., the frame

0

((glo ) e4 , (glo ) e3 , (glo ) eθ ) = ((glo) e4 , (glo) e3 , (glo) eθ ), where ((glo) e4 , (glo) e3 , (glo) eθ ) denotes the first global frame of M, i.e., the frame of Proposition 3.23. c) In the matching region, we have 0 ˇ (glo0 ) R) ˇ . , max sup u1+δdec −2δ0 dk ((glo ) Γ, 0≤k≤ksmall +kloss Match0

where

(glo0 )

ˇ and R (glo0 )

ˇ R

(glo0 ) ˇ

Γ

(glo0 ) ˇ

Γ are given by   2m = α, β, ρ + 3 , β, α , r   m 2Υ 2 = ξ, ω + 2 , κ − , ϑ, ζ, η, η, κ + , ϑ, ω, ξ , r r r

114

CHAPTER 3

with the Ricci coefficients and curvature components being the one associated to 0 0 0 the frame ((glo ) e4 , (glo ) e3 , (glo ) eθ ). d) Furthermore, we may also choose the global frame such that, in addition, one of the following two possibilities hold: 0 M ∩ { (ext) r ≥ 15m 4 }   0 0 0 ((glo ) e4 , (glo ) e3 , (glo ) eθ ) = (ext) Υ (ext) e04 , (ext) Υ−1(ext) e03 , (ext) e0θ .

i. We have on

(ext)

ii. We have on

(int)

M ∪ ( (ext) M ∩ { (ext) r ≤

0

0

15m0 4 })

0

((glo ) e4 , (glo ) e3 , (glo ) eθ ) = ((glo) e4 , (glo) e3 , (glo) eθ ). Remark 3.27. The global frame on M of Proposition 3.26 will be needed to derive decay estimates for the quantity q in Theorem M1 (stated in section 3.6.1).

3.6 3.6.1

PROOF OF THE MAIN THEOREM Main intermediate results

We are ready to state our main intermediary results. Theorem M1. Assume given a GCM admissible spacetime M as defined in section 3.1.2 verifying the bootstrap assumptions12 BA-E and BA-D for some sufficiently small  > 0. Then, if 0 > 0 is sufficiently small, there exists δextra > δdec such that we have the following estimates in M: n 1  o max sup ru 2 +δextra + u1+δextra |dk q| + ru1+δextra |dk e3 q| 0≤k≤ksmall +20

(ext) M

+

max

0≤k≤ksmall +20

sup u1+δextra |dk q| . 0 .

(int) M

Moreover, q also satisfies the following estimate: Z max u2+2δextra |dk q|2 0≤k≤ksmall +21 (int) M(≥u) Z + max u2+2δextra |dk e3 q|2 0≤k≤ksmall +20

. 20 .

Σ∗ (≥u)

Theorem M2. Under the same assumptions as above we have the following decay estimates for (ext) α: max

0≤k≤ksmall +20

sup (ext) M

 r2 (2r + u)1+δextra

1

+ r3 (2r + u) 2 +δextra



log(1 + u)   × |dk (ext) α| + r|dk e3 (ext) α|

. 0 .

12 Recall in particular that the conclusions of Theorem M0 hold under the bootstrap assumptions BA-E and BA-D.

115

MAIN THEOREM

Theorem M3. Under the same assumptions as above we have the following decay estimates for α: Z (int) Dksmall +16 [α] . 0 , max u2+2δextra |dk α|2 . 20 . 0≤k≤ksmall +18

Σ∗

Theorem M4. Under the same assumptions as above we also have the following decay estimates in (ext) M: (ext)

ˇ + Dksmall +8 [R]

(ext)

ˇ . 0 . Dksmall +8 [Γ]

Theorem M5. Under the same assumptions as above we also have the following ˇ and Γ ˇ in (int) M: decay estimates for R (int)

ˇ + Dksmall +5 [R]

(int)

ˇ . 0 . Dksmall +5 [Γ]

Note that, as an immediate consequence of Theorem M2 to Theorem M5, we have obtained, under the same assumptions as above, the following improvement of our bootstrap assumptions on decay: (Dec)

Nksmall +5 . 0 . 3.6.2

(3.6.1)

End of the proof of the main theorem

Definition 3.28 (Definition of ℵ(u∗ )). Let 0 > 0 and  > 0 be given small constants satisfying the constraint (3.3.3). Let ℵ(u∗ ) be the set of all GCM admissible spacetimes M defined in section 3.1.2 such that • u∗ is the value of u on the last outgoing slice C∗ , • u∗ satisfies (3.3.4), • the bootstrap assumptions (3.4.1) and (3.4.2) hold true, i.e., relative to the combined norms defined in section 3.2.3, we have (En)

(Dec)

Nklarge ≤ , Nksmall ≤ . Definition 3.29. Let U be the set of all values of u∗ ≥ 0 such that the spacetime ℵ(u∗ ) exists. The following theorem shows that U is not empty. Theorem M6. There exists δ0 > 0 small enough such that for sufficiently small constants 0 > 0 and  > 0 satisfying the constraints (3.3.3) and (3.3.4), we have [1, 1 + δ0 ] ⊂ U. In view of Theorem M6, we may define U∗ as the supremum over all value of u∗ that belongs to U. U∗ := sup u∗ . u∗ ∈U

Assume by contradiction that U∗ < +∞.

116

CHAPTER 3

Then, by the continuity of the flow, U∗ ∈ U. Furthermore, according to the consequence (3.6.1) of Theorem M2 to Theorem M5, the bootstrap assumptions on decay (3.4.2) on any spacetime of ℵ(U∗ ) are improved by (Dec)

Nksmall +5 . 0 . To reach a contradiction, we still need an extension procedure for spacetimes in ℵ(u∗ ) to larger values of u, as well as to improve our bootstrap assumptions on weighted energies (3.4.1). This is done in two steps. Theorem M7. Any GCM admissible spacetime in ℵ(u∗ ) for some 0 < u∗ < +∞ such that (Dec)

Nksmall +5 . 0 , has a GCM admissible extension (satisfying (3.3.4)), i.e., u0∗ > u∗ , initialized by Theorem M0, which verifies (Dec)

Nksmall . 0 . Remark 3.30. Recall that the definition of a GCM admissible spacetime in section 3.1.2 is such that T = {r = rT } for some rT satisfying     δH 3δH 2m0 1 + ≤ rT ≤ 2m0 1 + . (3.6.2) 2 2 All results obtained so far, in particular Theorems M0–M7, hold for any choice of rT satisfying (3.6.2), see Remark 8.2 for a more precise statement. It is at this stage, in Theorem M8 below, that we need to make a specific choice of rT in the context of a Lebesgue point argument required for the control of top order derivatives. This choice will be made in (8.3.2). Theorem M8. There exists a choice of rT satisfying (3.6.2) such that the GCM admissible spacetime exhibited in Theorem M7 satisfies in addition (En)

Nklarge . 0 and therefore belongs to ℵ(u0∗ ). In particular u0∗ belongs to U. In view of Theorem M8, we have reached a contradiction, and hence U∗ = +∞ so that the spacetime may be continued forever. This concludes the proof of the main theorem.

3.6.3 3.6.3.1

Conclusions The Penrose diagram of M

Complete future null infinity. We first deduce from our estimate that our spacetime M has a complete future null infinity I+ . The portion of null infinity of

117

MAIN THEOREM

M corresponds to the limit r → +∞ along the leaves Cu of the outgoing geodesic foliation of (ext) M. As Cu exists for all u ≥ 0 with suitable estimates, it suffices to prove that u is an affine parameter of I+ . To this end, recall from our main (Dec) theorem that the estimates Nksmall . 0 hold which implies in particular13   m sup ru1+δdec |ξ| + ω − 2 + r−1 |ς − 1| r (ext) M

. 0 .

(3.6.3)

As |m − m0 | . 0 m0 , see Lemma 3.16, m is bounded. We infer that lim

Cu ,r→+∞

ξ, ω = 0 for all 1 ≤ u < ∞.

In view of the identity D3 e3

= −2ωe3 + 2ξeθ ,

we infer that e3 is a null geodesic generator of I+ . Since we have e3 (u) = 2ς with |ς − 1| . 0 in view of (3.6.3), u is an affine parameter of I+ so that I+ is indeed complete. (Dec)

Existence of a future event horizon. Next, note that the estimates Nksmall . 0 also imply
! 2m 2 1 − 2 r sup u1+δdec κ + + κ − . 0 . r r (int) M In particular, considering the spacetime region r ≤ 2m0 (1 − δH /2) of (int) M, and in view of the estimate |m − m0 | . 0 m0 , we infer, for all r ≤ 2m0 (1 − δH /2), that r − 2m 2 + O(0 ) ≤ 2 (r − 2m0 + 2m0 − 2m) + O(0 ) r2 r 2m0 (−δH + 0 ) + O(0 ). r2

κ ≤ 2 ≤

Thus, since 0 < 0  δH  1, we deduce, sup    δH (int) M r≤2m 0 1− 2

κ ≤

−

≤

−

δH 2m0 1 −


δH 2 2

+ O(0 )

δH . 4m0

Thus, all 2-spheres S(u, s) of the ingoing geodesic foliation of (int) M which are located in the spacetime region r ≤ 2m0 (1 − δH /2) of (int) M are trapped. This implies that the past of I+ in M does not contain this region, and hence M contains the event horizon H+ of a black hole in its interior. Moreover, since the timelike hypersurface T is foliated by the outgoing null cones Cu of (ext) M, it is in the past of I+ . Hence, since T is one of the boundaries of (int) M, H+ is actually located in the interior of the region (int) M. 13 Using

also Proposition 3.17 for the control of ς.

118

1

CHAPTER 3

Asymptotic stationarity of M. Recall that we have introduced a vectorfield T in (ext) M as well as one in (int) M by T = e3 + Υe4 in

(ext)

M,

(int)

T = e4 + Υe3 in

M.

ˇ e3 (m), e4 m. Thus, We can easily express all components of (T ) π in terms of Γ, (Dec) making use of the estimate Nksmall . 0 of our main theorem, we deduce |(T) π| .

0 in ru1+δdec

(ext)

0 in u1+δdec

M and |(T) π| .

(int)

M.

H+ C⇤ C⇤

A

H+ In particular, T is an asymptotically Killing vectorfield H+and hence our spacetime M is asymptotically stationary. C⇤ C⇤ The above conclusions regarding I+ and H+ allow us to draw the Penrose C⇤ C⇤ diagram of M, see figure 3.3. H+ A A C⇤ C1 C1 C⇤ C1 C1 A ⌃⇤ ⌃⇤ C1

C1

C1

H+

⌃⇤

C⇤

I+

nt

T

3.6.3.2

)

M

tH ) M +

⇤

C )M

T

⇤

t (i n

M

C1

A

(int)

t) (e x M

1

T C⇤

(i n

I+

M

C

M

xt )

M

nt

)

T

I+

(e x t)

I+

⌃⇤

M

(ext)

xt )

M

T

(i

M

⌃⇤

(i

(e

⇤

T

C⇤ M

I+

(e

I+

⇤

+

⌃

1

⌃

A

C (int)

CH1

M

C1

)

C1

nt

C

C1

M

A

) xt

C⇤

(e

(i

(ext)

T

I+

Figure 3.3: The Penrose diagram of the spacetime M

Limits at null infinity and Bondi mass

Recall the following formula for the derivative of the Hawking mass in (ext) M, see Proposition 2.71,  Z  r 1 2 1 2 e4 (m) = − κϑ − κ ˇ ϑϑ + 2ˇ κρˇ + 2eθ (κ)ζ + 2κζ . 32π S 2 2

119

MAIN THEOREM

(Dec)

As a simple corollary of the decay estimates of our main theorem, i.e., Nksmall . 0 , we deduce 20

|e4 (m)| .

r2 u1+2δdec

.

(3.6.4)

Since r−2 is integrable, we infer the existence of a limit to m as r → +∞ along Cu MB (u)

lim m(u, r) for all 1 ≤ u < +∞

=

r→+∞

where MB (u) is the so-called Bondi mass. Next, we recall the following formula in 1 e4 (ϑ) + κϑ 2

=

(ext)

M, see Proposition 2.63,

1 2 /d?2 ζ − κϑ + 2ζ 2 . 2

(Dec)

In view of Nksmall . 0 , we deduce |e4 (rϑ)| .

0 1 r2 u 2 +δdec

.

Since r−2 is integrable, we infer the existence of a limit to rϑ as r → +∞ along Cu Θ(u, ·)

=

lim rϑ(r, u, ·) for all 1 ≤ u < +∞.

r→+∞

(Dec)

On the other hand, in view of Nksmall . 0 again, 0 , u1+δdec

r|ϑ| .

on

(ext)

M.

We infer that |Θ(u, ·)| . 3.6.3.3

0 for all 1 ≤ u < +∞. u1+δdec

The spheres at null infinity are round

The Gauss curvature is given by the formula K

1 1 = −ρ − κκ + ϑϑ. 4 4

Thus, in view of our estimates in (ext) M, K − 1 . 2 r

0 1 r3 u 2 +δdec

so that lim r2 K = 1.

r→+∞

In particular the spheres at null infinity are round.

120 3.6.3.4

CHAPTER 3

A Bondi mass formula

Using the formula for e3 (m) in (ext) M, see Proposition 2.71, together with the (Dec) estimates Nksmall . 0 , we deduce Z 2 e3 (m) + r κϑ . 64π S

20 ru

3 2 +2δdec

and hence Z 2 e3 (m) + 1 (rϑ) 8|S| S

.

20 ru

3 2 +2δdec

.

Letting r → +∞ along Cu , and using that the spheres at null infinity are round, we infer in view of the definition of MB and Θ Z 1 e3 (MB )(u) = − Θ2 (u, ·) for all 1 ≤ u < +∞. 8 S2 Since e3 (u) = 2ς and e3 is orthogonal to the spheres foliating I+ , we infer e3 = 2ς ∂u . Thus, we obtain the following Bondi mass type formula Z ς ∂u MB (u) = − Θ2 (u, ·) for all 1 ≤ u < +∞, 16 S2 with ς satisfying (3.6.3). 3.6.3.5

Final Bondi mass

In view of the estimate |Θ(u, ·)| .

0 for all 1 ≤ u < +∞, u1+δdec

and the control for ς in (3.6.3), we infer that |∂u MB (u)| .

20 u2+2δdec

for all 1 ≤ u < +∞.

In particular, since u−2−2δdec is integrable, the limit along I+ exists MB (+∞) = lim MB (u) u→+∞

and is the so-called final Bondi mass. We denote it as m∞ , i.e., m∞ = MB (+∞). Control of m − m∞ . We have as a consequence of the above estimate for ∂u MB and the definition of m∞ |MB (u) − m∞ | .

20 1+2δ dec u

for all 1 ≤ u < +∞.

121

MAIN THEOREM

Also, recall from (3.6.4) that we have obtained in |e4 (m)| .

(ext)

M

20 2 1+2δ dec r u

which yields, together with the definition of MB (u), by integration in r at fixed u |m(r, u) − MB (u)| .

20 ru1+2δdec

in

(ext)

M.

We infer sup u1+2δdec |m − m∞ | . 20 .

(3.6.5)

(ext) M

Also, recall the following formula for the derivative of the Hawking mass in see Proposition 2.71 in the context of an outgoing geodesic foliation,  Z  r 1 2 1 2 e3 (m) = − κϑ − κ ˇ ϑϑ + 2ˇ κρˇ − 2eθ (κ)ζ + 2κζ . 32π S 2 2

(int)

M,

(Dec)

Together with the estimates Nksmall . 0 , we deduce |e3 (m)| .

20 u2+2δdec

on

(int)

M

and hence by integration in r at fixed u, for r ∈ [2m0 (1 − δH ), rT ],   m(r, u) − m rT , u .

20 u2+2δdec

m0 δH

on

(int)

According to (3.6.5), since {r = rT } = T = (ext) M ∩ (int) M ⊂ u = u in T by the initialization of u,   u1+2δdec m rT , u − m∞ . 20 .

M.

(ext)

M, and since

We deduce sup u1+2δdec |m − m∞ | .

20 .

(3.6.6)

(int) M

Combining (3.6.5) and (3.6.6) with the estimate sup |m − m0 | .

0 m0 ,

M

in the statement of our main theorem (see also Lemma 3.16), we infer that |m∞ − m0 | . 0 m0 . In particular we deduce that m∞ > 0 since 0 can be made arbitrarily small.

122

CHAPTER 3

3.6.3.6

Coordinates systems on

(ext)

M and

(int)

M

In view of Proposition 3.17, and together with the control of the averages κ, κ (Dec) provided by Lemma 3.15, the control of κ ˇ provided by the estimates Nksmall . 0 , and the control of m − m∞ obtained in (3.6.5), we infer for the spacetime metric g on (ext) M in the (u, r, θ, ϕ) coordinates system     0 g = gm∞ , (ext) M + O 1+δ (dr, du, rdθ)2 , r2 (sin θ)2 (dϕ)2 u dec where gm∞ , (ext) M denotes the Schwarzschild metric of mass m∞ > 0 in outgoing Eddington-Finkelstein coordinates, i.e.,     2m∞ gm∞ , (ext) M = −2dudr − 1 − (du)2 + r2 (dθ)2 + (sin θ)2 (dϕ)2 . r Also, in view of Proposition 3.18, and together with the control of the averages κ, (Dec) κ provided by Lemma 3.15, the control of κ ˇ provided by the estimates Nksmall . 0 , and the control of m − m∞ obtained in (3.6.6), we infer for the spacetime metric g on (int) M in the (u, r, θ, ϕ) coordinates system    0 2 2 2 2 g = gm∞ , (int) M + O (dr, du, rdθ) , r (sin θ) (dϕ) u1+δdec where gm∞ , (ext) M denotes the Schwarzschild metric of mass m∞ > 0 in ingoing Eddington-Finkelstein coordinates, i.e.,     2m∞ gm∞ , (int) M = 2dudr − 1 − (du)2 + r2 (dθ)2 + (sin θ)2 (dϕ)2 . r Asymptotic of the future event horizon. We show below that H+ is located in the following region of (int) M !   √ √ 0 0 2m 1 − 1+δ ≤ r ≤ 2m 1 + on H+ for any 1 ≤ u < +∞.(3.6.7) δdec u dec u1+ 2 Note first that the lower bound follows from the fact that √

sup  (int) M

 r≤2m 1−

√

0 u1+δdec



κ ≤ ≤

−

0 1+δdec

u

 2 + O √ 0 m 1 − u1+δdec √ 0 − < 0. 1+δ 2m0 u dec



0



u1+δdec

Concerning the upper bound, we need to show that any 2-sphere    √  0  S(u1 ) := S u1 , r = 2m 1 + , 1 ≤ u1 < +∞ δ 1+ dec u1 2

(3.6.8)

is in the past of I+ . Since (ext) M is in the past of I+ , it suffices to show that the forward outgoing null cone emanating from any 2-sphere (3.6.8) reaches (ext) M in

123

MAIN THEOREM

finite time. Assume, by contradiction, that there exists an outgoing null geodesic, denoted by γ, perpendicular to S(u1 ), that does not reach (ext) M in finite time. Let e04 be the geodesic generator of γ. In view of Lemma 2.87 on general null frame transformation, and denoting by (e4 , e3 , eθ ) the null frame14 of (int) M, we look for e04 under the form   1 e04 = λ e4 + f eθ + f 2 e3 , 4 and the fact that e04 is geodesic implies the following transport equations along γ for f and λ in view of Lemma 2.92 (applied15 with f = 0): λ−1 e04 (f ) +

κ

 + 2ω f

2 λ−1 e04 (log(λ))

=

−2ξ + E1 (f, Γ),

=

2ω + E2 (f, Γ),

where E1 and E2 are given schematically by E1 (f, Γ) E2 (f, Γ)

1 = − ϑf + l.o.t., 2 1 1 = f ζ − f 2 ω − ηf − f 2 κ + l.o.t. 2 4

Here, l.o.t. denote terms which are cubic or higher order in f and Γ denotes the Ricci coefficients w.r.t. the original null frame (e3 , e4 , eθ ) of (int) M. We then proceed as follows. 1. First, we initialize f and λ as follows on the γ ∩ S(u1 ): λ = 1 on γ ∩ S(u1 ).

f = 0,

2. Then, we initiate a continuity argument by assuming for some  u1 < u2 < u1 + that we have √ 0 |f | ≤ 1 +δ , u12 dec

Υ≥

√

0

1+ 2u1

δdec 2

,

u1 0

 δdec 2

0 < λ < +∞ on γ(u1 , u2 ) ∩

(int)

M (3.6.9)

where γ(u1 , u2 ) denotes the portion of γ in u1 ≤ u ≤ u2 . 3. We have 1 2 λ−1 e04 (u) = e4 (u) + f 2 e3 (u) = . 4 ς Relying on our control of the ingoing geodesic foliation of 14 Recall (int) M.

15 That

that we assume by contradiction that γ does not reach

is, we keep the direction of e3 fixed.

(int)

(ext) M

M, the above

and hence stays in

124

CHAPTER 3

assumption for f and the transport equation for f , we obtain on γ(u1 , u2 ) ∩ (int) M sup γ(u1 ,u2 )∩ (int) M

|f | . .

0 1+δdec u1 1−

δdec 2

1+

δdec 2

0

u1

(u2 − u1 )

which improves our assumption in (3.6.9) on f . 4. We have in view of the control of f λ−1 e04 (r)

1 = e4 (r) + f 2 e3 (r) = Υ + O 4

!

0 dec u1+δ 1

.

This yields λ−1 e04 (log(Υ))

2m r 2 e4 (r)

=

2m r2 Υ

− 2r λ−1 e04 (m) Υ  

+O

=

0 1+δdec

u1

.

Υ

Thanks to our assumption on the lower bound of Υ, we infer λ−1 e04 (log(Υ)) and since we are in

(int)

=

√ 2m (1 + O( 0 )) 2 r

M λ−1 e04 (log(Υ)) ≥

1 . 3m0

Integrating from u = u1 , we deduce   √ 0 u − u1 Υ ≥ exp δ √ 1+ dec 3m0 (1 + 0 )u1 2 which is an improvement of our assumption in (3.6.9) on Υ. 5. In view of the control of f and of the ingoing geodesic foliation of rewrite the transport equation for λ as λ−1 e04 (log(λ))

=

2ω + E2 (f, Γ)

=

2m − 2 +O r

0

!

dec u1+δ 1

On the other hand, since we have obtained above λ−1 e04 (log(Υ))

=

√ 2m (1 + O( 0 )) 2 r

.

(int)

M, we

125

MAIN THEOREM

we immediately infer λ−1 e04 (log(λ)Υ2 ) > 0,

√ λ−1 e04 (log(λ) Υ) < 0.

Integrating from u = u1 , this yields √

(1 +

√

√

!2

0

Υ

δ 1+ dec 2

−2

0 )u

≤λ≤

(1 +

√

! 12

0

1

δ 1+ dec 2

Υ− 2 .

0 )u

Since Υ has an explicit lower bound in view of our previous estimate, as well as an explicit upper bound since we are in (int) M, this yields an improvement of our assumptions in (3.6.9) for λ. 6. Since we have improved all our bootstrap assumptions (3.6.9), we infer by a continuity argument the following bound       δdec √ 2 0 u − u1 u 1  ∩ (int) M. Υ ≥ exp on γ u1 , u1 + δ √ 1+ dec 3m0 0 2 (1 + 0 )u1 Now, in this u interval, we may choose     δdec u1 u3 := u1 + 3m0 1 + log 2 0 for which we have Υ ≥ 1. This is a contradiction since Υ = O(δH ) in (int) M. Thus, we deduce that γ reaches (ext) M before u = u3 , a contradiction to our assumption on γ. This concludes the proof of (3.6.7).

3.7

THE GENERAL COVARIANT MODULATION PROCEDURE

The role of this section is to give a short description of the results concerning our General Covariant Modulation (GCM) procedure, which is at the heart of our proof. We will apply it in (ext) M under our main bootstrap assumptions BA-E and BA-D. The results stated in this section will be proved in Chapter 9. 3.7.1

Spacetime assumptions for the GCM procedure

To state our results, which are local in nature, it is convenient to consider axially symmetric polarized spacetime regions R foliated by two functions (u, s) such that • On R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. • We denote by (e3 , e4 , eθ ) the null frame adapted to the outgoing geodesic foliation (u, s) on R. ◦

• We denote by S a fixed sphere of R ◦

S ◦

◦ ◦

:= S(u, s)

(3.7.1)

◦

and by r the area radius of S, where S(u, s) denote the 2-spheres of the outgoing geodesic foliation (u, s) on R.

126

CHAPTER 3

• In adapted coordinates (u, s, θ, ϕ) with b = 0, see Proposition 2.75, the spacetime metric g in R takes the form, with Ω = e3 (s), b = e3 (θ), 2 1 g = −2ςduds + ς Ωdu + γ dθ − ςbdu + e2Φ dϕ2 , 2 2



2

(3.7.2)

where θ is chosen such that b = e4 (θ) = 0. • The spacetime metric induced on S(u, s) is given by g/ = γdθ2 + e2Φ dϕ2 .

(3.7.3)

• The relation between the null frame and coordinate system is given by e4 = ∂s ,

e3 =

2 ∂u + Ω∂s + b∂θ , ς

eθ = γ −1/2 ∂θ .

(3.7.4)

◦

• We denote the induced metric on S by ◦

◦

g/ = γ dθ2 + e2Φ dϕ2 . ◦

◦

◦ ◦

Definition 3.31. Let 0 < δ ≤  two sufficiently small constants. Let (u, s) real numbers so that ◦

1 ≤ u < +∞,

◦

4m0 ≤ s < +∞.

(3.7.5)

◦ ◦

We define R = R(δ, ) to be the region n ◦ R := |u − u| ≤ δR ,

o ◦ |s − s| ≤ δR ,

◦ ◦
− 1 2

δR := δ 

,

(3.7.6)

◦

such that assumptions A1–A3 below, with constant  on the background foliation ◦

of R, are verified. The smaller constant δ controls the size of the GCM quantities as it will be made precise below. Consider the renormalized Ricci and curvature components associated to the (u, s) geodesic foliation of R:  

2 2Υ m ˇ ˇ:= κ Γ ˇ , ϑ, ζ, η, κ − , κ + ,κ ˇ , ϑ, ξ, ω ˇ , ω − 2 , Ω, Ω+Υ , ς +1 , r r r   ˇ : = α, β, ρˇ, ρ + 2m , β, α . R r3 Since our foliation is outgoing geodesic we also have ξ = ω = 0,

η + ζ = 0.

(3.7.7)

127

MAIN THEOREM

ˇ = Γg ∪ Γb where We decompose Γ   2 2Υ Γg = κ ˇ , ϑ, ζ, κ ˇ, κ − , κ + , r r n

o m ˇ r−1 ςˇ, r−1 Ω + Υ , r−1 ς − 1 . Γb = η, ϑ, ξ, ω ˇ , ω − 2 , r−1 Ω, r

(3.7.8)

Given an integer smax , we assume the following:16 A1. For k ≤ smax , we have on R ◦

kΓg kk,∞ . r−2 ,

(3.7.9)

◦

kΓb kk,∞ . r−1 ,

and ◦

kα, β, ρˇ, µ ˇkk,∞ . r−3 , ◦

ke3 (α, β)kk−1,∞ . r−4 ,

(3.7.10)

◦

kβkk,∞ . r−2 , ◦

kαkk,∞ . r−1 . A2. We have, with m0 denoting the mass of the unperturbed spacetime, m ◦ sup − 1 . . (3.7.11) R m0 A3. The metric coefficients are assumed to satisfy the following assumptions in R, for all k ≤ smax

 

γ

eΦ ◦

r 2 − 1, b, −1 + kΩ + Υk∞,k + kς − 1k∞,k . . (3.7.12)

r r sin θ ∞,k We will assume, in addition, that there exists scalar functions C = C(u, s), M = M (u, s) such that the following small GCM conditions hold true on R, κ − 2 + dk κ ˇ + r dk−1 ( /d?1 κ − CeΦ ) r ◦ +r2 dk−1 ( /d?1 µ − M eΦ ) . δr−2 for all k ≤ smax ,(3.7.13) r

−2

Z ηeΦ . S

16 In

◦

δ,

r

−2

Z ◦ ξeΦ . δ.

(3.7.14)

S

applications, smax = ksmall + 4 in Theorem M7, and smax = klarge + 5 in Theorem M0 and Theorem M6.

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CHAPTER 3

Also,   2 2m + Ω − 1 − ς r SP Additionally we may assume on R Z Z ◦ ◦ Φ Φ r βe . δ, r eθ (κ)e . δ, S

3.7.2

S

◦

. δ.

(3.7.15)

Z ◦ Φ r eθ (κ)e . δ.

(3.7.16)

S

Deformations of surfaces ◦

◦

Definition 3.32. We say that S is an O() Z-polarized deformation of S if there ◦

exists a map Ψ : S → S of the form ◦  ◦ ◦ ◦ Ψ(u, s, θ, ϕ) = u + U (θ), s + S(θ), θ, ϕ

(3.7.17)

where U, S are smooth functions defined on the interval [0, π] of amplitude at most ◦ . We denote by ψ the reduced map defined on the interval [0, π], ◦

◦

ψ(θ) = (u + U (θ), s + S(θ), θ).

(3.7.18)

We restrict ourselves to deformations which fix the south pole, i.e., U (0) = S(0) = 0. 3.7.3

(3.7.19)

Adapted frame transformations

We consider general null transformations introduced in Lemma 2.87,   1 e04 = λ e4 + f eθ + f 2 e3 , 4     1 1 1 1 e0θ = 1 + f f eθ + f e4 + f 1 + f f e3 , 2 2 2 4      1 1 1 1 e03 = λ−1 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 . 2 16 4 4

(3.7.20)

◦

Definition 3.33. Given a deformation Ψ : S → S we say that a new frame (e03 , e04 , e0θ ), obtained from the standard frame (e3 , e4 , eθ ) via the transformation (3.7.20), is S-adapted if we have e0θ = eS θ =

1 ψ# (∂θ ) (γ S )1/2

(3.7.21)

where ψ# (∂θ ) is the push-forward defined by the deformation map ψ. The condition translates into the following relations between the functions U, S

129

MAIN THEOREM

defining the deformation and the transition functions (f, f ).
S # 1/2

#

ς ∂θ U = (γ ) ∂θ S −

f

#



 1 # 1 + (f f ) , 4


1/2 # ς# # 1 Ω ∂θ U = (γ S )# f , 2 2  # 1 2 S # # # 2 (γ ) = γ + (ς ) Ω + b γ (∂θ U )2 4

(3.7.22)

− 2ς # ∂θ U ∂θ S − (γςb)# ∂θ U,

U (0) = S(0) = 0. 3.7.4

GCM results

Theorem 3.34 (GCMS-I). Consider the region R as above, verifying the assump◦

tions A1–A3 and the small GCM conditions17 (3.7.13). Let S denote the sphere ◦

◦ ◦

S = S(u, s). For any fixed Λ, Λ ∈ R verifying, ◦ ◦
2

|Λ|, |Λ| . δ r

,

(3.7.23) ◦

1. There exists a unique GCM sphere S = S(Λ,Λ) , which is a deformation18 of S, S S and an adapted null frame eS 3 , eθ , e4 , such that the following GCM conditions are 19 verified /d2S,? /d1S,? κS = /d2S,? /d1S,? µS = 0,

κS =

2 . rS

(3.7.24)

In addition Z

Z

Φ

f e = Λ, S

f eΦ = Λ,

(3.7.25)

S

where (f, f ) belong to the triplet (f, f , λ = ea ) which denote the change of frame ◦

coefficients from the frame of S to the one of S. 2. The transition functions (f, f , log λ) verify

(f, f , log λ) h

k (S)

◦

. δ,

k ≤ smax + 1.

(3.7.26)

3. The area radius rS and Hawking mass mS of S verify S ◦ ◦ r − r . δ,

◦ S ◦ m − m . δ.

(3.7.27)

The precise version of Theorem 3.34 and its proof are given in section 9.4. The next result requires stronger assumptions for Γb than those made in A1. 17 Here,

the other assumptions (3.7.14) and (3.7.15) are not needed. the sense of Definition 3.32. 19 ΓS , RS denote the Ricci and curvature components with respect to the adapted frame on S. 18 In

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CHAPTER 3

A1-Strong. For k ≤ smax ,



◦



. r−2 ,

Γg

Γb k,∞





Γb

◦

k,∞

. r−1 ,

k,∞

◦

1

. () 3 r−2 .

(3.7.28)

Theorem 3.35 (GCMS-II). In addition to the assumptions of Theorem 3.34 we also assume that A1-Strong and (3.7.16) hold true. Then, ◦

1. There exists a unique GCM sphere S, which is a deformation of S, such that in addition to (3.7.24) the following GCM conditions also hold true on S. Z Z S Φ β S eΦ = 0, eS (3.7.29) θ (κ )e = 0. S

S

2. The transition functions (f, f , log λ) verify the estimates (3.7.26). 3. The area radius rS and Hawking mass mS of S verify (3.7.27). The precise version of Theorem 3.35 and its proof are given in 9.7. Theorem 3.36 (GCMH). Consider the region R as above, verifying the assumptions A1–A3 and the small GCM conditions (3.7.13)–(3.7.15). Let the sphere ◦

◦ ◦

S0 = S0 [u, s, Λ0 , Λ0 ] be the deformation of S constructed in Theorem GCMS-I above. There exists a smooth spacelike hypersurface Σ0 ⊂ R passing through S0 , a scalar function uS defined on Σ0 , whose level surfaces are topological spheres denoted by S, and a smooth collection of constants ΛS , ΛS verifying ΛS0 = Λ0 ,

ΛS0 = Λ0 ,

such that the following conditions are verified: 1. The surfaces S of constant uS verify all the properties stated in Theorem GCMSI for the prescribed constants ΛS , ΛS . In particular they come endowed with null S S frames (eS 4 , eθ , e3 ) such that i. For each S the GCM conditions (3.7.24), (3.7.25) hold with Λ = ΛS , Λ = ΛS . ii. The transversality conditions hold true on each S. ξ S = 0,

ω S = 0,

η S + ζ S = 0.

(3.7.30)

2. We have, for some constant cΣ0 , uS + rS = cΣ0 ,

along

Σ0 .

(3.7.31)

3. Let ν S be the unique vectorfield tangent to the hypersurface Σ0 , normal to S, and S normalized by g(ν S , eS 4 ) = −2. There exists a unique scalar function a on Σ0 S such that ν is given by S S ν S = eS 3 + a e4 .

The following normalization condition holds true at the south pole SP of every

131

MAIN THEOREM

sphere S, i.e., at θ = 0, aS

SP

= −1 −

2mS . rS

(3.7.32)

4. Under the additional transversality condition20 on Σ0 S eS 4 (u ) = 0,

e4 (rS ) =

rS S κ = 1, 2

(3.7.33)

the Ricci coefficients η S , ξ S are well defined and verify Z Z S Φ η e = ξ S eΦ = 0. S

(3.7.34)

S

5. The transition functions (f, f , log λ) verify the estimates (3.7.26). 6. The area radius rS and Hawking mass mS of S verify (3.7.27). The precise version of Theorem 3.36 and its proof are given in section 9.8. 3.7.5

Main ideas

Both theorems GCMS-II and GCMH are based on Theorem GSMS-I. They are heavily based on the transformation formulas for the Ricci and curvature coefficients recorded in Proposition 2.90. 3.7.5.1

Sketch of the proof of Theorems GSMS-I and GSMS-II ◦

A given deformation Ψ : S → S is fixed by the parameters U, S and transition functions F = (f, f , λ) connected by the system (3.7.22). Making use of the transformation formulas one can show that the GCM conditions (3.7.24)-(3.7.25) hold true if and only if the transition functions F verify a coercive nonlinear elliptic Hodge system of the form DΨ F = B(Ψ), where the operator DΨ depends on the deformation Ψ and the right-hand side B depends on both Ψ and the background foliation (see Proposition 9.33 for the precise form of the system). To find a desired GSMS deformation we have to solve a coupled system between the transport type equations in (3.7.22) and the elliptic coercive system DΨ F = 0 of Proposition 9.33. The actual proof is thus based on an iteration procedure for a sequence of de◦

◦

formation spheres S(n) of S given by the maps Ψ(n) = (U (n) , S (n) ) : S → S(n) and the corresponding transition functions (f (n) , f (n) , λ(n) ). The iteration procedure for the quintets Q(n) = (U (n) , S (n) , f (n) , f (n) , λ(n) ), starting with the trivial quintet Q(0) corresponding to the zero deformation, is described in section 9.4.3. The main steps in the proof are as follows. 1. Given the triplet (f (n) , f (n) , λ(n) ) the pair (U (n) , S (n) ) defines the deformation ◦

sphere S(n) and the corresponding pullback map #n : S → S(n) according to 20 Here the average of κS is taken on S. In view of the GCM conditions (3.7.24) we deduce S eS 4 (r ) = 1.

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CHAPTER 3

the equation (3.7.22). 2. Given the pair Ψ(n) = (U (n) , S (n) ) and the deformation sphere S(n) we define the triplet (f (n+1) , f (n+1) , λ(n+1) ) by solving the corresponding elliptic system DΨ(n) F (n+1) = B(Ψ(n) ). This step is based on the crucial a priori estimates of section 9.4.1. 3. Given the new pair (f (n+1) , f (n+1) ) we make use of the equations (3.7.22) to find a unique new map (U (n+1) , S (n+1) ) and thus the new deformation sphere S(n + 1). 4. The convergence of the iterates Q(n) , described in section 9.4.5 in the boundedness Proposition 9.42 and the contraction Proposition 9.43. The latter requires ◦

us to carefully compare the iterates Q(n) , Q(n+1) by pulling them back to S. One has to be particularly careful with the behavior of the iterates on the axis of symmetry. Theorem GSMS-II, which is an easy consequence of Theorem is proved R SGSMS-I R Φ in section 9.7 and the transformation formulas which relate β e to Λ = f eΦ S S R S S Φ R Φ and S eθ (κ )e to Λ = S f e . One can show that there exist choices of Λ, Λ such R R S Φ that S β S eΦ = S eS θ (κ )e = 0. 3.7.5.2

Sketch of the proof of Theorem GCMH

The proof of Theorem GCMH makes use of Theorem GCMS-I to construct Σ0 as a union of GCMS spheres. Step 1. Theorem GCMS-I allows us to construct, for every value of the parameters (u, s) in R (i.e., such that the background spheres S(u, s) ⊂ R) and every real numbers (Λ, Λ), a unique GCM sphere S[u, s, Λ, Λ], as a Z-polarized deformation ◦ ◦ of S(u, s). In particular (3.7.24) and (3.7.25) are verified and S0 = S0 [u, s, Λ0 , Λ0 ]. Step 2. We look for functions Ψ(s), Λ(s), Λ(s) such that 1. We have ◦

◦

Ψ(s) = u,

◦

Λ(s) = Λ0 ,

◦

Λ(s) = Λ0 .

2. The resulting hypersurface Σ0 = ∪s S[Ψ(s), s, Λ(s), Λ(s)] verifies uS + rS = cΣ0 ,

along

Σ0 .

3. The additional GCM conditions (3.7.32) and (3.7.34) of Theorem GCMH are verified. These conditions lead to a first order differential system for Ψ(s), Λ(s), Λ(s), with ◦ prescribed initial conditions at s which allows us to determine the desired surface. The proof is given in detail in section 9.8.

133

MAIN THEOREM

3.8

OVERVIEW OF THE PROOF OF THEOREMS M0–M8

In this section, we provide a brief overview of the proof of Theorems M0–M8. In addition to the null frame adapted to the outgoing foliation of (ext) M and to the null frame adapted to the ingoing foliation of (int) M, we have also introduced two global frames on M = (int) M ∪ (ext) M as well as associated scalars r and m in section 3.5. Unless otherwise specified, when we discuss a particular spacetime region, i.e., (ext) M, (int) M or M, it should be understood that the frame as well as r and m are the ones corresponding to that region. 3.8.1

Discussion of Theorem M0

Step 1. Recall our GCM conditions on S∗ = Σ∗ ∩ C∗ Z Z eθ (κ)eΦ = 0, βeΦ = 0. S∗

S∗

Recall that ν = e3 +a∗ e4 is the unique tangent vectorfield to Σ∗ which is orthogonal to eθ and normalized by g(ν, e4 ) = −2. Using the null structure equation for e3 (κ) and e3 (β), as well as e4 (κ) and e4 (β), we obtain transport equations along Σ∗ in the ν direction for Z Z eθ (κ)eΦ and βeΦ = 0. S

S

Integrating these transport equations in ν, we propagate the control on S∗ to Σ∗ . In particular, we obtain the following estimates on S1 = Σ∗ ∩ C1 , Z Z  Φ Φ eθ (κ)e + r βe . 2 + . 0 , (3.8.1) r S1 S1 where we used in the last inequality the dominance condition of r on Σ∗ , see (3.3.4). Step 2. We consider the transition functions (f, f , λ) from the frame of the initial data layer to the frame of (ext) M. Since • S1 is a sphere of (ext) M in the initial data layer, • S1 is a sphere of the GCM hypersurface Σ∗ , • the estimate (3.8.1) holds on S1 , we can invoke a corollary of the GCM procedure of section 3.7.4 to obtain a first improved bound for (f, f , λ) on S1 with O(0 ) smallness constant. After further improvements, leading in particular to a r−1 gain for f compared to f and λ, this ultimately leads to   sup r|d≤klarge +4 f | + |d≤klarge +4 (f , log λ)| + |m − m0 | . 0 . (3.8.2) S1

Step 3. Relying on the transport equations21 in e4 for (f, f , λ), see Corollary 21 The control of f on C requires in fact a more subtle treatment, see Step 10 and Step 11 of 1 the proof of Theorem M0.

134

CHAPTER 3

2.93, and Proposition 2.71 for m, we propagate (3.8.2) to C1 , and then, proceeding similarly in the e3 direction to propagate the estimates to C 1 , we finally obtain   sup r|d≤klarge +1 f | + |d≤klarge +1 (f , log λ)| + |m − m0 | . 0 . (3.8.3) C1 ∪C 1

Together with the control of the initial data layer foliation and the transformation formulas of Proposition 2.90, we then obtain the desired estimates on C1 ∪ C 1 for the curvature components. Remark 3.37. The fact that f and λ display a r loss compared to f in (3.8.3) does not affect the desired estimates for the curvature components on C1 ∪C 1 , see Remark 4.4. See also Remark 4.5 for a heuristic explanation of this a priori anomalous behavior. 3.8.2

Discussion of Theorem M1

Here are the main steps in the proof of Theorem M1. Step 1. Consider the global frame on M constructed in Proposition 3.26 and the definition of q on M with respect to that frame, see section 2.3.3 for the definition of q with respect to any null frame. According to Theorem 2.108 we have 2 q + V q = N,

V = κκ

(3.8.4)

where the nonlinear term N = Err[g q] is a long expression of terms quadratic, ˇ R ˇ involving various powers of r. Making use of the symbolic or higher order, in Γ, notation introduced in Definition 2.94 we have, see (2.4.8),   Err[2 q] = r2 d≤2 (Γg · (α, β)) + e3 r3 d≤2 (Γg · (α, β)) + d≤1 (Γg · q) + l.o.t. ˇ R). ˇ where the terms denoted by l.o.t. are higher order in (Γ, Remark 3.38. Recall from Remark 2.109 that the above good structure of the error term Err[2 q] only holds in a frame for which ξ = 0 for r ≥ 4m0 and η ∈ Γg . This is why, in Theorem M1, q is defined relative to the global frame of Proposition 3.26, see also Remark 2.110. Step 2. We follow the Dafermos-Rodnianski version of the vectorfield method to derive desired decay estimates. We recall that, in the context of a wave equation of the form (Sch) ψ = 0 on Schwarzschild spacetime, their strategy consists in the following: • Start by deriving Morawetz energy type estimates for ψ with nondegenerate flux energies and the usual degeneracy of bulk integrals at r = 3m. • Derive rp -weighted estimates for 0 < p < 2 and use them, in conjunction with the Morawetz estimates, to derive decay estimates. • The decay estimates obtained by using the standard rp -weighted approach are too weak to be useful in our nonlinear approach. We improve them by making use of a recent variation of the Dafermos-Rodnianski approach due to Angelopoulos, Aretakis and Gajic [5] which is based on first commuting the wave equation

135

MAIN THEOREM

(Sch) ψ = 0 with r2 (e4 + r−1 ) and then repeating the process described for the resulting new equation. This procedure allows us to derive the improved decay estimates consistent with our decay norms. • Derive estimates for higher derivatives by commuting with T, r d/, the redshift vectorfield, and re4 . Step 3. The estimates mentioned in Step 2 have to be adapted to the case of our equation (3.8.4). There are four main differences to take into account: • The application of the vectorfield method to our context produces various nontrivial commutator terms which have to be absorbed. This is taken care of by ˇ R, ˇ as well as, in some cases, by integration by our bootstrap assumption for Γ, parts. • The presence of the potential V is mostly advantageous but various modifications have to be nevertheless made, especially near the trapping region.22 • The presence of the nonlinear term N is the most important complication. The precise null structure of N is essential and various integrations by parts are needed. • The quadratic terms involving η in N can only be treated provided the definition of q is done with respect to the global frame on M constructed in Proposition 3.26, for with η behaves better in powers of r−1 . 3.8.3

Discussion of Theorem M2

Recall from section 2.3.3 that q is defined with respect to a general null frame as follows:     1 2 4 2 q = r e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω − 8ω κ + κ α 2 which yields the following transport equation for α   1 e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω 2 − 8ω κ + κ2 α 2

=

q . r4

Recall also that q, controlled in Theorem M1, is defined w.r.t. the global frame of Proposition 3.26 whose normalization is such that, in particular, ω is a small quantity. Also, since we have e3 (r) =

r κ + l.o.t. 2

we infer e3 (e3 (r2 α))

=

q + l.o.t. r2

Integrating twice this transport equation from C1 where we control the initial data — and in particular α — in view of Theorem M0, and using the decay for q provided 22 At the linear level, on a Schwarzschild spacetime, this step was also treated (minus the improved decay) in the paper [26].

136

CHAPTER 3

by Theorem M1, we deduce23  2  1 r (2r + u)1+δextra sup + r3 (2r + u) 2 +δextra |d≤ksmall +20 α| . log(1 + u) (ext) M  3  1 r (2r + u)1+δextra 4 +δextra 2 sup + r (2r + u) |d≤ksmall +19 e3 (α)| . log(1 + u) (ext) M

0 , 0 .

Now that we control α in the global frame of Proposition 3.26, we need to go back to the frame of (ext) M. By invoking the relationships between our various frames of (ext) M, see Proposition 3.26 and Proposition 3.20, and the transformation formula for α, we infer h i (ext) Dksmall +20 (ext) α . 0 and hence the conclusion of Theorem M2. 3.8.4

Discussion of Theorem M3

Here are the main steps in the proof of Theorem M3. Step 1. To derive decay estimates for α in M, we first recall the following Teukolsky-Starobinsky identity, see (2.3.15),   3 3 2 2 7 ? ? e3 (r e3 (rq)) + 2ωr e3 (rq) = r /d2 /d1 /d1 /d2 α + κρe4 α − κρe3 (α) + l.o.t. 2 2 where l.o.t. denotes terms which are quadratic of higher, and where all quantities are defined w.r.t. the global frame of Proposition 3.26. Then, introducing the vectorfield  1 ˇ −κ ˇ e3 , Te = e4 − κ + κΩ ˇΩ κ we rewrite the identity as 6mTeα + r4 /d?2 /d?1 /d1 /d2 α

=

 1 2 2 e (r e (rq)) + 2ωr e (rq) + l.o.t. (3.8.5) 3 3 3 r3

As it turns out, see Remark 6.11, this is a forward parabolic equation on each hypersurface of constant r in (int) M. Step 2. Thanks to • the control in (int) M of the RHS of (3.8.5) which follows from the decay estimates of Theorem M1 for q, as well as the bootstrap assumptions for the quadratic and higher order terms, • the control of α on C 1 — i.e., of the initial data of (3.8.5) — provided by Theorem M0, • parabolic estimates for the forward parabolic equation (3.8.5), 23 Recall

that δextra has been introduced in Theorem M1 and satisfies δextra > δdec .

137

MAIN THEOREM

(int)

we obtain the desired decay estimates for α in

M.

Step 3. It remains to control α on Σ∗ . Recall that ν denotes the unique tangent vectorfield to Σ∗ which can be written as ν = e3 + ae4 . The Teukolsky-Starobinsky identity of Step 1 can then be written as 6mνα + r4 /d?2 /d?1 /d1 /d2 α

=

 1 2 2 e (r e (rq)) + 2ωr e (rq) + l.o.t. (3.8.6) 3 3 3 r3

where l.o.t. denotes terms which are quadratic of higher, as well as terms which are linear but display additional decay in r. This is a forward parabolic equation along Σ∗ . To obtain the desired decay for α along Σ∗ , one then proceeds as in Step 2, using in addition, for the linear term with extra decay in r, the behavior (3.3.4) of r on Σ∗ . 3.8.5

Discussion of Theorem M4

Here are the main steps in the proof of Theorem M4. Step 1. We derive decay estimates for the spacelike GCM hypersurface Σ∗ . More precisely, thanks to • the GCM conditions on Σ∗ κ=

2 , /d?2 /d?1 κ = 0, /d?2 /d?1 µ = 0, r

Z

ηeΦ = 0,

S

Z

ξeΦ = 0,

S

• the control of q in (ext) M, established in Theorem M1, and hence in particular on Σ∗ , • the control of α of the outgoing geodesic foliation in (ext) M, established in Theorem M2, and hence in particular on Σ∗ , • the control of α on Σ∗ , established in Theorem M3, • the dominance condition (3.3.4) of r on Σ∗ −2 r Σ∗ ≥ 0 3 u1+δdec , • the identity (2.3.11) relating q to derivatives of ρ, i.e.,   3 3 4 ? ? q = r /d2 /d1 ρ + ρκϑ + ρκϑ + · · · , 4 4 • elliptic estimates for Hodge operators on the 2-spheres foliating Σ∗ , we infer the control with improved decay of all Ricci and curvature components on the spacelike hypersurface Σ∗ . Step 2. We derive decay estimates for the outgoing geodesic foliation of More precisely: 1

(ext)

M.

• First, we propagate the estimates involving only u− 2 −δdec decay in u from Σ∗ to (ext) M. • We then focus on the harder to recover estimates, i.e., the ones involving u−1−δdec

138

CHAPTER 3

decay in u. We proceed as follows. – We first propagate the main GCM quantities κ ˇ, µ ˇ, and a renormalized quantity involving κ ˇ (see the quantity Ξ in Lemma 7.36) from Σ∗ to (ext) M.

– We then recover the estimates involving u−1−δdec decay in u on T . To this end, we use that we control the main GCM quantities, α from Theorem M3 (since T belongs both to (ext) M and (int) M), q and α from Theorem M1– M2, and the estimates are then derived somewhat in the spirit of the ones on Σ∗ , in particular by relying on elliptic estimates for Hodge operators on the 2-spheres foliating T . – To recover the remaining estimates in (ext) M involving u−1−δdec decay in u, we integrate the transport equations in e4 forward from T , which concludes the proof of Theorem M4. 3.8.6

Discussion of Theorem M5

Here are the main steps in the proof of Theorem M5. Step 1. We first derive decay estimates for the ingoing geodesic foliation of on the timelike hypersurface T . More precisely, thanks to

(int)

M

• the fact that the null frame of (int) M is defined on T as a simple conformal renormalization of the null frame of (ext) M in view of its initialization, see section 3.1.2, • the control of the outgoing geodesic foliation of (ext) M on T obtained in Theorem M4, ˇ (ext) Γ) ˇ to ((int) R, ˇ (int) Γ) ˇ this allows us to transfer the decay estimates for ((ext) R, on T . Step 2. We derive on (int) M decay estimates for the ingoing geodesic foliation of (int) M. More precisely, thanks to • the improved decay estimates for α in (int) M derived in Theorem M3, ˇ and R ˇ on T derived in Step 1, • the improved decay estimates for Γ • the null structure equations and Bianchi identities, ˇ and R ˇ corresponding to the ingoing we infer O(0 u−1−δdec ) decay estimates for Γ (int) geodesic foliation of M which concludes the proof of Theorem M5. 3.8.7

Discussion of Theorem M6

Step 1. Using a) The control of the initial data layer, b) Theorem GCMS-II of section 3.7.4, c) Theorem GCMH of section 3.7.4, we produce a smooth hypersurface Σ∗ in the initial data layer starting from a GCM sphere S∗ , and satisfying all the required properties for the future spacelike boundary of a GCM admissible spacetime, according to item 3 of Definition 3.2.

139

MAIN THEOREM

Step 2. We then consider the outgoing geodesic foliation initialized on Σ∗ which foliates the region we denote (ext) M, to the past of Σ∗ , and included in the outgoing part (ext) L0 of the initial data layer. In order to control it, we consider the transition functions (f, f , λ) from the background frame of the initial data layer to the frame of (ext) M. These functions satisfy transport equations in e4 with the right-hand side depending on (f, f , λ) and the Ricci coefficients of the background foliation. Integrating the transport equations from Σ∗ , where (f, f , λ) are under control as a by-product of the use of Theorem GCMH in Step 1, we obtain the control of (f, f , λ) in (ext) M. Using the transformation formulas of Proposition 2.90, and using the control of the initial data layer, we then infer the desired control (i.e., with 0 smallness constant and suitable r-weights) for the Ricci coefficients and curvature components of the foliation of (ext) M. Step 3. (ext) M terminates on a timelike hypersurface T of constant area radius.24 We then consider the ingoing geodesic foliation initialized on T according to item 4 of Definition 3.2, which foliates the region we denote (int) M, included in the ingoing part (int) L0 of the initial data layer. Proceeding as in Step 2, relying on transport equations in e3 instead of e4 , we then derive the desired control (i.e., with 0 smallness constant) for the Ricci coefficients and curvature components of the foliation of (int) M, thus concluding the proof of Theorem M6. 3.8.8

Discussion of Theorem M7

From the assumptions of Theorem M7 we are given a GCM admissible spacetime M = M(u∗ ) ∈ ℵ(u∗ ) verifying the following improved bounds, for a universal constant C > 0, (Dec)

Nksmall +5 (M) ≤ C0 provided by Theorems M1–M5. We then proceed as follows. Step 1. We extend M by a local existence argument, to a strictly larger spacetime M(extend) , with a naturally extended foliation and the following slightly increased bounds (Dec)

Nksmall +5 (M(extend) ) ≤ 2C0 but which may not verify our admissibility criteria. Step 2. Using a) The control of the extended spacetime M(extend) , b) Theorem GCMS-II of section 3.7.4, c) Theorem GCMH of section 3.7.4, e ∗ in M(extend) \ M startwe produce a small piece of smooth GCM hypersurface Σ ing from a GCM sphere Se∗ .

24 With

respect to the foliation of

(ext) M.

140

CHAPTER 3

e ∗ all the Step 3. By a continuity argument based on a priori estimates, we extend Σ way to the initial data layer, while ensuring that it remains in M(extend) \ M and satisfying all the required properties for the future spacelike boundary of a GCM admissible spacetime, according to item 3 of Definition 3.2. e ∗ which Step 4. We then consider the outgoing geodesic foliation initialized on Σ f included in the outgoing part of M(extend) . foliates the region we denote (ext) M, In order to control it, we consider the transition functions (f, f , λ) from the backf These functions ground frame of the initial data layer to the frame of (ext) M. satisfy transport equations in e4 with the right-hand side depending on (f, f , λ) and the Ricci coefficients of the background foliation. Integrating the transport e ∗ , where (f, f , λ) are under control as a by-product of the use of equations from Σ f Using the Theorem GCMH in Step 2, we obtain the control of (f, f , λ) in (ext) M. transformation formulas of Proposition 2.90, and using the control of the initial data layer, we then derive the desired control (i.e., with 0 smallness constant and suitable u and r weights) for the Ricci coefficients and curvature components of the f foliation of (ext) M. f terminates on a timelike hypersurface Te of constant area radius.25 Step 5. (ext) M We then consider the ingoing geodesic foliation initialized on Te according to item 4 f included in the ingoing of Definition 3.2, which foliates the region we denote (int) M, (extend) part of M . Proceeding as in Step 4, relying on transport equations in e3 instead of e4 , we then derive the desired control (i.e., with 0 smallness constant and suitable u-weights) for the Ricci coefficients and curvature components of the f thus concluding the proof of Theorem M7. foliation of (int) M, 3.8.9

Discussion of Theorem M8

So far, we have only improved our bootstrap assumptions on decay estimates. We ˇ now improve our bootstrap assumptions on energies and weighted energies for R ˇ relying on an iterative procedure recovering derivatives one by one.26 and Γ Step 0. Let Im0 ,δH the interval of R defined by      δH 3δH Im0 ,δH := 2m0 1 + , 2m0 1 + . 2 2

(3.8.7)

Recall that T = {r = rT }, where rT ∈ Im0 ,δH , and note, see also Remark 3.30, that the results of Theorems M0–M7 hold for any rT ∈ Im0 ,δH . It is at this stage that we need to make a specific choice of rT in the context of a Lebesgue point argument. More precisely, we choose rT such that we have Z Z ˇ2 = ˇ 2. |d≤klarge R| inf |d≤klarge R| (3.8.8) {r=rT }

25 With

r0 ∈Im0 ,δH

{r=r0 }

f respect to the foliation of (ext) M. also [33] for a related strategy to recover higher order derivatives from the control of lower order ones. 26 See

141

MAIN THEOREM

In view of this definition, and since T = {r = rT }, we infer that Z Z ˇ2 . ˇ 2.   |d≤klarge R| |d≤klarge R| (ext) M

T

(3.8.9)

r∈Im0 ,δH

Remark 3.39. From now on, we may thus assume that the spacetime M satisfies the conclusions of Theorem M0 and Theorem M7, as well as (3.8.9), and our goal (En) is to prove Theorem M8, i.e., to prove that Nklarge . 0 holds. Step 1. The O(0 ) decay estimates derived in Theorem M7 imply in particular the following (non-sharp) consequence (En)

.

Nksmall

0 ,

where we recall27 (En)

Nk

(ext)

=

ˇ + Rk [R]

(ext)

ˇ + Gk [Γ]

(int)

ˇ + Rk [R]

(int)

ˇ Gk [Γ].

This allows us to initialize our iteration scheme in the next step. Step 2. Next, for J such that ksmall ≤ J ≤ klarge − 1, consider the iteration assumption (En)

NJ

.

B [J],

(3.8.10)

where B [J]

:=

J X

`(J)

(0 )`(j) B 1−`(j) + 0

j=ksmall −2

B

 Z := 

B,

`(j) := 2ksmall −2−j ,(3.8.11)

 12 

(ext) M

r∈Im0 ,δH

ˇ 2 .  |d≤klarge R|

(3.8.12)

In view of Step 1, (3.8.10) holds for J = ksmall . From now on, we assume that (3.8.10) holds for J such that ksmall ≤ J ≤ klarge − 2, and our goal is to show that this also holds for J + 1 derivatives. Step 3. Using the Teukolsky wave equations for α and α, as well as a wave equation for ρˇ, see Proposition 8.14, we derive Morawetz type estimates for J + 1 derivatives (En) of these quantities in terms of O(B [J] + 0 NJ+1 ). Step 4. Relying on Bianchi identities, we also derive Morawetz type estimates for J +1 derivatives for β and β. As a consequence, we obtain Morawetz type estimates (En)

for J + 1 derivatives of all curvature components in terms of O(B [J] + 0 NJ+1 ).

27 See sections 3.2.1 and 3.2.2 for the definition of our norms measuring energies for curvature components and Ricci coefficients.

142

CHAPTER 3

Step 5. As a consequence of Step 4, we immediately obtain, for any r0 ≥ 4m0 , (int)

ˇ + RJ+1 [R]

(ext)

ˇ RJ+1 [R]

(ext)

≤

(En)

10 0 ˇ R≥r J+1 [R] + O(r0 (B [J] + 0 NJ+1 )).

Step 6. Relying on the Bianchi identities, we derive rp -weighted estimates for J +1 derivatives of curvature on r ≥ r0 with r0 ≥ 4m0 . We obtain (ext)

0 ˇ R≥r J+1 [R] .

1

(ext)

r0δB

(En) 10 0 ˇ G≥r k [Γ] + r0 (B [J] + 0 NJ+1 ).

Step 7. Next, we estimate the Ricci coefficients of (ext) M. To control them, we rely on the null structure equations in (ext) M. Using the null structure equations in (ext) M and the GCM conditions on Σ∗ , we derive the following weighted estimates for J + 1 derivatives of the Ricci coefficients: (ext)

ˇ . GJ+1 [Γ]

(ext)

(En)

ˇ + B [J] + 0 N RJ+1 [R] J+1 .

Together with the estimates of Step 5 and Step 6, we infer for a large enough choice of r0 (ext)

ˇ + GJ+1 [Γ]

(int)

ˇ + RJ+1 [R]

(ext)

ˇ . RJ+1 [R]

(En)

B [J] + 0 NJ+1 .

Step 8. Next, we estimate the Ricci coefficients of (int) M. Using the information on T induced by Step 7 and the null structure equations in (int) M, we derive (int)

ˇ . GJ+1 [Γ]

(int)

ˇ + B [J] + 0 N(En) + RJ+1 [R] J+1

Z

J+1 (ext)

T

|d

(

ˇ 2 R)|

 12 .

We need to deal with the last term. Relying on a trace theorem in the spacetime region (ext) M(r ∈ Im0 ,δH ), and the fact that J + 2 ≤ klarge , we obtain Z T

ˇ 2 |dJ+1 ((ext) R)|

 12 .

 Z 

 41 (ext) M

 r∈Im0 ,δH

ˇ 2  ( (ext) RJ+1 [R]) ˇ 12  |dklarge R|

ˇ + (ext) RJ+1 [R]. Step 9. The last estimate of Step 7 and the two estimates of Step 8 yield, for 0 > 0 small enough, (En)

NJ+1

.

 Z  B [J] +

 14 (ext) M

 r∈Im0 ,δH

ˇ 2  |dklarge R|



(En)

B [J] + 0 NJ+1

 12

.

In view of the definition (3.8.11) of B [J], we infer that (En)

NJ+1

.

B [J + 1]

which is the iteration assumption (3.8.10) for J + 1 derivatives. We deduce that (3.8.10) holds for all J ≤ klarge − 1, and hence (En)

Nklarge −1

. B [klarge − 1].

143

MAIN THEOREM

Step 10. Relying on the conclusion of Step 9, and arguing as in Step 3 to Step 7, we obtain the conclusion of Step 7 for J = klarge − 1, i.e., (ext)

ˇ + Gklarge [Γ]

(int)

ˇ + Rklarge [R]

(ext)

(En)

ˇ . B [klarge − 1] + 0 N Rklarge [R] klarge .

We then infer that (En)

B [klarge − 1] . 0 + 0 Nklarge which yields, together with the last estimate of Step 9, (ext)

ˇ + Gklarge [Γ]

(int)

ˇ + Rklarge [R]

(ext)

ˇ . Rklarge [R]

(En)

0 + 0 Nklarge .

ˇ Arguing as for the first estimate of Step 11. It remains to recover (int) Gklarge [Γ]. Step 8 with J = klarge − 1, we have (int)

ˇ Gklarge [Γ]

.

ˇ + B [klarge − 1] + 0 N(En) Rklarge [R] klarge Z  12 ˇ 2 + |dklarge ((ext) R)| . (int)

T

Thanks to the outcome of Step 10, we deduce that (int)

ˇ Gklarge [Γ]

Z

(En)

. 0 + 0 Nklarge +

T

ˇ 2 |dklarge ((ext) R)|

 12

and hence, for 0 > 0 small enough, using again the last estimate of Step 10, (En) Nklarge

Z . 0 + T

klarge (ext)

|d

(

ˇ 2 R)|

 12 .

It remains to estimate the last term of the RHS of the previous inequality. It is at this stage that we use the choice of rT , or rather its consequence (3.8.9), which implies Z T

ˇ 2 |dklarge ((ext) R)|

 12

(En)

. 0 + 0 Nklarge

so that we finally obtain, for 0 > 0 small enough, (En)

Nklarge

. 0

which concludes the proof of Theorem M8.

3.9

STRUCTURE OF THE REST OF THE BOOK

The rest of this book is devoted to the proof of Theorems M0–M8, as well as our GCM procedure. More precisely, 1. Theorem M0, together with other first consequences of the bootstrap assump-

144

CHAPTER 3

tions, is proved in Chapter 4. Theorem M1 is proved in Chapter 5. Theorems M2 and M3 are proved in Chapter 6. Theorems M4 and M5 are proved in Chapter 7. Theorems M6, M7 and M8 are proved in Chapter 8. Our GCM procedure is described in details in Chapter 9. Chapter 10 contains estimates for Regge-Wheeler type wave equations used in Theorem M1. 8. Many of the long calculations are to be found in the appendices. 2. 3. 4. 5. 6. 7.

Chapter Four Consequences of the Bootstrap Assumptions 4.1

PROOF OF THEOREM M0

According to the statement of Theorem M0 we consider given the initial layer L0 = (ext) L0 ∪ (int) L0 as defined in Definition 3.1. We also assume that the initial layer norm verifies sup

Ik . 0

(4.1.1)

k≤klarge +5

where Ik =(ext) Ik + (ext)

 ρ + 2m0 + r2 |β| + r|α| r3 (ext) L 0
! 2m0 2 1 − 2 2 r sup r |ϑ| + κ − + |ζ| + κ + r r (ext) L 0   m 0 sup r |ϑ| + ω − 2 + |ξ| r (ext) L 0 " γ sup r 2 − 1 + r|b| + |Ω + Υ| + |ς − 1| r (ext) L (ext) r ≥4m 0( 0 0) Φ # e +r − 1 , r sin θ

=

I0

+ + +

(int)

I0

= +

I00

=

Ik + I0k and  7 sup r 2 +δB (|α| + |β|) + r3 (int)

  2m0 |α| + |β| + ρ + 3 + |β| + |α| r (int) L 0 "
0 2 1 − 2m r κ + sup |ϑ| + κ − + |ζ| + r (int) L 0 # m0 + ω + 2 + |ξ| , r sup

sup (int) L ∩ (ext) L 0 0


|f | + |f | + | log(λ−1 0 λ)| ,

2 + |ϑ| r

λ0 = (ext) λ0 = 1 −

2m0 (ext) r L0

,

with Ik the corresponding higher derivatives norms obtained by replacing each component by d≤k of it. In the definition of I00 above, (f, f , λ) denote the transition functions of Lemma 2.87 from the frame of the outgoing part (ext) L0 of the initial data layer to the frame of the ingoing part (int) L0 of the initial data layer in the

146

CHAPTER 4

region (int) L0 ∩ (ext) L0 . We divide the proof of Theorem M0 in the following steps. Step 1. We have the following lemma. Lemma 4.1. We have Z  Φ e4 eθ (κ)e S Z  Φ e4 βe ZS  Φ e3 βe S

(ext)

M Z   = − κeθ (κ) + 4Kζ − ϑeθ (κ) + 2eθ (ζ 2 ) eΦ , S  Z  1 1 = − κβ + ζα − ϑβ eΦ , 2 2 S Z Z Z 1 1 Φ Φ = − eθ (κκ)e + 3ρ ηe + eθ (ϑϑ)eΦ 4 S 4 S S ! Z 1 1 + κβ + 2ωβ + 3η ρˇ − ϑβ + ξα − ϑβ eΦ 2 2 S  Z  +Err e3 βeΦ ,

on

S

and Z  e3 eθ (κ)eΦ

Z = κe3

S

Φ



ζe S

−κ

Z

Φ

Z (

βe + S

S

1 −κ ˇ β − κ2 ζ + 6ρξ 2 )

1 −2ωeθ (κ) − ϑ(eθ (κ) − κζ) + Err[ /d2 /d?2 ξ] eΦ 2  Z  Z     3 1 +Err e3 eθ (κ)eΦ + κ ˇ e3 (ζ) + κ − ϑ ζ eΦ 2 2 S   S  Z  Z  3 1 −ˇ κ e3 (ζ) + κ − ϑ ζ eΦ − κErr e3 ζeΦ . 2 2 S S Proof. We have in

(ext)

M, see Proposition 2.63,

1 e4 (κ) + κ κ = −2 /d1 ζ + 2ρ − 12 ϑ ϑ + 2ζ 2 . 2 Together with the following commutation relation [eθ , e4 ]

=

1 (κ + ϑ)eθ , 2

we infer 1 1 1 e4 (eθ (κ)) + κeθ (κ) + ϑeθ (κ) + κeθ (κ) = 2 /d?1 /d1 ζ + 2eθ (ρ) − eθ (ϑ ϑ) + 2eθ (ζ 2 ). 2 2 2 Also, we have in view of Proposition 2.74 the following identity 3 e3 (eθ (κ)) − κe3 (ζ) = −2 /d2 /d?2 ξ − κβ + κ2 ζ − κeθ κ + 6ρξ − 2ωeθ (κ) + Err[ /d2 /d?2 ξ]. 2

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Next, in view of Z  e4 eθ (κ)eΦ S Z  e4 βeΦ ZS  e3 βeΦ S

147

Corollary 2.65, we have in (ext) M    Z  3 1 = e4 (eθ (κ)) + κ − ϑ eθ (κ) eΦ 2 2 S    Z  3 1 = e4 (β) + κ − ϑ β eΦ , 2 2 S     Z  Z  3 1 = e3 (β) + κ − ϑ β eΦ + Err e3 βeΦ , 2 2 S S

and Z  Z  e3 eθ (κ)eΦ − κe3 ζeΦ S S     Z  Z  3 1 e3 (eθ (κ)) + κ − ϑ eθ (κ) eΦ + Err e3 eθ (κ)eΦ 2 2 S     Z S  Z  3 1 Φ −κ e3 (ζ) + κ − ϑ ζ e − κErr e3 ζeΦ 2 2 S S    Z  3 1 e3 (eθ (κ)) − κe3 (ζ) + κ − ϑ (eθ (κ) − κζ) eΦ 2 2 S  Z  Z     3 1 +Err e3 eθ (κ)eΦ + κ ˇ e3 (ζ) + κ − ϑ ζ eΦ 2 2 S   S  Z  Z  1 3 −ˇ κ e3 (ζ) + κ − ϑ ζ eΦ − κErr e3 ζeΦ . 2 2 S S

=

=

Together with the above identities for e4 (eθ (κ)) and e3 (eθ (κ)), as well as the Bianchi identities of Proposition 2.63 for e4 (β) and e3 (β), we infer Z e4

eθ (κ)e

Φ



Z = S

S

1 1 1 κeθ (κ) − κeθ (κ) + 2 /d?1 /d1 ζ + 2eθ (ρ) − eθ (ϑ ϑ) 2 2 2 !

−ϑeθ (κ) + 2eθ (ζ 2 ) eΦ , Z e4

βeΦ



Z  =

S

Z e3

S Φ

βe S



 1 1 − κβ + /d2 α + ζα − ϑβ eΦ , 2 2

! 1 1 κβ + eθ (ρ) + 2ωβ + 3ηρ − ϑβ + ξα − ϑβ eΦ 2 2 S  Z  +Err e3 βeΦ , Z

=

S

148

CHAPTER 4

and Z e3

eθ (κ)e

Φ



Z − κe3

S

Φ



ζe S

Z

1 1 − 2 /d2 /d?2 ξ − κβ − κ2 ζ + 6ρξ − 2ωeθ (κ) − ϑ(eθ (κ) − κζ) 2 2 S !  Z  +Err[ /d2 /d?2 ξ] eΦ + Err e3 eθ (κ)eΦ

=

S

     Z  3 1 3 1 Φ + κ ˇ e3 (ζ) + κ− ϑ ζ e −κ ˇ e3 (ζ) + κ − ϑ ζ eΦ 2 2 2 2 S S  Z  −κErr e3 ζeΦ . Z





S

Using in particular the fact that /d?2 (eΦ ) = 0, that /d?2 is the adjoint of /d2 , and the identity /d?1 /d1 = /d2 /d?2 + 2K, we deduce Z e4

eθ (κ)e

Φ



Z

1 1 1 κeθ (κ) − κeθ (κ) + 4Kζ + 2eθ (ρ) − eθ (ϑ ϑ) 2 2 2 !

=

S

S

−ϑeθ (κ) + 2eθ (ζ 2 ) eΦ , Z

 Z  1 1 e4 βe = − κβ + ζα − ϑβ eΦ , 2 2 S S Z  Z Z Z 1 e3 βeΦ = eθ (ρ)eΦ + 3ρ ηeΦ + κβ + 2ωβ + 3η ρˇ 2 S S S S !  Z  1 Φ Φ −ϑβ + ξα − ϑβ e + Err e3 βe , 2 S and Z  Φ e3 eθ (κ)e S

Φ



Z = κe3

Φ

ζe S

 −κ

Z

Φ

Z

βe + S

S

1 −κ ˇ β − κ2 ζ + 6ρξ 2 !

1 −2ωeθ (κ) − ϑ(eθ (κ) − κζ) + Err[ /d2 /d?2 ξ] eΦ 2  Z  Z     3 1 Φ +Err e3 eθ (κ)e + κ ˇ e3 (ζ) + κ − ϑ ζ eΦ 2 2 S S     Z  Z  3 1 Φ Φ −ˇ κ e3 (ζ) + κ − ϑ ζ e − κErr e3 ζe . 2 2 S S Finally, from the identity (2.1.20) for eθ (K) and the formula for K, we have Z Z Z 1 1 Φ Φ eθ (ρ)e = − eθ (κκ)e + eθ (ϑϑ)eΦ . 4 4 S S S

149

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

We deduce Z  Z   Φ e4 eθ (κ)e = − κeθ (κ) + 4Kζ − ϑeθ (κ) + 2eθ (ζ 2 ) eΦ , S

S

and Z

Φ

e3

βe S



1 4 Z

Z

eθ (κκ)eΦ + 3ρ

Z

S

S

ηeΦ +

1 4

Z

eθ (ϑϑ)eΦ ! 1 1 + κβ + 2ωβ + 3η ρˇ − ϑβ + ξα − ϑβ eΦ 2 2 S  Z  +Err e3 βeΦ

= −

S

S

which concludes the proof of Lemma 4.1. Step 2. Using the transport equations of Lemma 4.1 and the bootstrap assumptions on decay for k = 0, 1 derivatives in (ext) M, we infer in that region, and in particular on Σ∗ Z Z Z  1 1 2 Φ Φ Φ e4 e (κ)e . e (κ)e + ζe + 2 1+δ , θ θ 2 r S r r u dec S SZ Z  1 2 e4 βeΦ . βeΦ + 2 1+δ , r r u dec ZS Z S Z Z  2 Φ e3 eθ (κκ)eΦ + r−3 ηeΦ + 1 βeΦ + βe . r ru1+δdec , S S S Z Z ZS   1 1 Φ Φ Φ e3 e3 eθ (κ)e . ζe + βe r r S S Z Z S Z 1 1 1 2 + 3 ξeΦ + 2 ζeΦ + 2 eθ (κ)eΦ + 2+2δ . dec r r r u S S S Recall the following GCM conditions Z 2 κ= , ηeΦ = 0, r S We deduce on Σ∗ Z  Φ e4 e (κ)e θ SZ  e4 βeΦ ZS  Φ e3 βe S Z  Φ e3 e (κ)e θ S

. . . .

Z

ξeΦ = 0 on Σ∗ .

S

Z 1 2 Φ ζe + 2 1+δ , 2 r r u dec Z S 1 2 βeΦ + 2 1+δ , r S r u dec Z Z 1 1 2 Φ Φ eθ (κ)e + βe + 1+δ , r S r S ru dec Z Z  1 1 e3 ζeΦ + βeΦ r r Z S Z S 1 1 2 Φ Φ + 2 ζe + 2 eθ (κ)e + 2+2δ . dec r r u S S

150

CHAPTER 4

Also, projecting both Codazzi on eΦ , using /d?2 (eΦ ) = 0 and the fact that /d?2 is the adjoint of /d2 , and using also the GCM condition for κ on Σ∗ , we have on Σ∗ Z Z Z Z 1 1 1 βeΦ = − /d?1 κeΦ − ζκeΦ + ϑ ζeΦ , 2 S 2 S 2 S S Z Z Z r Φ Φ ζe = r βe + ϑ ζeΦ . 2 S S S Together with the bootstrap assumptions on decay for k = 0, 1 derivatives in (ext) M, we infer on Σ∗ Z Z   Z 2 Φ Φ e3 e3 βeΦ +  ζe . r βe + u1+δdec , S S SZ Z 2  ζeΦ . r βeΦ + u1+δdec , Z S Z S Z 2 βeΦ . eθ (κ)eΦ + 1 ζeΦ + 3  r u 2 +δdec S ZS Z S 2 2 . eθ (κ)eΦ + βeΦ + 3 +δ + 1+δ . ru dec u 2 dec S S We have thus on Σ∗ Z Z  1 2 Φ Φ e4 e (κ)e . βe θ r + r2 u1+δdec , SZ ZS  1 2 Φ e4 . βeΦ + βe r r2 u1+δdec , ZS ZS Z  1 1 2 Φ Φ Φ e3 βe . eθ (κ)e + βe + 1+δ , r S r S ru dec S Z Z Z  Z  1 1 Φ Φ Φ Φ e3 e (κ)e . e βe + βe + e (κ)e θ θ 3 r r S S S S 2 2 + 1+δ + 2+2δ . dec ru dec u

(4.1.2)

In view of the behavior (3.3.4) of r on Σ∗ , and plugging the third equation in the fourth, we infer on Σ∗ Z  Φ e4 eθ (κ)e . S Z  Φ e4 βe . S

Z  Φ e3 βe . S Z  Φ e3 eθ (κ)e . S

2

Z 2 βeΦ + r2 u1+δdec , dec u1+δ S ∗ 2 Z 03 2 Φ βe + r2 u1+δdec , dec u1+δ S ∗ 2 Z Z 1 03 2 Φ Φ e (κ)e + βe + θ u1+δdec ru1+δdec , r S S ∗ 2 2 Z Z 03 03 2 Φ Φ βe + e (κ)e + θ u1+δdec u2+2δdec . dec u1+δ S S ∗ ∗ 03

151

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Step 3. Let ν∗ the unique tangent vector to Σ∗ which can be written as ν∗ = e3 + ae4 where a is a scalar function on Σ∗ . Recall that there exists a constant c∗ such that Σ∗ = {u + r = c∗ }. We infer ν∗ (u + r) = 0 and hence 0 = e3 (u + r) + ae4 (u + r) =

2 r r + (κ + A) + a κ ς 2 2

which yields 2

= −ς

a

+ 2r (κ + A) . r 2κ

In view of the GCM condition on κ and the definition of the Hawking mass m, we have on Σ∗ κ=

2 , r

a

=

κ=−

2Υ r

and hence, we have on Σ∗ 2 r − + Υ − A. ς 2

The bootstrap assumptions on decay for k = 0 derivatives in (ext) M, the definition (2.2.22) of A, and the estimates for ς and Ω yield the rough estimate1 |a| . 1. Together with the fact that ν∗ = e3 + ae4 and the estimates of Step 2, we infer Z  Φ ν∗ eθ (κ)e . S Z  Φ ν∗ βe . S

2

03 dec u1+δ ∗ 2

03 dec u1+δ ∗

2 Z Z 3 0 βeΦ + 0 eθ (κ)eΦ + u1+δdec u2+2δdec , S S ∗ Z Z 0 βeΦ + 1 eθ (κ)eΦ + r ru1+δdec . S S

Step 4. We assume on Σ∗ the following bootstrap assumptions recovered at the end of this step Z Z u1+δdec eθ (κ)eΦ + ruδdec βeΦ ≤ . (4.1.3) S

S

1 The estimates for Ω and ς are proved later in Proposition 3.17. Since the proof does not rely on Theorem M0, we may use it here.

152

CHAPTER 4 2

This implies, using also the behavior (3.3.4) of r on Σ∗ , and the fact that  = 03 , Z  0 Φ ν∗ eθ (κ)e . 2+2δ , dec u SZ Z  (4.1.4) 1 0 Φ Φ ν∗ βe . eθ (κ)e + 1+δ . r S ru dec S Now, recall that we have the following GCM on the last sphere S∗ = Σ∗ ∩ C∗ of Σ∗ Z Z eθ (κ)eΦ = βeΦ = 0. S∗

S∗

Integrating backward from S∗ the estimate for eθ (κ) in (4.1.4) yields on Σ∗ Z 0 eθ (κ)eΦ . . u1+δdec S

Plugging in the estimate for β in (4.1.4), we infer on Σ∗ Z  0 Φ ν∗ βe . ru1+δdec . S

Integrating backward from S∗ yields on Σ∗ Z βeΦ . S

0 . ruδdec

We have therefore obtained Z Z u1+δdec eθ (κ)eΦ + ruδdec βeΦ . 0 S

S

which is an improvement of the bootstrap assumptions (4.1.3). In particular, at the first sphere S1 = Σ∗ ∩ C1 of Σ∗ , we have obtained Z Z Φ Φ eθ (κ)e + r βe . 0 . (4.1.5) S1

S1

Remark 4.2. Note that the only bootstrap assumption used in the proof of Theorem M0 is the bootstrap assumption BA-D on decay for k = 0, 1 derivatives. Indeed, to obtain (4.1.5), we have only used, in Steps 1–4, the bootstrap assumption BA-D on decay for k = 0, 1 derivatives, while, from now on, we will only rely on (4.1.5) and the assumptions (4.1.1) on the initial data layer. This observation will allow us to use the conclusions of Theorem M0, not only for the bootstrap spacetime M in Theorem M1–M5, but also for the extended spacetime in the proof of Theorem M8, where the only assumption is the one on decay (which is established for the extended spacetime in Theorem M7). Step 5. On the sphere S1 = Σ∗ ∩ C1 of Σ∗ , we have in view of the GCM conditions

153

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

of Σ∗ and (4.1.5) κ=

2 , r

/d?2 /d?1 κ = 0,

Z

/d?2 /d?1 µ = 0,

S1

Z eθ (κ)eΦ + r

S1

βeΦ . 0 .

(4.1.6)

We consider the transition functions (f, f , λ) from the frame of the outgoing part (ext) L0 of the initial data layer to the frame of (ext) M. We assume the following bootstrap assumptions along C1   sup kf kh4 (S) + r−1 k(f , log(λ))kh4 (S) ≤ . (4.1.7) S⊂C1

In particular, the estimate (4.1.7) allows us to apply Lemma 9.15 with δ1 =  which yields  sup ( (ext) r)−1 | (ext) r − (ext) rL0 | + |u − uL0 | S1  +( (ext) r)−1 | (ext) s − (ext) sL0 | . . (4.1.8) In particular, since u = 1 on S1 , and condition in r, we infer sup |uL0 − 1| . , S1

(ext)

r=

(ext)

s on Σ∗ verifies the dominant

inf (ext) sL0 ≥ S1

1 − 23  . 2 0

Since (ext) L0 contains the region {4m0 ≤ (ext) sL0 < +∞} ∪ {0 ≤ uL0 ≤ 2}, we infer that the sphere S1 is included in (ext) L0 . We will not only improve the bootstrap assumption (4.1.7), but also gain derivatives iteratively. To this end, for 4 ≤ j ≤ klarge + 5, we consider the following iteration assumption kf khj (S1 ) + r−1 k(f , log(λ))khj (S1 )

≤ .

(4.1.9)

Note that (4.1.9) holds true for j = 4 in view of (4.1.7), and our goal is to show that (4.1.9) holds with j replaced by j + 1. Since • • • •

S1 is a sphere of (ext) M in (ext) L0 , S1 is a sphere of the GCM hypersurface Σ∗ , the estimate (4.1.6) holds on S1 , the estimate (4.1.9) holds on S1 , ◦

◦

we can invoke Corollary 9.51 with the choice  = δ = 0 , δ1 = , smax = j, and with the background foliation being the one of the outgoing part (ext) L0 of the initial data layer. We obtain S1

r−1 k(f, f , λ − λ )khj+1 (S1 )

.

0

− 1| . 0 + r−1 sup (ext) r −

(ext)

(4.1.10)

and S1

|λ

S1

rL0 .

(4.1.11)

154

CHAPTER 4

Remark 4.3. In order to prove the iteration assumption (4.1.9) with j replaced by j + 1, we need in particular to improve the estimate for f in (4.1.10) by r−1 . Obtaining this improvement is the focus of Steps 6 to 8. Remark 4.4. The anomalous behavior for f and λ in (4.1.7), i.e., the fact that they display a r loss compared to f , does not affect the desired estimates for the curvature components, see (4.1.22). This is due to the fact that, in the change of frame formulas for the curvature components, λ and f are multiplied by terms that decay faster in r. Remark 4.5. In view of (4.1.8), while |u − uL0 | .  on S1 , we have |s − sL0 | . r on S1 . This, as well as the anomalous behavior of f mentioned above, shows that the sphere S1 is a large deformation, along the outgoing direction, of spheres of the initial data layer (ext) L0 . This reflects the fact that S1 (and Σ∗ ) captures the center of mass frame of the limiting Schwarzschild solution, while the initial data layer foliation captures the center of mass frame of the initial Schwarzschild solution. The behavior of s − sL0 , as well as the one of f , is consistent with the presence of a Lorentz boost between these two center of mass frames. From now on, we denote the frame, Ricci coefficients and curvature components associated to the frame of (ext) M with a prime, while the frame, Ricci coefficients and curvature components associated to the frame of (ext) L0 are unprimed. From the following transformation formula of Proposition 2.90,   3 1 β 0 = λ β + ρf + f α + l.o.t. , 2 2 together with the estimate (4.1.10) for f to estimate the linear term ρf , the estimate (4.1.9) for (f, f , λ) to estimate the other terms, and the estimates (4.1.1) for the outgoing part (ext) L0 of the initial data layer,2 we have, since j ≤ klarge + 5, max r2 k d/0k β 0 kL2 (S1 )

k≤j−1

.

0 .

(4.1.12)

Also, we have ρ0

3 = ρ + ρf f + f β + f β + l.o.t. 2

Differentiating with respect to e0θ , and using the decomposition of e0θ , we infer      1 1 1 1 e0θ (ρ0 ) = 1 + f f eθ + f e4 + f 1 + f f e3 ρ 2 2 2 4   3 +e0θ ρf f + f β + f β + l.o.t. 2   1 1 3 = eθ (ρ) + f e4 (ρ) + f e3 (ρ) + e0θ ρf f + f β + f β + l.o.t. 2 2 2 2 We use, here and in the remainder of the proof, property 6 of Lemma 9.15 to control the hj (S1 ) norm of the Ricci coefficients and curvature components of the initial data foliation of (ext) L in terms of their sup norm. 0

155

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Together with the estimate (4.1.10) for f and f to estimate the linear terms f e4 (ρ) and f e3 (ρ), the estimate (4.1.9) for (f, f , λ) to estimate the other terms, and the estimates (4.1.1) for the curvature components and the Ricci coefficients of the outgoing part (ext) L0 of the initial data layer, we have, using also the behavior (3.3.4) of r on Σ∗ and the fact that S1 ⊂ Σ∗ , as well as an elliptic estimate and the fact that j ≤ klarge + 5, k

max r2 k d/0 ρˇ0 kL2 (S1 )

.

k≤j−1

0 .

(4.1.13)

Step 6. Recall the definition of the mass aspect function µ0 : µ0

=

1 − /d1 0 ζ 0 − ρ0 + ϑ0 ϑ0 . 4

Together with the GCM conditions /d?2 0 /d?1 0 µ0 = 0 on Σ∗ , and the fact that S1 ⊂ Σ∗ , we infer /d?2 0 /d?1 0 /d1 0 ζ 0

= − /d?2 0 /d?1 0 ρ0 +

1 ?0 ?0 0 0 /d2 /d1 (ϑ ϑ ). 4

In view of the identity /d?1 0 /d1 0 = /d2 0 /d?2 0 + 2K 0 , we infer ( /d?2 0 /d2 0 + 2K 0 ) /d?2 0 ζ 0

=

− /d?2 0 /d?1 0 ρ0 +

1 ?0 ?0 0 0 /d2 /d1 (ϑ ϑ ) + 2e0θ (K 0 )ζ 0 . 4

Using the estimate for ρ0 of Step 5 and an elliptic estimate, k

max r2 k d/0 /d?2 0 ζ 0 kL2 (S1 )

k≤j−2

. 0 .

(4.1.14)

Note that the quadratic terms involving ϑ0 ϑ0 and e0θ (K 0 )ζ 0 are estimated using the transformation formulas,3 the estimates (4.1.9) for (f, f , λ), and the estimates (4.1.1) for the curvature components and the Ricci coefficients of the outgoing part (ext) L0 of the initial data layer. Step 7. Recall Codazzi for ϑ0 /d2 0 ϑ0

= −2β 0 − /d?01 κ0 + ζ 0 κ0 − ϑ0 ζ 0 .

We differentiate w.r.t. /d?2 0 and use the GCM condition κ0 = 2/r0 which holds on Σ∗ and S1 ⊂ Σ∗ to deduce /d?2 0 /d2 0 ϑ0

= −2 /d?2 0 β 0 + κ0 /d?2 0 ζ 0 − /d?2 0 (ϑ0 ζ 0 ).

Together with the estimate of Step 5 for β 0 , the estimate of Step 6 for /d?2 0 ζ 0 , dealing with the quadratic terms as above, and using an elliptic estimate, we infer k

max rk d/0 ϑ0 kL2 (S1 ) k≤j

. 0 .

3 In fact, in view of the identity K 0 = −ρ0 − 1 κ0 κ0 + 1 ϑ0 ϑ0 , the GCM conditions for κ0 and κ0 , 4 4 and the control of ρ0 in Step 5, we only need the transformation formulas for ϑ0 and ϑ0 . These formulas involve at most one angular derivative of f and f , and no transversal derivative.

156

CHAPTER 4

Next, recall the transformation formula   1 ϑ0 = λ ϑ − /d?2 0 (f ) + f (ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t. . 4 Together with the above estimate for ϑ0 , the estimate (4.1.9) for (f, f , λ), and the estimates (4.1.1) for the Ricci coefficients of the outgoing part (ext) L0 of the initial data layer, we infer k

max kr /d?2 0 ( d/0 f )kL2 (S1 ) k≤j

. 0 + 2 . 0 .

Together with a Poincar´e inequality, we infer 0k

max k d/ f kL2 (S1 )

k≤j+1

. 0 + r

−2

Z

S1

fe . Φ

(4.1.15)

Step 8. In view of the last estimate of Step 7, we need to control the ` = 1 mode of f . Recall from Lemma 4.1  Z   Z  1 0 0 1 0 0 Φ 0 0 0 Φ 0 0 e4 r βe = − κ ˇ β +ζ α − ϑβ e . 2 2 S S Transporting along C1 from S1 , using the control of the ` = 1 mode of β 0 in (4.1.6) on S1 , and using the bootstrap assumptions on (ext) M, we infer Z 0 Φ sup r β e . 0 + 2 . 0 . S⊂C1

S

In particular, consider the sphere S4m0 = C1 ∩ {r0 = 4m0 }. Then Z β 0 e Φ . 0 . S4m 0

Together with the transformation formula   3 1 0 β = λ β + ρf + f α + l.o.t. , 2 2 which we rewrite, multiply by eΦ , and integrate on S4m0 ,  0  Z Z Z 3m0 3m 3m Φ 0 Φ f e = − β e + − f eΦ r3 r0 3 S4m0 r0 3 S4m0 S4m0   Z Z 3 3 2m Φ + (λ − 1)ρf e + ρ + 3 f eΦ 2 S4m0 2 S4m0 r   Z 1 + λ β + f α + l.o.t. eΦ , 2 S4m0 the bootstrap assumptions (4.1.7) for (f, f , λ), and the control of the initial data

157

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

layer, we infer Z Φ fe . S4m 0

.

0 + 2 +  sup |r0 − r| + |m0 − m|




S4m0


0 +  sup |r0 − r| + |m0 − m| . S4m0

Now, the bootstrap assumptions (4.1.7) for (f, f , λ), together with the estimate for r − r0 in Lemma 9.15 with δ1 = , and the one for m0 − m in Corollary 9.19 with ◦  = , yields
sup |r0 − r| + |m0 − m| .  S4m0

and hence Z Φ fe . S4m

0 + 2 . 0 .

0

Next, recall from Corollary 2.93 that f satisfies the following transport equations along C1 λ−1 e04 (r0 f )

= E10 (f, Γ).

We deduce from Corollary 2.65 that      Z Z  1 0 1 0 0 −2 Φ 0 −2 0 0 e4 r fe = r e4 (f ) + κ +κ ˇ − ϑ f eΦ 2 2 S S    Z  3 0 1 0 0 −2 0 −1 0 0 = r r e4 (r f ) + κ ˇ − ϑ f eΦ 2 2 S    Z  3 0 1 0 −2 −1 = r0 r0 λ0 E10 (f, Γ) + κ ˇ − ϑ f eΦ . 2 2 S In view of the form of E10 in Corollary 2.93, the bootstrap assumption (4.1.7) for f , and the estimates (4.1.1) for the Ricci coefficients of the outgoing part (ext) L0 of the initial data layer, we have r2 |E10 (f, Γ)| . 0 + 2 . 0 on C1 . We deduce   Z Φ e4 r0 −2 fe . S

h   i 0 −1 0 0 2 (S) + kϑ kL2 (S) kf kL2 (S) . + sup r kˇ κ k L r2 S⊂C1

Using the bootstrap assumption (4.1.7) for f , and the bootstrap assumption on decay on (ext) M for κ ˇ 0 and ϑ0 , we infer   Z 0 + 2 0 Φ e4 r0 −2 f e . . 2. 2 r r S Integrating forward from r = 4m0 , and using the above estimate for the ` = 1 mode

158

CHAPTER 4

of f on S4m0 , we obtain Z sup r−2 f eΦ .

S⊂C1

0 .

S

k

Together with the estimate for d/0 f of Step 7, and since S1 ⊂ C1 , we deduce k

max k d/0 f kL2 (S1 )

k≤j+1

. 0 .

Together with (4.1.10), we infer S1

kf khj+1 (S1 ) + r−1 k(f , λ − λ )khj+1 (S1 )

. 0 .

In particular, the above estimate for (f, f ) allows us to use Lemma 9.15 with δ1 = 0 which yields 0 r sup − 1 . 0 . r S1 Together with (4.1.11), we infer kf khj+1 (S1 ) + r−1 k(f , log λ)khj+1 (S1 )

.

0 .

This implies the iteration assumption (4.1.9) for j + 1, for all 4 ≤ j ≤ klarge + 5. Thus, we have obtained kf khklarge +6 (S1 ) + r−1 k(f , log λ)khklarge +6 (S1 )

.

0 .

In view of the above estimate for (f, f , λ), and since S1 ⊂ Σ∗ , we may apply ◦

Corollary 9.53 with δ = 0 and smax = klarge + 5 which yields kd≤klarge +6 f kL2 (S1 ) + r−1 kd≤klarge +6 (f , log λ)kL2 (S1 ) +kd≤klarge +5 e03 (f , log λ)kL2 (S1 )

.

0 .

The above control of (f, f ), together with Lemma 9.15 for δ1 = 0 , and Corollary ◦

9.19 with δ1 =  = 0 , implies 0   0 m r sup − 1 + − 1 . 0 . m0 r S1 We have thus obtained on S1 kd≤klarge +6 f kL2 (S1 ) + r−1 kd≤klarge +6 (f , log(λ))kL2 (S1 ) (4.1.16) 0   0 m r +kd≤klarge +5 e03 (f , log(λ))kL2 (S1 ) + sup − 1 + − 1 . 0 . m r 0 S 1

159

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Finally, we will also need the following estimates on S1

 

≤k 2 0 0 0 large +5

r d κ − , κ ˇ , ϑ

2 r0 L (S1 )

 0 

r ≤klarge +5 +r−1 −1

d

2 r

. 0 .

(4.1.17)

L (S1 )

The estimates for κ0 in (4.1.17) follow from the GCM condition on κ0 , as well as Raychaudhuri for transversal derivatives. The estimate for ϑ0 in (4.1.17) follows from the transformation formula, the control (4.1.16) of (f, f , λ), and the control of the initial data layer. We obtain similarly the control of ξ 0 , ω 0 and η 0 on S1 , which in turn yields the control of Ω0 and ς 0 on S1 in view of Lemma 2.61, and finally the 0 control of rr in (4.1.17) relying on (4.1.16) and (2.2.21). Step 9. Recall from Corollary 2.93 that (f, log(λ)) satisfy the following transport equations along C1 λ−1 e04 (rf ) λ−1 e04 (log(λ))

=

E10 (f, Γ),

= E20 (f, Γ),

where, in view of the form of E10 , E20 in Corollary 2.93 and the estimates (4.1.1) for the Ricci coefficients of the outgoing part (ext) L0 of the initial data layer, we have |dk E10 (f, Γ)| + |dk E20 (f, Γ)| .

0 + |d≤k f |2 for k ≤ klarge + 5 on C1 . r2

Next, recall from Lemma 2.69 the following commutator identity     m m 2 e4 (m) [T, e4 ] = ω− 2 − κ− + e4 + (η + ζ)eθ r 2r r r while from Lemma 2.68, we have schematically

[ d/, e4 ] = κ ˇ , ϑ d/ + ζ, rβ . Together with the fact that λ−1 e04

= e4 + f eθ +

f2 e3 , 4

the commutator above identities for [T, e4 ] and [ d/, e4 ], as well as the estimates (4.1.1) for the Ricci coefficients and curvature components of the outgoing part (ext) L0 of the initial data layer, we infer, for k ≤ klarge + 5, |dk [T, λ−1 e04 ]h| + |dk [ d/, λ−1 e04 ]h| 0 ≤k+1 1 1 . |d h| + |d≤k (f dh)| + |d≤k (hdf )| + |d≤k (f 2 dh)| + |d≤k (hf df )|. 2 r r r By commuting first the transport equations in λ−1 e04 with (T, d/)k , and by using

160

CHAPTER 4

these transport equations to recover the e4 derivatives, we deduce λ−1 e04 (rdk f ) λ−1 e04 (dk log(λ))

=

0 E1,k (f, Γ),

0 = E2,k (f, Γ),

where we have 0 0 |E1,k (f, Γ)| + |E2,k (f, Γ)| .

0 + |d≤k f |2 for k ≤ klarge + 5 on C1 . r2

This allows us to propagate the estimates for (f, λ) in (4.1.10) on S1 to any sphere on C1 , and hence   sup kd≤klarge +5 f kL2 (S) + r−1 kd≤klarge +5 log λkL2 (S) . 0 . (4.1.18) S⊂C1

Step 10. Our next goal is to control f along C1 . We cannot proceed along the same lines as the control of (f, λ) in Step 9. Indeed, we cannot rely on the last transport equation along λ−1 e04 of Corollary 2.93, as it is not consistent with the control of f on S1 derived in Step 8. Instead, we first control α0 , κ0 and ϑ0 . Recall the following transformation formula α0

  3 = λ2 α + 2f β + f 2 ρ + l.o.t. 2

which does not depend on f . Together with the control of (f, λ) of Step 9 and the control of the initial data layer, we infer 5

sup r 2 +δB kd≤klarge +5 α0 kL2 (S)

.

0 .

S⊂C1

Next, recall e04



2 κ − 0 r 0



e04 (ˇ κ0 ) + κ0 κ ˇ0

  1 0 2 0 + κ κ − 0 = 2 r

=

1 1 02 − ϑ02 + κ ˇ , 4 2

1 0 2 1 0 2 1 02 − (ˇ κ ) − (ˇ κ ) − (ϑ − ϑ0 2 ), 2 2 2

and e04 (ϑ0 ) + κ0 ϑ0 = −2α0 . Proceeding as in Step 9, we commute first these transport equations with (T, d/)k , and use the transport equations to recover the e4 derivatives. By integrating the resulting transport equations from S1 where κ0 , κ ˇ 0 and ϑ0 are under control in view 0 of (4.1.17), and using the above control of α , we infer

 

≤k 2 0 0 0 large +5

sup r d κ − 0, κ ˇ,ϑ . 0 . (4.1.19)

2 r S⊂C1 L (S)

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

161

Also, we have, using in particular (2.2.21),   0  r λ−1 e04 (r0 ) e4 (r) f 2 e3 (r) −1 0 λ e4 log = − − r r r 4 r   −1 1 −1 0 λ 1 rf 2 = λ κ −κ − κ ˇ0 + κ ˇ− (κ + A) 2 2 2 8   −1 1 0 λ 1 rf 2 = /d1 (f ) + Err(κ, κ0 ) − κ ˇ0 + κ ˇ− (κ + A) 2 2 2 8 where we have also used the change of frame formula for κ0 . Proceeding as in Step 9, we commute first these transport equations with (T, d/)k , and use the transport equations to recover the e4 derivatives. By integrating the resulting transport 0 equations from S1 where rr is under control in view of (4.1.17), and using the 4 estimate of Step 9 for f and λ, the estimate of Step 10 for κ ˇ 0 , and the estimate (ext) for the initial data foliation layer on L0 , we infer

 0 

≤k r large +5

sup r−1 d log . 0 . (4.1.20)

r L2 (S) S⊂C1 Step 11. Recall Codazzi for ϑ0 /d2 0 ϑ0

= −2β 0 − /d?01 κ0 + ζ 0 κ0 − ϑ0 ζ 0 .

This yields ζ0

=

r0 2



  2 2β 0 + /d2 0 ϑ0 + /d?01 κ0 + ϑ0 ζ 0 − ζ 0 κ0 − 0 . r

Together with the control of κ0 , ϑ0 and r0 of Step 10, we infer sup kd≤klarge +4 ζ 0 kL2 (S)

.

S⊂C1

sup rkd≤klarge +4 β 0 kL2 (S) + 0

S⊂C1

+0 sup kd≤klarge +4 ζ 0 kL2 (S) S⊂C1

and hence, for 0 small enough, sup kd≤klarge +4 ζ 0 kL2 (S)

S⊂C1

.

sup rkd≤klarge +4 β 0 kL2 (S) + 0 .

S⊂C1

Recall the transformation formulas   3 1 0 β = λ β + ρf + f α + l.o.t. , 2 2 1 1 ζ 0 = ζ − e0θ (log(λ)) + (−f κ + f κ) + f ω − f ω + f e0θ (f ) 4 2 1 + (−f ϑ + f ϑ) + l.o.t. 4 Together with the control of f and λ from Step 9, the control of the initial data 4 Note

that the RHS of the transport equation does not depend on f .

162

CHAPTER 4

foliation layer on

(ext)

L0 , and the above control of ζ 0 , we infer

sup r−1 kd≤klarge +4 f kL2 (S)

.

S⊂C1

sup kd≤klarge +4 ζ 0 kL2 (S) + 0

S⊂C1

.

sup rkd≤klarge +4 β 0 kL2 (S) + 0

S⊂C1

. 0 + 0 sup r−1 kd≤klarge +4 f kL2 (S) S⊂C1

and hence, for 0 small enough, sup r−1 kd≤klarge +4 f kL2 (S)

. 0 .

S⊂C1

Together with the control of f and λ from Step 9, we have in particular   sup kd≤klarge +4 f kL2 (S) + r−1 kd≤klarge +4 (log(λ), f )kL2 (S) . 0 . S⊂C1

Note that this concludes the improvement of the bootstrap assumptions (4.1.7) on (f, f , λ). Also, using Sobolev, we infer   sup r|d≤klarge +2 f | + |d≤klarge +2 (f , log(λ))| .

0 .

(4.1.21)

C1

Step 12. In view of (4.1.21), the change of frame formulas of Proposition 2.90, and the estimates (4.1.1) for the curvature components of the outgoing part (ext) L0 of the initial data layer, we obtain ( h 7   i 9 max sup r 2 +δB |dk (ext) α| + |dk (ext) β| + r 2 +δB |dk−1 e3 ( (ext) α)| (4.1.22) 0≤k≤klarge

C1

 + sup r3 C1

  ) k (ext) 2m 0 2 k (ext) k (ext) d ρ + 3 + r |d β| + r|d α| . r

Also, according to Proposition 2.71, we have in (ext) (ext)

e4 (

(ext)

m)

=

r 32π

Z

(ext)

0 .

M

2 (ext) κ ˇ (ext) ρˇ + 2 (ext) eθ ( (ext) κ) (ext) ζ

S

! 1 (ext) (ext) 2 1 (ext) (ext) (ext) (ext) (ext) 2 κ( κ ˇ ϑ ϑ+2 − ϑ) − κ( ζ) , 2 2 which together with the bootstrap assumptions on (ext) M yields sup r2 (ext) e4 ( (ext) m) . 20 . C1

This allows us to propagate the estimates for (ext) m in (4.1.16) on S1 to any sphere on C1 , and hence (ext) m sup − 1 . 0 . (4.1.23) m0 C1

163

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS r0 r

Also, in view of the control

of Step 10, we have

(ext) r sup (ext) − 1 . 0 . r C L0

1

Step 13. Recall that • • • •

((ext) e4 , (ext) e3 , (ext) eθ ) denotes the null frame of (ext) M, ((int) e4 , (int) e3 , (ext) eθ ) denotes the null frame of (int) M, ( (ext) (e0 )3 , (ext) (e0 )4 , (ext) (e0 )θ ) denotes the null frame of ( (int) (e0 )3 , (int) (e0 )4 , (int) (e0 )θ ) denotes the null frame of

(ext) (int)

L0 , L0 .

Also, recall that the timelike hypersurface T is given by     δH 3δH (ext) T ={ r = rT } where 2m0 1 + ≤ rT ≤ 2m0 1 + 2 2 to that T ⊂ as follows (int)

(int)

L0 ∩

(ext)

e4 = λ (ext) e4 ,

L0 , and recall that the frame of (int)

e3 = λ−1

(ext)

(int)

e3 ,

(int)

eθ =

M is initialed on T

(ext)

eθ on T

where λ=

(ext)

λ=1−

2 (ext) m . (ext) r

Denoting • by (f, f , λ) the transition functions from the frame of the outgoing part (ext) L0 of the initial data layer to the frame of (ext) M as in Steps 5 to 12, • by (f 0 , f 0 , λ0 ) the transition functions from the frame of the ingoing part (int) L0 of the initial data layer to the frame of (int) M, ˜ the transition functions on (int) L0 ∩ (ext) L0 from the frame outgoing • by (f˜, f˜ , λ) (ext) part L0 of the initial data layer to the frame of the ingoing part (int) L0 of the initial data layer, we obtain, using also that C1 ∩ C 1 ⊂ T ,   sup |d≤klarge +2 (f 0 , f 0 , log(λ0 ))| . C1 ∩C 1

sup



C1 ∩C 1

|d≤klarge +2 (f, f , log(λ))|

+ sup



 ˜ |d≤klarge +2 (f˜, f˜ , log(Υ−1 λ))| 0



 |d≤klarge +2 log(Υ−1 0 Υ)|

C1 ∩C 1

+ sup



C1 ∩C 1

where we have denoted Υ0 = 1 −

2m0 , (ext) r L0

Υ=1−

2 (ext) m . (ext) r

˜ Together with the control of (f˜, f˜ , log(Υ−1 0 λ)) provided on

(int)

L0 ∩

(ext)

L0 by the

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CHAPTER 4

estimates (4.1.1), the estimates (4.1.21) for (f, f , λ), and the estimates and (ext) r − (ext) rL0 obtained in Step 12, we infer   sup |d≤klarge +2 (f 0 , f 0 , log(λ0 ))| . 0 .

(ext)

m − m0

C1 ∩C 1

Step 14. We propagate the estimate for (f 0 , f 0 , log(λ0 )) on C1 ∩ C 1 provided by Step 8 to C 1 using the analog of Corollary 2.93 in the ingoing direction e3 . We obtain the following estimate   sup |d≤klarge +2 (f , log λ)| + sup |d≤klarge +1 f | . 0 . C1

C1

Together with the change of frame formulas of Proposition 2.90, and the estimates (4.1.1) for the curvature components of the ingoing part (int) L0 of the initial data layer, we obtain "   k (int) 2m0 k (int) k (int) max sup |d α| + |d β| + d ρ+ 3 (4.1.24) 0≤k≤klarge C r 1 # +|dk (int) β| + |dk (int) α|

.

0 .

Also, since we have as a consequence of the initialization on T of the ingoing geodesic foliation of (int) M (int)

we infer from the control of

m=

(ext)

(ext)

m on C1 ∩ C 1

m provided by Step 12

| (int) m − m0 | . 0 on C1 ∩ C 1 . We then propagate, similarly to Step 12, this bound to C 1 and obtain sup (int) m − m0 . 0 . C1

Together with (4.1.22), (4.1.23) and (4.1.24), this concludes the proof of Theorem M0.

4.2

CONTROL OF AVERAGES AND OF THE HAWKING MASS

In this section, we prove Lemma 3.15 and Lemma 3.16. 4.2.1

Proof of Lemma 3.15

Step 1. We start with the control of ρ on M. Recall the identity (2.2.12) ρ+

2m r3

=

1 ϑϑ. 4

165

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Thus, in view of the bootstrap assumptions BA-D, BA-E, we have, 3 2m in (ext) M, ρ + 3 . 2 min{r−3 u− 2 −δdec , r−2 u−2−2δdec } r 2m in (int) M. ρ + 3 . 2 u−2−2δdec r Differentiating the equation with   2m e4 ρ + 3 r   2m e3 ρ + 3 r   2m eθ ρ + 3 r

respect to e3 , e4 we derive = = =

1 e4 (ϑ)ϑ + ϑe4 (ϑ) + l.o.t., 4 1 e3 (ϑ)ϑ + ϑe3 (ϑ) + l.o.t., 4 0.

Taking higher derivatives in e3 , e4 and making use of the bootstrap assumptions BA-D, BA-E, we derive, in (ext) M,   ≤k d small ρ + 2m . 2 min{r−3 u− 32 −δdec , r−2 u−2−2δdec }, r3   ≤k d large ρ + 2m . r−3 u−1/2−δdec , r3 and in

(int)

M,   ≤k d small ρ + 2m r3   ≤k d large ρ + 2m r3

. 2 u−2−2δdec , . 2 u−1−δdec .

In particular,   3 2m sup u 2 +δdec r3 d≤ksmall ρ + 3 r (ext) M   1 2m +δdec 3 ≤klarge 2 + sup u r d ρ+ 3 r (ext) M   3 2m +δdec ≤ksmall 2 sup u ρ+ 3 d r (int) M   1 2m +δdec ≤klarge 2 + sup u ρ+ 3 d r (int) M

. 0 ,

. 0 .

Step 2. Next, we proceed with the control of κ in (ext) M. Recalling Lemma 2.72, we start with     2 1 2 1 1 2 e4 κ − + κ κ− = − ϑ2 + κ ˇ . (4.2.1) r 2 r 4 2

166

CHAPTER 4

In view of Corollary 2.67 we deduce, from the first equation,      2 e4 r κ − = −r 14 ϑ2 + 12 κ ˇ2 . r

(4.2.2)

Making use of the GCM condition κ =

2 on Σ∗ , r

κ =

2 on Σ∗ , r

which yields

we deduce, integrating (4.2.2) with respect to r along Cu from Σ∗ , 2 1+δdec 3 sup u r κ − . 2 . 0 . r (ext) M Also, making use of the bootstrap assumptions BA-D, BA-E we easily deduce   ≤ksmall +1 2 sup u1+δdec r3 d% κ− . 2 . 0 , r (ext) M   1 2 +δdec 3 ≤klarge +1 2 sup u r d% κ− . 2 . 0 . r (ext) M We next commute (4.2.2) with e3 and derive          2 1 2 1 2 2 e4 e3 r κ − = e3 r ϑ + κ ˇ − [e3 , e4 ] r κ − r 4 2 r       1 2 1 2 2 = e3 r ϑ + κ ˇ − 2ω r κ − 4 2 r    2 −2ζ r κ − . r It is thus easy to see that we can prove estimates of the type   2 sup u1+δdec r3 d≤ksmall +1 κ − . 2 . 0 , r (ext) M   1 2 +δdec 3 ≤klarge +1 2 sup u r d κ− . 2 . 0 , r (ext) M provided that we can check that  1+δdec 3 ≤ksmall +1 sup u r e3 κ− (ext) M  ≤k 1 +1 sup u 2 +δdec r3 e3 large κ− (ext) M

 2 . r  2 . r

2 . 0 , 2 . 0 .

167

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

The difficulty in this case is to make sure that we can control terms of the type    1 k+1 2 1 k+1 2 ek+1 r e (ϑ ) + e (ˇ κ ) 3 4 3 2 3 ˇ R. ˇ To see this we note that using only at most k derivatives of Γ, e3 (ϑ2 )

=

e3 (ˇ κ2 )

=

ˇκ−Ω ˇκ e 3 ϑ2 − ( Ω ˇ )ϑ2 + κ ˇ ϑˇ2 , ˇκ−Ω ˇκ e3 κ ˇ 2 − (Ω ˇ )ˇ κ2 + κ ˇ κˇ2 ,

and 1 1 e3 (ϑ) + κϑ − 2ωϑ = −2 /d?2 ζ − κϑ + 2ζ 2 , 2 2 1 1 ˇ κ + Err[e3 κ (4.2.3) e3 κ ˇ + κκ ˇ = −2ˇ µ − κˇ κ + 2(ˇ ω κ + ωˇ κ) + Ωκ ˇ ], 2 2 1 1 ˇ κ κ. Err[e3 κ ˇ ] : = 2(ζ 2 − ζ 2 ) + 2(ˇ ωκ ˇ−ω ˇκ ˇ) − κ ˇκ ˇ− κ ˇκ ˇ − Ωˇ 2 2 We thus derive   2 sup u r d≤ksmall +1 κ − r (ext) M   1 2 + sup u 2 +δdec r3 d≤klarge +1 κ − . r (ext) M 1+δdec 3

0 .

Step 3. We next estimate κ in (ext) M making use of the identity (2.2.14) derived in connection to the Hawking mass   2Υ 2Υ 2 1 κ+ = κ− − κ ˇκ ˇ. r rκ r κ Thus, in view of the estimates for κ derived in Step 2 we easily infer that     3 1 2Υ 2Υ 2 +δdec r 2 d≤klarge sup u 2 +δdec r2 d≤ksmall κ + + sup u κ + . 0 r (ext) M r (ext) M as desired. Step 4. We estimate ω in (ext) M based on the following identity in Lemma 2.72         2 1 2 2 4 m 2m e3 κ − + κ κ− = 2ω κ − + ω− 2 +2 ρ+ 3 r 2 r r r r r   1 2 ˇ 1 − κ κ− Ω + 2ˇ ωκ ˇ − ϑϑ + 2ζ 2 2 r 2  1 ˇ 1 ˇ 4 (ˇ + Ω − ϑ2 + κ ˇ 2 − Ω(e κ) + κˇ κ) + κ ˇκ ˇ 2 2 1ˇ − Ωˇ κ, r

168

CHAPTER 4

which we rewrite as (         m r 2 1 2 2 2m ω− 2 = e3 κ − + κ κ− − 2ω κ − −2 ρ+ 3 r 4 r 2 r r r     1 2 ˇ 1 1ˇ + κ κ− Ω − 2ˇ ωκ ˇ + ϑϑ − 2ζ 2 − Ω − ϑ2 + κ ˇ2 2 r 2 2 ) 1 1ˇ ˇ +Ω(e4 (ˇ κ) + κˇ κ) − κ ˇκ ˇ + Ωˇ κ . (4.2.4) 2 r Using the estimates of ρ in Step 1, the estimates for κ in Step 2, as well as our bootstrap assumptions on decay and energy, we easily derive   1 m  m  sup u1+δdec r2 d≤ksmall ω − 2 + sup u 2 +δdec r2 d≤klarge ω − 2 . 0 . r r (ext) M (ext) M Remark 4.6. It is to estimate klarge derivatives of ω −mr−2 that we had to control klarge + 1 derivatives of κ − 2/r in Step 2. Step 5. We estimate Ω in (ext) M. First we need the control of Ω on Σ∗ . To this end, we recall that s is initialized on Σ∗ by s = r so that ν(s − r)

=

0 on Σ∗ ,

ν = e3 + ae4 ,

where the scalar function a is such that the vectorfield ν is tangent to Σ∗ . On the other hand, we have e4 (s) = 1 and e4 (r) =

r κ = 1 on Σ∗ 2

where we used the GCM condition κ = 2/r on Σ∗ . We infer e3 (s) = e3 (r) on Σ∗ and hence Ω

= e3 (r) on Σ∗ .

This yields Ω

= e3 (r) =

rκ r + A, 2 2

and hence, in view of the estimate for κ of Step 3, the fact that A contains only quadratic terms in view of the formula for A, and in view of the bootstrap assumptions on decay and energy, we infer   1 m  m  sup u1+δdec r d≤ksmall Ω − 2 + sup u 2 +δdec r d≤klarge Ω − 2 . 0 . r r Σ∗ Σ∗ Then, we use e4 (Ω) = −2ω and Corollary 2.66 to obtain e4 (Ω)

=

ˇ −2ω + κ ˇΩ

169

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

and hence e4 (Ω + Υ)

=

m m −2 ω − 2 + r r 



2 κ− r



ˇ − 2e4 (m) . +κ ˇΩ r

Commuting with d, integrating from Σ∗ where we have controlled Ω above, and using the estimates of Step 2 for κ, Step 4 for ω, the bootstrap assumptions, and the estimates for e4 (m) of Lemma 3.16 (which do not depend on the control of Ω), we infer   1 m  m  sup u1+δdec r d≤ksmall Ω − 2 + sup u 2 +δdec r d≤klarge Ω − 2 . 0 . r r (ext) M (ext) M Step 6. Next, we control (int) κ on the cylinder T . From the initialization of the frame of (int) M on T , we have (int)

r=

(ext)

r,

(int)

κ = Υ (ext) κ,

(int)

κ = Υ−1 (ext) κ on T .

Also, making use of the identity (2.2.14) derived in connection to the Hawking mass, we have   2Υ 2Υ 2 1 (ext) (ext) κ+ = κ− − (ext) (ext) κ ˇ (ext) κ ˇ. (ext) r r r κ κ We deduce (int)

κ+

2 r

=

Υ−1



(ext)

κ+

2Υ r

 =



2 r (ext) κ

(ext)

κ−

2 r



Υ−1 − (ext) (ext) κ ˇ (ext) κ ˇ on T . κ To derive higher tangential derivatives along T we remark that the vectorfield TT = e4 −

e4 (r) κ+A e3 = e4 − e3 , e3 (r) κ

together with eθ , spans the tangent space to T . The transversal derivatives, on the other hand, can be determined with help of the equation     2 1 2 1 1 2 e3 κ + + κ κ+ = − ϑ2 + κ ˇ r 2 r 4 2 adapted to the (int) M foliation. Making use of the estimates for (ext) κ in (ext) M derived in Step 2 and the bootstrap assumptions, we infer that     3 1 2 2 +δdec ≤ksmall +1 (int) +δdec ≤klarge +1 (int) 2 2 sup u κ+ + sup u κ+ d d r r T T   1 . 0 + sup u 2 +δdec dklarge +1 (ext) κ ˇ (ext) κ ˇ . T

Now, in view of the transport equations involving

(ext)

e4 ( (ext) κ ˇ ),

(ext)

e3 ( (ext) κ ˇ ),

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CHAPTER 4

(ext)

e4 ( (ext) κ ˇ ) and (ext) e3 ( (ext) κ ˇ ), as well as the bootstrap assumptions, we have   1 sup u 2 +δdec dklarge +1 (ext) κ ˇ (ext) κ ˇ T 1 . 0 + sup u 2 +δdec (ext) κ ˇ dklarge /d1 ( (ext) ζ) T 1 1 + sup u 2 +δdec (ext) κ ˇ dklarge /d1 ( (ext) ζ) + sup u 2 +δdec (ext) κ ˇ dklarge /d1 ( (ext) ξ) T T 1 1 . 0 + sup u 2 +δdec /d?1 (ext) κ ˇ dklarge (ext) ζ + sup u 2 +δdec /d?1 (ext) κ ˇ dklarge (ext) ζ T T 1 + sup u 2 +δdec /d?1 (ext) κ ˇ dklarge (ext) ξ T

.

0

where we have integrated /d1 by parts and used that /d?1 is its adjoint. We infer     ≤k ≤k 3 1 2 +δ +1 (int) +δ dec small dec d large +1 (int) κ + 2 . 0 . 2 sup u 2 d κ + + sup u r r T T Step 7. From now on, we only work with the frame of (int) M. Starting with the equation     2 1 2 1 1 2 e3 κ + + κ κ+ = − ϑ2 + κ ˇ r 2 r 4 2 and using the estimates of Step 5, we can then proceed precisely as in Step 2 (using the (int) M counterpart of the equations (4.2.3)) to derive     1 2 2 2 +δdec d≤klarge +1 sup u1+δdec d≤ksmall +1 κ + + sup u κ + . 0 . r (int) M r (int) M Step 8. Finally, we estimate the remaining averages in (int) M, i.e., κ and ω. To estimate κ we make use once more of the identity   2Υ 2Υ 2 1 κ− = − κ+ − κ ˇκ ˇ. r rκ r κ Making use of the estimates of κ in Step 5 as well as the bootstrap assumptions for κ ˇ and κ ˇ we easily derive     1 2Υ 2Υ 2 +δdec d≤klarge sup u1+δdec d≤ksmall κ − + sup u κ − . 0 . r (int) M r (int) M

171

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Step 9. To estimate ω we proceed as in Step 4 by making use of the identity (         m r 2 1 2 2 2m ω+ 2 = e4 κ + + κ κ+ − 2ω κ + −2 ρ+ 3 r 4 r 2 r r r     1 2 ˇ 1 1ˇ + κ κ+ Ω − 2ˇ ωκ ˇ + ϑϑ − 2ζ 2 − Ω − ϑ2 + κ ˇ2 2 r 2 2 ) 1 1 ˇ 4 (ˇ ˇκ . +Ω(e κ) + κˇ κ) − κ ˇκ ˇ + Ωˇ 2 r Thus, in view of the estimates of ρ in Step 1, the estimates for κ in Step 5, the estimates of κ above,5 as well as the bootstrap assumptions BA-D and BA-E, we deduce   1 m  m  sup u1+δdec d≤ksmall ω + 2 + sup u 2 +δdec d≤klarge ω + 2 . 0 . r r (int) M (int) M Step 10. It remains to estimate Ω in (int) M. First we need the control of Ω on T . To this end, we recall that s is initialized on T by s = r so that TT (s − r)

=

0 on T ,

TT = e4 −

κ+A e3 , κ

where the vectorfield has been introduced above and is tangent to T . On the other hand, we have e3 (s) = −1 and e3 (r) = rκ/2, and hence Ω

  κ − 2r κ+A r = e4 (r) + (−1 − e3 (r)) = (κ + A) 1 + on T . κ 2 κ

This yields Ω

=

  κ − 2r r (κ + A) 1 + on T , 2 κ

and hence, in view of the estimate for κ of Step 7, the estimate for κ of Step 8, the fact that A contains only quadratic terms in view of the formula for A, and in view of the bootstrap assumptions on decay and energy, we infer

1 sup u1+δdec d≤ksmall Ω − Υ + sup u 2 +δdec d≤klarge Ω − Υ . 0 . T

T

Then, we use the analog of the transport equation used to estimate Ω in i.e.,    m m 2 ˇ + 2e3 (m) . e3 (Ω − Υ) = 2 ω + 2 − κ+ +κ ˇΩ r r r r

(ext)

M,

Commuting with d, integrating from T where we have controlled Ω above, and using the estimates of Step 2 for κ, Step 4 for ω, the bootstrap assumptions, and 5 It is to estimate k 2 large derivatives of ω + m/r that we made sure to control klarge + 1 derivatives of κ + 2/r.

172

CHAPTER 4

the estimates for e4 (m) of Lemma 3.16 (which do not depend on the control of Ω), we infer

1 sup u1+δdec d≤ksmall Ω − Υ + sup u 2 +δdec d≤klarge Ω − Υ . 0 . (int) M

(int) M

This concludes the proof of Lemma 3.15. 4.2.2

Proof of Lemma 3.16

Step 1. We start with the control of e3 (m) and e4 (m) in Proposition 2.71 we have in (ext) M Z r e4 (m) = Err1 , 32π S

(ext)

M. According to (4.2.5)

and e3 (m)

=

Z Z
r
r ˇ + ς −1 Ωˇ 1 − ς −1 ςˇ Err1 + Ω ς Err1 32π S 32π S Z
r +ς −1 ςˇ 2ρˇ κ + 2ˇ ρκ + 2κ /d1 η + 2κ /d1 ξ + Err2 32π S Z −1 r ˇ (2ρˇ −ς (Ωˇ ς + Ως) κ + 2ˇ ρκ − 2κ /d1 ζ + Err2 ) 32π S h i m ˇ − ς −1 −ˇ ςκ ˇ + Ω ςˇκ ˇ + Ωςκ , r

(4.2.6)

where Err1

:=

Err1

:=

Err2

:=

Err2

:=

1 1 2ˇ κρˇ + 2eθ (κ)ζ − κϑ2 − κ ˇ ϑϑ + 2κζ 2 , 2 2
1 1 2ˇ ρκ ˇ − 2eθ (κ)η − 2eθ (κ)ξ − κ ˇ ϑϑ + 2κη 2 + 2κ η − 3ζ ξ − κϑ2 , 2 2 1 2 1 2 2ˇ ρκ ˇ − κϑ − κϑϑ + 2κζ , 2 2 
1 1 2 2ˇ ρκ ˇ + κ 2η − ϑϑ + 2κ η − 3ζ ξ − κϑ2 . 2 2

Thus, according to the bootstrap assumption BA-D on decay, we deduce |e4 (m)| . |e3 (m)| .

2 r−2 u−1−δdec , 2 u−2−2δdec .

Moreover, differentiating the equations with respect to e3 , e4 and making use of both bootstrap assumptions BA-D and BA-E on decay and energy, and integrating by part once the eθ derivative for the terms involving eθ (κ) and eθ (κ) when they contain top order derivatives, we infer that max sup r2 u1+δdec |dk e4 (m)| . 0 , (ext) M   1 sup r2 u 2 +δdec + ru1+δdec |dk e4 (m)| . 0 , 0≤k≤ksmall

max

0≤k≤klarge

(ext) M

173

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

as well as max

0≤k≤ksmall

sup u2+2δdec |dk e3 (m)| .

0 ,

sup u1+δdec |dk e3 (m)| .

0 ,

(ext) M

max

0≤k≤klarge

(ext) M

consistent with the statement of the lemma. Step 2. We derive the estimates on (int) M. According to the analogue of Proposition 2.71 in the situation of the incoming geodesic foliations of (int) M, and proceeding as in Step 1, we easily derive   max sup u1+δdec |dk e3 (m)| + |dk e4 (m)| . 2 . 0 . (4.2.7) 0≤k≤klarge

(int) M

(ext)

Step 3. We estimate m − m0 in have

M. First, recall from Theorem M0 that we

sup |m − m0 | . 0 m0 .

(4.2.8)

C1 ∪C 1

We start with the control in (ext) M. Note that (ext) M is covered by integral curves of e3 starting from C1 . Thus, integrating the e3 m equation and making use of the estimate supC1 |m − m0 | . 0 m0 as well as the fact that e3 (u) = 2, we easily deduce that sup |m − m0 | . 0 m0 + 2 . 0 m0 .

(ext) M

Step 4. We estimate |m − m0 | on T . In view of our initialization of the ingoing geodesic foliation of (int) M on T , (int)

κ (int) κ =

(ext)

κ (ext) κ on T .

Since the spheres of both foliations agree on T , we infer from the definition of the Hawking mass (int)

Using the estimate for

(ext)

m=

(ext)

m on T .

m we infer that

sup | (int) m − m0 | .

0 m0 .

T

Step 5. We estimate |m − m0 | on e3 (r) + 1

(int)

=

Thus, in view of the estimate for κ +

M. Note first that in   r r 2 κ+1= κ+ . 2 2 r 2 r

derived in Lemma 3.15

sup |e3 (r) + 1| . 2 .

(int) M

(int)

M

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CHAPTER 4

Thus integrating the estimate (4.2.7) in r ∈ [2m0 (1 − δH ), rT ], where we recall that rT ≤ 2m0 (1 + 2δH ), we derive sup |m − m0 | . 0 m0 .

(int) M

Since M =

(ext)

M∪

(int)

M we infer that sup |m − m0 | .

0 m0 .

M

This concludes the proof of Lemma 3.16.

4.3

CONTROL OF COORDINATES SYSTEMS

The goal of this section is to prove Propositions 3.17 and 3.18. In both cases, the first two claims, on the form of the spacetime metric in the corresponding coordinates system as well as on the expression of the coordinates vectorfield with respect to the null frame (e4 , e3 , eθ ), are already proved in Propositions 2.78 and ˇ Ω, ˇ ς, ς, γ, b, b 2.81. So we only focus on the third claim, i.e., on estimating Ω, Φ and e . The proof of Propositions 3.17 and 3.18 thus reduces to the proof of the following lemma. Lemma 4.7. Let θ ∈ [0, π] be the Z-invariant scalar on M defined by (2.2.52), i.e., θ = cot−1 (reθ (Φ)) .

(4.3.1)

Let b = e4 (θ),

b = e3 (θ),

γ=

1 . (eθ (θ))2

(4.3.2)

Then, we have max

0≤k≤ksmall

   γ   1 ru 2 +δdec + u1+δdec dk 2 − 1 + r dk b r (ext) M k
1+δdec k ˇ max sup u d Ω + d (ς − 1) + r dk b sup



0≤k≤ksmall

0≤k≤ksmall

. ,

(ext) M

sup u1+δdec

max

. ,

(int) M

 k k  ˇ + d (ς − 1) + dk γ − 1 d Ω 2 r ! k k + d b + d b

. .

Also, eΦ satisfies max

0≤k≤ksmall

sup



ru

   eΦ k +u − 1 . , d r sin θ  Φ  e 1+δdec k . . sup u d − 1 r sin θ (int) M

1 2 +δdec

(ext) M

max

0≤k≤ksmall

1+δdec

175

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Proof. We prove the estimates in to the reader.

(ext)

M. The proof in

(int)

M is similar and left

ˇ Recall that Step 1. We start with the estimate for Ω. ˇ =ξ /d?1 Ω so that the bootstrap assumptions for ξ imply on any 2-sphere of the foliation of (ext) M and for any k ≤ ksmall 1

ˇ L4 (S) + kdk r /d?1 Ωk ˇ L2 (S) . r2 sup |dk ξ| . ru−1−δdec . r 2 kdk r /d?1 Ωk S

In view of the commutation formulas of Lemma 2.68 and of Proposition 2.28, together with the bootstrap assumptions, we infer any k ≤ ksmall , schematically, [dk , r /d?1 ]

= O()d≤k + O(1)d≤k−1 ,

and hence, 1

ˇ L4 (S) + kr /d?1 dk Ωk ˇ L2 (S) r 2 kr /d?1 dk Ωk 1

ˇ L2 (S) + r 2 kd≤k Ωk ˇ L4 (S) . ru−1−δdec + kd≤k Ωk 1

ˇ L2 (S) + r 2 kd≤k−1 Ωk ˇ L4 (S) +kd≤k−1 Ωk −1−δdec ? ≤k ˇ ˇ L2 (S) . ru + kr /d1 d ΩkL2 (S) + kd≤k Ωk 1

1

ˇ L2 (S) + kd≤k Ωk ˇ 2 2 kd≤k−1 Ωk ˇ 22 , +kd≤k−1 Ωk L (S) L (S) where we used Gagliardo-Nirenberg on S. Together with the Poincar´e inequality of Corollary 2.37 for /d?1 , we deduce 1

ˇ L4 (S) + kr /d?1 dk Ωk ˇ L2 (S) + kdk Ωk ˇ L2 (S) . ru−1−δdec + kd≤k−1 Ωk ˇ L2 (S) . r 2 kr /d?1 dk Ωk By iteration, and using again Gagliardo-Nirenberg on S, we infer on any 2-sphere of the foliation of (ext) M and for any k ≤ ksmall ˇ L4 (S) + kdk Ωk ˇ L4 (S) kr /d?1 dk Ωk

1

. r 2 u−1−δdec ,

and thus, by Sobolev embedding max

0≤k≤ksmall

ˇ .  sup u1+δdec |dk Ω|

(ext) M

ˇ which is the desired estimate for Ω. Step 2. Next, we estimate ς. First, recall that we have eθ (log ς)

= η − ζ.

Since the bootstrap assumptions for η − ζ are at least as good as for ξ, we obtain,

176

CHAPTER 4

ˇ arguing as in Step 1, the following analog of the above estimate for Ω sup u1+δdec |dk ςˇ| . .

max

0≤k≤ksmall

(ext) M

Now that we control ςˇ, we turn to the estimate for ς. First, recall from the GCM on Σ∗ that we have u + r = cΣ∗ and a SP = −1 − 2m r , where ν = e3 + ae4 and ν is tangent to Σ∗ , with cΣ∗ a constant, and SP denoting the south pole of the spheres of Σ∗ . We deduce on the south poles of Σ∗   2 2m 0 = ν(u + r) = e3 (u) + e3 (r) + ae4 (r) = + e3 (r) − 1 + e4 (r) ς r and hence 2 −2 ς

r = − 2



2Υ κ+ r





2m +A− 1+ r

  2 on SP ∩ Σ∗ . κ− r

Together with the fact that ς = ς − ςˇ, the above control of ςˇ, the control of κ and ˇ in Step 1, the κ provided by Lemma 3.15, the formula for A, the control for Ω bootstrap assumptions on decay, and the fact that ς is constant on the sphere, we infer sup u1+δdec |dk (ς − 1)| .

max

0≤k≤ksmall Σ∗

.

Using ς = ς + ςˇ and the above estimates for ς and ςˇ, we obtain sup u1+δdec |dk (ς − 1)| .

max

0≤k≤ksmall Σ∗

.

Finally, recall e4 (ς)

=

0.

Commuting with d, using the bootstrap assumptions on decay and the above control for ς − 1 on Σ∗ , we infer max

0≤k≤ksmall

sup u1+δdec |dk (ς − 1)| . .

(ext) M

Remark 4.8. In (int) M, we analogously transport ς from the timelike hypersurface T . To estimate ς on T , one uses the following identity (in the frame of (int) M)

  κ − 2Υ + Υ κ + 2r κ+A 2 A 2 r −1 = − −1 − − on T . ς Υκ ς Υκ Υκ This identity follows from the definition of ς and ς, the identity for e3 (r) and e4 (r) in (int) M, the fact that u = u on T , and that T = {r = rT } so that the vectorfield TT = e4 − is tangent to T .

e4 (r) κ+A e3 = e4 − e3 e3 (r) κ

177

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Step 3. We make the auxiliary bootstrap the end of Step 5 Φ e ≤ 2r,

assumption which will be recovered at eθ (eΦ ) ≤ 2.

(4.3.3)

We start with the estimate for eΦ . Recall from (2.2.53) that the following identity holds eΦ r sin θ

=

√

1+a

(4.3.4)

where a has been introduced in (2.2.54) by a=

e2Φ + (eθ (eΦ ))2 − 1. r2

In order to estimate eΦ , it thus suffices to estimate a. Step 4. Now, recall from Lemma 2.85 that a verifies the following identities on (ext) M, e4 (a)

=

eθ (a)

=

e3 (a)

=

  (ˇ κ − ϑ)e2Φ + 2eθ (eΦ ) β − e4 (Φ)ζ eΦ , 2 r      2m 1 4Υ 1 2Φ 2eθ (Φ)e ρ+ 3 + κκ + 2 − ϑϑ , r 4 r 4     κ ˇ − A − ϑ e2Φ + 2eθ (eΦ ) β + e3 (Φ)ζ + ξe4 (Φ) eΦ . 2 r

Together with our bootstrap assumptions on decay for in (ext) M for κ ˇ , ϑ, κ ˇ , ϑ, β, ˇ and the bootstrap assumption (4.3.3), we infer β, ρ, ζ, ξ and Ω max

1≤k≤ksmall

sup



 1 ru 2 +δdec + u1+δdec dk a

. .

(ext) M

In particular, we deduce sup (ext) M

 1  ru 2 +δdec + u1+δdec |ˇ a| . .

Step 5. To estimate a we make use of equation (2.1.13) according to which 

2 eθ (eΦ ) = 1

on the axis of symmetry.

Since e2Φ also vanishes there we infer that a = 0 on the axis. Therefore, on the ˇ = −a, i.e., axis, a a = −ˇ a|axis and therefore, |a| . |ˇ a| .

 ru

1 2 +δdec

+ u1+δdec

.

178

CHAPTER 4

We conclude that max

0≤k≤ksmall

sup



 1 ru 2 +δdec + u1+δdec dk a . .

(4.3.5)

(ext) M

In view of (4.3.4) and (4.3.5), we immediately infer   1   eΦ max sup ru 2 +δdec + u1+δdec dk − 1 . 0≤k≤ksmall (ext) M r sin θ

.

Together with (4.3.5) and the definition of a, this implies Φ e = (1 + O())r sin θ ≤ 3r , 2 r 2Φ e 3 eθ (eΦ ) = 1 − + a ≤ | cos θ| + O() ≤ , 2 r 2

(4.3.6)

which is an improvement of the bootstrap assumption (4.3.3) which hence holds everywhere on (ext) M. Step 6. We now prove the estimates for b, b and γ. Recall from Lemma 2.84 that θ defined by (4.3.1) satisfies reθ (θ) e3 (θ) e4 (θ)

r2 (K − r12 ) , 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + A + ϑ) eθ (Φ) + rξe4 (Φ) + rζe3 (Φ) = − , 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + ϑ) eθ (Φ) − rζe3 (Φ) = − . 1 + (reθ (Φ))2 =

1+

In view of the definition of b, b and γ, we infer r2 (K − r12 ) , 1 + (reθ (Φ))2 rβ + 2r (−ˇ κ + A + ϑ) eθ (Φ) + rξe4 (Φ) + rζe3 (Φ) b = − , 1 + (reθ (Φ))2 κ + ϑ) eθ (Φ) − rζe3 (Φ) rβ + 2r (−ˇ b = − . 1 + (reθ (Φ))2

r √ γ

=

1+

Also, we have in view of the definition of a 1 + (reθ (Φ))2

=

1+

(eθ (eΦ ))2 e2Φ r2

=

r2 (1 + a) e2Φ

179

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

and hence r √ γ

=

= −

e r2

b = −

e2Φ r2

b

 r2 (K − r12 ) , 1+a  rβ + 2r (−ˇ κ + A + ϑ) eθ (Φ) + rξe4 (Φ) + rζe3 (Φ) , 1+a   rβ + 2r (−ˇ κ + ϑ) eθ (Φ) − rζe3 (Φ) . 1+a

e2Φ r2  2Φ

1+



ˇ The bootstrap assumptions on decay in (ext) M for κ ˇ , ϑ, κ ˇ , ϑ, β, β, ζ, ξ and Ω, the estimate (4.3.5) for a, the estimate (4.3.6), and the identity K−

1 r2

1 1 1 − κκ + ϑϑ − ρ − 2 4 4 r    1 4Υ 2m 1 − κκ + 2 − ρ + 3 + ϑϑ 4 r r 4

= =

imply max

0≤k≤ksmall

sup



ru

1 2 +δdec

+u

1+δdec

(ext) M

     r dk √ − 1 + r dk b . , γ

and max

0≤k≤ksmall

sup ru1+δdec dk b

. .

(ext) M

In particular, we also have max

0≤k≤ksmall

sup (ext) M



  γ  1 ru 2 +δdec + u1+δdec dk 2 − 1 r

These are the desired estimates for b, b and γ in of the lemma.

(ext)

. .

M. This concludes the proof

In this section, we also prove two useful lemmas concerning estimates on 2spheres of (ext) M and (int) M. Lemma 4.9. Let θ ∈ [0, π] be the Z-invariant scalar on M defined by (2.2.52). Then, we have on M reθ (Φ)

=

$ sin θ

where $ is a reduced 1-scalar satisfying sup |$| M

≤ 2.

Also, we have 1 sin θ

≤

2|reθ (Φ)| + 2 on M.

180

CHAPTER 4

Proof. The proof is similar on from (4.3.6) that

(ext)

M and

(int)

eθ (eΦ ) ≤

M so we focus on

(ext)

M. Recall

3 . 2

Furthermore, in view of Proposition 3.17, we have in particular Φ e sup − 1 . . (ext) M r sin θ Since we have $

=

r sin θeθ (Φ),

we deduce |$| =

r sin θ 3 |eθ (eΦ )| ≤ (1 + O()) ≤ 2, eΦ 2

which is the desired estimate for $. We now consider the upper bound for (sin θ)−1 . Recall the definition (2.2.54) of a a=

e2Φ + (eθ (eΦ ))2 − 1. r2

We infer r2 eθ (Φ)2

= =

r2 (eθ (eΦ ))2 e2Φ 1+a −1 e2Φ r2

=

=

   e2Φ 1 + a − (sin θ)2 1 + r2 (sin θ)2 − 1    e2Φ (sin θ)2 1 + r2 (sin − 1 2 θ)   e2Φ 2 2 (cos θ) + a − (sin θ) r2 (sin − 1 θ)2    2Φ e (sin θ)2 1 + r2 (sin θ)2 − 1

and hence r sin θ|reθ (Φ)| =

  e2Φ (cos θ)2 + a − (sin θ)2 r2 (sin − 1 2 θ) r .   e2Φ 1 + r2 (sin θ)2 − 1

Now, in view of (4.3.5), a satisfies in particular sup |a| . .

(ext) M

181

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Together with Φ e sup − 1 r sin θ (ext)

. ,

M

we infer p sin θ|reθ (Φ)| =

(cos θ)2 + O() p . 1 + O()

Thus, we deduce sin θ|reθ (Φ)|

≥

√

2 1 π 3π (1 + O()) ≥ for 0 ≤ θ ≤ and ≤ θ ≤ π. 2 2 4 4

On the other hand, we have sin θ ≥

√

2 π 3π on ≤ θ ≤ 2 4 4

and hence 1 sin θ

≤ 2|reθ (Φ)| + 2 on 0 ≤ θ ≤ π

which is the desired estimate. This concludes the proof of the lemma. Lemma 4.10. Let θ ∈ [0, π] be the Z-invariant scalar on M defined by (2.2.52). Then, for any reduced 1-scalar h, we have on any 2-sphere S on (ext) M and of (int) M



h |h| −1

sup Φ . r sup(|h| + | d/h|) and . r−1 khkh1 (S) .

eΦ 2 e S S L (S) Proof. The proof is similar on (ext) M and (int) M so we focus on (ext) M. Recall that the 2-surface S is parametrized by the coordinate θ ∈ [0, π], and that the axis corresponds to the two poles θ = 0 and θ = π. In view of Φ e sup − 1 . , (ext) M r sin θ we have |h| sup . r−1 sup |h| Φ S S∩{ π ≤θ≤ 3π } e 4

4

and



h



eΦ

3π L2 (S∩{ π 4 ≤θ≤ 4 })

. r−1 khkL2 (S)

which is the desired estimate for π/4 ≤ θ ≤ 3π/4. It remains to consider the portions 0 ≤ θ ≤ π/4 and 3π/4 ≤ θ ≤ π of S. These regions can be treated analogously, so we focus on 0 ≤ θ ≤ π/4. Recall from Remark 2.24 that any reduced scalar in sk , for k ≥ 1, must vanish on the axis of symmetry of Z, i.e., at the two poles. In particular, h must vanish at θ = 0. We

182

CHAPTER 4

deduce h heΦ = = eΦ e2Φ

Rθ 0

∂θ (eΦ h) = e2Φ

Rθp Rθ√ Φ γ S eθ (eΦ h) γe /d1 h 0 = 0 . e2Φ e2Φ

Since we have |γ| . r, we infer Rθ

|h| eΦ

0

.

eΦ | d/h| e2Φ

and since Φ e sup − 1 r sin θ (ext)

. ,

M

we deduce |h| eΦ

. r

Rθ −1 0

sin(θ0 )| d/h|dθ0 . (sin θ)2

This yields sup S∩{0≤θ≤ π 4}

|h| . r−1 sup | d/h| eΦ S

which is the desired sup norm estimate for 0 ≤ θ ≤ π/4. It remains to control the L2 norm on 0 ≤ θ ≤ π/4. We have, in view of the above,

2

h



eΦ 2 L

.

r−2

(S∩{0≤θ≤ π 4 })

Z

π 4

R

θ 0

sin(θ0 )| d/h|dθ0 (sin θ)4

0

.

r−1

Z

r−1

Z

π 4

0

.

0 π 4

0

.

r−1

Z

π 4

0

Z

Z

π 4

eΦ dθ !

θ

(sin(θ0 ))2 | d/h|2 dθ0

(sin θ)2 | d/h|2

Z θ

π 4

dθ (sin θ)2 !

dθ0 (sin(θ0 ))2

dθ

| d/h|2 sin θdθ

.

r−2

.

r−2 k d/hk2L2 (S)

0

2

| d/h|2 eΦ dθ

and hence



h



eΦ

L2 (S∩{0≤θ≤ π 4 })

. r−1 k d/hkL2 (S)

which is the desired L2 (S) estimate for 0 ≤ θ ≤ π/4. This concludes the proof of the lemma.

183

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

4.4

POINTWISE BOUNDS FOR HIGHER ORDER DERIVATIVES

The goal of this section is to prove Proposition 3.19. We deal first with the region r ≤ 4m0 as follows: 1. The curvature components and Ricci coefficients satisfy in view of the bootstrap assumptions on energy Z Z     ˇ 2 + |Γ| ˇ 2 + max ˇ 2 + |Γ| ˇ2 max |R| |R| ≤ 2 . k≤klarge

k≤klarge −1

(int) M

(ext) M(r≤4m ) 0

2. We first take the trace on the ingoing null cones foliating (int) M and the outgoing null cones foliating (ext) M(r ≤ 4m0 ) which loses one derivative. We thus obtain Z   ˇ 2 + |Γ| ˇ2 max sup |R| k≤klarge −1 1≤u≤u∗

Cu

Z +

max



sup

k≤klarge −2 1≤u≤u∗

Cu (r≤4m0 )

ˇ 2 + |Γ| ˇ2 |R|



. 2 .

3. We then take the trace on the 2-spheres S foliating the null cones in (ext) M(r ≤ 4m0 ) to infer   ˇ L2 (S) + kΓk ˇ L2 (S) max sup kRk k≤klarge −2

+

(int)

M and

(int) M

max

k≤klarge −3



sup (ext) M(r≤4m ) 0

ˇ 2 + |Γ| ˇ2 |R|



. .

4. Finally, using the Sobolev embedding on the 2-sphere S, which loses 2 derivatives, we deduce     ˇ + |Γ| ˇ + max ˇ + |Γ| ˇ max sup |R| sup |R| . , k≤klarge −4

k≤klarge −5

(int) M

which is the desired estimate in the region (ext)

It remains to consider the region

(ext) M(r≤4m

(int)

M∪

0)

(ext)

M(r ≤ 4m0 ).

M(r ≥ 4m0 ). We proceed as follows.

Step 1. The Ricci coefficients satisfy, in view of the bootstrap assumptions on energy, Z "   max r2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2 + (d≤k κ ˇ )2 + (d≤k ϑ)2 k≤klarge

Σ∗

# +

≤k

(d

≤k

2

η) + (d Z

+

sup λ≥4m0

2

≤k

ω ˇ ) + (d "

2

ξ)

  λ2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2

{r=λ}

#! +

λ2−δB (d≤k κ ˇ )2 + (d≤k ϑ)2 + (d≤k η)2 + (d≤k ω ˇ )2 + λ−δB (d≤k ξ)2

≤ 2 .

184

CHAPTER 4

We take the trace on the 2-spheres S foliating the timelike cylinders {r = r0 }, for r0 ≥ 4m0 , which loses a derivative, and infer in particular n
max sup r kdk κ ˇ kL2 (S) + kdk ζkL2 (S) + kdk ϑkL2 (S) k≤klarge −1

(ext) M(r≥4m ) 0

+r1−

δB 2

kdk κ ˇ kL2 (S) + kdk ηkL2 (S) + kdk ϑkL2 (S) o δB +kdk ω ˇ kL2 (S) + r− 2 kdk ξkL2 (S) . . Also, we take the trace on the 2-spheres S foliating the spacelike hypersurface Σ∗ , which loses a derivative, and infer in particular max

sup rkdk κ ˇ kL2 (S)

k≤klarge −1 Σ∗

.

.

Step 2. On can easily prove the following trace theorem   Z Z 5+δB k 2 max sup r (d α) . sup r4+δB (d≤klarge α)2 , k≤klarge −1

r≥4m0

1≤u≤u∗

S

Cu

which together with the bootstrap assumptions on energy for α in implies   Z max sup r5+δB (dk α)2 . 2 . k≤klarge −1

r≥4m0

r≥4m0

M(r ≥ 4m0 )

S

Step 3. Using the trace theorem   Z max sup r5 (dk β)2 . k≤klarge −1

(ext)

Z sup 1≤u≤u∗

S

r4 (d≤klarge β)2 ,

Cu

we infer, together with the bootstrap assumptions on energy for β in the region (ext) M(r ≥ 4m0 ),   Z max sup r5 (dk β)2 . 2 . (4.4.1) k≤klarge −1

r≥4m0

S

The power of r of the above estimate is not strong enough. To upgrade the estimate, recall that we have the Bianchi identity e4 (β) + 2κβ This yields   Z e4 r5+δB β2 S

/d2 α + ζα.

 e4 (r) 2 2βe4 (β) + κβ 2 + b β r S   Z 1 − δ 5 + δB 2 B = r5+δB − κβ 2 + 2β(r−1 d/α + ζα) − κ ˇβ 2 2 S Z

=

r5+δB



=

185

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

and hence 

5+δB

Z

2



1 − δB + 2

Z

e4 r β r5+δB κβ 2 S S   Z 5 + δB 2 4+δB = r 2β( d/α + rζα) − κ ˇβ 2 S Z  12 Z  12 Z 4+δB ≤1 2 4+δB 2 . r (d α) r β +  r4+δB β 2 S

S

S

where we used the pointwise estimates of Step 1 for κ ˇ and ζ. We infer   Z Z Z e4 r5+δB β2 + r4+δB β 2 . r4+δB (d≤1 α)2 . S

S

S

Integrating, from r ≥ 6m0 , we deduce Z Z 5+δB 2 sup r β + sup r4+δB β 2 r≥6m0 1≤u≤u∗ Cu (r≥6m0 ) S Z Z 4+δB ≤1 2 . sup r (d α) + β2 1≤u≤u∗

Cu

Sr=6m0

2

.  , where we used the bootstrap assumptions on energy for α in (ext) M(r ≥ 4m0 ) and the non-sharp estimate (4.4.1) for β. Using again (4.4.1), we obtain Z Z sup r5+δB β 2 + sup r4+δB β 2 . 2 . r≥4m0

S

1≤u≤u∗

Cu (r≥4m0 )

To discuss higher order derivatives, recall from Lemma 2.68 the following commutator, written in schematic form, [ d/, e4 ]

= (ˇ κ, ϑ) d/ + (ζ, rβ).

Also, recall from Lemma 2.69 the following commutator,  
m
[T, e4 ] = ω − rm2 − 2r κ − 2r + e4 (m) e4 + (η + ζ)eθ . r In view of the estimates of Step 1 for klarge − 1 derivatives of κ ˇ , ϑ, ζ, η, ω ˇ , the pointwise estimates for β in (4.4.1), the control of κ in Lemma 3.15, and the control of e4 (m) in Lemma 3.16, we infer, schematically,

 

k

. O(r−2 )kd≤k+1 βkL2 (S) for k ≤ klarge − 2.

d [ d/, e4 ]β, [T, e4 ]β 2 L (S)

Thus, commuting the Bianchi identity for e4 (β) with T and d/ together with the above commutator estimate, using the Bianchi identity to recover the e4 derivatives,

186

CHAPTER 4

we obtain for higher order derivatives max

k≤klarge −1

sup r

!

Z

2

r

1≤u≤u∗

S

4+δB

k

2

(d β)

Cu (r≥4m0 )

r4+δB (d≤klarge α)2

sup 1≤u≤u∗

.

k

(d β) + sup

r≥4m0

Z .

Z

5+δB

Cu (r≥4m0 )

2 .

Step 4. Recall from Proposition 2.73 that we have 3 3 e4 ρˇ + κˇ ρ + ρˇ κ = 2 2 Err[e4 ρˇ]

/d1 β + Err[e4 ρˇ],

3 1 = − κ ˇ ρˇ + κ ˇ ρˇ − 2 2

This yields  Z  4 2 e4 r (ˇ ρ) = S

=



1 ϑα + ζβ 2



 +

 1 ϑα + ζβ . 2

  e4 (r) 2 2 r 2ˇ ρe4 (ˇ ρ) + κˇ ρ +4 ρˇ r ZS   r4 − 3ρˇ κρˇ + 2ˇ ρ(r−1 d/β + Err[e4 ρˇ]) + κ ˇ ρˇ2 Z

4

S

and hence " Z  12 # e4 r4 (ˇ ρ)2 S

Z r

.

4



2

(ρˇ κ) + (r

−1

2

2

2 2

d/β) + (Err[e4 ρˇ]) + κ ˇ ρˇ

 12

.

S

Using the estimates of Steps 1, 2 and 3 for κ ˇ , ζ, ϑ, α and β, and the control of ρ in Lemma 3.15, we infer " Z  12 #  Z  12   4 2 4 2 e4 r (ˇ ρ) . r (ˇ ρ) . δB + 3 r2 S S r2+ 2 Integrating from r = 4m0 , we control kˇ ρkL2 (S) from the control in r ≤ 4m0 , we infer Z 4 sup r ρˇ2 . 2 . r≥4m0

S

Next, commuting the equation for e4 (ˇ ρ) with T and d/ together with the commutator estimate of Step 3, using the equation for e4 (ˇ ρ) to recover the e4 derivatives, we obtain similarly for higher order derivatives Z 4 max sup r (dk ρˇ)2 . 2 . k≤klarge −2 r≥4m0

S

187

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Step 5. Recall from Proposition 2.73 that we have the following transport equations in the e4 direction, 1 1 κ+ κ ˇκ = e4 κ ˇ + κˇ 2 2 Err[e4 κ ˇ]

=

−2 /d1 ζ + 2ˇ ρ + Err[e4 κ ˇ ],     1 1 1 1 2 2 − κ ˇκ ˇ− κ ˇκ ˇ + − ϑϑ + 2ζ − − ϑϑ + 2ζ . 2 2 2 2

This yields  Z   Z  e4 (r) 2 e4 r (ˇ κ)2 = r 2ˇ κe4 (ˇ κ) + κˇ κ2 + κ ˇ r S S   Z   1 = r 2ˇ κ − κ ˇ κ − 2 /d1 ζ + 2ˇ ρ + Err[e4 κ ˇ] + κ ˇκ ˇ2 2 S and hence, using the estimates of Steps 1 and 4 for κ ˇ , ζ, ϑ and ρˇ, and the control of κ and κ in Lemma 3.15, we infer  Z  2 e4 r (ˇ κ)

 r2

.

S

Z

2

rˇ κ + r

− 32

Z

S

2

 12

rˇ κ S

and hence e4

 Z  12 ! 2 r (ˇ κ)

 r2

.

S

 Z  12 3 2 r (ˇ κ) + r− 2 . S

Integrating backward from Σ∗ , where κ ˇ is under control in view of Step 1, we infer Z sup r2 κ ˇ 2 . 2 . r≥4m0

S

Next, commuting the equation for e4 (ˇ κ) with T and d/ together with the commutator estimate of Step 3, using the equation for e4 (ˇ κ) to recover the e4 derivatives, we obtain similarly for higher order derivatives Z max sup r4 (dk κ ˇ )2 . 2 . k≤klarge −2 r≥4m0

S

Step 6. In view of Codazzi for ϑ, and the estimates of Step 1 on ζ and ϑ, and of Step 3 on κ ˇ in (ext) M(r ≥ 4m0 ), we infer max

k≤klarge −2

rkdk βkL2 (S)

sup (ext) M(r≥4m

. .

0)

Step 7. In view of the null structure equation for e3 (κ), and the estimates of Step 1 on ω ˇ , ζ, η and ϑ, and of Step 3 on κ ˇ in (ext) M(r ≥ 4m0 ), we infer max

k≤klarge −3

sup (ext) M(r≥4m

0)

kdk ξkL2 (S)

. .

Step 8. In view of the Bianchi identity for e3 (β), and the estimates of Step 1 on ω ˇ , ζ, and η, the estimates of Step 2 on ρˇ, of Step 3 on κ ˇ and of Step 5 on ξ in

188 (ext)

CHAPTER 4

M(r ≥ 4m0 ), we infer max

k≤klarge −3

sup (ext) M(r≥4m ) 0

kdk αkL2 (S)

.

.

Step 9. Gathering the estimates for Step 1 to Step 8, we have obtained n 5 δB
max sup r 2 + 2 kdk αkL2 (S) + kdk βkL2 (S) k≤klarge −1

(ext) M(r≥4m

k

0)


+ r kd κ ˇ kL2 (S) + kd ζkL2 (S) + kdk ϑkL2 (S) + kdk ηkL2 (S) o +kdk ϑkL2 (S) + kdk ω ˇ kL2 (S) n
+ max sup r2 kdk µkL2 (S) + kdk ρˇkL2 (S) k≤klarge −2

k

(ext) M(r≥4m

0)


o +r kdk κ ˇ kL2 (S) + kdk βkL2 (S) n o + max sup kdξkL2 (S) + kdk αkL2 (S) . . k≤klarge −3

(ext) M(r≥4m

0)

Using the Sobolev embedding on the 2-sphere S which loses 2 derivatives, and in view of the previous estimate on (ext) M(r ≤ 4m0 ), we infer n 7 δB

max sup r 2 + 2 |dk α| + |dk β| + r3 |dk µ ˇ| + |dk ρˇ| k≤klarge −5 M
+r2 |dk κ ˇ | + |dk ζ| + |dk ϑ| + |dk κ ˇ | + |dk β|
o +r |dk η| + |dk ϑ| + |dk ω ˇ | + |dξ| + |dk α| .  which is the desired estimate on Proposition 3.19.

4.5

(ext)

M(r ≥ 4m0 ). This concludes the proof of

PROOF OF PROPOSITION 3.20

Let (e4 , e3 , eθ ) the outgoing geodesic null frame of frame (e04 , e03 , e0θ ) of (ext) M provided by

(ext)

M. We will exhibit another

1 e04 = e4 + f eθ + f 2 e3 , 4 1 0 eθ = eθ + f e3 , 2 e03 = e3 ,

(4.5.1)

where f is such that f = 0 on Σ∗ ∩ C∗ ,

η 0 = 0 on Σ∗ ,

ξ 0 = 0 on

(ext)

M.

(4.5.2)

The desired estimates for the Ricci coefficients and curvature components with respect to the new frame (e04 , e03 , e0θ ) of (ext) M will be obtained using • the change of frame formulas of Proposition 2.90, applied to the change of frame

189

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

from (e4 , e3 , eθ ) to (e04 , e03 , e0θ ), • the estimates for f on (ext) M, • the estimates for the Ricci coefficients and curvature components with respect to the outgoing geodesic frame (e4 , e3 , eθ ) of (ext) M provided by the bootstrap assumptions on decay and Proposition 3.19. Step 1. We start by deriving an equation for f on (ext) M. In view of the condition ξ 0 = 0 on (ext) M, see (4.5.2), in view of ξ = ω = 0 and η = −ζ satisfied by the outgoing geodesic foliation of (ext) M, and in view of Lemma 2.91, we have 1 e04 (f ) + κf 2

1 1 3 = − f ϑ − f 2η − f 2ζ 2 2 2 1 3 1 3 1 1 + f κ + f ω + f 3 ϑ + f 4 ξ on 8 2 8 8

(ext)

M.

(4.5.3)

We also derive an equation for f on Σ∗ . In view of the condition η 0 = 0 on Σ∗ , see (4.5.2), and in view of Lemma 2.91, we have e03 (f )

=

1 −2η + 2f ω + f 2 ξ on Σ∗ . 2

Now, since u + r is constant on Σ∗ , the following vectorfield 0 νΣ := e03 + a0 e04 , ∗

a0 := −

e03 (u + r) , e04 (u + r)

is tangent to Σ∗ . We compute in view of the above 0 νΣ (f ) ∗

= e03 (f ) + a0 e04 (f ) 1 = −2η + 2f ω + f 2 ξ + a0 2

(

1 1 1 3 − κf − f ϑ − f 2 η − f 2 ζ 2 2 2 2 ) 1 1 1 1 + f 3κ + f 3ω + f 3ϑ + f 4ξ . 8 2 8 8

Using (4.5.1), we have a0

= = =

e03 (u + r) e04 (u + r) e3 (u + r)
− e4 + f eθ + 14 f 2 e3 (u + r) −

−r

2κ

+

2 ς + 1 2 4f

r 2 (κ + A) 2 r ς + 2 (κ +


A)

(4.5.4)

190

CHAPTER 4

and hence 0 νΣ (f ) ∗

1 = −2η + 2f ω + f 2 ξ 2 ( 2 r 1 1 ς + 2 (κ + A)
− κf − f ϑ −r 1 2 2 r 2 2 κ + f + (κ + A) 2 4 ς 2

(4.5.5)

1 3 1 1 1 1 − f 2η − f 2ζ + f 3κ + f 3ω + f 3ϑ + f 4ξ 2 2 8 2 8 8

) on

(ext)

M.

Step 2. Next, we estimate f on Σ∗ . Introducing an integer kloss and a small constant δ0 > 0 satisfying 16 ≤ kloss ≤

δdec (klarge − ksmall ), 3

δ0 =

kloss , klarge − ksmall

we assume the following local bootstrap assumption √  |d≤ksmall +kloss +2 f | ≤ on u1 ≤ u ≤ u∗ 1 +δ −2δ0 dec ru 2

(4.5.6)

where 1 ≤ u1 < u∗ . Since f = 0 on Σ∗ ∩ C∗ in view of (4.5.2), (4.5.6) holds for u1 close enough √ to u∗ , and our goal is to prove that we may in fact choose u1 = 1 and replace  with  in (4.5.6). In view of the estimates for the Ricci coefficients and curvature components with respect to the outgoing geodesic frame (e4 , e3 , eθ ) of (ext) M provided by Proposition 3.19, (4.5.5) yields 0 νΣ (f ) ∗

=

|dk h| . r−1 (|d≤k f | + |d≤k f |4 ) for k ≤ klarge − 5.

−2η + h,

Using commutator identities, using also (4.5.3) and (4.5.4), and in view of (4.5.6), we infer √  0 k ≤k for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ . |νΣ∗ ( d/ f )| . | d/ η| + 1 +δ −2δ0 2 dec r u2 0 Since f = 0 on Σ∗ ∩ C∗ in view of (4.5.2), and since νΣ is tangent to Σ∗ , we ∗ 0 deduce on Σ∗ , integrating along the integral curve of νΣ ∗ k

| d/ f | .

Z

u∗

≤k

| d/

η| +

√



1 2 +δdec −2δ0

Z

u∗

1 0 (u0 )r 2 νΣ ∗

u u u for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ .

191

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Since 0 νΣ (u) ∗

e03 (u) + a0 e04 (u)

=

2 r ς + 2 (κ + A)
e3 (u) − r 1 2 2 r 2κ + 4f ς + 2 (κ + A) 2 r 2 f2 ς + 2 (κ + A)
− ς 2ς 2r κ + 14 f 2 2ς + 2r (κ + A)

= =



 1 2 e4 + f eθ + f e3 u 4

we have 0 νΣ (u) ∗

=

2 + O()

and hence Z

u∗

√

Z

| d/ f | .

≤k

| d/

η| +



u∗

1 2 r u u u for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ .

k

1 2 +δdec −2δ0

Together with the behavior (3.3.4) of r on Σ∗ , we infer Z u∗  | d/k f | . | d/≤k η| + for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ . 1 +δ ru 2 dec −2δ0 u Next, we estimate η. We have by interpolation, since kloss ≤ klarge − ksmall , kloss +4 −ksmall

1− k

k d/≤ksmall +kloss +4 ηkL2 (S)

large . k d/≤ksmall ηkL2 (S)

k

kloss +4 −ksmall

k d/≤klarge ηkLlarge 2 (S)

and hence, using δ0 > 0, we have Z u∗ k d/≤ksmall +kloss +4 ηkL2 (S) u

! 12

Z

0 1+δ0

u

. Σ∗ (≥u)

.

1

| d/

Z

≤ksmall +kloss +4

0 2+2δdec

2

η|

≤ksmall

2

u | d/ η| 1 u 2 +δdec −2δ0 Σ∗ +4 Z  2(k kloss large −ksmall ) ≤klarge 2 × | d/ η|

 12 − 2(k

kloss +4 large −ksmall )

Σ∗

where we have used the fact that kloss + 4 δ0 (1 + δdec ) + klarge − ksmall 2

 =

1+

4 kloss



1 (1 + δdec ) + 2

and 1 1 4kloss + δdec − 2δ0 = + δdec − ≥ δdec > 0 2 2 klarge − ksmall

 δ0 ≤ 2δ0

,

192

CHAPTER 4

since 16 ≤ kloss ≤ 18 (klarge − ksmall ) and δdec > 0 is small. Now, recall from the bootstrap assumptions on decay and energy for η along Σ∗ that we have Z Z u2+2δdec |d≤ksmall η|2 + |d≤klarge η|2 ≤ 2 . Σ∗

Σ∗

We deduce Z

u∗

k d/≤ksmall +kloss +4 ηkL2 (S)

u

.

 1

u 2 +δdec −2δ0

.

Together with the Sobolev embedding on the 2-spheres S foliating Σ∗ , as well as the behavior (3.3.4) of r on Σ∗ , we infer Z u∗  | d/≤ksmall +kloss +2 η| . . 1 +δ ru 2 dec −2δ0 u Plugging in the above estimate for f , we infer 

| d/k f | .

ru

for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ .

1 2 +δdec −2δ0

Together with (4.5.3) and (4.5.4), we recover e4 and e3 derivatives to deduce |dk f | .



for k ≤ ksmall + kloss + 2, u1 ≤ u ≤ u∗ .

1

ru 2 +δdec −2δ0

This is an improvement of the bootstrap assumption (4.5.6). Thus, we may choose u1 = 1, and f satisfies the following estimate 

|dk f | .

ru

for k ≤ ksmall + kloss + 2 on Σ∗ .

1 2 +δdec −2δ0

Together with (4.5.4), as well as the behavior (3.3.4) of r on Σ∗ , we infer |dk−1 e03 f | . |dk−1 η| +

 r2

 for k ≤ ksmall + kloss + 2 on Σ∗ . ru1+δdec −2δ0

.

Collecting the two above estimates, we obtain |dk f | . |dk−1 e03 f | .

 ru

1 2 +δdec −2δ0



ru1+δdec −2δ0

Step 3. Next, we estimate f on assumption ≤ksmall +kloss +2

|d

f| ≤

for k ≤ ksmall + kloss + 2 on Σ∗ ,

(ext)

M. We assume the following local bootstrap √

ru

(4.5.7)

for k ≤ ksmall + kloss + 2 on Σ∗ .

1 2 +δdec −2δ0



+ u1+δdec −2δ0

on r ≥ r1

(4.5.8)

where r1 ≥ 4m0 . In view of the control of f on Σ∗ provided by (4.5.7), (4.5.8) holds for r1 sufficiently√large, and our goal is to prove that we may in fact choose r1 = 4m0 and replace  with  in (4.5.8).

193

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

Recall (4.5.3) 1 e04 (f ) + κf 2

1 1 3 = − f ϑ − f 2η − f 2ζ 2 2 2 1 3 1 3 1 1 + f κ + f ω + f 3 ϑ + f 4 ξ on 8 2 8 8

(ext)

M.

In view of the estimates for the Ricci coefficients and curvature components with respect to the outgoing geodesic frame (e4 , e3 , eθ ) of (ext) M provided by Proposition 3.19,   k d − 1 f ϑ − 1 f 2 η − 3 f 2 ζ + 1 f 3 κ + 1 f 3 ω + 1 f 3 ϑ + 1 f 4 ξ 2 2 2 8 2 8 8 1

. r−2 u− 2 |d≤k f | + r−1 (|d≤k f |2 + |d≤k f |4 ) for k ≤ klarge − 5. Using commutator identities, using also (4.5.3), and in view of (4.5.8), we infer6   1 e04 ( d/, T )k f + κ( d/, T )k f 2

≤

 r3 u1+δdec −2δ0

for k ≤ ksmall + kloss + 2, r ≥ r1 .

Integrating backwards from Σ∗ where we have (4.5.7), we deduce7 |( d/, T )k f |



≤

ru

1 2 +δdec −2δ0

+ u1+δdec −2δ0

for k ≤ ksmall + kloss + 2, r ≥ r1 .

Together with (4.5.3), we recover the e4 derivatives and obtain |dk f |

≤

 ru

1 2 +δdec −2δ0

+ u1+δdec −2δ0

for k ≤ ksmall + kloss + 2, r ≥ r1 .

This is an improvement of the bootstrap assumption (4.5.8). Thus, we may choose r1 = 4m0 , and we have |dk f | .

 ru

1 2 +δdec −2δ0

+

u1+δdec −2δ0

for k ≤ ksmall + kloss + 2 on

(ext)

M.

Also, commuting once (4.5.3) with e03 , using the commutator identity [e03 , e04 ] = 2ω 0 e04 − 2ω 0 e03 + (η 0 − η 0 )e0θ , and proceeding as above to integrate backward from Σ∗ where e03 f is under control from (4.5.7), we also obtain |dk−1 e03 f | . 6 Note

 ru1+δdec −2δ0

for k ≤ ksmall + kloss + 2 on

that δdec − 2δ0 = δdec −

2kloss δdec ≥ >0 klarge − ksmall 3

where we have used the definition of δ0 and the upper bound on kloss . 7 Note that (4.5.7) yields |dk f |

.

 u1+δdec −2δ0

in view of the behavior (3.3.4) of r on Σ∗ .

for k ≤ ksmall + kloss + 2 on Σ∗

(ext)

M.

194

CHAPTER 4

Collecting the two above estimates, we obtain  for k ≤ ksmall + kloss + 2 on 1 ru 2 +δdec −2δ0 + u1+δdec −2δ0  |dk−1 e03 f | . 1+δ −2δ0 for k ≤ ksmall + kloss + 2 on (ext) M, ru dec |dk f | .

(ext)

M, (4.5.9)

which is the desired estimate for f . Step 4. In view of Proposition 2.90 applied to our particular case, i.e., a triplet (f, , f , λ) with f = 0 and λ = 1, and the fact that the frame (e4 , e3 , eθ ) is outgoing geodesic, we have ξ 0 = ξ, 1 1 ζ 0 = ζ − f κ − f ω − f ϑ + l.o.t., 4 4 1 0 0 η = η + e3 (f ) − f ω + l.o.t., 2 1 1 0 η = −ζ + κf + f ϑ + l.o.t., 4 4 1 κ0 = κ + /d1 0 (f ) + f (ζ + η) − f 2 κ − f 2 ω + l.o.t., 4 κ0 = κ + f ξ + l.o.t., ϑ0 = ϑ − /d?2 0 (f ) + f (ζ + η) − f 2 ω + l.o.t., ϑ0 = ϑ + f ξ + l.o.t.,

1 1 ω 0 = f ζ − κf 2 − ωf 2 + l.o.t., 8 4 1 ω 0 = ω + f ξ, 2 and 3 α0 = α + 2f β + f 2 ρ + l.o.t., 2 3 β 0 = β + ρf + l.o.t., 2 0 ρ = ρ + f β + l.o.t., 1 β 0 = β + f α, 2 α0 = α,

(4.5.10)

where the lower order terms denoted by l.o.t. are linear with respect to ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ and α, β, ρ, β, α, and quadratic or higher order in f , and do not contain derivatives of the latter. Together with the estimates (4.5.9) for f on (ext) M, and the estimates for the Ricci coefficients and curvature components with respect to the outgoing geodesic frame (e4 , e3 , eθ ) of (ext) M provided by the

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

195

bootstrap assumptions on decay and Proposition 3.19, we immediately infer (   1 max sup r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk (Γ0g \ {η 0 })| 0≤k≤ksmall +kloss +1

(ext) M

+ru1+δdec −2δ0 |dk Γ0b |   2 0 2Υ 0 0 0 2 1+δdec −2δ0 k−1 0 0 +r u e3 κ − , κ + ,ϑ ,ζ ,η d r r   1 + r3 (u + 2r) 2 +δdec −2δ0 + r2 (u + 2r)1+δdec −2δ0 |dk (α0 , β 0 )|   1 + r3 (2r + u)1+δdec + r4 (2r + u) 2 +δdec −2δ0 |dk−1 e03 (α0 )|   1 + r3 u1+δdec + r4 u 2 +δdec −2δ0 |dk−1 e03 (β 0 )|   1 + r3 u 2 +δdec −2δ0 + r2 ru1+δdec −2δ0 |dk ρˇ0 | )   1+δdec −2δ0 2 k 0 k 0 +u r |d β | + r|d α | . (4.5.11) where we have introduced the notation   2 2Υ −1 0 Γ0g \ {η 0 } = rω 0 , κ0 − , ϑ0 , ζ 0 , η 0 , κ0 + , r (e4 (r) − 1), r−1 e0θ (r), e04 (m) . r r Note also, in view of the above transformation formula for ω 0 , i.e., ω0

=

1 1 f ζ − κf 2 − ωf 2 + l.o.t., 8 4

that we have in fact a gain of r−1 for ω 0 compared to (4.5.11), i.e.,   1 max sup r3 u 2 +δdec −2δ0 + r2 u1+δdec −2δ0 |dk ω 0 | . . (4.5.12) 0≤k≤ksmall +kloss +1

(ext) M

We now focus on estimating η 0 . Proceeding as for the other Ricci coefficients would yield for η 0 the same behavior as η and hence a loss of r−1 compared to the desired estimate. Instead, we rely on the following null structure equation which follows from Proposition 2.56 and the fact that ξ 0 = 0 1 e04 (η 0 − ζ 0 ) + κ0 (η 0 − ζ 0 ) 2

=

1 2 /d?01 ω 0 − ϑ0 (η 0 − ζ 0 ). 2

Next, • we commute with d/0 and T 0 , and we rely on the corresponding commutator identities, • we use the above equation for e04 (η 0 ) to recover the e04 derivatives, • we rely on the estimates (4.5.11), as well as the estimate (4.5.12) for ω 0 ,

196

CHAPTER 4

which allows us to derive 0 k 0 e4 (d (η − ζ 0 )) + 1 κ0 dk (η 0 − ζ 0 ) 2

.

 1 r4 u 2 +δdec −2δ0

+ r3 u1+δdec −2δ0  ≤k 0 + 2 |d (η − ζ 0 )|, k ≤ ksmall + kloss . r

Integrating backwards from Σ∗ where η 0 = 0 in view of (4.5.2), and using the control ζ 0 provided by (4.5.11), we infer   1 max sup r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk η 0 | 0≤k≤ksmall +kloss (ext) M   1 . + max sup r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk ζ 0 | 0≤k≤ksmall +kloss

(ext) M

. . Also, commuting first the equation for e04 (η 0 − ζ 0 ) with e03 , using the commutator identity [e03 , e04 ] = 2ω 0 e04 − 2ω 0 e03 + (η 0 − η 0 )e0θ , and proceeding as above to integrate backward from Σ∗ , we also obtain max

0≤k≤ksmall +kloss

. .

+

max

sup r2 u1+δdec −2δ0 |dk−1 e03 η 0 |

(ext) M

0≤k≤ksmall +kloss

sup r2 u1+δdec −2δ0 |dk−1 e03 ζ 0 |

(ext) M

.

Thus, together with (4.5.11), we infer (   1 max sup r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk Γ0g | 0≤k≤ksmall +kloss

(ext) M

+ru1+δdec −2δ0 |dk Γ0b |   2 2Υ 0 0 0 0 +r2 u1+δdec −2δ0 dk−1 e03 κ0 − , κ0 + ,ϑ ,ζ ,η ,η r r   1 3 +δ −2δ 2 1+δ + r (u + 2r) 2 dec 0 + r (u + 2r) dec −2δ0 |dk (α0 , β 0 )|   1 + r3 (2r + u)1+δdec + r4 (2r + u) 2 +δdec −2δ0 |dk−1 e03 (α0 )|   1 + r3 u1+δdec + r4 u 2 +δdec −2δ0 |dk−1 e03 (β 0 )|   1 + r3 u 2 +δdec −2δ0 + r2 ru1+δdec −2δ0 |dk ρˇ0 | )   1+δdec −2δ0 2 k 0 k 0 . . +u r |d β | + r|d α | Together with the fact that ξ 0 = 0 in view of (4.5.2), this concludes the proof of Proposition 3.20.

197

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

4.6 4.6.1

EXISTENCE AND CONTROL OF THE GLOBAL FRAMES Proof of Proposition 3.23

To match the frame of (int) M and a conformal renormalization of the frame of (ext) M, we will need to introduce a cut-off function. Definition 4.11. Let ψ : R → R a smooth cut-off function such that 0 ≤ ψ ≤ 1, ψ = 0 on (−∞, 0] and ψ = 1 on [1, +∞). We define ψm0 ,δH as follows: ( ψm0 ,δH (r) =

1

if

0

if


r ≥ 2m0 1 + 32 δH ,
r ≤ 2m0 1 + 12 δH ,

and ψm0 ,δH (r) = ψ

r − 2m0 1 + 12 δH 2m0 δH


!

    1 3 on 2m0 1 + δH ≤ r ≤ 2m0 1 + δH . 2 2

We are now ready to define the global frame of the statement of Proposition 3.23. Definition 4.12 (Definition of the global frame). We introduce a global null frame defined on (ext) M∪ (int) M and denoted by ((glo) e4 , (glo) e3 , (glo) eθ ). The global frame is defined as follows: 1. In

(ext)

M \ Match, we have

((glo) e4 , (glo) e3 , (glo) eθ ) = 2. In

(int)



(ext)

 Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ .

M \ Match, we have ((glo) e4 , (glo) e3 , (glo) eθ ) =



(int)

 e4 , (int) e3 , (int) eθ .

3. It remains to define the global frame on the matching region. We denote by (f, f , λ) the reduced scalars such that we have in the matching region 

(ext)

e4

=

(ext)

eθ

=

(ext)

e3

=

 1 2(int) λ e4 + f eθ + f e3 , 4     f (int) f f (int) 1 f (int) 1 + ff eθ + e4 + 1+ e3 , 2 2 2 4     f f (int) 1 1 λ−1 1 + f f + f 2 f 2 (int) e3 + f 1 + eθ 2 16 4 ! f 2 (int) + e4 , 4 (int)

where we recall that the frame of

(int)

(ext)

M has been extended to

(int)

M, see section

198

CHAPTER 4

3.5.1. Then, in the matching region, the global frame is given by   1 2 (glo) e4 = λ0 (int) e4 + f 0(int) eθ + f 0 (int) e3 , 4 !   0 f (int) f 0 f 0 (int) 1 0 0 (int) f0 (glo) eθ = 1+ f f eθ + e4 + 1+ e3 , 2 2 2 4 !   f 0 f 0 (int) 1 0 0 1 0 2 0 2 (int) 0 (glo) 0 −1 e3 = λ 1+ f f + f f e3 + f 1 + eθ 2 16 4 ! 2 f 0 (int) + e4 , 4 where f 0 = ψm0 ,δH ( (int) r)f,

f 0 = ψm0 ,δH ( (int) r)f ,

λ0 = 1 − ψm0 ,δH ( (int) r) + ψm0 ,δH ( (int) r)(ext) Υλ.

(4.6.1)

Remark 4.13. Recall that the smooth cut-off function ψ in Definition 3.22, allowing to define ψm0 ,δH , is such that we have in particular ψ = 0 on (−∞, 0] and ψ = 1 on [1, +∞). The following two special cases correspond to the properties (d) i. and (d) ii. of Proposition 3.23. • If the cut-off ψ in Definition 3.22 is such that ψ = 1 on [1/2, +∞), then   ((glo) e4 , (glo) e3 , (glo) eθ ) = (ext) Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ on (ext) M. • If the cut-off ψ in Definition 3.22 is such that ψ = 0 on (−∞, 1/2], then   ((glo) e4 , (glo) e3 , (glo) eθ ) = (int) e4 , (int) e3 , (int) eθ on (int) M. Definition 4.14 (Global area radius and Hawking mass). We define an area radius and a Hawking mass on (ext) M ∪ (int) M as follows: • On

(ext)

M \ Match, we have (glo)

• On

(int)

r=

(ext)

r,

(glo)

m=

(ext)

(int)

r,

(glo)

m=

(int)

m.

M \ Match, we have (glo)

r=

m.

• On the matching region, we have (glo) (glo)

r

=

m

=

(1 − ψm0 ,δH ( (int) r)) (int) r + ψm0 ,δH ( (int) r) (ext) r,

(1 − ψm0 ,δH ( (int) r)) (int) m + ψm0 ,δH ( (int) r) (ext) m.

The following two lemmas provide the main properties of the global frame. Lemma 4.15. We have in

(ext)

M \ Match the following relations between the

199

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

quantities in the respective frames: (glo)

α = Υ2(ext) α,

(glo)

(glo)

β = Υ−1(ext) β,

β = Υ(ext) β,

(glo)

(glo)

α = Υ−2(ext) α,

2m 2m (ext) = ρ+ 3 , 3 r r (glo) ξ = 0, (glo) ξ = Υ−2(ext) ξ, ρ+

(glo)

ζ = −(glo) η = (ext) ζ, (glo) η = (ext) η,   m m (ext) 2 e4 (m) (glo) ω+ 2 =− κ− + , r 2r r r   m m (ext) 2Υ (glo) −1 (ext) ω=Υ ω− 2 + κ− r 2Υr r !  e (m) m (ext) ˇ (ext) 3 (ext) (ext) ˇ + Ω κ− Ω κ ˇ − , 2Υr Υr     2Υ 2 2 2Υ (glo) (ext) (glo) −1 (ext) κ− =Υ κ− , κ+ =Υ κ+ , r r r r (glo)

κ ˇ = Υ(ext) κ ˇ,

(glo)

κ ˇ = Υ−1(ext) κ ˇ,

(glo)

ϑ = Υ(ext) ϑ,

(glo)

ϑ = Υ−1(ext) ϑ.

Proof. The proof follows immediately from the change of frame formula with the choice (f = 0, f = 0, λ = Υ), the fact that eθ (Υ) = 0, and the fact that the frame of (ext) M is outgoing geodesic and thus satisfies in particular ξ = ω = 0 and η = −ζ. Lemma 4.16 (Control of the global frame in the matching region). In the matching region, the following estimates hold for the global frame:8 " ˇ (glo) R) ˇ max sup u1+δdec dk ((glo) Γ, 0≤k≤ksmall −2

+

Match∩ (int) M

1+δdec

sup

u

Match∩ (ext) M

Z +

max

0≤k≤klarge −1

Match

k (glo) ˇ (glo) ˇ Γ, R) d (

#

1 k (glo) ˇ (glo) ˇ 2 2 Γ, R) d (

. 

and Z Match

1 klarge (glo) ˇ (glo) ˇ 2 2 ( Γ, R) d

.

Z 1 klarge (ext) ˇ 2 2 + ( R) . d T

Remark 4.17. The quantities associated to the global frame can be estimated as follows: • In (int) M \ Match, the global frame coincides with the frame of (int) M, and hence, the quantities associated to the global frame satisfy the same estimates as the bootstrap assumptions for the frame of (int) M. 8 We only need the first estimate for the proof of Proposition 3.23, but the second estimate will be needed in the proof of Theorem M8.

200

CHAPTER 4

• In (ext) M \ Match, estimates for the quantities associated to the global frame follow from the identities of Lemma 4.15 together with the bootstrap assumptions for the frame of (ext) M. • In Match, the estimates for the quantities associated to the global frame are provided by Lemma 4.16. The proof of Proposition 3.23 easily follows from Definition 4.12, Remark 4.13, and Lemma 4.16. Thus, from now on, we focus on the proof of Lemma 4.16 which is carried out in the next section. 4.6.2

Proof of Lemma 4.16

In this section, we prove Lemma 4.16. To ease the exposition, the quantities associated to the frame of (int) M are unprimed, the quantities associated to the frame of (ext) M are primed, and the quantities associated to the global frame are doubleprimed. Step 1. Let (e3 , eθ , e4 ) denote the null frame of (int) M (and its extension) and (e03 , e0θ , e04 ) the null frame of (ext) M (and its extension). We denote by (f, f , λ) the reduced scalars such that   1 2 0 e4 = λ e4 + f eθ + f e3 , 4     f ff 1 f 0 eθ = 1 + f f eθ + e4 + 1+ e3 , 2 2 2 4 !     f2 1 1 2 2 1 2 0 −1 e3 = λ 1 + f f + f f e3 + f + f f eθ + e4 . 2 16 4 4 Together with the initialization of the frame of (ext) M and 3.1.2 (where the spheres coincide), we have in particular f = f = 0,

λ = Υ−1 on T .

(int)

M on T in section (4.6.2)

Also, recall from section 3.5.1 that in order for (e03 , e0θ , e04 ) to be defined everywhere on (int) M ∩ Match, we need — in addition to the above initialization of (f, f , λ) on T — to initialize it also on C ∗ ∩ Match by f = f = 0,

λ = Υ−1 on C ∗ ∩ Match.

(4.6.3)

Step 2. Next, we control the change of frame (f, f , λ) from (e3 , eθ , e4 ) to (e03 , e0θ , e04 ) in the region (int) M ∩ Match. To this end, we rely on the transport equation of Lemma 2.92 together with the fact that ω 0 = ξ 0 = ζ 0 + η 0 = 0. Then, (f , f, log(λ)) satisfy the following transport equations: κ  λ−1 e04 (f ) + + 2ω f = −2ξ + E1 (f, Γ), 2 λ−1 e04 (log(λ)) = 2ω + E2 (f, Γ), κ λ−1 e04 (f ) + f = −2(ζ + η) + 2e0θ (log(λ)) + 2f ω + E3 (f, f , Γ), 2

201

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

where E1 , E2 and E3 are given by E1 (f, Γ) E2 (f, Γ) E3 (f, f , Γ)

1 = − ϑf + l.o.t., 2 1 1 = f ζ − f 2 ω − ηf − f 2 κ + l.o.t., 2 4 1 0 = −f eθ (f ) − f ϑ + l.o.t. 2

Here, l.o.t. denotes terms which are cubic or higher order in f, f (or in f only in the ˇ and do not contain derivatives of these quantities, where case of E1 and E2 ) and Γ ˇ Γ and Γ denote the Ricci coefficients and renormalized Ricci coefficients w.r.t. the original null frame (e3 , e4 , eθ ). We rewrite the transport equation for log(λ) as λ−1 e04 (log (Υλ)) = λ−1 e04 (log(λ)) + λ−1 e04 (log(Υ))   1 1 e4 + f eθ + f 2 e3 Υ = 2ω + E2 (f, Γ) + Υ 4  m 2 m(e4 (r) − Υ) 2 e4 (m) = 2 ω + 2 + E2 (f, Γ) + − 2 Υ r Υ r  r  1 1 2 − f eθ + f e3 Υ. Υ 4 In view of the above transport equations for f , f and λ, the initialization (4.6.2) and (4.6.3) for (f, f , λ) on T ∪ (C ∗ ∩ Match), and the control of Γ induced by the bootstrap assumptions on (int) M, we easily deduce max sup u1+δdec dk (f, log(Υλ)) 0≤k≤ksmall

+

(int) M∩Match

max

0≤k≤ksmall −1

k d (f, log(Υλ)) 2

max

 12

(int) M∩Match

Z +

. ,

(int) M∩Match

Z 0≤k≤klarge

u1+δdec dk f

sup

k 2 d f

max

0≤k≤klarge −1

 12 . .

(int) M∩Match

Step 3. We need to improve the number of derivatives in the top order estimate for (f, f , log(λ)). To this end, note first in view of the transformation formulas of Proposition 2.90 and the control of (f, f , log(λ)) provided by Step 2, we have in particular Z

k 0 2 ˇ d R

max

0≤k≤klarge −1

 12 . .

(int) M∩Match

Relying on this estimate, the control of the Ricci coefficients associated to the outgoing null frame (e04 , e03 , e0θ ) on T ∪ ( (int) M ∩ Match), and the null structure

202

CHAPTER 4

equations, we infer Z

k 0 2 ˇ d Γ

max

0≤k≤klarge −1

 12 .

.

(int) M∩Match

We refer to section 8.9 for a completely analogous proof where the Ricci coefficients are recovered in (int) M based on the control of the curvature components. In view of the transformation formulas of Proposition 2.90, which can be written schematically as   ˇ ∂ f, f , log(λ) = F (f, f , λ, Γ), the control of (f, f , log(λ)) provided by Step 2, and the above control of Γ0 , we infer Z

k d (f, f , log(Υλ)) 2

max

0≤k≤klarge

 12 . .

(int) M∩Match

Step 4. We still need to control one more derivative of (f, f , log(λ)). Repeating the process of Step 3, we use again the transformation formulas of Proposition 2.90 and then the final estimate of Step 3 for (f, f , log(λ)) yields the following control for the curvature components: Z

k 0 2 ˇ d R

max

0≤k≤klarge

 12 . .

(int) M∩Match

Arguing as in Step 3, we infer9 Z

k 0 2 ˇ d Γ

max

0≤k≤klarge

 12

(int) M∩Match

Z 1 klarge (ext) ˇ 2 2 . + ( R) . d T

Using again the transformation formulas of Proposition 2.90, this yields the following control for (f, f , log(λ)) Z

k d (f, f , log(Υλ)) 2

max

0≤k≤klarge +1

.

.

u1+δdec dk (f, f , log(Υλ)) .

,

(int) M∩Match

We have finally obtained for (f, f , λ) in max

0≤k≤ksmall −1

sup

(int)

M ∩ Match

(int) M∩Match

Z

k d (f, f , log(Υλ)) 2

max

0≤k≤klarge

 12

 12 .

,

(int) M∩Match

9 In Step 3, there is no term corresponding to the one integrated on T . This is due to the fact that for k ≤ klarge − 1, we have thanks to the bootstrap assumptions on energy and a trace estimate Z 1 k (ext) ˇ 2 2 max R) . . d ( 0≤k≤klarge −1

T

203

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

and Z

k d large +1 (f, f , log(Υλ)) 2

 12 .

(int) M∩Match

Z 1 klarge (ext) ˇ 2 2 + ( R) . d T

Step 5. In addition to the estimate of (f, f , λ) in (int) M ∩ Match of Step 4, we need to estimate (f, f , λ) in (ext) M ∩ Match. To this end, we first control in (ext) M ∩ Match the reduced scalar (f 0 , f 0 , λ0 ) satisfying e3

=

eθ

=

e4

=

  1 2 λ0 e03 + f 0 e0θ + f 0 e04 , 4     1 0 0 0 1 1 1 2 1 + f f eθ + f 0 e03 + f 0 + f f 0 e04 , 2 2 2 4      1 1 1 2 1 2 −1 2 2 λ0 1 + f 0 f 0 + f 0 f 0 e04 + f 0 + f 0 f 0 e0θ + f 0 e03 . 2 16 4 4

Together with the initialization of the frame of (ext) M and section 3.1.2 (where the spheres coincide), we have in particular f 0 = f 0 = 0,

(int)

M on T in

λ0 = Υ−1 on T .

Also, recall from section 3.5.1 that in order for (e3 , eθ , e4 ) to be defined everywhere on (ext) M ∩ Match, we need — in addition to the above initialization of (f, f , λ) on T — to initialize it also on C∗ ∩ Match by f 0 = f 0 = 0,

λ0 = Υ−1 on C∗ ∩ Match.

(4.6.4)

ˇ R) ˇ in Arguing similarly to Steps 1–4, we estimate (f 0 , f 0 , λ0 ) and (Γ, We obtain ˇ R) ˇ max sup u1+δdec dk (Γ, 0≤k≤ksmall −2

M∩Match.

(ext) M∩Match

Z +

(ext)

k ˇ R) ˇ 2 d (Γ,

max

 12 .

,

.

,

u1+δdec dk (f 0 , f 0 , log(Υ0 λ0 )) .

,

0≤k≤klarge −1

(ext) M∩Match

Z

k ˇ 2 d large R

 12

(ext) M∩Match

max

0≤k≤ksmall −1

sup (ext) M∩Match

Z

k 0 0 d (f , f , log(Υ0 λ0 )) 2

max

0≤k≤klarge

 12 .

,

(ext) M∩Match

and Z

k ˇ 2 d large Γ

 12

Z 1 klarge (ext) ˇ 2 2 . + ( R) , d

 12

Z 1 klarge (ext) ˇ 2 2 . + ( R) . d

(ext) M∩Match

Z

k d large +1 (f 0 , f 0 , log(Υ0 λ0 )) 2 (ext) M∩Match

T

T

204

CHAPTER 4

Step 6. As mentioned above, in addition to the estimate of (f, f , λ) of Step 4 in (int) M ∩ Match, we need to estimate (f, f , λ) in (ext) M ∩ Match. To this end, we derive simple algebraic relations between (f, f , λ) and (f 0 , f 0 , λ0 ) of Step 5. On the one hand, we have from the definition of (f, f , λ) g(e04 , e3 )

= −2λ,

g(e04 , eθ )

  ff = −f 1 + , g(e0θ , e3 ) = −f , 4   ff 0 −1 . g(e3 , eθ ) = λ f 1 + 4

g(e0θ , e4 )

= λf,   ff 1 2 2 0 −1 + f f , g(e3 , e4 ) = −2λ 1+ 2 16

On the other hand, we have from the definition of (f 0 , f 0 , λ0 ) g(e3 , e0θ ) = λ0 f 0 , g(eθ , e04 ) = −f 0 , ! ! f 0f 0 f 0f 0 1 02 02 0 0 0 0 −1 g(eθ , e3 ) = −f 1 + , g(e4 , e3 ) = −2λ 1+ + f f , 4 2 16 ! f 0f 0 0 0 −1 0 g(e4 , eθ ) = λ f 1 + . 4 g(e3 , e04 ) = −2λ0 ,

We immediately infer λ0 = λ,

f 0 = −λf,

f 0 = −λ−1 f .

In view of the estimates of Step 5, we infer max

0≤k≤ksmall −1

u1+δdec dk (f, f , log(Υλ))

sup Z

k d (f, f , log(Υλ)) 2

max

0≤k≤klarge

. ,

(ext) M∩Match

 12 . ,

(ext) M∩Match

and Z

k d large +1 (f, f , log(Υλ)) 2

 12 .

.

(ext) M∩Match

Together with Step 4, this yields max

0≤k≤ksmall −1

max

0≤k≤ksmall −1

u1+δdec dk (f, f , log(Υλ)) (int) M∩Match sup u1+δdec dk (f, f , log(Υλ)) sup

. , . ,

(ext) M∩Match

Z

k d (f, f , log(Υλ)) 2

max

0≤k≤klarge

 12 . ,

Match

and Z Match

k d large +1 (f, f , log(Υλ)) 2

 12

Z 1 klarge (ext) ˇ 2 2 . + ( R) . d T

Step 7. Next, we estimate r0 − r and m0 − m. Note first that in view of the initial-

205

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

ization of the foliations of (ext) M and (int) M on T , as well as the initializations (4.6.3) on C ∗ ∩ Match and (4.6.4) on C∗ ∩ Match, we have r0 = r, We start with the region e04 (r0 )

=

e03 (r0 )

=

m0 = m on T ∪ Match.

(4.6.5)

(int)

M ∩ Match. We have   r0 0 r0 2 κ =1+ κ0 − 0 , 2 2 r   0 r 0 r0 2Υ0 r0 (κ + A0 ) = −Υ0 + κ0 + 0 + A0 , 2 2 r 2

which together with the identities for e04 (m0 ) and e03 (m0 ) in the outgoing foliation of (ext) M and the control of the foliation of (ext) M in (int) M ∩ Match established in Step 4 yields, using also e0θ (r0 ) = e0θ (m0 ) = 0, max sup u1+δdec dk (e04 (r0 ) − 1, e03 (r0 ) + Υ0 , e0θ (r0 ), 0≤k≤ksmall −2 (int) M∩Match e04 (m0 ), e03 (m0 ), e0θ (m0 )) . , Z k 0 0 max d (e4 (r ) − 1, e03 (r0 ) + Υ0 , e0θ (r0 ), e04 (m0 ), 0≤k≤klarge −1

(int) M∩Match

2

e03 (m0 ), e0θ (m0 ))

! 12 .

.

On the other hand, we have in view of the decomposition of e04 , e03 and e0θ of Step 1   1 2 0 e4 (r) = λ e4 + f eθ + f e3 r 4   r 1 2 = λ (κ + A) + f e3 (r) 2 4       r 2Υ r 1 2 = 1 + λΥ − 1 + λ κ− + A + f e3 (r) , 2 r 2 4

e04 (m)

= =

  1 λ e4 + f eθ + f 2 e3 m 4   1 2 λ e4 (m) + f e3 (m) , 4

206

CHAPTER 4

e03 (r)

!    f2 1 1 2 2 1 2 = λ 1 + f f + f f e3 + f + f f eθ + e4 r 2 16 4 4 !   f2 1 1 2 2 −1 = λ e3 (r) + f f + f f e3 (r) + e4 (r) 2 16 4     r 2 1 1 = −Υ + λ−1 (λΥ − 1) + λ−1 κ+ + f f + f 2 f 2 e3 (r) 2 r 2 16 ! f2 + e4 (r) , 4

e03 (m)



−1

= λ

!    f2 1 1 2 2 1 2 1 + f f + f f e3 + f + f f eθ + e4 m 2 16 4 4 !   f2 1 1 2 2 1 + f f + f f e3 (m) + e4 (m) , 2 16 4 

−1

= λ−1

e0θ (r)

    f ff 1 f 1 + f f eθ + e4 + 1+ e3 r 2 2 2 4   f ff f e4 (r) + 1+ e3 (r), 2 2 4

 = =

and e0θ (r)

    f ff 1 f 1 + f f eθ + e4 + 1+ e3 m 2 2 2 4   f ff f e4 (m) + 1+ e3 (m). 2 2 4

 = =

Together with the identities for e4 (m) and e3 (m) in the ingoing foliation of (int) M, the final estimates of Step 6 for f and λ, and the bootstrap assumptions for the foliation of (int) M, we infer max sup u1+δdec dk (e04 (r) − 1, e03 (r) + Υ, e0θ (r), e04 (m), 0≤k≤ksmall −2 (int) M∩Match e03 (m), e0θ (m)) . , Z k 0 max d (e4 (r) − 1, e03 (r) + Υ, e0θ (r), e04 (m), 0≤k≤klarge −1

(int) M∩Match

2

e03 (m), e0θ (m))

! 12 . .

207

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

We deduce max

0≤k≤ksmall −2

sup (int) M∩Match

u1+δdec dk (e04 (r0 − r), e0θ (r − r0 ), d(m0 − m))

Z max

0≤k≤klarge −1

(int) M∩Match

k 0 0 d (e4 (r − r), e0θ (r − r0 ), d(m0 − m)) 2

. ,

 12 . .

In particular, we have sup (int) M∩Match

u1+δdec |(e04 (r0 − r), e04 (m0 − m))| . ,

and together with the initialization (4.6.5), we integrate the transport equation from T ∪ ( (int) M ∩ Match) and obtain sup (int) M∩Match

u1+δdec |(r0 − r, m0 − m)| . .

Together with the above estimates, and recovering the e03 (r0 − r) using  0      m m e03 (r0 − r) = e03 (r0 ) + Υ0 − e03 (r) + Υ + 2 − , r0 r we infer max

0≤k≤ksmall −1

sup (int) M∩Match

u1+δdec dk (r0 − r, m0 − m)

Z max

0≤k≤klarge

(int) M∩Match

max

sup (ext) M∩Match

Z max

0≤k≤klarge

 12 . .

(ext)

M ∩ Match, we infer u1+δdec dk (r0 − r, m0 − m) .

Finally, arguing similarly in the region 0≤k≤ksmall −1

k 0 d (r − r, m0 − m) 2

. ,

k 0 d (r − r, m0 − m) 2

,

 12 .

,

u1+δdec dk (r0 − r, m0 − m) . u1+δdec dk (r0 − r, m0 − m) .

,

(ext) M∩Match

and hence max

0≤k≤ksmall −1

sup (int) M∩Match

max

0≤k≤ksmall −1

sup (ext) M∩Match

Z max

0≤k≤klarge

Match

k 0 d (r − r, m0 − m) 2

,

 12 .

.

Step 8. Recall from Definition 4.12 that we have defined the global null frame (e004 , e003 , e00θ ) as • In (int) M \ Match, (e004 , e003 , e00θ ) = (e4 , e3 , eθ ). • In (ext) M \ Match, (e004 , e003 , e00θ ) = (Υe04 , Υ−1 e03 , e0θ ). • In Match, (e004 , e003 , e00θ ) is given by the change of frame formula starting from

208

CHAPTER 4

(e4 , e3 , eθ ) and with change of frame coefficients (f 00 , f 00 , λ00 ) given by f 00 = ψf,

f 00 = ψf ,

λ00 = 1 − ψ + ψΥ0 λ,

see (4.6.1). Also, recall that we have defined r00 and m00 as r00 = (1 − ψ)r + ψr0 ,

m00 = (1 − ψ)m + ψm0 .

Step 9. In view of the transformation formulas of Proposition 2.90, we have schematically ˇ 00 , R ˇ 00 ) = (Γ, ˇ R) ˇ + d(f 00 , f 00 , λ00 − 1) + f 00 + f 00 + (λ00 − 1) + (r00 − r) + (m00 − m). (Γ In view of the definition of (f 00 , f 00 , λ00 ) and (r00 , m00 ) in Step 8, we infer ˇ 00 , R ˇ 00 ) = (Γ, ˇ R) ˇ + d(f, f , Υλ − 1) + f + f + (Υλ − 1) + (r0 − r) + (m0 − m). (Γ ˇ the estimates for Together with the bootstrap assumptions in (int) M for (Γ, R), (ext) ˇ (Γ, R) in M provided by Step 5, the estimates for (f, f , λ) provided by Step 6 in Match, and the estimates for r0 − r and m0 − m provided by Step 7, we deduce ˇ 00 , R ˇ 00 ) . , max sup u1+δdec dk (Γ 0≤k≤ksmall −2 (int) M∩Match ˇ 00 , R ˇ 00 ) . , max sup u1+δdec dk (Γ 0≤k≤ksmall −2

(ext) M∩Match

Z

k 00 00 2 ˇ ,R ˇ ) d (Γ

max

0≤k≤klarge −1

 12 .

,

.

Z 1 klarge (ext) ˇ 2 2 + ( R) . d

Match

Z

k ˇ 00 , R ˇ 00 ) 2 d large (Γ

Match

 12

T

Since the double-primed quantities correspond to the quantities associated to the global frame, this concludes the proof of Lemma 4.16. 4.6.3

Proof of Proposition 3.26

To match the first global frame of M of Proposition 3.26 with a conformal renormalization of the second frame of (ext) M of Proposition 3.20, we will need to introduce a cut-off function. Definition 4.18. Let ψ : R → R a smooth cut-off function such that 0 ≤ ψ ≤ 1, ψ = 0 on (−∞, 0] and ψ = 1 on [1, +∞). We define ψm0 as follows: ( 1 if r ≥ 4m0 , ψm0 (r) = 0 0 if r ≤ 7m 2 ,

209

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

and 0 2 r − 7m 2 m0

ψm0 (r) = ψ


! on

7m0 ≤ 2

(ext)

r ≤ 4m0 .

We are now ready to define the second global frame, i.e., the global frame of the statement of Proposition 3.26. Definition 4.19 (Definition of the second global frame). We introduce a global 0 0 0 null frame defined on (ext) M ∪ (int) M and denoted by ((glo ) e4 , (glo ) e3 , (glo ) eθ ). The second global frame is defined as follows: 1. In

(ext)

M ∩ { (ext) r ≥ 4m0 }, we have 0

0

0

((glo ) e4 , (glo ) e3 , (glo ) eθ ) =



(ext)

 Υe04 , (ext) Υ−1 e03 , e0θ ,

where (e04 , e03 , e0θ ) denotes the second frame of (ext) M, i.e., the one constructed in Proposition 3.20. 0 2. In (int) M ∪ ( (ext) M ∩ { (ext) r ≤ 7m 2 }), we have   0 0 0 ((glo ) e4 , (glo ) e3 , (glo ) eθ ) = (glo) e4 , (glo) e3 , (glo) eθ ,
where (glo) e4 , (glo) e3 , (glo) eθ denotes the first global frame of M of Proposition 3.26. 3. It remains to define the global frame on the matching region Match0 . We denote by f the reduced scalar introduced in Proposition 3.20 such that we have in (ext) M e04

=

e0θ

=

e03

=

1 e4 + f (ext) eθ + f 2(ext) e3 , 4 f (ext) (ext) eθ + e3 , 2 (ext) e3 . (ext)

Then, in the matching region Match0 , the second global frame of M is given by   1 0 2 0 (glo) (glo0 ) 0 0 −1 (glo) 0(glo) e4 = Υ Υ e4 + f eθ + f Υ e3 , 4 f0 (glo0 ) eθ = (glo) eθ + Υ0 (glo) e3 , 2 (glo0 ) (glo) e3 = e3 , where f 0 = ψm0 ( (ext) r)f,

Υ0 = 1 − ψm0 ( (ext) r) + ψm0 ( (ext) r) (ext) Υ.

(4.6.6)

Remark 4.20. Recall that the smooth cut-off function ψ in Definition 3.25, allowing to define ψm0 ,δH , is such that we have in particular ψ = 0 on (−∞, 0] and ψ = 1 on [1, +∞). The following two special cases correspond to the properties (d) i. and (d) ii. of Proposition 3.26.

210

CHAPTER 4

• If the cut-off ψ in Definition 3.25 is such that ψ = 1 on [1/2, +∞), then   0 0 0 (ext) ((glo ) e4 , (glo ) e3 , (glo ) eθ ) = Υe04 , (ext) Υ−1 e03 , e0θ   15m0 (ext) (ext) on M r≥ . 4 • If the cut-off ψ in Definition 3.25 is such that ψ = 0 on (−∞, 1/2], then   0 0 0 (glo) ((glo ) e4 , (glo ) e3 , (glo ) eθ ) = e4 , (glo) e3 , (glo) eθ   15m0 on (int) M ∪ (ext) M (ext) r ≤ . 4 0

0

0

Remark 4.21. When dealing with the second global frame ((glo ) e4 , (glo ) e3 , (glo ) eθ ), the area radius and Hawking mass that we use are the ones corresponding to the first global frame, i.e., (glo) r and (glo) m. The following two lemmas provide the main properties of the second global frame of M. Lemma 4.22. We have in (ext) M(r ≥ 4m0 ) the following relations between the 0 0 0 quantities in the second global frame of M, i.e., ((glo ) e4 , (glo ) e3 , (glo ) eθ ), and the second frame of (ext) M, i.e., (e04 , e03 , e0θ ), (glo0 )

α = Υ2 α0 ,

(glo0 )

α = Υ−2 α0 ,

0 2m 2m = ρ0 + 3 , (glo ) β = Υ−1 β 0 , r3 r 0 0 0 (glo0 ) ξ = 0, (glo ) ξ = Υ−2 ξ 0 , (glo ) ζ = −(glo ) η = ζ 0 ,

(glo0 )

β = Υβ 0 ,

(glo0 )

ρ+

m m e0 (m) = Υω 0 + 2 (1 − e04 (r)) + 4 , 2 r r r     m m e0 (r) e0 (m) (glo0 ) ω = Υ−1 ω 0 − 2 + 2 1 − 3 − 3 , r r Υ Υr     0 2Υ 2 2 2Υ (glo0 ) κ− = Υ κ0 − , (glo ) κ + = Υ−1 κ0 + , r r r r (glo0 )

η = η0 ,

(glo0 )

(glo0 )

ϑ = Υϑ0 ,

ω+

(glo0 )

ϑ = Υ−1 ϑ0 .

Proof. The proof follows immediately from the change of frame formula with the choice (f = 0, f = 0, λ = Υ), the fact that eθ (Υ) = 0, and the fact that the frame (e04 , e03 , e0θ ) is such that ξ 0 = 0 and η 0 = −ζ 0 . Lemma 4.23 (Control of the second global frame in the matching region). In the matching region, the following estimate holds for the second global frame 0 ˇ (glo0 ) R) ˇ . . max sup u1+δdec −2δ0 dk ((glo ) Γ, 0≤k≤ksmall +kloss Match0

Remark 4.24. The quantities associated to the second global frame can be estimated as follows: 0 • In (int) M ∪ (ext) M( (ext) r ≤ 7m 2 ), the second global frame coincides with the first global frame, and hence, the quantities associated to the second global frame

211

CONSEQUENCES OF THE BOOTSTRAP ASSUMPTIONS

satisfy the same estimates as the corresponding quantities for the first global frame. • In (ext) M( (ext) r ≥ 4m0 ), estimates for the quantities associated to the second global frame follow from the identities of Lemma 4.22 together with the estimates of Proposition 3.20 for the second frame of (ext) M. • In Match0 , the estimates for the quantities associated to the global frame are provided by Lemma 4.23. The proof of Proposition 3.26 easily follows from Definition 4.19, Remark 4.20, and Lemma 4.23. Thus, from now on, we focus on the proof of Lemma 4.23 which is carried out below. Proof of Lemma 4.23. Recall from Definition 4.19 that we have in the matching region Match0   1 2 −1 (glo0 ) e4 = Υ0 Υ0 (glo) e4 + f 0(glo) eθ + f 0 Υ0 (glo) e3 , 4 0 0 f (glo ) eθ = (glo) eθ + Υ0 (glo) e3 , 2 (glo0 ) e3 = (glo) e3 , where f 0 = ψm0 ( (ext) r)f, Now, since

(ext)

r≥

7m0 2

Υ0 = 1 − ψm0 ( (ext) r) + ψm0 ( (ext) r) (ext) Υ. on Match0 , we also have in that region

((glo) e4 , (glo) e3 , (glo) eθ ) = ((ext) Υ (ext) e4 , ((ext) Υ)−1 (ext) e3 , (ext) eθ ). We deduce on Match0 (glo0 )

(glo0 ) (glo0 )



(ext)

 1 2 e4 + f 00(ext) eθ + f 00 (ext) e3 , 4

e4

=

(ext)

Υ

eθ

=

(ext)

eθ +

e3

=

((ext) Υ)−1 (ext) e3 ,

f 00 2

(ext)

e3 ,

where f 00

Υ0 ((ext) Υ)−1 f 0   = 1 − ψm0 ( (ext) r) + ψm0 ( (ext) r) (ext) Υ ((ext) Υ)−1 ψm0 ( (ext) r)f. =

In view of the transformation formulas of Proposition 2.90, we deduce, schematically,  0    (glo ) ˇ (glo0 ) ˇ (ext) ˇ (ext) ˇ Γ, R = Γ, R + df + f. Together with the bootstrap assumptions on decay and Proposition 3.19 for

(ext) ˇ

Γ

212 and

CHAPTER 4

(ext)

ˇ and the estimate (3.4.11) for f , we infer R, 0 ˇ (glo0 ) R) ˇ . max sup u1+δdec −2δ0 dk ((glo ) Γ, 0≤k≤ksmall +kloss Match0

which concludes the proof of Lemma 4.23.



Chapter Five Decay Estimates for q (Theorem M1) The goal of this chapter is to prove Theorem M1, i.e., to derive decay estimates for the quantity q for k ≤ ksmall + 20 derivatives. To this end, we will make use of the wave equation satisfied by q (see (2.4.7)) 2 q + κκ q =

N,

(5.0.1)

where N contains only quadratic or higher order terms. Now, in order to have a suitable right-hand side N , recall from the discussion in Remarks 2.109 and 2.110 that q is defined relative to the global null frame of Proposition 3.26 for which ξ = 0 for r ≥ 4m0 and η ∈ Γg . For such a global fame, N is given schematically by, see (2.4.8),   N = r2 d≤2 (Γg · (α, β)) + e3 r3 d≤2 (Γg · (α, β)) + d≤1 (Γg · q) + l.o.t. (5.0.2) 5.1

PRELIMINARIES

5.1.0.1

Smallness constants

Recall from the beginning of section 3.3.2 the constant m0 and the main small constants δH , δB , δdec ,  and 0 such that • The constant m0 > 0 is the mass of the initial Schwarzschild spacetime relative to which our initial perturbation is measured. • The integer klarge which corresponds to the maximum number of derivatives of the solution. • The size of the initial data layer norm is measured by 0 > 0. • The size of the bootstrap assumption norms are measured by  > 0. • δH > 0 measures the width of the region |r − 2m0 | ≤ 2m0 δH where the redshift estimate holds and which includes in particular the region (int) M. ˇ and R. ˇ • δdec is tied to decay estimates in u, u for Γ p • δB is involved in the r-power of the r -weighted estimates for curvature. Recall also that these constants satisfy in view of (3.3.1), (3.3.2), and (3.3.3) 0 < δH , δdec , δB  min{m0 , 1},

δB > 2δdec ,

0 ,   min{δH , δdec , δB , m0 , 1}, and 2

 = 03 .

klarge 

1 , δdec

214

CHAPTER 5

We will need the following additional small constants in this chapter: • δextra > 0, tied to the decay of q, and is chosen such that δextra > δdec , • δ > 0 for various degeneracies, ˇ R) ˇ • δ0 > 0 which comes from interpolating between k ≤ ksmall derivatives of (Γ, ˇ R), ˇ see Lemma 5.1, and k ≤ klarge derivatives of (Γ, • q0 > 0 which will allow us to recover the fact that the decay for q in Theorem M1 has an extra gain u−(δextra −δdec ) compared to the expected behavior inferred from the bootstrap assumptions. We will choose δextra such that δdec < δextra < 2δdec , δB ≥ 2δextra , δ and δ0 such that 0 < , 0  δ, δ0  δdec , δextra , δH , m0 , 1,

(5.1.1)

2δdec < q0 < 4δdec − 4δ0 − 4δ.

(5.1.2)

and q0 such that1

5.1.1

The foliation of M by τ

Recall that the spacetime M is decomposed as M = (int) M ∪ (ext) M and that u is an outgoing optical function on (ext) M while u is an ingoing optical function. In this chapter, we rely on the global frame (e3 , e4 , eθ , eϕ ) defined in section 3.5, and r and m denote the corresponding scalar functions associated to it. Also, we define the trapping region Mtrap as   5m0 7m0 Mtrap := ≤r≤ . (5.1.3) 2 2  Also, let (trap) M = M \ (trap) M the complement of (trap) M in M. We foliate our spacetime domain M by Z-invariant hypersurfaces Σ(τ ) which are: • Incoming null in (int) M, with e3 as null incoming generator. We denote this portion (int) Σ(τ ). • Strictly spacelike in (trap) M. We denote this portion by (trap) Σ. • Outgoing null in M>4m0 . We denote this portion by Σ>4m0 (τ ). 1 This

will allow us to choose in the proof of Theorem M1, see (5.2.10), δextra =

q0 − δ 2

which satisfies the desired estimate δextra > δdec for δ > 0 small enough.

215

DECAY ESTIMATES FOR q (THEOREM M1)

• The parameter τ of Σ(τ ) can be chosen, smoothly, such that  in M>4m0 ,   u u + r in Mtrap , τ :=   u in (int) M. • In particular, the unit normal in the region Mtrap , i.e., the normal to satisfies2 −2 ≤ g(NΣ , e4 ) ≤ −1, 5.1.2

−2 ≤ g(NΣ , e3 ) ≤ −1 on Mtrap .

(5.1.4)

(trap)

Σ,

(5.1.5)

Assumptions for Ricci coefficients and curvature

Recall from Remark 2.110 that q is defined, according to equation (2.3.10) in Lemma 2.96, relative to the global frame of Proposition 3.26 for which η ∈ Γg with the notation   m 2Υ (0) Γg = Γg = ξ, ϑ, ω + 2 , κ − , η, η, ζ, A , r r   2 (0) Γb = Γb = ϑ, κ + , A, ω, ξ , r where we recall that Υ=1−

2m , r

A=

2 e4 (r) − κ, r

A=

2 e3 (r) − κ. r

Note also that ξ vanishes in (ext) M away from the matching region of Proposition 3.26, and in particular for r ≥ 4m0 . For higher derivatives we write n o Γ(1) = dξ, dϑ, reθ ω, reθ (κ), dη, dη, dζ, dA , g n o (1) Γb = dϑ, reθ (κ), dξ, dA, reθ ω, dξ , and for s ≥ 2, Γ(s) g

=

ds−1 Γ(1) g ,

Moreover we denote n o (0) (1) (s) Γ≤s = Γ , Γ , . . . , Γ , g g g g 2N Σ

(s)

(1)

Γb = ds−1 Γb .

o n (0) (1) (s) Γ≤s = Γ , Γ , . . . , Γ . b b b b

is given in view of its definition by NΣ

= =

  1 e4 (r)e3 + (e3 (u) + e3 (r))e4 √ p 2 e4 (r)(e3 (u) + e3 (r))   1 (1 + O())e3 + (2 − Υ + O())e4 √ p 2 2 − Υ + O()

where we used the bootstrap assumptions.

216

CHAPTER 5

With these notations, we may now state the estimates satisfied by the Ricci coefficients and curvature components. Lemma 5.1. Consider the global frame of Proposition 3.26 and the above definition3 of Γg and Γb . Let an integer kloss and a small constant δ0 > 0 satisfying4 δdec (klarge − ksmall ), 3

16 ≤ kloss ≤

δ0 =

kloss . klarge − ksmall

(5.1.6)

Then, the Ricci coefficients and curvature components with respect to the global frame of Proposition 3.26 satisfy ξ = 0 on r ≥ 4m0 , ( max

sup

0≤k≤ksmall +kloss M



 1 r2 τ 2 +δdec −2δ0 + rτ 1+δdec −2δ0 |dk Γg | + rτ 1+δdec −2δ0 |dk Γb |

 7 δB   1 + r 2 + 2 + r3 τ 2 +δdec −2δ0 + r2 τ 1+δdec −2δ0 |dk α| + |dk β|   1 + r3 τ 2 +δdec −2δ0 + r2 τ 1+δdec −2δ0 |dk ρˇ| )   1+δdec −2δ0 2 k k +τ r |d β| + r|d α| . , ( max

sup

0≤k≤ksmall +kloss M

r2 τ 1+δdec −2δ0 |dk−1 e3 (Γg )| 3

1+δdec −2δ0

+r (τ + 2r)



k−1

|d

 e3 (α)| + |d e3 (β)| k

) . .

Proof. In r ≥ 4m0 , the global frame of Proposition 3.26 coincides with a conformal renormalization of the second frame of (ext) M, see Proposition 3.20. The estimates there follow immediately from the ones of Proposition 3.20. In the matching region 7/2m0 ≤ r ≤ 4m0 , the estimates are stated in Proposition 3.26. Finally, for (ext) M(r ≤ 7/2m0 ) and (int) M, the estimates follow directly from interpolation between the bootstrap assumptions on decay for k ≤ ksmall and the pointwise estimates of Proposition 3.19 for k ≤ klarge − 5. 5.1.3

Structure of nonlinear terms

The following lemma will be important in what follows.

3 Recall 4 Recall

in particular that the global frame of Proposition 3.26 is such that η ∈ Γg . that we have   1 klarge + 1. 0 < δdec  1, δdec klarge  1, ksmall = 2

In particular, we have δdec (klarge − ksmall )  1 and hence there exists an integer kloss satisfying the required constraints. We will in fact choose kloss = 33, see (5.2.3).

217

DECAY ESTIMATES FOR q (THEOREM M1)

Lemma 5.2. For the solution q to the wave equation (5.0.1), the structure of the error term N can be written schematically as follows: N

=

Ng + e3 (rNg ) + Nm [q]

(5.1.7)

where Ng = r2 d≤2 (Γg · (α, β)),

(5.1.8)

Nm [q] = d≤1 (Γg · q).

Moreover, for every k ≤ klarge − 3 we have, schematically, dk N

=

d≤k Ng + e3 (dk (rNg )) + dk Nm [q].

(5.1.9)

Remark 5.3. In fact, (5.1.7) and (5.1.9) also contain lower order terms which are strictly better in powers of r and contain at most the same number of derivatives. For convenience, we drop them in the rest of the proof of Theorem M1. Proof. For k = 0, this is an immediate consequence of (5.0.2). For the higher derivatives we write dk (e3 (rNg )) = e3 (dk (rNg )) + [dk , e3 ](rNg ). In view of the formula for [e3 , d/] of Lemma 2.68, and the commutator formula for [e3 , e4 ], we have, schematically,   1 [e3 , e3 ] = 0, [ d/, e3 ] = Γb d + Γb , [re4 , e3 ] = + Γb d. r In view of our assumptions i d (Γb ) ≤ r−1 ,

i ≤ klarge − 4,

Γb is at least as good as r−1 , and hence, we deduce, schematically, [d, e3 ]

=

1 1 d+ . r r

On the other hand, we have, schematically, [d, r] = r and hence, for k ≤ klarge − 3, [dk , e3 ](rNg )

=

X i+j≤k−1

= as desired.

d≤k Ng

di



1 1 d+ r r



dj (rNg )

218

CHAPTER 5

5.1.4

Main quantities

We restrict our attention to the region M(τ1 , τ2 ) = M ∩ {τ1 ≤ τ ≤ τ2 }. For a given ψ ∈ s2 (M) we introduce the following quantities, for 0 ≤ τ1 < τ2 ≤ τ∗ . 5.1.4.1

Morawetz bulk quantities

Consider the vectorfields, T :=

1 (e4 + Υe3 ) , 2

R :=

1 (e4 − Υe3 ) . 2

(5.1.10)

1

1

10 10 Let θ a smooth bump function equal 1 on |Υ| ≤ δH vanishing for |Υ| ≥ 2δH and define the modified vectorfields,

˘ := θ 1 (e4 − e3 ) + (1 − θ)Υ−1 R = R 2 1 T˘ := θ (e4 + e3 ) + (1 − θ)Υ−1 T = 2

i 1 h˘ θe4 − e3 , 2 i 1 h˘ θe4 + e3 , 2

(5.1.11)

where θ˘ = θ + Υ−1 (1 − θ). Note that ( 1 10 1 for |Υ| ≤ δH , ˘ θ= 1 −1 10 Υ for |Υ| ≥ 2δH .

(5.1.12)

Remark 5.4. Note that ˘ + T˘ = e4 , R ˘ + T˘ = Υ−1 e4 , R

˘ + T˘ = e3 , −R ˘ + T˘ = e3 , −R

in

(int)

M,

in M>4m0 .

We define the quantities Z Mor[ψ](τ1 , τ2 ) : =

1 ˘ 2 1 |Rψ| + 4 |ψ|2 3 r r M(τ1 ,τ2 )  2   3m 1 1 + 1− |∇ / ψ|2 + 2 |T˘ψ|2 , r r r Z Morr[ψ](τ1 , τ2 ) : = Mor[ψ](τ1 , τ2 ) + r−1−δ |e3 (ψ)|2 ,

(5.1.13)

M>4m0 (τ1 ,τ2 )

with m = m(τ, r) = m(u, r) the Hawking mass in M. The constant δ > 0 is a sufficiently small quantity. An equivalent definition for Morr[ψ](τ1 , τ2 ) is given

219

DECAY ESTIMATES FOR q (THEOREM M1)

below: "

Z Morr[ψ](τ1 , τ2 ) = (trap) M(τ

1 ,τ2 )

|Rψ|2 + r−2 |ψ|2 

3m + 1− r Z +



(trap)

M(τ1 ,τ2 )

2 

5.1.4.2



(trap)

# (5.1.14)

h
r−3 |e4 ψ|2 + r−1 |ψ|2 + r−1 |∇ / ψ|2 + r−1−δ |e3 ψ|2

where

1 |∇ / ψ| + 2 |T ψ|2 r 2

M denotes the complement of

(trap)

i

M.

Weighted bulk quantities

Define, for 0 < p < 2, B˙ p ; R [ψ](τ1 , τ2 ) : =

Z M≥R (τ1 ,τ2 )


rp−1 p|ˇ e4 (ψ)|2 + (2 − p)|∇ / ψ|2 + r−2 |ψ|2 , (5.1.15)

Bp [ψ](τ1 , τ2 ) : = Morr[ψ](τ1 , τ2 ) + B˙ p ; 4m0 [ψ](τ1 , τ2 ). The bulk quantity Bp [ψ](τ1 , τ2 ) is equivalent to5 Z τ2 Bp [ψ](τ1 , τ2 ) ' Mp−1 [ψ](τ )dτ τ1

where  2   3m m2 ˘ 2 2 −2 2 2 ˘ Mp−1 [ψ](τ ) = |Rψ| + r |ψ| + 1 − |∇ / ψ| + 2 |T ψ| r r Σ≤4m0 (τ ) Z
+ rp−1 p|e4 (ψ)|2 + (2 − p)|∇ / ψ|2 + r−2 |ψ|2 Z

Σ≥4m0 (τ )

Z + Σ≥4m0 (τ )

r−1−δ |e3 ψ|2 .

Remark 5.5. Note that, for δ ≤ p ≤ 2 − δ, Bp [ψ](τ1 , τ2 ) :

= Morr[ψ](τ1 , τ2 ) + B˙ p ; 4m0 [ψ](τ1 , τ2 )

5 This equivalence follows from the coarea formula and the fact that the lapse of the τ -foliation is controlled uniformly from above and below.

220

CHAPTER 5

is equivalent to Bp [ψ](τ1 , τ2 ) ' Morr[ψ](τ1 , τ2 ) Z
+ rp−1 |ˇ e4 (ψ)|2 + |∇ / ψ|2 + r−2 |e3 ψ|2 + r−2 |ψ|2 . M≥4m0 (τ1 ,τ2 )

Indeed, Z M≥4m0 (τ1 ,τ2 )

rp−3 |e3 ψ|2 .

Z M≥4m0 (τ1 ,τ2 )

r−1−δ |e3 ψ|2 .


Therefore, since r2 |ˇ e4 (ψ)|2 + |∇ / ψ|2 . |dψ|2 , we have Bp [ψ](τ1 , τ2 ) ' Morr[ψ](τ1 , τ2 ) Z
+ rp−3 |dψ|2 + |ψ|2 .

(5.1.16)

M≥4m0 (τ1 ,τ2 )

5.1.4.3

Basic energy-flux quantity

The basic energy-flux quantity on a hypersurface Σ(τ ) is defined by  Z 1 1 E[ψ](τ ) = (NΣ , e3 )2 |e4 ψ|2 + (NΣ , e4 )2 |e3 ψ|2 2 2 Σ(τ )  2 −2 2 +|∇ / ψ| + r |ψ| .

(5.1.17)

Here NΣ denotes a choice for the normal to Σ so that in particular we have ( NΣ = e3 on (int) Σ, NΣ = (5.1.18) NΣ = e4 on (ext) Σ, and, in view of (5.1.5), (NΣ , e3 ) ≤ −1 and (NΣ , e4 ) ≤ −1 5.1.4.4

on

(trap)

Σ.

(5.1.19)

Weighted energy-flux type quantities

We have

E˙ p ; R [ψ](τ )

Z     rp |ˇ e4 ψ|2 + r−2 |ψ|2 for p ≤ 1 − δ,   Σ≥R (τ ) := (5.1.20) Z      rp |ˇ e4 ψ|2 + r−p−1−δ |ψ|2 for p > 1 − δ,  Σ≥R (τ )

and Ep [ψ](τ )

:= E[ψ](τ ) + E˙ p ; 4m0 [ψ](τ ).

(5.1.21)

221

DECAY ESTIMATES FOR q (THEOREM M1)

Here eˇ4 denotes the first order operator eˇ4 ψ = r−1 Υ−1 e4 (rψ).

(5.1.22)

Remark 5.6. To control the weighted quantities (5.1.21), it will be convenient to introduce in (ext) M(r ≥ 4m0 ) the following renormalized frame e04 = Υ−1 e4 ,

e03 = Υe3 ,

e0θ = eθ .

In particular, this yields eˇ4 ψ = r−1 e04 (rψ). Note also that we have the following alternate form eˇ4 ψ = e04 ψ + r−1 ψ +

e04 (r) − 1 ψ r

where e04 (r) − 1 = Υ−1 e4 (r) − 1 = O(r−1 ) in view of our assumption on Γg . 5.1.4.5

Flux quantities

The boundary of M(τ1 , τ2 ) is given by ∂M(τ1 , τ2 )

=

Σ(τ1 ) ∪ Σ(τ2 ) ∪ A(τ1 , τ2 ) ∪ Σ∗ (τ1 , τ2 ).

Our basic flux quantity along the spacelike hypersurfaces A and Σ∗ is given by Z   −1 F [Ψ](τ1 , τ2 ) := δH |e4 Ψ|2 + δH |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 A(τ1 ,τ2 ) Z   + |e4 Ψ|2 + |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 , (5.1.23) Σ∗ (τ1 ,τ2 )

with A(τ1 , τ2 ) = A ∩ M(τ1 , τ2 ) and Σ∗ (τ1 , τ2 ) = Σ∗ ∩ M(τ1 , τ2 ). 5.1.4.6

Weighted flux quantities F˙p [ψ](τ1 , τ2 ) :=

Z Σ∗ (τ1 ,τ2 )

  rp |e4 ψ|2 + |∇ / ψ|2 + r−2 |ψ|2 ,

Fp [ψ](τ1 , τ2 ) := F [ψ](τ1 , τ2 ) + F˙p [ψ](τ1 , τ2 ). 5.1.4.7

Weighted quantities for the inhomogeneous term N

Recall the decomposition (5.1.7) for the inhomogeneous term N N

=

Ng + e3 (rNg ) + Nm [q].

(5.1.24)

222

CHAPTER 5

We define, for p ≥ δ, Z Ip [Ng ](τ1 , τ2 )

2

τ2

= τ1

dτ kNg kL2 ( (trap) Σ(τ ))

Z +

M(τ1 ,τ2 )

r3+δ |e3 (Ng )|2 .



(trap)



(trap)

M(τ1 ,τ2 )

Z +

+

r2+p |Ng ||e3 (Ng )| +



(trap)

Z M(τ1 ,τ2 )

r1+p |Ng |2

Z sup τ ∈[τ1 ,τ2 ]

2 rp+2 Ng

Σ(τ )

(5.1.25)

Remark 5.7. While Nm [q] is present in the decomposition of the inhomogeneous term N , (5.1.25) only contains a norm for Ng . In fact, Nm [q] will always be absorbed by the left-hand side wherever it appears. 5.1.4.8

Higher order derivative quantities

We define the higher order derivative quantities E s [ψ], Mors [ψ], Morrs [ψ], Eps [ψ], Bbs [ψ], Mps [ψ], F s [ψ], Fps [ψ], Ips [Ng ] by the obvious procedure, Qs [ψ] =

X

Q[dk ψ].

k≤s

Remark 5.8. Note that in view of Remark 5.5 we can also write, equivalently, for p < 2 − δ, Z Bps [ψ](τ1 , τ2 ) = M orrs [ψ](τ1 , τ2 ) + rp−3 |d≤1+s ψ|2 . (5.1.26) M>4m0 (τ1 ,τ2 )

5.1.4.9

Decay norms

We introduce s Ep,d [ψ] : = sup (1 + τ )d Eps [ψ](τ ), 0≤τ ≤τ∗

s Bp,d [ψ]

: = sup (1 + τ )d Bps [ψ](τ, τ∗ ), 0≤τ ≤τ∗

' sup (1 + τ )d 0≤τ ≤τ∗

s Fp,d [ψ] : = sup (1 + 0≤τ ≤τ∗

s Ip,d [Ng ]

Z

τ∗

s Mp−1 [ψ](τ 0 )dτ 0 ,

τ τ )d Fps [ψ](τ, τ∗ ),

: = sup (1 + τ )d Ips [Ng ](τ, τ∗ ). 0≤τ ≤τ∗

(5.1.27)

223

DECAY ESTIMATES FOR q (THEOREM M1)

5.2

PROOF OF THEOREM M1

Recall that we have to prove for k ≤ ksmall + 20 1

|dk q| . 0 r−1 (1 + τ )− 2 −δextra , 1

|dk q| . 0 r− 2 (1 + τ )−1−δextra ,

|dk e3 (q)| . 0 r−1 (1 + τ )−1−δextra , and Z (int) M(τ,τ

∗)

|dk e3 q|2 +

Z Σ∗ (τ,τ∗ )

|dk e3 q|2

.

20 (1 + τ )−2−2δextra ,

for some constant δextra such that δdec < δextra < 2δdec . 5.2.1

Flux decay estimates for q

The following result establishes decay of flux estimates for q. Theorem 5.9. Let 0 < q0 < 1 be a fixed number and s ≤ ksmall + 25. Then, for all δ > 0, with a constant C depending only on s, δ and q0 such that for all δ ≤ p ≤ 2 − δ, we have s s s Ep,2+q [q] + Bp,2+q [q] + Fp,2+q [q] 0 −p 0 −p 0 −p

.

s+4 s+5 Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ], 0 0 +2,0 0 −δ

(5.2.1)

s where we recall that the decay norms Ip,d [Ng ] are defined by s Ip,d [Ng ]

=

sup (1 + τ )d Ips [Ng ](τ, τ∗ ).

0≤τ ≤τ∗

Theorem 5.9 will be proved in section 5.4.3. s To prove Theorem M1 we have to eliminate the norms Ip,d [Ng ] on the right-hand side of Theorem 5.9. Proposition 5.10. Let s ≤ ksmall + 30 and assume q0 < 4δdec − 4δ0

(5.2.2)

where δ0 =

33 33 = k klarge − ksmall klarge − b large 2 c−1

(5.2.3)

is the small constant appearing in Lemma 5.1. Then, the following estimates hold true, s Iqs0 +2,0 [Ng ] + Iδ,2+q [Ng ] . 0 −δ

4 .

The proof of Proposition 5.10 is postponed to section 5.2.3. Together with Theorem 5.9, Proposition 5.10 immediately yields the proof of the following corollary.

224

CHAPTER 5

Corollary 5.11. In addition to the assumptions of Theorem 5.9 we assume 2δdec < q0 < 4δdec − 4δ0

(5.2.4)

where δ0 > 0 is given by (5.2.3). Then for a sufficiently small bootstrap constant  > 0, for all s ≤ ksmall + 25 and for all δ ≤ p ≤ 2 − δ, we have s+4 s s s Ep,2+q [q] + Bp,2+q [q] + Fp,2+q [q] . Eqs+2 [ˇq](0) + E2−δ [q](0) + 4 . 0 −p 0 −p 0 −p 0

5.2.2

Proof of Theorem M1 2/3

Since  = 0 , and in view of the control on q at τ = 0 provided by Theorem M0, we immediately deduce from Corollary 5.11, for all 0 < q ≤ q0 , δ ≤ p ≤ 2 − δ, and s ≤ ksmall + 25, s s s Ep,2+q [q] + Bp,2+q [q] + Fp,2+q [q] . 20 . 0 −p 0 −p 0 −p

(5.2.5)

We will also need the following two propositions concerning L2 estimates on spheres. Proposition 5.12. On any S = S(τ, r) ⊂ Σ(τ ), for s ≤ ksmall + 25, Z
1 s
1 s (1 + τ )1+q0 |q(s) |2 . E1+δ,1+q [q] 2 E1−δ,1+q [q] 2 0 −δ 0 +δ

(5.2.6)

Sr

and r

−1

2+q0 −δ

Z

(1 + τ )

Sr

|q(s) |2

.

s Eδ,2+q [q]. 0 −δ

Proposition 5.13. We have for s ≤ ksmall + 25 Z s (1 + τ )2+q0 −δ |e3 d≤s q|2 . Fδ,2+q [q]. 0 −δ

(5.2.7)

(5.2.8)

Σ∗ (τ,τ∗ )

Also, on any S = S(τ, r) ⊂ Σ(τ ), for s ≤ ksmall + 23, we have Z s+1 s+2 (1 + τ )2+q0 −δ |e3 d≤s q|2 . 20 + Fδ,2+q [q] + Eδ,2+q [q]. 0 −δ 0 −δ

(5.2.9)

Sr

The proof of Proposition 5.12 is postponed to section 5.4.4, and the proof of Proposition 5.13 is postponed to section 5.4.5. We now conclude the proof of Theorem M1. Indeed, in view of (5.2.5), Propo-

225

DECAY ESTIMATES FOR q (THEOREM M1)

sition 5.12 and Proposition 5.13, we infer for s ≤ ksmall + 25 Z (1 + τ )2+q0 −δ |d≤s+1 q|2 . 20 , (int) M(τ,τ ) ∗ Z 1+q0 (1 + τ ) |q(s) |2 . 20 , Sr Z r−1 (1 + τ )2+q0 −δ |q(s) |2 . 20 , Sr Z (1 + τ )2+q0 −δ |e3 d≤s q|2 . 20 , Σ∗ (τ,τ∗ )

and for s ≤ ksmall + 23 2+q0 −δ

Z

(1 + τ )

S

|ds e3 q|2

20 .

.

In view of the standard Sobolev inequality on the 2-surfaces S, i.e., kψkL∞ (S)

r−1 k(r∇ / )≤2 ψkL2 (S) ,

.

we immediately infer for s ≤ ksmall + 23 1

|q(s) | . 0 r−1 (1 + τ )− 2 − |q

(s)

| . 0 r

− 12

−1−

(1 + τ )

q0 2

,

q0 −δ 2

,

and for s ≤ ksmall + 21 |ds e3 (q)| . 0 r−1 (1 + τ )−1−

q0 −δ 2

.

Recall that q0 > 2δdec and that δ > 0 can be chosen arbitrarily small so that we have q0 − δ > 2δdec . In particular, we may choose δextra :=

q0 − δ , 2

δextra > δdec ,

which together with the above estimates for q implies for s ≤ ksmall + 25 Z 2+q0 −δ (1 + τ ) |d≤s+1 q|2 . 20 , (int) M(τ,τ ) ∗ Z 2+2δextra (1 + τ ) |e3 d≤s q|2 . 20 , Σ∗ (τ,τ∗ )

for s ≤ ksmall + 23 1

|q(s) | . 0 r−1 (1 + τ )− 2 −δextra , 1

|q(s) | . 0 r− 2 (1 + τ )−1−δextra , and for s ≤ ksmall + 21 |ds e3 (q)| .

0 r−1 (1 + τ )−1−δextra

(5.2.10)

226

CHAPTER 5

as desired. This concludes the proof of Theorem M1. 5.2.3

Proof of Proposition 5.10

Recall that 2

τ2

Z Ip [Ng ](τ1 , τ2 )

dτ kNg kL2 ( (trap) Σ(τ ))

= τ1

Z +

r



(trap)

M(τ1 ,τ2 )

Z +



(trap)

2+p

M(τ1 ,τ2 )

Z +



(trap)

M(τ1 ,τ2 )

r1+p |Ng |2

Z

|Ng ||e3 (Ng )| +

sup τ ∈[τ1 ,τ2 ]

2 rp+2 Ng

Σ(τ )

r3+δ |e3 (Ng )|2

and s Ip,d [Ng ]

sup (1 + τ )d Ips [Ng ](τ, τ∗ ).

=

0≤τ ≤τ∗

Since we have rδ (1 + τ )2+q0 −δ

.

r2+q0 + (1 + τ )2+q0 ,

and Z

2



(trap)

M(τ,τ∗ )

≤s

r |d

≤s

e3 (Ng )||d

Ng | .

Z 

(trap)

M(τ,τ∗ )

r|d≤s Ng |2

Z +



(trap)

M(τ,τ∗ )

r3 |d≤s e3 (Ng )|2 ,

we infer s Iqs0 +2,0 [Ng ] + Iδ,2+q [Ng ] 0 −δ "Z  . sup r4+q0 |d≤s+1 Ng |2 + (trap)

0≤τ ≤τ∗

+(1 + τ )2+q0

M(τ,τ∗ )

Z

Z



(trap)

Σ(τ 0 )

2+q0

M(τ,τ∗ )

r|d≤s Ng |2 +

∗]

Σ(τ 0 )



(trap)

M(τ,τ∗ )

2 r4+q0 d≤s Ng r3+δ |d≤s e3 (Ng )|2

!

+ sup ∗]

sup τ 0 ∈[τ,τ

Z τ 0 ∈[τ,τ

(5.2.11) Z

2 r2 d≤s Ng

Z

+(1 + τ )

τ

τ∗

0

≤s

dτ kd

2 # Ng kL2 ( (trap) Σ(τ 0 ))

.

In order to prove Proposition 5.10, it suffices to estimate the right-hand side of (5.2.11). To this end, we will estimate separately the terms with highest power of r, i.e., the first two terms, and the terms with highest power of τ , i.e., the four last terms.

227

DECAY ESTIMATES FOR q (THEOREM M1)

5.2.3.1

Terms with highest power of r in (5.2.11)

We estimate the first two terms of (5.2.11). Recall from Lemma 5.2 that Ng

=

r2 d≤2 (Γg · (α, β)).

Recall from Lemma 5.1 we have max

0≤k≤klarge −3

sup

7

r2+

δB 2



(ext) M(r≥4m ) 0

|dk α| + |dk β|



. .

We infer for s ≤ klarge − 6 7

|d≤s+1 Ng | . r− 2 −

δB 2

and hence, for s ≤ klarge − 6, we deduce Z  r4+q0 |d≤s+1 Ng |2 + (trap)

.

2

M(τ,τ∗ )

Z

Σ(τ 0 )

r−3−δB +q0 (r2 d≤s+2 Γg )2 .

sup τ 0 ∈[τ,τ∗ ]

Σ(τ 0 )

Since we also have for s ≤ klarge − 6 Z sup (r2 d≤s+3 Γg )2 . 2 , r0 ≥4m0

τ 0 ∈[τ,τ∗ ]

2 r4+q0 d≤s Ng

M(τ,τ∗ )

Z

+2

Z sup

r−3−δB +q0 (r2 d≤s+3 Γg )2



(trap)

|r2 d≤s+3 Γg |

{r=r0 }

Z

(d≤s+3 Γg )2 . 2 ,

Mr≤4m0

sup |r2 d≤s+2 Γg | . , M

we deduce Z 

(trap)



M(τ,τ∗ )

. 4 1 +

r4+q0 |d≤s+1 Ng |2 +

Z r≥4m0

dr r1+δB −q0

Z sup τ 0 ∈[τ,τ∗ ]

Σ(τ 0 )

2 r4+q0 d≤s Ng

 .

Since q0 < 4δdec and δB ≥ 4δdec , we have q0 < δB and hence, we obtain for s ≤ klarge − 6 Z Z 2 4+q0 ≤s+1 2  r |d Ng | + sup r4+q0 d≤s Ng . 4 . (trap)

M(τ,τ∗ )

τ 0 ∈[τ,τ∗ ]

Σ(τ 0 )

This is the desired control of the terms with highest power of r in (5.2.11).

228

CHAPTER 5

5.2.3.2

Terms with highest power of τ in (5.2.11)

We estimate the four last terms of (5.2.11). In view of Lemma 5.1 with kloss = 33, so that 33 33 δ0 = = , k klarge − ksmall − 2 klarge − b large c − 3 2 we have ≤ksmall +33 Γg d ≤ksmall +33 Γg d ≤S+32 e3 Γg d ≤ksmall +33 (α, β) d ≤ksmall +33 (α, β) d ≤S+32 e3 (α, β) d

.

r−2 τ −1/2−δdec +2δ0 ,

.

r−1 τ −1−δdec +2δ0 ,

.

r−2 [τ −1−δdec ]1−δ0 . r−2 τ −1−δdec +2δ0 ,

.

r−3 (τ + r)−1/2−δdec +2δ0 ,

. r−2 (τ + r)−1−δdec +2δ0 , 1

. r−3− 2 δ0 [τ −1−δdec ]1−δ0 . r−3 τ −1−δdec +2δ0 .

In particular, together with the bootstrap assumption for k ≤ ksmall , the pointwise bound 7

|d≤klarge −5 α| + |d≤klarge −5 β| . r− 2 −

δB 2

and since Ng = r2 d≤2 (Γg · (α, β)), we infer for s ≤ ksmall + 30 |ds Ng | . 2 r−3 τ −1−2δdec +2δ0

|ds Ng | . 2 r−1 τ −2−2δdec +2δ0 , 3

|ds e3 (Ng )| . 2 r−3 τ − 2 −2δdec +2δ0 , 7

|ds e3 (Ng )| . 2 r− 2 −

δB 2

τ −1−δdec +2δ0 .

(5.2.12)

229

DECAY ESTIMATES FOR q (THEOREM M1)

Using these 4 bounds and interpolation, we infer for δ > 0 Z

(1 + τ )2+q0



(trap)

2 ≤s

r d

+ sup Σ(τ 0 )

2+q0

τ∗

Z

. 4 (1 + τ )2+q0

7

×(r− 2 −

δB 2

Z

M(τ,τ∗ )

r3+δ |d≤s e3 (Ng )|2

2 Ng kL2 ( (trap) Σ(τ 0 ))

r(r−3 τ 0



(trap)

2+q0

−1−2δdec +2δ0 1+δ

)

(r−1 τ 0

−2−2δdec +2δ0 1−δ

)

M(τ,τ∗ )

Z

r3+δ (r−3 τ 0



(trap)

τ0

≤s

dτ kd

τ

+ (1 + τ )



(trap)

2 Ng

0

+(1 + τ )

4

Z

!

Z τ 0 ∈[τ,τ∗ ]

M(τ,τ∗ )

r|d≤s Ng |2 +

− 32 −2δdec +2δ0 2−2δ

)

M(τ,τ∗ )

−1−δdec +2δ0 2δ

) Z

+4 (1 + τ )2+q0 +4 (1 + τ )2+q0 Z . 4 (1 + τ )2+q0

Z

τ∗

−1−2δdec +2δ0 2

)

Σ(τ 0 )

τ0

−2−2δdec +2δ0

dτ 0

2

τ

r−3−δδB τ 0



(trap)

r−4 τ 0

sup τ 0 ∈[τ,τ∗ ]

Z

−3−4δdec +δ+4δ0 +2δδdec

M(τ,τ∗ )

Z

+4 (1 + τ )2+q0 +4 (1 + τ )2+q0

r2 (r−3 τ 0

sup τ 0 ∈[τ,τ∗ ]

τ∗

τ0

−2−4δdec +4δ0

Σ(τ 0 ) −2−2δdec +2δ0

dτ 0

2

τ

and since δ > 0, we obtain (1 + τ )

Z

2+q0

(trap)

Z + sup .  (1 + τ )

Σ(τ 0 )

≤s

M(τ,τ∗ )

2

Ng | +

Z



2 Ng



(trap)

! 2 ≤s

r d

τ 0 ∈[τ,τ∗ ] 4

r|d



+ (1 + τ )2+q0

M(τ,τ∗ )

Z τ

q0 −4δdec +δ+4δ0 +2δδdec

τ∗

r3+δ |d≤s e3 (Ng )|2

dτ 0 kd≤s Ng kL2 ( (trap) Σ(τ 0 ))

.

As we have q0 < 4δdec − 4δ0 , there exists δ > 0 small enough such that q0 − 4δdec + δ + 4δ0 + 2δδdec ≤ 0,

2

230

CHAPTER 5

and hence (1 + τ )

2+q0

Z

≤s

r|d



(trap)

M(τ,τ∗ )

Z

2

Ng | +



(trap)

M(τ,τ∗ )

r3+δ |d≤s e3 (Ng )|2 !

Z

2 ≤s



+ sup

r d

τ 0 ∈[τ,τ∗ ]

+(1 + τ )2+q0

Z

τ∗

τ

Σ(τ 0 )

2 Ng

dτ 0 kd≤s Ng kL2 ( (trap) Σ(τ 0 ))

2

4 .

.

This is the desired control of the terms with highest power of τ in (5.2.11). Together with (5.2.11) and the above control of the terms with highest power of r, we infer s Iqs0 +2,0 [Ng ] + Iδ,2+q [Ng ] 0 −δ "Z  . sup r4+q0 |d≤s+1 Ng |2 + (trap)

0≤τ ≤τ∗

+(1 + τ )2+q0

M(τ,τ∗ )

τ 0 ∈[τ,τ∗ ]

Z

Z



(trap)

+ sup Σ(τ 0 )

2+q0

M(τ,τ∗ )

r|d≤s Ng |2 +

Σ(τ 0 )



(trap)

M(τ,τ∗ )

2 r4+q0 d≤s Ng r3+δ |d≤s e3 (Ng )|2

!

Z τ 0 ∈[τ,τ∗ ]

Z sup

2 r2 d≤s Ng τ∗

Z

+(1 + τ )

τ

0

≤s

dτ kd

2 # Ng kL2 ( (trap) Σ(τ 0 ))

. 4 which is the desired estimate. This concludes the proof of Proposition 5.10.

5.3

IMPROVED WEIGHTED ESTIMATES

The goal of this section is to prove the two following theorems on improved weighted estimates. Theorem 5.14. Assume q verifies the following wave equation, see (5.0.1), 2 q + κκ q

= N

with N given, in view of Lemma 5.2, by N = Ng + e3 (rNg ) + Nm [q]. Then, for any δ ≤ p ≤ 2 − δ, 0 ≤ s ≤ ksmall + 30, sup τ ∈[τ1 ,τ2 ]

E sp [q](τ ) + Bps [q](τ1 , τ2 ) + Fps [q](τ1 , τ2 )

. E sp [q](τ1 ) + Ips+1 [Ng ](τ1 , τ2 ).

(5.3.1)

231

DECAY ESTIMATES FOR q (THEOREM M1)

The next result deals with weighted estimates for the quantity ˇq =

f2 eˇ4 q,

(5.3.2)

where f2 is a fixed smooth function of r defined as follows, ( r2 for r ≥ 6m0 , f2 (r) = 0 for r ≤ 4m0 .

(5.3.3)

Theorem 5.15. Assume q verifies equation, see (5.0.1), 2 q + κκ q =

N

with N = Ng + e3 (rNg ) + Nm [q] as in Lemma 5.2. Then, for any −1 + δ < q ≤ 1 − δ, 0 ≤ s ≤ ksmall + 29, sup τ ∈[τ1 ,τ2 ]

.

E sq [ˇq](τ ) + Bqs [ˇq](τ1 , τ2 )

s+2 E sq [ˇq](τ1 ) + E s+1 q+1 [q](τ1 ) + Iq+2 [Ng ](τ1 , τ2 ).

(5.3.4)

Remark 5.16. Note that in (5.3.1) and (5.3.4), the term Nm [q] does not appear in the right-hand side since it turns out that it can be absorbed by the left-hand side. The proof of Theorem 5.14 is postponed to section 5.3.2, and the proof of Theorem 5.15 is postponed to section 5.3.3. These proofs will rely on weighted energy flux estimates introduced in the next section. 5.3.1

Basic and higher weighted estimates for wave equations

Assume given a spacetime M verifying the bootstrap assumptions with small constant  > 0. The proof of Theorem 5.14 and Theorem 5.15 will rely on estimates stated below for solutions ψ ∈ s2 (M) of the equation 2 ψ = V ψ + N, 5.3.1.1

V = −κκ.

(5.3.5)

Basic weighted estimates

Theorem 5.17. Recall the definitions in (5.1.21), (5.1.15). The following holds for any 0 ≤ s ≤ ksmall + 30. For all δ ≤ p ≤ 2 − δ, we have sup τ ∈[τ1 ,τ2 ]

Eps [ψ](τ ) + Bps [ψ](τ1 , τ2 ) + Fps [ψ](τ1 , τ2 )

. Eps [ψ](τ1 ) + Jps [ψ, N ](τ1 , τ2 ),

(5.3.6)

232

CHAPTER 5

where, for p ≥ δ, we have introduced the notation Z Jp,R [ψ, N ](τ1 , τ2 ) : = rp eˇ4 ψN , M≥R (τ1 ,τ2 )  Z τ2 2 Jp [ψ, N ](τ1 , τ2 ) : = dτ kN kL2 ( (trap) Σ(τ )) Z τ1  + r1+δ |N |2 + Jp,4m0 [ψ, N ](τ1 , τ2 ), (trap)

(5.3.7)

M(τ1 ,τ2 )

and Jps [ψ, N ](τ1 , τ2 )

X

:=

Jp [dk ψ, dk N ](τ1 , τ2 ).

k≤s

The proof of Theorem 5.17 is postponed to section 10.4.5. 5.3.1.2

Higher weighted estimates

The next result deals with weighted estimates for the quantity ψˇ =

f2 eˇ4 ψ,

where f2 is a fixed smooth function of r defined as follows, ( r2 for r ≥ 6m0 , f2 (r) = 0 for r ≤ 4m0 .

(5.3.8)

(5.3.9)

Theorem 5.18. The following holds for any −1+δ < q ≤ 1−δ, 0 ≤ s ≤ ksmall +29, sup τ ∈[τ1 ,τ2 ]

ˇ ) + B s [ ψ](τ ˇ 1 , τ2 ) . E s [ ψ](τ ˇ 1 ) + Jˇs [ ψ, ˇ N ](τ1 , τ2 ) E sq [ ψ](τ q q q (5.3.10) s+1 + E s+1 max(q,δ) [ψ](τ1 ) + Jmax(q,δ) [ψ, N ],

where we have introduced the notation ˇ N ](τ1 , τ2 ) Jˇq [ ψ,



  3 2 ˇ := Jq,4m0 ψ, r e4 N + N (τ1 , τ2 ) r   Z
3 q+2 ˇ = r eˇ4 ψ · e4 N + N , r M≥4m0 (τ1 ,τ2 )

and ˇ N ](τ1 , τ2 ) := Jˇqs [ ψ,

X

ˇ dk N ](τ1 , τ2 ). Jˇq [dk ψ,

k≤s

The proof of Theorem 5.18 is postponed to section 10.4.6. We now proceed to the proof of Theorem 5.14 and Theorem 5.15 in the next two sections. The proofs will follow from the structure of the nonlinear term N of q provided by Lemma 5.2 and the use of Theorem 5.17 and Theorem 5.18.

233

DECAY ESTIMATES FOR q (THEOREM M1)

5.3.2

Proof of Theorem 5.14

Applying Theorem 5.17 to the equation for q, with N given by Lemma 5.2, we derive corresponding estimates with the norm Jps [q, N ](τ1 , τ2 ) on the right-hand side, i.e., for 0 ≤ s ≤ ksmall + 30, and for δ ≤ p ≤ 2 − δ, sup τ ∈[τ1 ,τ2 ]

E sp [q](τ ) + Bps [q](τ1 , τ2 ) + Fps [q](τ1 , τ2 )

. E sp [q](τ1 ) + Jps [q, N ](τ1 , τ2 ).

(5.3.11)

To prove Theorem 5.14, it suffices, in view of (5.3.11), to estimate Jps [q, N ](τ1 , τ2 ). Recall that, see (5.3.7) and (5.1.25), Z Ip [N ](τ1 , τ2 )

2

τ2

dτ kN kL2 ( (trap) Σ(τ ))

= τ1

Z +

M(τ1 ,τ2 )

M(τ1 ,τ2 )

r3+δ |e3 (Ng )|2

Z +



(trap)



(trap)

r2+p |Ng ||e3 (Ng )| +



(trap)

Z +

M(τ1 ,τ2 )

r1+p |N |2

Z sup τ ∈[τ1 ,τ2 ]

2 rp+2 N

Σ(τ )

and Z p Jp,R [q, N ] = r eˇ4 (q)N , M≥R (τ1 ,τ2 )  Z τ2 2 Z Jp [q, N ](τ1 , τ2 ) = dτ kN kL2 ( (trap) Σ(τ )) + τ1 s +Jp,4m0 [q, N ](τ1 , τ2 ),

Jps [q, N ](τ1 , τ2 )

=

X



(trap)

M(τ1 ,τ2 )

r1+δ |N |2

Jp [dk q, dk N ].

k≤s

Recall also from (5.1.9) dk N

= d≤k Ng + e3 (dk (rNg )) + dk Nm [q]

(5.3.12)

and consider separately the three terms. Case of Nm [q]. Recall that Nm [q] = d≤1 (Γg · q). We have, schematically, X dk Nm [q] = d1+k (Γg · q) = d≤i Γg d≤j q. i+j=k+1

We make use of the following consequence of the bootstrap assumptions, valid for k ≤ klarge − 5, ≤k d Γg ≤ r−2 to deduce k d Nm [q] .

r−2 d≤k+1 q .

(5.3.13)

234

CHAPTER 5

We deduce s Jp,4m [q, Nm [q]](τ1 , τ2 ) 0

XZ

.

M≥4m0 (τ1 ,τ2 )

k≤s



.

rp eˇ4 q(k) dk Nm [q]

XZ k≤s

2 rp−3 d1+k q .

M≥4m0 (τ1 ,τ2 )

Thus, recalling Remark 5.8, we infer s Jp,4m [q, Nm [q]](τ1 , τ2 ) . Bps [q](τ1 , τ2 ). 0

Next, we estimate in view of (5.3.13) Z Z  r1+δ |dk Nm [q]|2 .  (trap)

M(τ1 ,τ2 )

which yields, using again Remark 5.8, Z  r1+δ |dk Nm [q]|2 (trap)



(trap)

.

M(τ1 ,τ2 )

(5.3.14)

rδ−3 |d≤k+1 q|2

Bδs [q](τ1 , τ2 ).

(5.3.15)

M(τ1 ,τ2 )

We next estimate the integral Z τ2 dτ kdk Nm [q]kL2 ( (trap) Σ(τ )) . τ1

In view of the definition of Nm [q] = d≤1 (Γg · q), X dk Nm [q] = d≤k+1 (Γg · q) = d≤i Γg d≤j q i+j=k+1

= Now, since

k+1 2

j≤(k+1)/2

d

≤k+1

Γg d

≤ ksmall we have j≤(k+1)/2 Γg d

q + dj≤(k+1)/2 q d≤k+1 Γg = J1 + J2 .

. (1 + τ )−1−δdec .

Hence, kJ1 k2L2 ( (trap) Σ(τ ))

Z = (trap) Σ(τ )

.

j≤(k+1)/2 2 ≤k+1 2 Γg d q d

 (1 + τ )−2−2δdec E s [q](τ ), 2

i.e., kJ1 kL2 ( (trap) Σ(τ ))

. (1 + τ )−1−δdec (E s [q](τ ))

1/2

.

235

DECAY ESTIMATES FOR q (THEOREM M1)

For J2 we write Z

kJ2 k2L2 ( (trap) Σ(τ ))

j≤(k+1)/2 2 ≤k+1 2 q d Γg d

= (trap) Σ(τ )

sup

.

(trap) Σ(τ )

Z

j≤(k+1)/2 q d

!2 Z (trap) Σ(τ )

Z ≤(k+1)/2+2 2 q d

. (trap) Σ(τ )

(trap) Σ(τ )

≤k+1 2 Γg d

≤k+1 2 Γg d

or, since (k + 1)/2 + 2 ≤ s, kJ2 kL2 ( (trap) Σ(τ )) .

"Z

≤s 2 d q

(trap) Σ(τ )

#1/2 "Z (trap) Σ(τ )

≤k+1 2 Γg d

#1/2 .

In view of the above estimates for J1 and J2 , we deduce, for all k ≤ s ≤ klarge − 5, Z τ2 dτ kdk Nm [q]kL2 ( (trap) Σ(τ )) τ1

. 

sup Z

1/2

"Z

τ2

+

dτ τ1

. 

(E s [q](τ ))

τ1 ≤τ ≤τ2

sup

(trap) Σ(τ )

s

(E [q](τ ))

≤s 2 d q

#1/2 "Z (trap) Σ(τ )

Z

1/2

+



(trap)

τ1 ≤τ ≤τ2

M(τ1 ,τ2 )

≤s 2 d Γg

≤s 2 d q

! 12

#1/2

Z Mr≤4m0

≤s 2 d Γg

!1/2

Making use of the following consequence of the bootstrap assumptions Z Mr≤4m0

as well as the fact that Z 

(trap)

M(τ1 ,τ2 )

≤s 2 d Γg

≤s 2 d q

!1/2 . ,

. Morrs [q](τ1 , τ2 ),

we deduce Z

τ2

τ1

.



2

dτ kdk Nm [q]kL2 ( (trap) Σ(τ ))

sup

2

E s [q](τ ) + 2 Morrs [q](τ1 , τ2 )

(5.3.16)

τ1 ≤τ ≤τ2

which together with (5.3.15) and (5.3.14) yields for any p ≥ δ Jps [q, Nm [q]](τ1 , τ2 ) .

2

sup τ1 ≤τ ≤τ2

E s [q](τ ) + Bps [q](τ1 , τ2 ).

(5.3.17)

.

236

CHAPTER 5

Case of Ng . We write, as before, XZ

s Jp,4m [q, Ng ](τ1 , τ2 ) . 0

rp eˇ4 q(k) dk Ng

M≥4m0 (τ1 ,τ2 )

k≤s

!1/2

X Z

.



eˇ4 q

(k) 2



M≥4m0 (τ1 ,τ2 )

k≤s

×

r

p−1

!1/2

Z r

2 d Ng

p+1 k



.

M≥4m0 (τ1 ,τ2 )

Therefore, s Jp,4m [q, Ng ](τ1 , τ2 ) . 0

Bps [q](τ1 , τ2 )


1/2


1/2 Ips [Ng ](τ1 , τ2 )

. δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips [Ng ](τ1 , τ2 ) where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate. Also, we have in view of the definition of Ips [N ](τ1 , τ2 ) and the fact that p ≥ δ τ2

Z

τ1

≤s

dτ kd

2 Ng kL2 ( (trap) Σ(τ ))

Z +



(trap)

M(τ1 ,τ2 )

r1+δ |d≤s Ng |2

.

Ips [Ng ](τ1 , τ2 ).

Therefore, Jps [q, Ng ](τ1 , τ2 )

Z

τ2

= Zτ1 + .

dτ kd≤s Ng kL2 ( (trap) Σ(τ ))



(trap)

M(τ1 ,τ2 )

2

s r1+δ |d≤s Ng |2 + Jp,4m [q, Ng ](τ1 , τ2 ) 0

Iδs [Ng ](τ1 , τ2 ) + δ1−1 Ips [Ng ](τ1 , τ2 ) + δ1 Bps [q](τ1 , τ2 ),

i.e., Jps [q, Ng ](τ1 , τ2 ) . δ1−1 Ips [Ng ](τ1 , τ2 ) + δ1 Bps [q](τ1 , τ2 ).

(5.3.18)

Case of e3 (rNg ). First, note that we have Z

τ2

dτ kd

τ1

Z

τ2

. τ1

≤s

2 e3 (rNg )kL2 ( (trap) Σ(τ ))

≤s+1

dτ kd

Z +



(trap)

M(τ1 ,τ2 )

2 Ng kL2 ( (trap) Σ(τ ))

Z +



(trap)

M(τ1 ,τ2 )

Z +



(trap)

M(τ1 ,τ2 )

r1+δ |d≤s e3 (rNg )|2

r1+δ |d≤s Ng |2

r3+δ |d≤s e3 (Ng )|2

where we used the fact that |d≤s e3 (r)| . 1 and |d≤s r| . r. Hence, we infer in view

237

DECAY ESTIMATES FOR q (THEOREM M1)

of the definition of Ips [N ](τ1 , τ2 ) and the fact that p ≥ δ Z

τ2

τ1

.

dτ kd≤s e3 (rNg )kL2 ( (trap) Σ(τ ))

2

Z +



(trap)

M(τ1 ,τ2 )

r1+δ |d≤s e3 (rNg )|2

Ips+1 [Ng ](τ1 , τ2 ).

(5.3.19)

We then estimate Jp,4m0 [q(k) , e3 (dk (rNg ))](τ1 , τ2 ),

k ≤ s.

To this end, we introduce a smooth cut-off function φ0 vanishing for r ≤ 4m0 and equal to 1 for r ≥ 8m0 . Then, we have Z (k) k p (k) k Jp,4m0 [q , d (rNg )](τ1 , τ2 ) = r eˇ4 q e3 d (rNg ) M(τ1 ,τ2 ) .

Jp,4m0 [q(k) , φ0 dk (rNg )](τ1 , τ2 ) +Jp,4m0 [q(k) , (1 − φ0 )rNg ](τ1 , τ2 ).(5.3.20)

In view of the fact that 1 − φ0 is supported in r ≤ 8m0 , we easily obtain

.

Jp,4m0 [q(k) , (1 − φ0 )rNg ](τ1 , τ2 )  1/2
1 s s sup E [q](τ ) + Bp [q](τ1 , τ2 ) Ips+1 [Ng ](τ1 , τ2 ) 2 τ1 ≤τ ≤τ2

and hence Jp,4m0 [q(k) , (1 − φ0 )rNg ](τ1 , τ2 )   . δ1 sup E s [q](τ ) + Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 ) (5.3.21) τ1 ≤τ ≤τ2

where δ1 > 0 is chosen sufficiently small so that we can later absorb the terms δ1 supτ1 ≤τ ≤τ2 E s [q](τ ) and δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate. It remains to estimate the terms Jp,4m0 [q(k) , φ0 e3 (dk (rNg ))](τ1 , τ2 ),

k≤s

which is supported for r ≥ 4m0 . Note that e3 (rNg ) behaves like rNg and therefore the same sequence of estimates as for Ng would lead to a loss of r−1 . For this reason we need to integrate by parts in e3 . Proposition 5.19. The following estimate holds true, for all k ≤ s ≤ klarge − 5, X Jp,4m0 [q(k) , φ0 e3 (dk (rNg ))](τ1 , τ2 ) k≤s

. δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 )

(5.3.22)

for a sufficiently small δ1 > 0. We postponed the proof of Proposition 5.19 to the end of the section. We are

238

CHAPTER 5

now in position to conclude the proof of Theorem 5.14. Proof of Theorem 5.14. (5.3.21) and (5.3.22) yield X Jp,4m0 [q(k) , e3 (dk (rNg ))](τ1 , τ2 ) . δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 ). k≤s

Together with (5.3.17), (5.3.18) and (5.3.19), we infer Jps [q, N ](τ1 , τ2 ) . (δ1 + )Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 ) + 2

sup

E s [q](τ ).

τ1 ≤τ ≤τ2

In view of (5.3.11), this concludes the proof of Theorem 5.14. The proof of Proposition 5.19 will rely in particular on the following identity. Lemma 5.20. The following hold true for any ψ ∈ s2 : • We have, schematically, e3 e4 (rψ)

=

−r2 ψ + r4 / 2 ψ + r−1 dψ.

(5.3.23)

• The following identity holds true, schematically, e3 e4 (rdk ψ) = −d≤k (r2 ψ) + r4 / 2 (d≤k ψ) + r−1 d≤k+1 ψ.

(5.3.24)

Proof. We start with the following identity for ψ ∈ s2 , see Definition 2.103,   1 1 2 ψ = −e3 e4 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ 2 2 from which we deduce r2 ψ

    1 1 = −re3 e4 ψ + r 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ . 2 2

On the other hand, re3 e4 ψ

= e3 (re4 ψ) − (e3 r)e4 ψ = e3 (e4 (rψ) − e4 (r)ψ) − (e3 r)e4 ψ = e3 e4 (rψ) − e4 (r)e3 ψ − (e3 r)e4 ψ − (e3 e4 r)ψ.

Hence, r2 ψ

= + =

=

−e3 e4 (rψ) + e4 (r)e3 ψ + (e3 r)e4 ψ + (e3 e4 r)ψ + r4 / 2ψ   1 1 r 2ω − κ e4 ψ − rκe3 ψ + 2rηeθ ψ 2 2     1 1 −e3 e4 (rψ) + r4 / ψ + e4 r − rκ e3 ψ + e3 r − rκ + 2rω e4 ψ 2 2 +2rηeθ ψ r r −e3 e4 (rψ) + r4 / ψ + Ae3 ψ + (A + 4ω) e4 ψ + 2rηeθ ψ, 2 2

239

DECAY ESTIMATES FOR q (THEOREM M1)

i.e., e3 e4 (rψ)

r r = −r2 ψ + r4 / 2 ψ + Ae3 ψ + (A + 4ω) e4 ψ + 2rηeθ ψ 2 2

or, schematically, in view of the definition of dψ and of |ω| + r|Γg | + |Γb | . r−1 ,   e3 e4 (rψ) = −r2 ψ + r4 / 2 ψ + rΓg + Γb + r−1 e3 ψ = −r2 ψ + r4 / 2 ψ + r−1 dψ which is (5.3.23). To derive the identity for higher derivatives we write, schematically, dk e3 e4 (rψ)

= −dk (r2 ψ) + dk (r4 / 2 ψ) + dk (rΓg dψ).

We write dk e3 e4 (rψ) dk (r4 / 2 ψ)

= e3 e4 (rdk ψ) + [dk , e3 e4 r]ψ = e3 e4 (rdk ψ) + [dk , e3 ]dψ + e3 [dk , e4 r]ψ, =

r4 / 2 dk ψ + [dk , r4 / ]ψ = r4 / 2 dk ψ + [dk , r−1 ]d2 ψ + r−1 [dk , r2 4 / ]ψ.

In view of the identities for [e3 , d/] and [e4 , d/] of Lemma 2.68, the identities of Proposition 2.28 for commutation formulas involving /dk and /d?k derivatives, and the commutator formula for [e3 , e4 ], we have schematically [ d/, r2 4 / ] = d/ + 1,

[e3 , e3 ] = 0,

[e3 , d/] = Γb d + Γb ,

[e3 , e4 r] = (r−1 + Γg )d, 2

[e4 r, d/] = (r ξ + rΓg )d + rΓg .

In view of the estimates for Γg , Γb , and the fact that ξ = 0 for r ≥ 4m0 , we infer [dk , e3 ] = r−1 d≤k ,

[dk , r2 4 / ] = d≤k+1 ,

[dk , r−1 ] = r−1 d≤k−1

and hence dk e3 e4 (rψ)

=

e3 e4 (rdk ψ) + e3 [dk , e4 r]ψ + r−1 d≤k+1 ψ,

dk (r4 / 2 ψ)

=

r4 / 2 dk ψ + r−1 d≤k+1 ψ.

Also, we have [re4 , e4 r] = [re4 , e4 ]r + e4 [re4 , r] = −e4 (r)e4 r − e4 re4 (r) = −2e4 r + r−1 d and we infer by induction, schematically, [(re4 )j , e4 r] = e4 r(re4 )≤j−1 + r−1 d≤j so that, together with −1 ≤k−j [dk−j d , & , e4 r] = r

we infer [dk , e4 r]

=

≤j−1 k−j [(re4 )j dk−j d& + r−1 d≤k . & , e4 r] = e4 r(re4 )

240

CHAPTER 5

We deduce e3 e4 (r(re4 )j dk−j & ψ)

=

−(re4 )j dk−j / 2 (dk ψ) + r−1 d≤k+1 ψ & (r2 ψ) + r4 +e4 r(re4 )≤j−1 dk−j & ψ.

We infer by induction on j k−j e3 e4 (r(re4 )j d& ψ)

= −(re4 )≤j dk−j / 2 (d≤k ψ) + r−1 d≤k+1 ψ & (r2 ψ) + r4

and hence e3 e4 (rdk ψ) = −d≤k (r2 ψ) + r4 / 2 (d≤k ψ) + r−1 d≤k+1 ψ which is (5.3.24). This concludes the proof of Lemma 5.20. We now are in position to prove Proposition 5.19. Proof of Proposition 5.19. We integrate by parts Z   (k) k p (k) k Jp,4m0 [q , φ0 d (rNg )](τ1 , τ2 ) . e3 φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) (5.3.25) + |Bpk (τ1 )| + |Bpk (τ2 )| Z p (k) k + Div(e3 )φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) where Div(e3 ) denotes the spacetime divergence of e3 , and where the boundary terms are bounded by Z |Bpk (τ1 )| . rp |ˇ e4 q(k) | |dk (rNg )|, Σ(τ ) Z 1 k |Bp (τ2 )| . rp |ˇ e4 q(k) | |dk (rNg )|. Σ(τ2 )

We estimate |Bpk (τ )|

Z . Σ(τ )

.

rp |ˇ e4 q(k) | |dk (rNg )|

Z Σ(τ )

.

rp |ˇ e4 q(k) |2

1/2  Z Σ(τ )

 1/2  Z Epk [q](τ ) Σ(τ )

rp |dk (rNg )|2

rp+2 |dk Ng |2

1/2

1/2

.

We deduce, with δ1 > 0 a sufficiently small constant, for any τ ∈ [τ1 , τ2 ], Z k Bp (τ1 ) . δ1 sup Epk [q](τ ) + δ −1 sup rp+2 |Ng≤k |2 , 1 τ1 ≤τ ≤τ2 τ1 ≤τ ≤τ2 Σ(τ ) Z (5.3.26) k p+2 ≤k 2 Bp (τ2 ) . δ1 sup Epk [q](τ ) + δ −1 sup r |N | . g 1 τ1 ≤τ ≤τ2

τ1 ≤τ ≤τ2

Σ(τ )

241

DECAY ESTIMATES FOR q (THEOREM M1)

Next, notice that Div(e3 ) = κ − 2ω so that |Div(e3 )| . r−1 . Together with the fact that e3 (Φ0 (r)) is supported in 4m0 ≤ r ≤ 8m0 , the fact that |e3 (r)| . 1 and rˇ e4 q(k) = e4 (rq(k) ) + O(r−1 )e4 (q(k) ), we infer

.

Z   p (k) k e3 φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) Z p (k) k + Div(e3 )φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) Z φ0 (r)rp−1 e3 e4 (rq(k) )dk (rNg ) M(τ1 ,τ2 ) Z + rp−1 |ˇ e4 (q(k) )||dk (rNg )| M≥4m0 (τ1 ,τ2 )

Z + M4m0 ≤r≤8m0 (τ1 ,τ2 )

.

|ˇ e4 (q(k) )||dk (rNg )|

Z p−1 (k) k φ0 (r)r e3 e4 (rq )d (rNg ) M(τ1 ,τ2 ) 1 !2 Z Z + M≥4m0 (τ1 ,τ2 )

rp−1 |ˇ e4 (q(k) )|2

M≥4m0 (τ1 ,τ2 )

! 12 rp+1 |d≤k Ng |2

and hence Z   p (k) k e3 φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) Z p (k) k + Div(e3 )φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) Z
1/2 s
1 . φ0 (r)rp−1 e3 e4 (rq(k) )dk (rNg ) + Bps [q](τ1 , τ2 ) Ip [Ng ](τ1 , τ2 ) 2 M(τ1 ,τ2 ) which yields Z   e3 φ0 (r)rp eˇ4 q(k) dk (rNg ) M(τ1 ,τ2 ) Z p (k) k + Div(e3 )φ0 (r)r eˇ4 q d (rNg ) M(τ1 ,τ2 ) k −1 s s . L + δ1 Bp [q](τ1 , τ2 ) + δ1 Ip [Ng ](τ1 , τ2 )

(5.3.27)

242

CHAPTER 5

where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate, and where we have introduced the notation Z
k L : = φ0 (r)rp−1 e3 e4 rq(k) dk (rNg ). (5.3.28) M(τ1 ,τ2 )

It remains to estimate the term Lk . Making use of Lemma 5.20, we deduce Z k L = φ0 (r)rp−1 e3 e4 (rq(k) )dk (rNg ) M(τ1 ,τ2 ) Z = − φ0 (r)rp−1 d≤k (r2 q)dk (rNg ) M(τ1 ,τ2 ) Z + φ0 (r)rp 4 / 2 (d≤k q)dk (rNg ) M(τ1 ,τ2 ) Z + φ0 (r)rp−2 d≤k+1 q dk (rNg ) M(τ1 ,τ2 )

= Lk1 + Lk2 + Lk3 . We first estimate Lk3 as follows: Z k L3 . rp−2 |d≤k+1 q| |dk (rNg )| M≥4m0 (τ1 ,τ2 )

.

Z

r

p−3 ≤k+1



d

2 1/2  q

M≥4m0 (τ1 ,τ2 )

Z M≥4m0 (τ1 ,τ2 )

In view of Remark 5.8 we thus deduce Z k
 L3 . Bpk [q] 1/2 M≥4m0 (τ1 ,τ2 )

.


1/2 Bpk [q](τ1 , τ2 )

rp+1 |d≤k Ng |2

rp+1 |d≤k Ng |2

1/2

.

1/2


1/2 Ipk [Ng ](τ1 , τ2 )

and hence k L3

. δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips [Ng ](τ1 , τ2 )

(5.3.29)

where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate. We now estimate the term Z k L2 = φ0 (r)rp 4 / 2 (dk q)dk (rNg ) M(τ1 ,τ2 )

243

DECAY ESTIMATES FOR q (THEOREM M1)

by first performing another integration by parts in the angular directions Z k L2 . rp−2 dk+1 q dk+1 (rNg ) M≥4m0 (τ1 ,τ2 )

!1/2

Z r

.

p−3 k+1



d

2 q

r

M≥4m0 (τ1 ,τ2 )

Bpk [q](τ1 , τ2 )

.

!1/2

Z

p+1 ≤k+1



d

2 Ng

M≥4m0 (τ1 ,τ2 )


1/2

Ipk+1 [Ng ](τ1 , τ2 )


1/2

.

Hence, δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 )

k L2 .

(5.3.30)

where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate. It remains to estimate the term Z k L1 = − φ0 (r)rp−1 d≤k (r2 q)dk (rNg ). M(τ1 ,τ2 )

Making use of the equation verified by q, i.e., 2 q = −κκq + N , we deduce dk (r2 q)

=

−dk (rκκq) + dk (rN ).

Recall (5.1.9) dk N

=

d≤k Ng + e3 (dk (rNg )) + dk Nm [q].

d≤k (rN )

=

rd≤k N + d≤k−1 N

=

rd≤k Ng + re3 (d≤k (rNg )) + rd≤k Nm [q]

We infer

and hence |dk (r2 q)| . r−1 d≤k q + r d≤k Ng | + r2 d≤k e3 (Ng )| + r dk Nm [q]| . r−1 d≤k+1 q + r d≤k Ng | + r2 d≤k e3 (Ng )|. (5.3.31) Note that we have used in the last inequality the form of Nm [q] = d≤1 (Γg q) and the fact that |Γg | ≤ r−2 . We deduce, using (5.3.31), Z Z k L1 . rp−1 |d≤k+1 q |d≤k Ng | + rp+1 |d≤k Ng |2 M≥4m0 (τ1 ,τ2 )

Z + M≥4m0 (τ1 ,τ2 )

.

Bpk [q](τ1 , τ2 )

M≥4m0 (τ1 ,τ2 )

rp+2 |d≤k e3 (Ng )||d≤k Ng |


1/2


1/2 Ipk [Ng ](τ1 , τ2 ) + Ipk [Ng ](τ1 , τ2 ).

We deduce k L1

. δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips [Ng ](τ1 , τ2 )

(5.3.32)

244

CHAPTER 5

where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [q](τ1 , τ2 ) by the left-hand side of our main estimate. Together with (5.3.29) and (5.3.30) we deduce k L . δ1 Bpk [q](τ1 , τ2 ) + δ −1 Ipk [Ng ](τ1 , τ2 ). (5.3.33) 1 Together with (5.3.25), (5.3.26) and (5.3.28), we infer X Jp,4m0 [q(k) , φ0 dk (rNg )](τ1 , τ2 ) . δ1 Bps [q](τ1 , τ2 ) + δ1−1 Ips+1 [Ng ](τ1 , τ2 ) k≤s

which concludes the proof of Proposition 5.19. 5.3.3

Proof of Theorem 5.15

We apply Theorem 5.18 to the case when ψ = q. Hence, E sq [ˇq](τ2 ) + Bqs [ˇq](τ1 , τ2 ) . E sq [ˇq](τ1 ) + Jˇqs [ˇq, N ](τ1 , τ2 ) s+1 + E s+1 max(q,δ) [q](τ1 ) + Jmax(q,δ) [q, N ](τ1 , τ2 ).

(5.3.34)

Also, recall that ˇq = f2 eˇ4 q, where f2 is a fixed smooth function of r defined as follows, ( r2 for r ≥ 6m0 , f2 (r) = 0 for r ≤ 4m0 .

(5.3.35)

In particular, ˇq is supported in r ≥ 4m0 , and hence, in view of Remark 5.5, Z Bq [ˇq](τ1 , τ2 ) ' rq−3 |dˇq|2 , (5.3.36) M≥4m0 (τ1 ,τ2 )

where we have used the fact that −1 + δ ≤ q ≤ 1 − δ. First, notice that the proof of Theorem 5.14 yields s+1 Jmax(q,δ) [q, N ](τ1 , τ2 ) .

sup τ1 ≤τ ≤τ2

s+1 E s+1 [q](τ ) + Bmax(q,δ) [q](τ1 , τ2 )

s+2 +Imax(q,δ) [Ng ](τ1 , τ2 ).

Hence, using Theorem 5.14, together with the fact that max(q, δ) ≤ 1 − δ, we infer s+1 s+1 s+2 Jmax(q,δ) [q, N ](τ1 , τ2 ) . Emax(q,δ) [q](τ1 ) + Imax(q,δ) [Ng ](τ1 , τ2 ).

Since q ≥ −1 + δ, we have max(q, δ) ≤ δ ≤ q + 1 and thus s+1 s+1 s+2 Jmax(q,δ) [q, N ](τ1 , τ2 ) . Eq+1 [q](τ1 ) + Iq+1 [Ng ](τ1 , τ2 ).

(5.3.37)

245

DECAY ESTIMATES FOR q (THEOREM M1)

It only remains to estimate the term X Jˇqs [ˇq, N ](τ1 , τ2 ) = Jˇq [dk ˇq, dk N ](τ1 , τ2 ) k≤s

with Jˇq [ˇq, N ](τ1 , τ2 )

= =

   3 2 Jq,4m0 ˇq, r e4 N + N (τ1 , τ2 ) r   Z
3 q+2 r eˇ4 ˇq · e4 N + N . r M≥4m0 (τ1 ,τ2 )

We rewrite in the equivalent form Jˇq [dk ˇq, dk N ](τ1 , τ2 )

Z = M≥4m

0



rq rˇ e4 dk ˇq dk+1 N . (τ1 ,τ2 )

(5.3.38)

Using the identity (5.1.9), we have dk+1 N = d≤k+1 Ng + e3 (d≤k+1 rNg ) + dk+1 Nm [q]. The integral due to d≤k+1 Ng is treated as follows: Z Jˇq [dk ˇq, dk Ng ](τ1 , τ2 ) . rq rˇ e4 dk ˇq d≤k+1 Ng M≥4m0 (τ1 ,τ2 )

.

Z

2 1/2 rq−3 rˇ e4 dk ˇq

M≥4m0 (τ1 ,τ2 )

×

Z

2 1/2 rq+3 d≤k+1 Ng .

M≥4m0 (τ1 ,τ2 )

Therefore, Jˇqs [ˇq, Ng ](τ1 , τ2 ) . .


1/2 s+1
1/2 Bqs [ˇq](τ1 , τ2 ) Iq+2 [Ng ](τ1 , τ2 ) s+1 δ1 Bqs [ˇq](τ1 , τ2 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 )

(5.3.39)

where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bqs [q](τ1 , τ2 ) by the left-hand side of our main estimate.

246

CHAPTER 5

The integral due to dk+1 Nm [q] is treated as follows: Jˇq [dk ˇq, dk Nm [q]](τ1 , τ2 ) Z . rq rˇ e4 dk ˇq dk+1 Nm [q] M≥4m0 (τ1 ,τ2 )

Z

rq+1 eˇ4 dk ˇq d≤k+2 q d≤k+2 Γg

. M≥4m0 (τ1 ,τ2 )

Z

1 rq−1 τ − 2 −δdec +2δ0 eˇ4 dk ˇq d≤k+2 q

.  M≥4m0 (τ1 ,τ2 )

! 12

Z

2 rq τ −1−2δdec +4δ0 eˇ4 dk ˇq

.  M≥4m0 (τ1 ,τ2 )

 . 

sup τ1 ≤τ ≤τ2

Z

! 12 2 rq−2 d≤k+2 q

M≥4m0 τ1 ,τ2 )

 12   12 s+1 Eqs [ˇq](τ ) Bq+1 [q](τ1 , τ2 )

where we have used |Γg | . r−2 τ −1/2−δdec +2δ0 and 2δ0 < δdec . Since δ ≤ q+1 ≤ 2−δ and s ≤ ksmall + 29, we have in view of Theorem 5.14 s+1 Bq+1 [q](τ1 , τ2 ) .

s+1 s+2 Eq+1 [q](τ1 ) + Iq+1 [Ng ](τ1 , τ2 ).

We infer X

Jˇq [dk ˇq, dk Nm [q]](τ1 , τ2 )

k≤s

.

2

sup τ1 ≤τ ≤τ2

s+1 s+2 Eqs [ˇq](τ ) + Eq+1 [q](τ1 ) + Iq+1 [Ng ](τ1 , τ2 ).

(5.3.40)

It remains to estimate the integral due to e3 (d≤k+1 rNg ). We proceed as in Proposition 5.19 by integration by parts, and obtain in particular the following analog of (5.3.27): s+1 Jˇq [dk ˇq, dk e3 (rNg )](τ1 , τ2 ) . P k + δ1 Bqs [ˇq](τ1 , τ2 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 ) (5.3.41) where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bqs [ˇq](τ1 , τ2 ) by the left-hand side of our main estimate, and where we have introduced the notation P k for the analog of Lk in (5.3.28), i.e.,6 Z
P k := rq e3 e4 rdk ˇq d≤k+1 (rNg ). (5.3.42) M(τ1 ,τ2 )

As in Lemma 5.20, e3 e4 (rdk ˇq)

= −d≤k (r2 ˇq) + r4 / 2 (d≤k ˇq) + r−1 d≤k+1 ˇq.

(5.3.43)

6 Recall that ˇ q is localized in r ≥ 4m0 so that we don’t need in (5.3.42) the cut-off function φ0 (r) introduced in Proposition 5.19.

DECAY ESTIMATES FOR q (THEOREM M1)

247

We infer P

k

Z


rq e3 e4 rdk ˇq d≤k+1 (rNg )

= M(τ1 ,τ2 )

Z

= − rq d≤k (r2 ˇq)d≤k+1 (rNg ) M(τ1 ,τ2 ) Z + rq+1 4 / 2 (d≤k ˇq)d≤k+1 (rNg ) M(τ1 ,τ2 ) Z + rq−1 d≤k+1 ˇq d≤k+1 (rNg ) M(τ1 ,τ2 )

= P1k + P2k + P3k . The last two terms on the right can be treated exactly as the corresponding terms in the treatment of Lk . This yields to the following analog of (5.3.29) and (5.3.30) k P3 . δ1 Bqs [ˇq](τ1 , τ2 ) + δ −1 I s+1 [Ng ](τ1 , τ2 ), 1 q+2 (5.3.44) k P2 . δ1 Bqs [ˇq](τ1 , τ2 ) + δ −1 I s+2 [Ng ](τ1 , τ2 ), 1 q+2 where δ1 > 0 is chosen sufficiently small so that we can later absorb the term δ1 Bps [ˇq](τ1 , τ2 ) by the left-hand side of our main estimate. It thus only remains to consider the term analogous to Lk1 , i.e., Z P1k = rq dk (r2 ˇq)d≤k+1 (rNg ). M(τ1 ,τ2 )

Now, in view of Proposition 10.47, q verifies, schematically, 2 ˇq = r−2 d≤1 ˇq + r−2 d≤2 q + rd≤1 N so that dk (r2 ˇq)

= r−1 d≤k+1 ˇq + r−1 d≤k+2 q + r2 d≤k+1 N = r−1 d≤k+1 ˇq + r−1 d≤k+2 q + r2 d≤k+1 Ng + r2 d≤k+1 Nm [q] +r2 d≤k+1 e3 (rNg ).

We infer the following decomposition of P1k : Z   k P1 = rq−1 d≤k+1 ˇq + d≤k+2 q d≤k+1 (rNg ) M(τ1 ,τ2 ) Z + rq+2 d≤k+1 Nm [q]d≤k+1 (rNg ) M(τ1 ,τ2 ) Z   + rq+2 d≤k+1 Ng + d≤k+1 e3 (rNg ) d≤k+1 (rNg ) M(τ1 ,τ2 )

=

k P11

k k + P12 + P13 .

248

CHAPTER 5

k P11 is estimated as Jˇqs [ˇq, Ng ](τ1 , τ2 ), see (5.3.39), and hence k |P11 | .

Bqs [ˇq](τ1 , τ2 )


1/2 s+1 Iq+2 [Ng ](τ1 , τ2 )


1/2

s+1 s . δ1 Bqs [ˇq](τ1 , τ2 ) + Bmax(q,δ) [q](τ1 , τ2 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 )

which in view of Theorem 5.14 yields s+1 s+1 δ1 Bqs [ˇq](τ1 , τ2 ) + Emax(q,δ) [q](τ1 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 ). (5.3.45)

k |P11 | .

k Next, P12 is estimated as follows: Z k |P12 | . rq+3 |d≤k+1 Nm [q]||d≤k+1 Ng | M≥4m0 (τ1 ,τ2 )

Z . M≥4m0 (τ1 ,τ2 )

rq+3 |d≤k+2 Γg ||d≤k+2 q||d≤k+1 Ng |

Z

1



.

M≥4m0 (τ1 ,τ2 )

rq+1 τ − 2 −δdec +2δ0 |d≤k+2 q||d≤k+1 Ng | ! 12

Z

rq−2 |d≤k+2 q|2



.

M≥4m0 (τ1 ,τ2 )

× 

.



! 12

Z r

q+4 −1−2δdec +4δ0

τ

M≥4m0 (τ1 ,τ2 )

 12

s+1 Bq+1 [q](τ1 , τ2 )

|d

sup

r Σ(τ )

Ng |

2

! 12

Z τ ∈[τ1 ,τ2 ]

≤k+1

q+4

≤k+1

|d

Ng |

2

where we have used |Γg | . r−2 τ −1/2−δdec +2δ0 and 2δ0 < δdec . We infer s+1 s+1 k |P12 | . Bq+1 [q](τ1 , τ2 ) + Iq+2 [Ng ](τ1 , τ2 ).

(5.3.46)

k Finally, P13 is estimated as follows: Z   k |P13 | . rq+3 |d≤k+1 Ng | + |d≤k+1 e3 (rNg )| |d≤k+1 Ng | M≥4m0 (τ1 ,τ2 )

Z . M≥4m0 (τ1 ,τ2 )

rq+3 |d≤k+1 Ng |2

Z + M≥4m0 (τ1 ,τ2 )

.

rq+4 |d≤k+1 e3 (Ng )||d≤k+1 Ng |

s+1 Iq+2 [Ng ](τ1 , τ2 ).

Together with (5.3.45) and (5.3.46), we infer |P1k |

≤

k k k |P11 | + |P12 | + |P13 |

s+1 s+1 s+1 . δ1 Bqs [ˇq](τ1 , τ2 ) + Emax(q,δ) [q](τ1 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 ) + Bq+1 [q](τ1 , τ2 ).

249

DECAY ESTIMATES FOR q (THEOREM M1)

Together with (5.3.44), we deduce |P k |

≤

|P1k | + |P2k | + |P3k |

s+1 s+2 s+1 . δ1 Bqs [ˇq](τ1 , τ2 ) + Emax(q,δ) [q](τ1 ) + δ1−1 Iq+2 [Ng ](τ1 , τ2 ) + Bq+1 [q](τ1 , τ2 ).

Together with (5.3.34), (5.3.37), (5.3.39), (5.3.40) and (5.3.41), this concludes the proof of Theorem 5.15.

5.4

DECAY ESTIMATES

In this section we prove the decay estimates. In particular: • • • • •

In section 5.4.1, we prove first flux decay estimates for q. In section 5.4.2, we prove flux decay estimates for ˇq. In section 5.4.3, we prove Theorem 5.9. In section 5.4.4, we prove Proposition 5.12 on pointwise decay estimates for q. In section 5.4.5, we prove Proposition 5.13 on flux estimates on Σ∗ and on improved pointwise estimates for e3 (q). The decay estimates rely on the norms (5.1.27) which we recall below. s Ep,d [ψ] = sup (1 + τ )d Eps [ψ](τ ), 0≤τ ≤τ∗

s Bp,d [ψ] = sup (1 + τ )d 0≤τ ≤τ∗

Z

τ∗ s Mp−1 [ψ](τ )dτ,

τ

s Fp,d [ψ] = sup (1 + τ )d Fps [ψ](τ ), 0≤τ ≤τ∗

s Ip,d [Ng ]

5.4.1

= sup (1 + τ )d Ips [Ng ](τ, τ∗ ). 0≤τ ≤τ∗

First flux decay estimates

The goal of this section is to prove the following flux decay estimates for q. Theorem 5.21. Assume q verifies all the estimates of Theorem 5.14. Then the following estimates hold true for all s ≤ ksmall + 30 and for all δ ≤ p ≤ 2 − δ: s−[2−δ−p]

s−[2−δ−p]

s−[2−δ−p]

Ep,2−δ−p [q] + Bp,2−δ−p [q] + Fp,2−δ−p [q]

s+1 s+1 s . E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ].

(5.4.1)

Here, [x] denotes the least integer greater or equal to x. Proof. We make use of Theorem 5.14 according to which we have, for δ ≤ p ≤ 2−δ, and 0 ≤ k ≤ ksmall + 30, Eps [q](τ2 ) + Bps [q](τ1 , τ2 ) + Fps [q](τ1 , τ2 ) . Eps [q](τ1 ) + Ips+1 [Ng ](τ1 , τ2 ) which we write in the form Z τ2 s Eps (τ2 ) + Mp−1 (τ )dτ . Eps (τ1 ) + Ips+1 [Ng ](τ1 , τ2 ), τ1

δ ≤ p ≤ 2 − δ. (5.4.2)

250

CHAPTER 5

In particular, s E2−δ (τ ) +

Z

τ s M1−δ (λ)dλ

τ /2

s+1 s . E2−δ (τ /2) + I2−δ, 0 [Ng ].

By the mean value theorem there exists τ0 ∈ [τ /2, τ ] such that s M1−δ (τ0 ) .

 1 s s+1 E2−δ (τ /2) + I2−δ, [N ] . g 0 τ

Since7 s−1 s E1−δ (τ ) . M1−δ (τ ),

we deduce s−1 E1−δ (τ0 ) .

 1 s s+1 E2−δ (τ /2) + I2−δ, 0 [Ng ] . τ

Moreover, applying (5.4.2) again for p = 1 − δ, we deduce Z τ s−1 s−1 s−1 s E1−δ (τ ) + M−δ (λ)dλ . E1−δ (τ0 ) + (1 + τ )−1 I1−δ,1 [Ng ] τ0   s+1 s s . (1 + τ )−1 E2−δ (τ /2) + I2−δ, [N ] + I [N ] . g g 1−δ,1 0 In particular,   s−1 s+1 s s E1−δ (τ ) . (1 + τ )−1 E2−δ (τ /2) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] .

(5.4.3)

Interpolating with s+1 s s E2−δ (τ ) . E2−δ (τ /2) + I2−δ, 0 [Ng ]

by using Eps

.

p2 −p

p−p1

(Eps1 ) p2 −p1 (Eps2 ) p2 −p1 ,

p1 ≤ p ≤ p2 ,

we deduce s−1 s−1 E1s−1 (τ ) . (E1−δ (τ ))1−δ (E2−δ (τ ))δ   s+1 s s . (1 + τ )−1+δ E2−δ (τ /2) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] .

The same inequality holds for τ replaced by τ /2, i.e.,   s+1 s s E1s−1 (τ /2) . (1 + τ )−1+δ E2−δ (τ /4) + I2−δ, [N ] + I [N ] . g g 1−δ,1 0

(5.4.4)

We now repeat the procedure starting this time with the inequality (5.4.2) for 7 Note that the loss of derivative is due to the degeneracy of the bulk integral in the trapping region.

251

DECAY ESTIMATES FOR q (THEOREM M1)

p = 1, E1s−1 (τ ) +

Z

τ

M0s−1 (λ)dλ

. E1s−1 (τ /2) + I1s [Ng ](τ /2, τ )

τ /2 s . E1s−1 (τ /2) + (1 + τ )−1+δ I1,1−δ [Ng ].

Thus, in view of (5.4.4), Z τ M0s−1 (λ)dλ τ /2

  s+1 s s s . (1 + τ )−1+δ E2−δ (τ /4) + I2−δ, [N ] + I [N ] + I [N ] g g g 1−δ,1 1,1−δ 0 or, since s E2−δ (τ /4)

s+1 s . E2−δ (0) + I2−δ,0 [Ng ],

we infer that Z

τ

M0s−1 (λ)dλ

. B(1 + τ )−1+δ

τ /2

where B:

s+1 s s s = E2−δ (0) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] + I1,1−δ [Ng ].

(5.4.5)

Repeating the mean value argument, we can find τ1 ∈ [τ /2, τ ] such that Z 1 τ M0s−1 (τ1 ) . M s−1 (λ)dλ . B(1 + τ )−2+δ . τ τ /2 0 We now make use of the fact that the energy norm E s−1 is comparable with M0s−1 everywhere except in the trapping region where we lose a derivative. Thus E s−2 (τ1 ) . M0s−1 (τ1 ) and therefore, E s−2 (τ1 ) . B(1 + τ )−2+δ .

(5.4.6)

We would like now to compare E s−2 (τ ) with E s−2 (τ1 ) using the usual version of the energy inequality and thus derive a similar estimate for the former. Unfortunately,8 we don’t have a closed energy inequality for E and we therefore have instead to rely on Eδ for which we have the inequality Eδs−2 (τ ) . Eδs−2 (τ1 ) + Iδs−1 [Ng ](τ1 , τ ).

(5.4.7)

8 The loss of δ is due to the fact that we are on a perturbation of Schwarzschild rather than on Schwarzschild.

252

CHAPTER 5

We also have in view of (5.4.3)   s−2 s+1 s s E1−δ (τ1 ) . (1 + τ )−1 E2−δ (0) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] . Interpolating this last inequality with (5.4.6) we deduce, for δ > 0 sufficiently small, Eδs−2 (τ1 ) . . .


1−2δ s−2
δ E s−2 (τ1 ) 1−δ E1−δ (τ1 ) 1−δ s−2 s−1 (1 + τ )−2+2δ (B + E1−δ (0) + I1−δ,0 [Ng ])

(1 + τ )−2+2δ B.

Thus, in view of (5.4.7), s−1 Eδs−2 (τ ) . Eδs−2 (τ1 ) + Iδs−1 [Ng ](τ1 , τ ) . (1 + τ )−2+2δ (B + Iδ,2−2δ [Ng ]),

i.e., Eδs−2 (τ )
s+1 s−1 s s s . (1 + τ )−2+2δ E2−δ (0) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] + I1,1−δ [Ng ] + Iδ,2−2δ [Ng ] . We infer s−2 Eδ,2−2δ

.

s+1 s−1 s s s E2−δ (0) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ] + I1,1−δ [Ng ] + Iδ,2−2δ [Ng ]




which can be written in the shorter form (by interpolation of the middle terms) s−2 Eδ,2−2δ

.

s+1 s+1 s E2−δ (0) + I2−δ, 0 [Ng ] + Iδ,2−2δ [Ng ].

.

s+1 s s E2−δ (0) + I2−δ, 0 [Ng ] + I1−δ,1 [Ng ]

(5.4.8)

Also, (5.4.3) yields s−1 E1−δ,1

.

s+1 s+1 s E2−δ (0) + I2−δ, 0 [Ng ] + Iδ,2−2δ [Ng ],

(5.4.9)

while from Theorem 5.14, we have s E2−δ,0

.

s+1 s E2−δ (0) + I2−δ, 0 [Ng ].

(5.4.10)

Interpolating (5.4.8) and (5.4.9), as well as (5.4.9) and (5.4.10), we infer for all s ≤ ksmall + 30 and for all δ ≤ p ≤ 2 − δ s−[2−δ−p]

s+1 s+1 s Ep,2−δ−p [q] . E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ].

(5.4.11)

Finally, making use of Theorem 5.14 between τ and τ∗ , we have in particular Bps−[2−δ−p] [q](τ, τ∗ ) + Fps−[2−δ−p] [q](τ, τ∗ ) . Eps−[2−δ−p] [q](τ ) + Ips+1−[2−δ−p] [Ng ](τ, τ∗ )   s−[2−δ−p] s+1 . (1 + τ )−(2−δ−p) Ep,2−δ−p [q] + Ip,2−δ−p [Ng ]

253

DECAY ESTIMATES FOR q (THEOREM M1)

and hence, we infer for all s ≤ ksmall + 30 and for all δ ≤ p ≤ 2 − δ s−[2−δ−p]

s−[2−δ−p]

s−[2−δ−p]

s+1 s+1 Bp,2−δ−p [q] + Fp,2−δ−p [q] . Ep,2−δ−p [q] + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ].

Together with (5.4.11), this concludes the proof of Theorem 5.21. 5.4.2

Flux decay estimates for ˇq

The goal of this section is to prove the following flux decay estimates for ˇq. Theorem 5.22. The following estimates hold for all q0 − 1 ≤ q ≤ q0 , where q0 is a fixed number δ < q0 ≤ 1 − δ, and s ≤ ksmall + 28: s+2 s+3 s s Eq,q [ˇq] + Bq,q [ˇq] . Eqs0 [ˇq](0) + E2−δ [q](0) + Iqs+3 [Ng ] + Iδ,2+q [Ng ]. 0 −q 0 −q 0 +2,0 0 −δ

Proof. Since δ < q0 ≤ 1 − δ, according to Theorem 5.15, ˇq = f2 eˇ4 q verifies, for any q0 − 1 ≤ q ≤ q0 and any s ≤ ksmall + 29, Eqs [ˇq](τ2 ) + Bqs [ˇq](τ1 , τ2 ) .

s+1 s+2 Eqs [ˇq](τ1 ) + Eq+1 [q](τ1 ) + Iq+2 [Ng ](τ1 , τ2 ).

According to the definition of our decay norms above we have s+2 s+2 Iq+2 [Ng ](τ1 , τ2 ) . (1 + τ1 )q−q0 Iq+2,q [Ng ]. 0 −q

(5.4.12)

Also, according to the definition 5.1.27 for the decay norms for q we also have s+1 s+2 Eq+1 [q](τ1 ) . (1 + τ1 )q−q0 Eq+1,q [q]. 0 −q

We deduce,9 for all q0 − 1 ≤ q ≤ q0 , Z τ2 s s Eq [ˇq](τ2 ) + Mqs [ˇq](τ ) . Eqs [ˇq](τ1 ) + (1 + τ1 )q−q0 E˜q,q 0 −q

(5.4.13)

τ1

where s+1 s+2 s E˜q,q := Eq+1,q [q] + Iq+2,q [Ng ]. 0 −q 0 −q 0 −q

(5.4.14)

In particular, Eqs0 [ˇq](τ2 ) +

Z

τ2

τ1

Mqs0 −1 [ˇq](τ )dτ

. Eqs0 [ˇq](τ1 ) + E˜qs0 ,0 .

(5.4.15)

By the mean value theorem we deduce that there exists τ0 ∈ [τ1 , τ2 ] such that Mqs0 −1 [ˇq](τ0 ) .

    1 1 Eqs0 [ˇq](τ1 ) + E˜qs0 ,0 . Eqs0 ,0 [ˇq] + E˜qs0 ,0 . τ2 − τ1 τ2 − τ1

9 Note that it is important in what follows that the r q weighted estimates hold also for negative values of q.

254

CHAPTER 5

Thus also, Eqs0 −1 [ˇq](τ0 ) .

  1 Eqs0 ,0 [ˇq] + E˜qs0 ,0 . τ2 − τ1

(5.4.16)

We now make use of (5.4.13) to compare the quantities Eq [ˇq] for negative weights (q = q0 − 1) at different values of τ . Eqs0 −1 [ˇq](τ0 ) + (1 + τ0 )−1 E˜qs0 −1,1 .

Eqs0 −1 [ˇq](τ2 ) .

Combining this with (5.4.16) we deduce Eqs0 −1 [ˇq](τ2 ) .

  1 Eqs0 ,0 [ˇq] + E˜qs0 ,0 + (1 + τ0 )−1 E˜qs0 −1,1 . τ2 − τ1

Applying this inequality for τ2 = τ ≤ τ∗ , τ1 = 12 τ , τ0 ∈ [τ1 , τ2 ], we deduce   (5.4.17) Eqs0 −1 [ˇq](τ ) . (1 + τ )−1 Eqs0 ,0 [ˇq] + E˜qs0 ,0 + E˜qs0 −1,1 . We now interpolate this last inequality with the following immediate consequence of (5.4.15): Eqs0 [ˇq](τ ) . Eqs0 ,0 [ˇq] + E˜qs0 ,0 to deduce, for all q0 − 1 ≤ q ≤ q0 ,   Eqs [ˇq](τ ) . (1 + τ )q−q0 Eqs0 ,0 [ˇq] + E˜qs0 ,0 + E˜qs0 −1,1 , i.e., s Eq,q [ˇq] 0 −q

. Eqs0 ,0 [ˇq] + E˜qs0 ,0 + E˜qs0 −1,1 .

s In view of the definition of E˜q,q , this yields, for all q0 − 1 ≤ q ≤ q0 , 0 −q s Eq,q [ˇq] . 0 −q

Eqs0 ,0 [ˇq] + Eqs+1 [q] + Eqs+1 [q] + Iqs+2 [Ng ] + Iqs+2 [Ng ]. 0 +1,0 0 ,1 0 +2,0 0 +1,1

On the other hand, we have, in view of Theorem 5.15, Eqs0 ,0 [ˇq] . Eqs0 [ˇq](0) + Eqs+1 [q] + Iqs+2 [Ng ] 0 +1,0 0 +2,0 and hence s Eq,q [ˇq] . Eqs0 [ˇq](0) + Eqs+1 [q] + Eqs+1 [q] + Iqs+2 [Ng ] + Iqs+2 [Ng ]. 0 −q 0 +1,0 0 ,1 0 +2,0 0 +1,1

Now, since δ < q0 ≤ 1 − δ, we have δ < q0 < q0 + 1 ≤ 2 − δ and thus, we may apply Theorem 5.21 to obtain for all q0 − 1 ≤ q ≤ q0 s+1 s+2 s+3 s+3 Eq+1,q [q] . E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ]. 0 −q

(5.4.18)

255

DECAY ESTIMATES FOR q (THEOREM M1)

We thus infer s Eq,q [ˇq] 0 −q

.

s+2 s+3 s+3 Eqs0 [ˇq](0) + E2−δ [q](0) + Iqs+3 [Ng ] + Iqs+3 [Ng ] + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ] 0 +2,0 0 +1,1

and hence, for all q0 − 1 ≤ q ≤ q0 , s+2 s+3 s Eq,q [ˇq] . Eqs0 [ˇq](0) + E2−δ [q](0) + Iqs+3 [Ng ] + Iδ,2−2δ [Ng ]. (5.4.19) 0 −q 0 +2,0

Finally, making use of Theorem 5.15 between τ and τ∗ , we have in particular Bqs [ˇq](τ, τ∗ ) . . .

s+1 s+2 Eqs [ˇq](τ ) + Eq+1 [q](τ ) + Iq+2 [Ng ](τ, τ∗ )   s+1 s+2 s (1 + τ )−(q0 −q) Eq,q [ˇ q ] + E [q] + I [N ] g q+1,q0 −q q+2,q0 −q 0 −q   s+2 s+3 s+3 −(q0 −q) s (1 + τ ) Eq,q0 −q [ˇq] + E2−δ [q](0) + Iq0 +2,0 [Ng ] + Iδ,2−2δ [Ng ]

where we used (5.4.18) in the last inequality. Hence, we infer for all s ≤ ksmall + 28 and for all q0 − 1 ≤ q ≤ q0 s+2 s+3 s s Bq,q [ˇq] . Eq,q [ˇq] + E2−δ [q](0) + Iqs+3 [Ng ] + Iδ,2−2δ [Ng ]. 0 −q 0 −q 0 +2,0

Together with (5.4.19), this concludes the proof of Theorem 5.22. 5.4.3

Proof of Theorem 5.9

In this section, we prove Theorem 5.9 by making use of Theorem 5.21 and Theorem 5.22. We start with the main estimate of Theorem 5.22 with q = −δ which we write in the form s E−δ [ˇq] . (1 + τ )−q0 −δ Cqs0

where Cqs0

s+2 s+3 := Eqs0 [ˇq](0) + E2−δ [q](0) + Iqs+3 [Ng ] + Iδ,q [Ng ]. 0 ,0 0 +2−δ

s In view of the definition (5.1.21) of E−δ [ˇq] and since ˇq = f2 eˇ4 q, Z
r−δ |ˇ e4 ˇq|2 + r−2 |ˇq|2 . (1 + τ )−q0 −δ Cqs0 . Σ≥4m0 (τ )

Hence, s E˙ 2−δ,4m [q] = 0

Z Σ≥4m0 (τ )

r2−δ |ˇ e4 q|2

. (1 + τ )−q0 −δ Cqs0 .

(5.4.20)

In view of the decay estimates (5.4.1) for q established in Theorem 5.21 we have E s (τ ) . 2+s B2−δ :

=

2+s (1 + τ )−2+2δ B2−δ , s+2 s+3 s+3 E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ].

256

CHAPTER 5

Thus, the quantity s s s E2−δ = E2−δ [q](τ ) = E˙ 2−δ,4m [q] + E s [q] 0

verifies s E2−δ


2+s (1 + τ )−q0 −δ Cqs0 + B2−δ .

.

(5.4.21)

s On the other hand, E2−δ verifies (5.4.2) for p = 2 − δ, i.e., s E2−δ (τ2 )

Z

τ2

+

s M1−δ (τ )dτ

τ1

s+1 s . E2−δ (τ1 ) + I2−δ [Ng ](τ1 , τ2 ).

Since s+1 s+1 I2−δ [Ng ](τ1 , τ2 ) . (1 + τ1 )−q0 −δ I2−δ,q [Ng ], 0 +δ

we infer s E2−δ (τ )

Z

+

s . E2−δ (τ /2)

τ

s M1−δ (τ 0 )dτ 0 τ /2 s+1 + I2−δ [Ng ](τ /2, τ )

  2+s s+1 . (1 + τ )−q0 −δ Cqs0 + B2−δ + I2−δ,q [Ng ] . 0 +δ

(5.4.22)

Following the same arguments as in the proof of Theorem 5.21 we deduce, for a τ0 ∈ [τ /2, τ ],   s−1 2+s s+1 E1−δ (τ0 ) . (1 + τ )−q0 −1−δ Cqs0 + B2−δ + I2−δ,q [N ] g +δ 0 and since s+1 s s E1−δ (τ ) . E1−δ (τ0 ) + I1−δ (τ0 , τ )[Ng ],

we infer that s E1−δ (τ )

. (1 + τ )−q0 −1−δ



(5.4.23)  3+s s+2 s+1 Cqs+1 + B + I [N ] + I [N ] . g g 2−δ 2−δ,q0 +δ 1−δ,1+q0 +δ 0

Interpolating with (5.4.21), i.e., s E2−δ

.

2+s (1 + τ )−q0 −δ Cqs0 + B2−δ




we deduce E1s

s s . (E1−δ )1−δ (E2−δ )δ   3+s s+2 s+1 . (1 + τ )−q0 −1 Cqs+1 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] . 0 0 +δ 0 +δ

Hence, E1s

.

  3+s s+2 s+1 (1 + τ )−q0 −1 Cqs+1 + B2−δ + I2−δ,q + I1−δ,1+q . (5.4.24) 0 0 +δ 0 +δ

257

DECAY ESTIMATES FOR q (THEOREM M1)

As in the proof of Theorem 5.21 we repeat the procedure starting with the inequality (5.4.2) for p = 1, Z τ E1s (τ ) + M0s (λ)dλ τ /2

. E1s (τ /2) + I1s+1 [Ng ](τ /2, τ )   3+s s+2 s+1 . (1 + τ )−q0 −1 Cqs+1 + B + I [N ] + I [N ] g g 2−δ 2−δ,q0 +δ 1−δ,1+q0 +δ 0 s+1 +(1 + τ )−1−q0 I1,1+q [Ng ] 0   3+s s+s s+1 s+1 −q0 −1 s+1 . (1 + τ ) Cq0 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] + I1,1+q [Ng ] 0 0 +δ 0 +δ

from which we infer that, for a τ0 ∈ [τ /2, τ ],  s+4 s+3 s+2 E s (τ0 ) . (1 + τ )−q0 −2 Cqs+2 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] 0 0 +δ 0 +δ  s+2 +I1,1+q [Ng ] . (5.4.25) 0 Interpolating (5.4.23) and (5.4.25) we deduce, for δ > 0 sufficiently small,
1−2δ 1−δ


δ s E1−δ (τ0 ) 1−δ  s+4 s+3 . (1 + τ )−2−q0 +δ Cqs+2 + B2−δ + I2−δ,q [Ng ] 0 0 +δ

Eδs (τ0 ) .

E s (τ0 )

 s+2 s+2 +I1−δ,1+q [Ng ] + I1,1+q [Ng ] . 0 0 +δ Thus, since we have, as in (5.4.7), Eδs (τ ) . Eδs (τ0 ) + Iδs+1 [Ng ](τ0 , τ ), we deduce  s+4 s+3 s+2 Eδs (τ ) . (1 + τ )−2−q0 +δ Cqs+2 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] 0 0 +δ 0 +δ  s+2 s+1 +I1,1+q [Ng ] + (1 + τ )−2−q0 +δ Iδ,2+q [Ng ], 0 0 −δ i.e.,  s+4 s+3 s+2 Eδs (τ ) . (1 + τ )−2−q0 +δ Cqs+2 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] 0 0 +δ 0 +δ  s+2 s+1 +I1,1+q [Ng ] + Iδ,2+q [Ng ] . 0 0 −δ By interpolating the middle terms we write   s+4 s+3 s+3 Eδs (τ ) . (1 + τ )−2−q0 +δ Cqs+2 + B + I [N ] + I [N ] . g g 2−δ 2−δ,q0 +δ δ,2+q0 −δ 0 We now recall Cqs0 2+s B2−δ :

s+2 s+3 := Eqs0 [ˇq](0) + E2−δ [q](0) + Iqs+3 [Ng ] + Iδ,q [Ng ] 0 +2,0 0 +2−δ

=

s+2 s+3 s+3 E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ].

258

CHAPTER 5

Hence, s+4 s+3 s+3 Cqs+2 + B2−δ + I2−δ,q [Ng ] + Iδ,2+q [Ng ] 0 0 +δ 0 −δ

= +

s+4 s+5 Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,q [Ng ] 0 0 +2,0 0 +2−δ

s+4 s+5 s+5 s+3 s+3 E2−δ [q](0) + I2−δ,0 [Ng ] + Iδ,2−2δ [Ng ] + I2−δ,q [Ng ] + Iδ,2+q [Ng ]. 0 +δ 0 −δ

We deduce s+4 s+5 s Eδ,2+q [q] . Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ].(5.4.26) 0 −δ 0 0 +2,0 0 −δ

We can also simplify the right-hand side of (5.4.24), 3+s s+2 s+1 Cqs+1 + B2−δ + I2−δ,q [Ng ] + I1−δ,1+q [Ng ] 0 0 +δ 0 +δ

s+4 s+5 . Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ]. 0 0 +2,0 0 −δ

Thus, (5.4.23) becomes s+4 s+5 s E1−δ,1+q . Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ]. (5.4.27) 0 +δ 0 0 +2,0 0 −δ

Similarly, (5.4.21) yields s+4 s+5 s E2−δ,q . Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ]. 0 −δ 0 0 +2,0 0 −δ

(5.4.28)

Interpolating (5.4.26) and (5.4.27), as well as (5.4.27) and (5.4.28), we infer for all s ≤ ksmall + 25 and for all δ ≤ p ≤ 2 − δ s+4 s+5 s Ep,2+q [q] . Eqs+2 [ˇq](0) + E2−δ [q](0) + Iqs+5 [Ng ] + Iδ,2+q [Ng ]. (5.4.29) 0 −p 0 0 +2,0 0 −δ

Finally, making use of Theorem 5.14 between τ and τ∗ , we have in particular Bps [q](τ, τ∗ ) + Fps [q](τ, τ∗ ) . E sp [q](τ ) + Ips+1 [Ng ](τ, τ∗ )   s+1 s . (1 + τ )−(2+q0 −p) Ep,2+q [q] + Ip,2+q [Ng ] 0 −p 0 −p and hence, we infer for all s ≤ ksmall + 25 and for all δ ≤ p ≤ 2 − δ s+5 s s s Bp,2+q [q] + Fp,2+q [q] . Ep,2+q [q] + Iqs+5 [Ng ] + Iδ,2+q [Ng ]. 0 −p 0 −p 0 −p 0 +2,0 0 −δ

Together with (5.4.29), this concludes the proof of Theorem 5.9.

259

DECAY ESTIMATES FOR q (THEOREM M1)

5.4.4

Proof of Proposition 5.12

Let χ be a smooth cut-off function vanishing for r ≤ 4m0 and equal to 1 for r ≥ 6m0 . To prove estimate (5.2.6) we consider the identity, Z  (s) 2 e4 χ(q ) Z  Sr  = e4 (χ(q(s) )2 ) + κχ(q(s) )2 ZSr   = χ(2q(s) e4 q(s) + 2r−1 (q(s) )2 ) + χ0 (q(s) )2 + χ(κ − 2r−1 )|q(s) |2 ZSr   = 2χq(s) eˇ4 q(s) + χ0 (q(s) )2 + O(r−2 )|q(s) |2 . Sr

Integrating between 4m0 and r for a fixed r ≥ 6m0 , we deduce, in view of the definitions of E[q(s) ](τ ) and of Ep [q(s) ](τ ), Z Z |q(s) |2 . |q(s) ||ˇ e4 q(s) | + E[q(s) ](τ ) Sr

Σ(τ )≥4m0

!1/2

Z r

.

1+δ

Σ(τ )≥4m0

|ˇ e4 q

(s) 2

|

!1/2

Z r Σ(τ )≥4m0

−1−δ

|q

(s) 2

|

+E[q(s) ](τ ) .


1/2
1/2 E1+δ [q(s) ](τ ) E1−δ [q(s) ](τ ) .

Clearly, this estimate also holds for r ≤ 6m0 . Together with the definition (5.1.27) s of Ep,d [q(s) ], we immediately infer (1 + τ )1+q0

Z Sr

|q(s) |2

.


1
1 s s E1+δ,1+q [q] 2 E1−δ,1+q [q] 2 0 +δ 0 −δ

which is the desired estimate (5.2.6). To prove (5.2.7) we start instead with the identity,   Z −1 (s) 2 e4 r χ(q ) Sr Z   e (r) Z 4 = r−1 e4 (χ(q(s) )2 ) + κχ(q(s) )2 − 2 χ(q(s) )2 r Sr Sr Z   = r−1 χ(2q(s) e4 q(s) + r−1 (q(s) )2 ) + χ0 (q(s) )2 + χ(κ − 2r−1 )|q(s) |2 Sr Z e4 (r) − 1 − χ(q(s) )2 r2 Sr Z   = 2r−1 χe4 (q(s) )q(s) + r−1 χ0 (q(s) )2 + O(r−2 )|q(s) |2 . Sr

Integrating between 4m0 and r for a fixed r ≥ 6m0 , we deduce, in view of the

260

CHAPTER 5

definitions of E[q(s) ](τ ) and of Ep [q(s) ](τ ), Z Z r−1 |ψ|2 . r−1 |q(s) ||e4 (q(s) )| + E[q(s) ](τ ) Sr

Σ(τ )≥4m0

!1/2

Z . 2 Σ(τ )≥4m0

|e4 (q

(s)

)|

2

!1/2

Z r Σ(τ )≥4m0

−2

|q

(s) 2

|

+E[q(s) ](τ ) . E[q(s) ](τ ) . Eδ [q(s) ](τ ). Clearly, this estimate also holds for r ≤ 6m0 . Together with the definition (5.1.27) s of Ep,d [q(s) ], we immediately infer r

−1

(1 + τ )

2+q0 −δ

Z Sr

|q(s) |2

s . Eδ,2+q [q] 0 −δ

which is the desired estimate (5.2.7). This concludes the proof of Proposition 5.12. 5.4.5

Proof of Proposition 5.13

Recall the following definitions: Z   −1 F [ψ](τ1 , τ2 ) = δH |e4 Ψ|2 + δH |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 A(τ1 ,τ2 ) Z   + |e4 Ψ|2 + |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 , Σ∗ (τ1 ,τ2 ) Z   F˙p [ψ](τ1 , τ2 ) = rp |e4 ψ|2 + |∇ / ψ|2 + r−2 |ψ|2 , Σ∗ (τ1 ,τ2 )

Fp [ψ](τ1 , τ2 ) F s [ψ](τ1 , τ2 )

= F [ψ](τ1 , τ2 ) + F˙p [ψ](τ1 , τ2 ), X = F [dk ψ](τ1 , τ2 ), k≤s

Fps [ψ](τ1 , τ2 )

X

=

Fp [dk ψ](τ1 , τ2 ),

k≤s s Fp,d [ψ]

sup (1 + τ )d Fps [ψ](τ, τ∗ ).

=

0≤τ ≤τ∗

We deduce s F s [q](τ, τ∗ ) ≤ Fδs [q](τ, τ∗ ) ≤ (1 + τ )−2−q0 +δ Fδ,2+q [q] 0 −δ

and hence in particular Z 2+q0 −δ (1 + τ ) Σ∗ (τ,τ∗ )



|e3 d≤s q|2 + r−2 |d≤s q|2

which yields the desired estimate (5.2.8).



s . Fδ,2+q [q] (5.4.30) 0 −δ

261

DECAY ESTIMATES FOR q (THEOREM M1)

Next, we focus on the proof of (5.2.9). We start with the following trace estimate sup ke3 d≤s qkL2 (S)

Σ∗ (τ,τ∗ )

. kνe3 d≤s qkL2 (Σ∗ (τ,τ∗ )) + ke3 d≤s qkL2 (Σ∗ (τ,τ∗ ))

where we recall that ν is tangent to Σ∗ , orthogonal to eθ and given by ν = e3 + ae4 ,

1 −2 ≤ a ≤ − . 2

We infer sup ke3 d≤s qkL2 (S)

Σ∗ (τ,τ∗ )

. ke3 e3 d≤s qkL2 (Σ∗ (τ,τ∗ )) + ke4 e3 d≤s qkL2 (Σ∗ (τ,τ∗ )) +ke3 d≤s qkL2 (Σ∗ (τ,τ∗ )) . ke3 d≤s+1 qkL2 (Σ∗ (τ,τ∗ )) + kr−1 d≤s+1 qkL2 (Σ∗ (τ,τ∗ )) +k[e4 , e3 ]d≤s qkL2 (Σ∗ (τ,τ∗ ))

. ke3 d≤s+1 qkL2 (Σ∗ (τ,τ∗ )) + kr−1 d≤s+1 qkL2 (Σ∗ (τ,τ∗ )) . In view of (5.4.30), we deduce n o sup (1 + τ )2+q0 −δ ke3 d≤s qk2L2 (S) . Σ∗

s+1 Fδ,2+q [q]. 0 −δ

(5.4.31)

Next, we extend (5.4.31) to r ≥ 4m0 . In view of (5.3.24), we have schematically e3 e4 (rdk q)

= −d≤k (r2 q) + r4 / 2 (d≤k q) + r−1 d≤k+1 q

= −d≤k (r2 q) + r−1 d≤k+2 q. Also, we have e4 (re3 (dk q))

= e3 e4 (rdk q) + [e4 , e3 ](rdk q) − e4 (e3 (r)dk q)

and hence, we infer schematically e4 (re3 (dk q))

= −d≤k (r2 q) + r−1 d≤k+2 q.

Now, recall (5.3.31) |dk (r2 q)| . r−1 d≤k+1 q + r d≤k Ng | + r2 d≤k e3 (Ng )|. We deduce |e4 (re3 (dk q))| .

r−1 d≤k+2 q + r d≤k Ng | + r2 d≤k e3 (Ng )|.

262

CHAPTER 5

Now, we have   Z e4 r−2 (re3 (d≤s q))2 Z  S  −2 = r 2e4 (re3 (d≤s q))re3 (d≤s q) + κ(re3 (d≤s q))2 S Z 2e4 (r) −2 − r (re3 (d≤s q))2 r S Z   = r−2 2e4 (re3 (d≤s q))re3 (d≤s q) + (κ − 2r−1 )(re3 (d≤s q))2 S Z e4 (r) − 1 −2 −2 r (re3 (d≤s q))2 r S and hence

.

  Z e4 r−2 (re3 (d≤s q))2 S Z (  r−2 r−1 d≤s+2 q + r d≤s Ng | + r2 d≤s e3 (Ng )| |re3 (d≤s q)| S

) +r .

r

−2

−2

(re3 (d

Z  S

+r

− 72

≤s

q))

2

 2 1 r− 2 d≤s+2 q + r2 d≤s Ng |2 + r4 d≤s e3 (Ng )|2

Z

(re3 (d≤s q))2 .

S

Together with (5.2.7), this yields   Z e4 r−2 (re3 (d≤s q))2 Z  S  11 7 . r−2 r 2 d≤s Ng |2 + r 2 d≤s e3 (Ng )|2 S Z 3 s+2 −4 +r (re3 (d≤s q))2 + r− 2 (1 + τ )−2−q0 +δ Eδ,2+q [q]. 0 −δ S

Now, recall from (5.2.12) that we have for s ≤ ksmall + 30 |ds Ng | . 2 r−3 τ −1−2δdec +2δ0

|ds Ng | . 2 r−1 τ −2−2δdec +2δ0 , 3

|ds e3 (Ng )| . 2 r−3 τ − 2 −2δdec +2δ0 , 7

|ds e3 (Ng )| . 2 r− 2 −

δB 2

τ −1−δdec +2δ0 .

263

DECAY ESTIMATES FOR q (THEOREM M1)

By interpolation, we infer Z   7 11 r−2 r 2 d≤s Ng |2 + r 2 d≤s e3 (Ng )|2 .

3

5

4 r− 2 τ − 2 −4δdec +4δ0

S

+4 r−1− . and hence   Z e4 r−2 (re3 (d≤s q))2

. r−4

S

+r

Z

20 r−1−

δB 2

δB 2

5

τ − 2 −3δdec +4δ0 5

τ − 2 −3δdec +4δ0

(re3 (d≤s q))2 + 20 r−1−

δB 2

5

τ − 2 −3δdec +4δ0

S − 32

s+2 (1 + τ )−2−q0 +δ Eδ,2+q [q]. 0 −δ

We integrate from Σ∗ . By Gronwall, and in view of (5.4.31), we deduce for r ≥ 4m0 Z s+1 s+2 (1 + τ )2+q0 −δ (e3 d≤s q)2 . 20 + Fδ,2+q [q] + Eδ,2+q [q]. 0 −δ 0 −δ Sr

On the other hand, we have by a trace estimate for r ≤ 4m0 Z s+2 2+q0 −δ (1 + τ ) (e3 d≤s q)2 . E0,2+q [q]. 0 −δ Sr

We finally deduce on M Z (1 + τ )2+q0 −δ (e3 d≤s q)2 Sr

s+1 s+2 . 20 + Fδ,2+q [q] + Eδ,2+q [q] 0 −δ 0 −δ

which is the desired estimate (5.2.9). This concludes the proof of Proposition 5.13.

Chapter Six Decay Estimates for α and α (Theorems M2, M3) In this chapter, we rely on the decay of q to prove the decay estimates for α and α. More precisely, we rely on the results of Theorem M1 to prove Theorem M2 and M3.

6.1 6.1.1

PROOF OF THEOREM M2 A renormalized frame on

(ext)

M

In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. We have the following control for the Ricci coefficients in that frame. Lemma 6.1. Consider the global null frame (e3 , e4 , eθ ) constructed in Proposition 3.26. Then, the Ricci coefficients satisfy the following estimates: max

0≤k≤ksmall +20

  1 m 2Υ sup u 2 r2 dk ω + 2 , κ − , ϑ, ζ, η, η r r (ext) M   k 2 +r d ξ, ω, κ + , ϑ r ! k + d (e4 (r) − Υ, e3 (r) + 1)

. .

Proof. This follows immediately from the stronger estimates of Lemma 5.1 with the choice kloss = 20. 6.1.2

A transport equation for α

To recover α from q, we derive below a transport equation for α where q is on the RHS. We are careful to avoid terms of the type e3 (ω) as they are anomalous w.r.t. decay in r. Indeed, they only decay linearly in r−1 while all comparable terms decay like r−2 in r.

265

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

Lemma 6.2. We have        α 2 1 2 α 2 κ e3 e3 − 8ω − 2 /d1 ξ − ϑ κ2 κ 2 κ2    q 4 1 2 = + 10ω + − /d1 ξ − 2(η − 3ζ)ξ + ϑ e3 α r4 κ 4 (    8 1 1 + − 2 /d1 ξ + 6κ − 24ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 ω + ϑ2 κ 2 2    ) 4 48 24 1 − e3 ((η − 3ζ)ξ) + 16 + ω − 2 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 ζξ α. κ κ κ 2 Proof. We compute   α e3 e3 κ2



 e3 α 2e3 (κ)α − κ2 κ3     1 e3 (κ) e3 (κ) 2 e e α − 4 e α − 2κ e α . 3 3 3 3 κ2 κ κ3

= e3 =

Now, recall the following null structure equation 1 1 e3 (κ) + κ2 + 2ω κ = 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 . 2 2 We infer e3 (κ) 1 1 = − κ − 2ω + κ 2 κ



1 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2



and  e3

e3 (κ) κ3



   1 ω 1 1 = e3 − − 2 2 + 3 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2κ κ κ 2    1 1 2 e3 (κ) ω + 2 / d ξ + 2(η − 3ζ)ξ − ϑ = + e −2 1 3 κ2 κ3 2 2κ2    1 1 1 1 = − + −2ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 4 2κ κ 2    ω 1 1 +e3 −2 2 + 3 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 κ κ 2

266

CHAPTER 6

and hence 2



κ e3 e3

α κ2



  e3 (κ) e3 (κ) 2 = e3 e3 α − 4 e3 α − 2κ e3 α κ κ3 1 = e3 e3 α + 2κe3 α + κ2 α 2    4 1 2 + 8ω − 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ e3 α κ 2 (    1 1 + − κ −2ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 κ 2    ) ω 1 1 2 2 −2κ e3 −2 2 + 3 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ α. κ κ 2

Next, recall from section 2.3.3 that q is defined with respect to a general null frame as follows:     1 2 4 2 q = r e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω − 8ω κ + κ α . 2 We infer κ2 e3 e3



α κ2

 =

   q 4 1 2 + 14ω − 2 / d ξ + 2(η − 3ζ)ξ − ϑ e3 α 1 r4 κ 2 ( 

 1 2 + 4e3 (ω) − 8ω + 10ω κ − 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ 2    ) ω 1 1 2 2 −2κ e3 −2 2 + 3 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ α. κ κ 2 2

We rewrite the following terms (    ) ω 1 1 2 2 4e3 (ω) − 2κ e3 −2 2 + 3 2 /d1 ξ − ϑ α κ κ 2     ω 2 1 = κ2 e3 8 2 − 3 2 /d1 ξ − ϑ2 α κ κ 2     α 2 1 2 −4κ2 ωe3 + 2 / d ξ − ϑ e3 (α) 1 κ2 κ 2     ω 2 1 = κ2 e3 8 2 − 3 2 /d1 ξ − ϑ2 α κ κ 2      2 1 1 + −4ω + 2 /d1 ξ − ϑ2 e3 α − 4κ2 ωe3 α κ 2 κ2

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

267

so that we obtain        α ω 2 1 2 2 κ e3 e3 − 8 2 − 3 2 /d1 ξ − ϑ α κ2 κ κ 2 (    q 4 1 2 = + 10ω − /d1 ξ + 2(η − 3ζ)ξ − ϑ e3 α + − 8ω 2 + 10ω κ r4 κ 4      ) 1 2 1 1 2 2 − 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ − 4κ e3 (η − 3ζ)ξ − 4κ ωe3 α 3 2 κ κ2 which we rewrite as        α 2 1 2 α 2 κ e3 e3 − 8ω − 2 /d1 ξ − ϑ 2 κ κ 2 κ2    q 4 1 = + 10ω − /d1 ξ + 2(η − 3ζ)ξ − ϑ2 e3 α 4 r κ 4 (   1 2 4 2 + − 8ω + 10ω κ − 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ − e3 ((η − 3ζ)ξ) 2 κ ) e3 (κ) e3 (κ) +12 2 (η − 3ζ)ξ + 8 ω α. κ κ Now, recall from above that we have   e3 (κ) 1 1 1 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 . = − κ − 2ω + κ 2 κ 2 We finally deduce        α 2 1 2 α κ2 e3 e3 − 8ω − 2 / d ξ − ϑ 1 2 κ κ 2 κ2    q 4 1 = + 10ω + − /d1 ξ − 2(η − 3ζ)ξ + ϑ2 e3 α 4 r κ 4 (    8 1 2 1 + − 2 /d1 ξ + 6κ − 24ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ ω + ϑ2 κ 2 2    ) 4 48 24 1 − e3 ((η − 3ζ)ξ) + 16 + ω − 2 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 ζξ α. κ κ κ 2 This concludes the proof of the lemma. 6.1.3

Estimates for transport equations in e3

The following lemma will be useful to integrate the transport equations in e3 . Lemma 6.3. Let p ∈

(ext)

M. Let γ[p] the unique integral curve of e3 starting from

268

CHAPTER 6

a point on C1 terminating at p. Then, we have for l ≥ 1 Z 1 1 . 1 1 2+l +δ 1+l 1+δ +δ 1+l extra extra extra r (2r + u) 2 + rl (2r + u)1+δextra γ[p] r 0 + r0 u0 u0 2 and Z

1 1

2 +δextra γ[p] r 0 u0 2

+

.

r0 u0 1+δextra

1 1

r(2r + u) 2 +δextra +

1 log(1+u) (2r

+ u)1+δextra

where (u, r) correspond to p and (r0 , u0 ) to a point on γ[p], and where the integration along γ[p] relies on a parametrization of γ[p] normalized with respect to e3 . Proof. Note first from the construction of (ext) M that γ[p] exists for any p ∈ (ext) M (i.e., any point p can be joined to C1 by an integral curve of e3 ), and γ[p] is included in (ext) M. Next, recall that the integration along γ[p] relies on a parametrization of γ[p] normalized with respect to e3 . To parametrize the integration by u or r, we will thus have to derive an upper bound for the corresponding Jacobian of the change of variable, i.e., for 1 , |e3 (u)| To this end, note that we have on e3 (u) =

1 . |e3 (r)|

(ext)

M

2 2 + O() 1 ≥ ≥ ≥1 ςΥ Υ Υ

since Υ ≤ 1 by definition. Also, we have on |e3 (r)|

≥

= ≥

Hence, we have obtained on

(ext)

(ext)

M in view of Lemma 6.1

1 − |e3 (r) + 1| 1 + O() 1 . 2

M

1 1 ≤ , |e3 (u)| 2

1 ≤ 1. |e3 (r)|

(6.1.1)

Next, since e3 (u) > 0 and e3 (r) < 0 in (ext) M, we have r0 ≥ r and 1 ≤ u0 ≤ u. We start with the proof of the first inequality. We consider two cases: • If r ≥ u, we have Z

1 1

2+l 0 2 +δextra γ[p] r 0 u

≤ .

1 r2+l

Z 0

u

+ r0 1+l u0 1+δextra 1

1 du0 u 2 −δextra . 1 |e3 (u0 )| u0 2 +δextra r2+l 1 1

r1+l (2r + u) 2 +δextra + rl (2r + u)1+δextra

,

269

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

where we used (6.1.1). • If r ≤ u, we separate the integral in r0 ≥ u, which coincides with 1 ≤ u0 ≤ u, and r ≤ r0 ≤ u and compute Z 1 1

r0 2+l u0 2 +δextra + r0 1+l u0 1+δextra 1 du0 1 0 2+l +δ 1+l 1+δ 0 r0 u0 2 extra + r0 u0 extra |e3 (u )| Z u 1 dr0 + 1 0 2+l 0 2 +δextra r r0 u + r0 1+l u0 1+δextra |e3 (r )| Z u 1 1 du0 1 2+l 0 +δextra u 0 |e3 (u )| u0 2   Z u Z u 1 1 dr0 1 1 dr0 + min , 1 u 2 +δextra r |e3 (r0 )| r0 2+l u1+δextra r |e3 (r0 )| r0 1+l   1 1 1 1 1 + min , 5 1 u 2 +δextra u 2 +δextra r1+l u1+δextra rl 1 , 1 +δ 1+l extra r (2r + u) 2 + rl (2r + u)1+δextra γ[p] u

Z =

.

. .

where we used (6.1.1). This proves the first inequality. The second inequality is obtained similarly as follows: • If r ≥ u, we have Z γ[p]

r 0 2 u0

1 2 +δextra

1 + r0 u0 1+δextra

≤ . .

1 r2 u

Z

u

0

1 du0 1 |e3 (u0 )| u0 2 +δextra

1 2 −δextra

r2 1 r(2r + u)

1 2 +δextra

+ (2r + u)1+δextra

,

where we used (6.1.1). • If r ≤ u, we separate the integral in r0 ≥ u, which coincides with 1 ≤ u0 ≤ u,

270

CHAPTER 6

and r ≤ r0 ≤ u and compute Z

1

2 1 +δextra γ[p] r 0 u0 2 Z u

+ r0 u0 1+δextra

du0 |e3 (u0 )| 0 + Z u 1 dr0 + 1 0 2 +δextra r r 0 u0 2 + r0 u0 1+δextra |e3 (r )| Z u 1 1 du0 . 1 3 0 u 0 |e3 (u )| u0 2 +δextra   Z u Z u 1 1 dr0 1 1 dr0 + min , 1 u 2 +δextra r |e3 (r0 )| r0 2 u1+δextra r |e3 (r0 )| r0 Z 1 00 ! 1 1 1 1 dr . + min , 1+δextra 5 1 +δ +δ extra extra r r u r00 u2 u2 u 1

=

.

1 r0 2 u0 2 +δextra

r0 u0 1+δextra

1 r(2r + u)

1 2 +δextra

(2r+u)1+δextra log(1+u)

+

,

where we used (6.1.1). This concludes the proof of the lemma. Corollary 6.4. Let ψ a solution of the following transport equation e3 (ψ) = h on

(ext)

M.

Let also 0 < u1 ≤ u∗ . Then: • If h and ψ satisfy for l ≥ 1 0

|h| .

1 r2+l u 2 +δextra

|ψ| .

0 3 l+ +δ extra 2 r

+ r1+l u1+δextra

on

(ext)

M(u ≤ u1 ),

on C1 ,

we have sup



(ext) M(u≤u ) 1

 1 r1+l (2r + u) 2 +δextra + rl (2r + u)1+δextra |ψ| . 0 .

• If h and ψ satisfy 0

|h| .

1 r2 u 2 +δextra

0

|ψ| .

r

3 2 +δextra

+ ru1+δextra

on

(ext)

M(u ≤ u1 ),

on C1 ,

we have  sup (ext) M(u≤u ) 1

1

r(2r + u) 2 +δextra +

(2r + u)1+δextra log(1 + u)

 |ψ| . 0 .

271

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

Proof. This follows immediately from Lemma 6.3. 6.1.4

Decay estimates for α

We start with an estimate for α on C1 . Lemma 6.5. We have 7

sup r 2 +δextra |dk α| +

max

0≤k≤ksmall +22 C1

max

9

sup r 2 +δextra |dk e3 α| .

0 .

0≤k≤ksmall +21 C1

Proof. Recall that on C1 , we have obtained in Theorem M0 ( h 7   i 9 max sup r 2 +δB |dk (ext) α| + |dk (ext) β| + r 2 +δB |dk−1 e3 ( (ext) α)| 0≤k≤klarge

C1

  ) k (ext) 2m0 2 k (ext) k (ext) + sup r d ρ + 3 + r |d β| + r|d α| . 0 . r C1 

3

Since we have chosen δB ≥ δextra , we deduce h 7 i 9 max sup r 2 +δextra |dk (ext) α| + r 2 +δextra |dk−1 e3 ( (ext) α)| 0≤k≤klarge C1

. 0 .

Next, recall that q is defined with respect to the global frame constructed in Proposition 3.26. In view of Proposition 3.26 and Proposition 3.20, and the change of frame formula for α in Proposition 2.90, we have   3 α = ( (ext) Υ)2 (ext) α + 2f (ext) β + f 2 (ext) ρ + l.o.t. (6.1.2) 2 where f satisfies,1 see (3.4.11), |dk f | .



, for k ≤ ksmall + 22 on (ext) M, ru + u  |dk−1 e3 f | . for k ≤ ksmall + 22 on (ext) M. ru 1 2

(6.1.3)

We easily infer max

7

sup r 2 +δextra |dk α| +

0≤k≤ksmall +22 C1

max

9

sup r 2 +δextra |dk e3 α| .

0≤k≤ksmall +21 C1

0 .

This concludes the proof of the lemma. Next, let 0 < u1 ≤ u∗ . We introduce the following bootstrap assumption for α 1 Here we use (3.4.11) with k loss = 20. Note also that the estimates we claim here for f are slightly weaker that those in (3.4.11).

272 on

CHAPTER 6

(ext)

M(u ≤ u1 ): max

0≤k≤ksmall +20

 r2 (2r + u)1+δextra

sup (ext) M(u≤u

log(1 + u)

1)

1

+ r3 (2r + u) 2 +δextra



  × |dk α| + r|dk e3 α|

(6.1.4) ≤ .

The goal of this section will be the following proposition, i.e., the improvement of these bootstrap assumptions. Proposition 6.6. We have max

0≤k≤ksmall +20

 r2 (2r + u)1+δextra

sup

log(1 + u)

(ext) M(u≤u ) 1

1

+ r3 (2r + u) 2 +δextra



  × |dk α| + r|dk e3 α| .

0 .

Proposition 6.6 will be proved at the end of this section. Based on the bootstrap assumptions (6.1.4), we estimate the RHS of the transport equation for α. Lemma 6.7. We have 



e3 e3

α κ2



 − F1

=

F2

where F1 and F2 satisfy max

0≤k≤ksmall +20

+

sup (ext) M(u≤u

max

1)

0≤k≤ksmall +20

  1 r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dk F1 | sup (ext) M(u≤u ) 1



 1 r2 u1+δextra + r3 u 2 +δextra |dk F2 | . 0 .

Proof. Recall that we have        α 2 1 2 α 2 κ e3 e3 − 8ω − 2 /d1 ξ − ϑ 2 κ κ 2 κ2    q 4 1 = + 10ω + − /d1 ξ − 2(η − 3ζ)ξ + ϑ2 e3 α 4 r κ 4 (    8 1 2 1 + − 2 /d1 ξ + 6κ − 24ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ ω + ϑ2 κ 2 2    ) 4 48 24 1 2 − e3 ((η − 3ζ)ξ) + 16 + ω − 2 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ ζξ α κ κ κ 2 which we rewrite as     α e3 e3 − F = 1 κ2

F2

273

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

where F1 and F2 are defined by    2 1 α F1 := 8ω − 2 /d1 ξ − ϑ2 κ 2 κ2 and F2

   q 1 4 1 2 + 10ω + − / d ξ − 2(η − 3ζ)ξ + ϑ e3 α 1 r4 κ2 κ2 κ 4 (    1 8 1 1 + 2 − 2 /d1 ξ + 6κ − 24ω + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 ω + ϑ2 κ κ 2 2    ) 4 48 24 1 − e3 ((η − 3ζ)ξ) + 16 + ω − 2 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 ζξ α. κ κ κ 2

:=

In view of the bootstrap assumptions (6.1.4) for α, the estimates of Lemma 6.1 for the Ricci coefficients, and using Theorem M2 to estimate q, we easily infer   1 max sup r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dk F1 | 0≤k≤ksmall +20

(ext) M(u≤u

1)

 . 

max

0≤k≤ksmall +20

sup (ext) M(u≤u ) 1

1 r2 (2r + u)1+δextra + r3 (2r + u) 2 +δextra log(1 + u)





 × |dk α| + r|dk e3 α| . 2 . 0 and max

0≤k≤ksmall +20

.

sup (ext) M(u≤u

max

0≤k≤ksmall +20

 1 r2 u1+δextra + r3 u 2 +δextra |dk F2 |



 1 u1+δextra + ru 2 +δextra |dk q|

1)

sup (ext) M(u≤u



1)

 +

max

0≤k≤ksmall +20



sup (ext) M(u≤u

1)

1 r2 (2r + u)1+δextra + r3 (2r + u) 2 +δextra log(1 + u)

 × |d α| + r|d e3 α| k

k

. 0 + 2 . 0 . This concludes the proof of the lemma. Lemma 6.8. For 0 ≤ k + j ≤ ksmall + 20, we have     α j e3 e3 d/k ej4 − F = F2, d/k ,ej 1, d /k ,e4 4 κ2



274

CHAPTER 6

where max

0≤l≤ksmall +20−k

+



sup (ext) M(u≤u ) 1

max

0≤l≤ksmall +20−k

 1 r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dl F1, d/k | 

sup (ext) M(u≤u

1)

 1 r2 u1+δextra + r3 u 2 +δextra |dl F2, d/k |

. 0 , and for j ≥ 1 max

0≤l≤ksmall +20−k−j



sup (ext) M(u≤u

1

r1+j (2r + u)1+δextra + r2+j (2r + u) 2 +δextra



1)

l

×|d F1, d/k ,ej | 4

+

max

0≤l≤ksmall +20−k−j

. 0 +



sup (ext) M(u≤u

max

1)

 1 r2+j u1+δextra + r3+j u 2 +δextra |dl F2, d/k ,ej | 4

 r2 (2r + u)1+δextra

sup

log(1 + u)  × | d/k (re4 )j−1 e3 α| + | d/k (re4 )j−2 e23 α| . 0≤j+k≤ksmall +20

 1 + r3 (2r + u) 2 +δextra r

(ext) M(u≤u ) 1



Proof. Recall from Lemma 6.7 that we have     α e3 e3 − F = 1 κ2

F2

where F1 and F2 satisfy   1 max sup r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dk F1 | 0≤k≤ksmall +20 (ext) M   1 + max sup r2 u1+δextra + r3 u 2 +δextra |dk F2 | . 0 . 0≤k≤ksmall +20

(ext) M

Differentiating with d/k , this yields       α α k k e3 e3 d/k + [ d / , e ] − d / F = 3 1 κ2 κ2

    α d/k F2 − [ d/k , e3 ] e3 − F 1 κ2

and hence     α e3 e3 d/k − F k 1, d / κ2

= F2, d/k

where F1, d/k := d/k F1 − [ d/k , e3 ]



α κ2

 ,

    α F2, d/k := d/k F2 − [ d/k , e3 ] e3 − F 1 . κ2

In view of Lemma 2.68, we have schematically [ d/, e4 ]

=

Γg d + Γg + rβ,

[ d/, e3 ]

=

Γb d + Γb + rβ.

275

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

Together with the estimates of Lemma 6.1 for the Ricci coefficients and curvature components as well as the bootstrap assumptions (6.1.4) for α on (ext) M, we infer   1 max sup r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dj F1, d/k | 0≤j≤ksmall +20−k (ext) M   1 + max sup r2 u1+δextra + r3 u 2 +δextra |dj F2, d/k | . 0 . 0≤j≤ksmall +20−k

(ext) M

Next, we consider the case j ≥ 1. We have the commutator [e4 , e3 ]

2ωe3 − 2ωe4 − 4ζeθ .

=

In view of the estimates of Lemma 6.1 for the Ricci coefficients, and in view of the bootstrap assumptions (6.1.4) for α, we infer after commutation by ej4 for 0 ≤ k + j ≤ ksmall + 20     α k j e3 e3 d/ e4 − F1, d/k ,ej = F2, d/k ,ej 4 4 κ2 where max

0≤l≤ksmall +20−k−j



sup (ext) M(u≤u

1

r1+j (2r + u)1+δextra + r2+j (2r + u) 2 +δextra



1)

l

×|d F1, d/k ,ej | 4

+

max

0≤l≤ksmall +20−k−j

. 0 +



sup (ext) M(u≤u

max

1)

sup

 1 r2+j u1+δextra + r3+j u 2 +δextra |dl F2, d/k ,ej | 4

 r2 (2r + u)1+δextra

log(1 + u)  × | d/k (re4 )j−1 e3 α| + | d/k (re4 )j−2 e23 α| . 0 . 0≤j+k≤ksmall +20

 1 + r3 (2r + u) 2 +δextra r

(ext) M(u≤u ) 1



This concludes the proof of the lemma. We are now ready to prove Proposition 6.6. Step 1. For 0 ≤ k ≤ ksmall + 20, recall from the above lemma with j = 0 that we have     α k e3 e3 d/ − F1, d/k = F2, d/k κ2 where max

0≤j≤ksmall +20−k

sup (ext) M(u≤u ) 1



 1 r2 u1+δextra + r3 u 2 +δextra |dj F2, d/k | .

Also, we have in view of Lemma 6.5   5 α sup r 2 +δextra e3 d/k − F k 1, d / 2 0≤k≤klarge −4 C1 κ max

. 0 .

0 .

276

CHAPTER 6

In view of Corollary 6.4, we immediately infer for any 0 ≤ k ≤ ksmall + 20   1 max sup r2 (2r + u) 2 +δextra + r(2r + u)1+δextra 0≤k≤ksmall +20

(ext) M(u≤u ) 1

  α × e3 d/k − F k 1, d / 2 κ

. 0 .

Since we have from the above lemma that   1 max sup r(2r + u)1+δextra + r2 (2r + u) 2 +δextra |dj F1, d/k | . 0 , 0≤j≤ksmall +20−k

(ext) M(u≤u ) 1

we deduce that we have for any 0 ≤ k ≤ ksmall + 20   1 max sup r2 (2r + u) 2 +δextra + r(2r + u)1+δextra 0≤k≤ksmall +20

(6.1.5)

(ext) M(u≤u ) 1

  α k × e3 d/ . 0 . κ2 Step 2. Next, note that we have in view of Lemma 6.5   3 α max sup r 2 +δextra d/k . 0 . 0≤k≤klarge −3 C1 κ2 Together with the transport equation (6.1.5), and in view of Corollary 6.4, we infer     1 (2r + u)1+δextra k α +δextra 2 max sup r(2r + u) + . 0 . d/ 0≤k≤ksmall +20 (ext) M(u≤u ) log(1 + u) κ2 1 In view of the control of κ provided by Lemma 6.1, we easily deduce   1 r2 (2r + u)1+δextra k max sup r3 (2r + u) 2 +δextra + d/ α . 0 . 0≤k≤ksmall +20 (ext) M(u≤u ) log(1 + u) 1 Together with (6.1.5), we infer   1 r2 (2r + u)1+δextra 3 +δ extra max sup r (2r + u) 2 + 0≤k≤ksmall +20 (ext) M(u≤u ) log(1 + u) 1   × | d/k α| + r| d/k e3 α| .

0 .

Step 3. Next, recall from section 2.3.3 that q is defined with respect to a general null frame as follows:     1 q = r4 e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω 2 − 8ω κ + κ2 α . 2 We infer e3 (e3 (α))

=

  q 1 2 2 − (2κ − 6ω)e (α) − −4e (ω) + 8ω − 8ω κ + κ α. 3 3 r4 2

277

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

Together with the above estimate for α and e3 α, we infer by iteration max

0≤k≤ksmall +20

 r2 (2r + u)1+δextra

sup

1

+ r3 (2r + u) 2 +δextra



log(1 + u)   × |( d/, e3 )k α| + r|( d/, e3 )k e3 α| . 0 .

(ext) M(u≤u ) 1

Step 4. Arguing as for Step 1, but with j ≥ 1, we infer the following analog of (6.1.5):   1 max sup r2 (2r + u) 2 +δextra + r(2r + u)1+δextra 0≤j+k≤ksmall +20

(ext) M(u≤u

1)

  α × e3 d/k (re4 )j κ2 . 0 +

max

sup

 r2 (2r + u)1+δextra

log(1 + u)  × | d/k (re4 )j−1 e3 α| + | d/k (re4 )j−2 e23 α| . 0≤j+k≤ksmall +20

 1 + r3 (2r + u) 2 +δextra r

(ext) M(u≤u ) 1



Step 5. Arguing as for Step 2, but with j ≥ 1, we infer the following analog of the last estimate of Step 2:   1 r2 (2r + u)1+δextra max sup r3 (2r + u) 2 +δextra + 0≤j≤ksmall +20 (ext) M(u≤u ) log(1 + u) 1   × | d/k (re4 )j α| + r| d/k (re4 )j e3 α|  r2 (2r + u)1+δextra  1 . 0 + max sup + r3 (2r + u) 2 +δextra r 0≤j+k≤ksmall +20 (ext) M(u≤u ) log(1 + u) 1   × | d/k (re4 )j−1 e3 α| + | d/k (re4 )j−2 e23 α| . Step 6. Arguing as for Step 3, but with j ≥ 1, we infer the following analog of the last estimate of Step 3: max

 r2 (2r + u)1+δextra

sup

1

+ r3 (2r + u) 2 +δextra



log(1 + u)  × |( d/, e3 )k (re4 )j α| + r|( d/, e3 )k (re4 )j e3 α|  r2 (2r + u)1+δextra  1 . 0 + max sup + r3 (2r + u) 2 +δextra r 0≤j+k≤ksmall +20 (ext) M(u≤u ) log(1 + u) 1   × |( d/, e3 )k (re4 )j−1 e3 α| + r|( d/, e3 )k (re4 )j−2 e23 α| . 0≤j+k≤ksmall +20

(ext) M(u≤u

1)



Step 7. Arguing by iteration on j, noticing that the last estimate of Step 3 corresponds to the desired estimate for j = 0, and in view of the estimate derived

278

CHAPTER 6

in Step 6, we finally obtain max

0≤j+k≤ksmall +20

 r2 (2r + u)1+δextra

sup (ext) M(u≤u

log(1 + u)

1)

1

+ r3 (2r + u) 2 +δextra





 × |( d/, e3 )k (re4 )j α| + r|( d/, e3 )k (re4 )j e3 α|

. 0

and hence max

 r2 (2r + u)1+δextra

sup

0≤k≤ksmall +20

(ext) M(u≤u ) 1

log(1 + u)

1

+ r3 (2r + u) 2 +δextra



  × |dk α| + r|dk e3 α| .

0 .

This concludes the proof of Proposition 6.6. 6.1.5

End of the proof of Theorem M2

First, note in view of the estimates for α on C1 provided by Lemma 6.5 that the bootstrap assumptions (6.1.4) for α hold by continuity for some sufficiently small u1 > 0. Then, we may in view of Proposition 6.6 choose u1 = u∗ . We deduce therefore max

0≤k≤ksmall +20

sup

 r2 (2r + u)1+δextra log(1 + u)

(ext) M

 1 + r3 (2r + u) 2 +δextra   × |dk α| + r|dk e3 α|

. 0 .

Next, recall from (6.1.2) and (6.1.3) that we have   3 α = ( (ext) Υ)2 (ext) α + 2f (ext) β + f 2 (ext) ρ + l.o.t. 2 where f satisfies |dk f | .



, for k ≤ ksmall + 22 on (ext) M, ru + u  |dk−1 e3 f | . for k ≤ ksmall + 22 on (ext) M. ru 1 2

Together with bootstrap assumptions for max

0≤k≤ksmall +20

sup (ext) M

(ext)

 r2 (2r + u)1+δextra

β and

(ext)

ρ, we easily infer 1

+ r3 (2r + u) 2 +δextra



log(1 + u)   × |dk (ext) α| + r|dk e3 (ext) α|

This concludes the proof of Theorem M2.

. 0 .

279

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

6.2

PROOF OF THEOREM M3

Theorem M3 contains decay estimates for α in with the estimate on (int) M before moving to 6.2.1

Estimate for α in

(int)

(int)

M and on Σ∗ . We first proceed M.

(ext)

M

Recall that q, controlled in Theorem M1, is defined with respect to the global frame of Proposition 3.26. Recall also that we may choose the global null frame to coincide with the ingoing geodesic null frame of (int) M in (int) M (see property (b) in Proposition 3.26 together with property (d) ii. in Proposition 3.23). Thus, in this section, as we only work on (int) M, the null frame (e4 , e3 , eθ ) denotes both the frame of (int) M and the global frame with respect to which q is defined. We start with the following definition. Definition 6.9. In (int) M

(int)

M, we define with respect to the ingoing geodesic frame of  1 Te := e4 − κ + A e3 . κ

The estimate for α in

(int)

(6.2.1)

M relies on the following proposition.

Proposition 6.10. Let 0 ≤ k ≤ ksmall + 17. Then, α satisfies in 6mTe(dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

=

(int)

M

Fk

where Fk satisfies Z max

0≤k≤ksmall +17

(int) M

u2+2δdec |d≤1 Fk |2

. 20 .

Remark 6.11. In view of the definition of Te, we have  1 Te(r) = e4 (r) − κ + A e3 (r) = 0 κ so that Te is tangent to the hypersurfaces of constant r. In particular, (Te, eθ ) spans the tangent space of hypersurfaces of constant r. Therefore, in view of Proposition 6.10, α and its derivatives satisfy on each hypersurface of constant r in (int) M, i.e., on {r = r0 } for 2m0 (1 − δH ) ≤ r ≤ rT , a forward parabolic equation. Furthermore, since we have Te(u) = 2/ς = 2 + O(), u plays the role of time in this forward parabolic equation. We also derive estimates for the control of the parabolic equation appearing in the statement of Proposition 6.10. Lemma 6.12. Let f and h reduced 2-scalars such that   6mTe + r4 /d?2 /d?1 /d1 /d2 f = h. Then, for any real number n ≥ 0 and any r0 such that 2m0 (1 − δH ) ≤ r0 ≤ rT , we

280

CHAPTER 6

have Z

n

sup 1≤u≤u∗

(1 + u )f

Z

2

2

f +

.n

S(r=r0 ,u)

2

S(r=r0 ,1)

Z

u∗

Z 1

Z

(1 + un−2 )(df )2

S(r=r0 ,u)

(1 + un )(d≤1 h)2 .

+ (int) M

We are now in position to control α in (int) M. Recall from Proposition 6.10 that α satisfies in (int) M for 0 ≤ k ≤ ksmall + 17 6mTe(dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

= Fk .

Applying Lemma 6.12 with n = 2 + 2δdec , f = dk α and h = Fk , we infer for any r0 such that 2m0 (1 − δH ) ≤ r0 ≤ rT Z sup (1 + u2+2δdec )(dk α)2 1≤u≤u∗

S(r=r0 ,u)

Z

(dk α)2 + 2

. S(r=r0 ,1)

Z

Z

u∗

1

Z

(1 + u2δdec )(dk+1 α)2

S(r=r0 ,u)

(1 + u2+2δdec )(d≤1 Fk )2 .

+ (int) M

Together with the bounds for α on C 1 provided by Theorem M0, the bootstrap assumptions on decay and energy for α in (int) M, and the bound for Fk provided by Proposition 6.10, we infer for 0 ≤ k ≤ ksmall + 17 in (int) M Z sup (1 + u2+2δdec )(dk α)2 . 20 . 1≤u≤u∗

S(r=r0 ,u)

In particular, we have obtained max

0≤k≤ksmall +17

sup u1+δdec kdk αkL2 (S)

. 0 .

(int) M

Using the Sobolev embedding on 2-surface and the fact that r is bounded on we infer max

0≤k≤ksmall +15

sup u1+δdec |dk α| .

(int)

M,

0

(int) M

and hence (int)

Dksmall +15 [α] . 0

(6.2.2)

which is the desired estimate for α in (int) M. The proof of Proposition 6.10 will be given in section 6.2.3, and the proof of Lemma 6.12 in section 6.2.4. But first we show, in the next section, how to conclude the proof of Theorem M3 by controlling α on Σ∗ .

281

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

6.2.2

Estimate for α on Σ∗

Recall that q, controlled in Theorem M1, is defined with respect to the global frame of Proposition 3.26. We will first control α in this frame, before coming back to (ext) M at the end of the argument. We start with the following definition. Definition 6.13. In Σ∗ , we define, with respect to the global frame of Proposition 3.26, νe := e3 + ae4 ,

(6.2.3)

where the scalar function a is uniquely defined so that νe is tangent to Σ∗ . The estimate for α on Σ∗ relies on the following proposition. Proposition 6.14. Let 0 ≤ k ≤ ksmall + 18. Then, α satisfies on Σ∗ 6me ν (dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

=

Fk

.

20 .

where Fk satisfies Z max

0≤k≤ksmall +18

Σ∗

u2+2δdec |Fk |2

Remark 6.15. Since νe is tangent to Σ∗ , and since (e ν , eθ ) spans the tangent space of Σ∗ , in view of Proposition 6.14, α and its derivatives satisfy on Σ∗ a forward parabolic equation. Furthermore, since we have νe(u) = 2 + O(), u plays the role of time in this forward parabolic equation. We also derive estimates for the control of the parabolic equation appearing in the statement of Proposition 6.10. Lemma 6.16. Let f and h reduced 2-scalars such that   6me ν + r4 /d?2 /d?1 /d1 /d2 f = h. Then, for any real number n ≥ 0, we have Z Z Z (1 + un )f 2 .n f 2 + 2 Σ∗

Σ∗ ∩C1

(1 + un−2 )(df )2 +

Σ∗

Z

(1 + un )h2 .

Σ∗

Using the lemma we are in position to control α on Σ∗ . According to Proposition 6.14 α satisfies in Σ∗ , for 0 ≤ k ≤ ksmall + 18, 6me ν (dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

=

Fk .

Applying Lemma 6.16 with n = 2 + 2δdec , f = dk α and h = Fk , we infer Z Z Z 2+2δdec k 2 k 2 2 (1 + u )(d α) . (d α) +  (1 + u2δdec )(dk+1 α)2 Σ∗ Σ∗ ∩C1 Σ∗ Z + (1 + u2+2δdec )(Fk )2 . Σ∗

Together with the bounds for α on C 1 provided by Theorem M0, the bootstrap

282

CHAPTER 6

assumptions on decay and energy for α in (ext) M, and the bound for Fk provided by Proposition 6.14, we infer Z (1 + u2+2δdec )(dk α)2 . 20 . Σ∗

In particular, we have obtained Z

(1 + u2+2δdec )(dk α)2

max

0≤k≤ksmall +18

. 0 .

Σ∗

Now, recall that α in the above estimate is defined with respect to the global frame of Proposition 3.26. In view of Proposition 3.26 and Proposition 3.20, and the change of frame formula for α in Proposition 2.90, we have α

=

( (ext) Υ)−2 (ext) α.

Hence, we immediately infer Z max

0≤k≤ksmall +18

(1 + u2+2δdec )(dk(ext) α)2

.

0

Σ∗

which is the desired estimate in Σ∗ . Together with (6.2.2), this concludes the proof of Theorem M3. The proof of Proposition 6.14 will be given in section 6.2.5, and the proof of Lemma 6.16 will be given in section 6.2.6. 6.2.3

Proof of Proposition 6.10

In this section we derive as corollary of the Teukolsky-Starobinsky identity, see Proposition 2.101, a parabolic equation for α. Corollary 6.17. The quantity α satisfies in

=

(int)

M the following equation:

6mTeα + r4 /d?2 /d?1 /d1 /d2 α  1 2 2 e (r e (rq)) + 2ωr e (rq) − r−3 Err[T S] 3 3 3 r3     3 4 2m 2 − r ρ + 3 κ − 3mr κ + e4 α 2 r r       3 4 2m 3mr 2 6m − − r ρ+ 3 κ+ κ+ κ + 3mrˇ κ+ A e3 α 2 r κ r κ

where the vectorfield Te is defined by (6.2.1). Proof. According to Proposition 2.3.15, we have    3  e3 (r2 e3 (rq)) + 2ωr2 e3 (rq) = r7 /d?2 /d?1 /d1 /d2 α + ρ κe4 − κe3 α + Err[T S]. 2

283

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

This yields  3 4  r ρ κe4 − κe3 α + r4 /d?2 /d?1 /d1 /d2 α 2

=

 1 2 2 e (r e (rq)) + 2ωr e (rq) 3 3 3 r3 −r−3 Err[T S].

Now, we have in view of the definition of Te  3 4  r ρ κe4 − κe3 − 6mTe 2      3 4 2m 2 = r ρ + 3 κ − 3mr κ + e4 2 r r       3 2m 3mr 2 6m + − r4 ρ + 3 κ + κ+ κ + 3mrˇ κ+ A e3 . 2 r κ r κ We infer 6mTeα + r4 /d?2 /d?1 /d1 /d2 α  1 2 2 = e (r e (rq)) + 2ωr e (rq) − r−3 Err[T S] 3 3 3 r3     3 4 2m 2 r ρ + 3 κ − 3mr κ + e4 α − 2 r r       3 4 2m 3mr 2 6m − − r ρ+ 3 κ+ κ+ κ + 3mrˇ κ+ A e3 α. 2 r κ r κ This concludes the proof of the corollary. Corollary 6.18. α satisfies in

(int)

M

6mTeα + r4 /d?2 /d?1 /d1 /d2 α

= F

where F satisfies Z max

0≤k≤ksmall +18

(int) M

u2+2δdec |dk F |2

. 20 .

Proof. In view of Corollary 6.17, α satisfies 6mTeα + r4 /d?2 /d?1 /d1 /d2 α

= F

with F F1

 1 2 2 e (r e (rq)) + 2ωr e (rq) + F1 , 3 3 3 r3      3 4 2m 2 := −r−3 Err[T S] − r ρ + 3 κ − 3mr κ + e4 α 2 r r       3 2m 3mr 2 6m − − r4 ρ + 3 κ + κ+ κ + 3mrˇ κ+ A e3 α. 2 r κ r κ :=

Using the bootstrap assumptions in

(int)

M for decay and energies, and in view of

284

CHAPTER 6

the fact that F1 contains only quadratic or higher order terms, we easily derive 3

max

0≤k≤ksmall +18

3

sup u 2 + 2 δdec |dk F1 | . 2 . 0 .

(int) M

In view of the definition of F , this yields Z max u2+2δdec |dk F |2 . 20 + 0≤k≤ksmall +18

(int) M

Z max

0≤k≤ksmall +20

(int) M

u2+2δdec |dk q|2 .

Together with Theorem M1, and the fact that δextra > δdec , we infer Z max u2+2δdec |dk F |2 . 20 . 0≤k≤ksmall +18

(int) M

This concludes the proof of the corollary. We are now ready to prove Proposition 6.10. In view of Corollary 6.18, α satisfies 6mTeα + r4 /d?2 /d?1 /d1 /d2 α

= F.

Commuting with dk , we infer 6mTe(dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

=

Fk

where Fk is defined by Fk

:= −6m[dk , Te]α − 6

k X j=1

dj (m)dk−j Teα − [dk , r /d?2 ]r /d?1 r /d1 r /d2 α

−r /d?2 [dk , r /d?1 ]r /d1 r /d2 α k +d F.

− r /d?2 r /d?1 [dk , r /d1 ]r /d2 α − r /d?2 r /d?1 r /d1 [dk , r /d2 ]α

Note that we have schematically ˇ [d, d/] = Γd,

  ˇ+Γ ˇ d, [Te, d/] = dΓ

285

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

as well as [Te, re4 ]

= =

=

=

=

    1 1 e4 (r)e4 − κ + A [e3 , re4 ] + re4 κ + A e3 κ κ  r   r 1 κ + A e4 − κ+A κe4 + r − 2ωe3 + 4ζeθ 2 κ 2     1 κ A e3 e3 + re4 +re4 κ κ        1 κ 2r A e3 + re4 κ + A ω + re4 κ κ κ  4r  − κ + A ζeθ κ (   2m κ 2r  m  2m(e4 (r) − Υ) − +Υ + κ ω+ − + 2e4 (m) r κ κ r r    )  κ 2r 1 4r  +re4 + Υ + Aω + re4 A e3 − κ + A ζeθ κ κ κ κ   ˇ+Γ ˇ d, dΓ

and [Te, e3 ]

   1 [e4 , e3 ] + e3 κ+A e3 κ      κ 1 = 2ω + e3 + e3 A e3 − 4ζeθ κ κ (    m  2m(e3 (r) + 1) 2e3 (m) κ = 2 ω+ 2 − + + e3 +Υ r r2 r κ  ) 1 +e3 A e3 − 4ζeθ κ   ˇ+Γ ˇ d. = dΓ =

Together with the bootstrap assumptions in (int) M for decay and energies, and in view of the fact that F1 contains only quadratic or higher order terms, we easily derive k X 3 3 + δ max sup u 2 2 dec dj (m)dk−j Teα − 6m[dk , Te]α − 6 0≤k≤ksmall +18 (int) M j=1

k

−[d , r /d?2 ]r /d?1 r /d1 r /d2 α −r /d?2 r /d?1 [dk , r /d1 ]r /d2 α

− r /d?2 [dk , r /d?1 ]r /d1 r /d2 α

−r /d?2 r /d?1 r /d1 [dk , r /d2 ]α . 2 .

286

CHAPTER 6

In view of the definition of Fk , this yields Z max u2+2δdec |Fk |2 . 4 + 0≤k≤ksmall +18

Z max

0≤k≤ksmall +18

(int) M

(int) M

u2+2δdec |dk F |2 .

Together with the estimate for F of Corollary 6.18, we infer Z max u2+2δdec |Fk |2 . 4 + 20 . 20 . 0≤k≤ksmall +18

(int) M

This concludes the proof of Proposition 6.10. 6.2.4

Proof of Lemma 6.12

In this section we prove Lemma 6.12, i.e., we derive estimates for the control of the parabolic equation appearing in the statement of Proposition 6.10. To this end, we first start with a Poincar´e inequality. Lemma 6.19. We have Z f /d?2 /d?1 /d1 /d2 f S

≥ 24

Z

(1 + O())K 2 f 2 .

S

Proof. We have /d?2 /d?1 /d1 /d2

=

/d?2 (−4 / 1 + K) /d2

= − /d?2 4 / 1 /d2 + K /d?2 /d2 + /d?1 (K) /d2   = −4 / 2 /d?2 /d2 + 4 / 2 /d?2 − /d?2 4 / 1 /d2 + K /d?2 /d2 + /d?1 (K) /d2   = ( /d?2 /d2 − 2K) /d?2 /d2 + 3K /d?2 − /d?1 (K) /d2 + K /d?2 /d2 + /d?1 (K) /d2 =

( /d?2 /d2 )2 + 2K /d?2 /d2 .

Recall also the Poincar´e inequality for /d2 which holds for any reduced 2-scalar f Z Z | /d2 f |2 ≥ 4 Kf 2 . S

Then, we easily infer Z f /d?2 /d?1 /d1 /d2 f S

S

Z

Z

2Kf /d?2 /d2 f Z Z ≥ 42 (1 + O())K 2 f 2 + 8 (1 + O())K 2 f 2 S ZS ≥ 24 (1 + O())K 2 f 2 =

f ( /d?2 /d2 )2 f

+

S

S

S

where we also used the estimates for the Gauss curvature   1 K = 2 + O 2 , reθ (K) = O 2 , r r r which follow from the bootstrap assumptions.

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

287

The following identity will be useful. Lemma 6.20. We have for any reduced scalar f Z   Z  Z Z  κ 2 κ ˇf 2 Af e3 (f ) + κ ˇf 2 − Te f2 = 2 f Tef + κ κ S S S Z   Z S  1 − A (2f e3 (f ) + κf 2 ) + Err e4 f2 . κ S S Proof. Recall from the definition of Te that  1 Te = e4 − κ + A e3 . κ We infer, in view of the analog of Proposition 2.64 for an ingoing geodesic foliation, Z  2 e T f ZS    Z 1 2 2 = e4 f − κ + A e3 f κ S S  Z   Z  Z 1 2 2 2 = (2f e4 (f ) + κf ) + Err e4 f − κ+A (2f e3 (f ) + κf ) κ S S S   Z  Z   2 = 2f Tef + κ + A f e3 (f ) + κf 2 + Err e4 f2 κ S S   Z 1 2 − κ+A (2f e3 (f ) + κf ) κ S  Z  Z Z  2 κ 2 2 e = 2 fTf + Af e3 (f ) + κ ˇf − κ ˇf κ κ S S Z   Z S  1 − A (2f e3 (f ) + κf 2 ) + Err e4 f2 . κ S S This concludes the proof of the lemma. We are now ready to prove Lemma 6.12. Recall from Lemma 6.20 that we have Z   Z  Z Z  2 κ Te f2 = 2 f Tef + Af e3 (f ) + κ ˇf 2 − κ ˇf 2 κ κ S S S Z   Z S  1 − A (2f e3 (f ) + κf 2 ) + Err e4 f2 . κ S S In view of the equation satisfied by f , we infer Z  Z  Z  1 1 2 4 ? ? e T f = − r f /d2 /d1 /d1 /d2 f + hf 3m 3m S S S   Z  Z 2 κ 2 2 + Af e3 (f ) + κ ˇf − κ ˇf κ κ S S Z   Z  1 2 2 − A (2f e3 (f ) + κf ) + Err e4 f . κ S S

288

CHAPTER 6

Now, from the definition of Te, we have Te(u) = 2/ς. We deduce  Z  Z  un n 2 4 ? ? e T u f + r f /d2 /d1 /d1 /d2 f 3m S S Z    Z  Z un 2 κ = hf + un Af e3 (f ) + κ ˇ f 2 − un κ ˇf 2 3m κ κ S S Z   Z  S Z n 2 u 2 n 2 (2f e3 (f ) + κf ) + u Err e4 f + nun−1 f 2. − A κ ς S S S This yields in view of the bootstrap assumptions  Z  Z  un Te un f2 + r4 f /d?2 /d?1 /d1 /d2 f 3m S S Z Z un . khkL2 (S) kf kL2 (S) + un−1 |f ||d≤1 f | + nun−1 f 2. 3m S S Next, we rely on the Poincar´e inequality of Lemma 6.19 to deduce  Z  Z Z Z Z n 2 n 2 n 2 2 n−2 2 n−1 e T u f +u f . u h + u (df ) + nu f 2. S

S

S

S

S

Integrating in u between 1 and u∗ , and recalling that Te(u) = 2/ς, we infer for any r0 such that 2m0 (1 − δH ) ≤ r0 ≤ rT ! Z Z Z u

n

u0 f 2

un f 2 +

S(r=r0 ,u)

Z

u

Z

2

!

Z

0n 2

f +

. S(r=r0 ,1)

+2

u

Z

du0

S(r=r0 ,u0 )

1

du0

u h 1

S(r=r0

,u0 )

!

Z u

0 n−2

du0 + n

(df )2

S(r=r0 ,u0 )

1

u

Z

!

Z

0 n−1 2

u

du0 .

f

S(r=r0 ,u0 )

0

In particular, we have for n = 0 Z

2

sup 1≤u≤u∗

Z

u∗

!

Z

f +

f

S(r=r0 ,u)

Z

2

Z

1 u∗

S(r=r0 ,1)

Z

! 2

h 1

du

S(r=r0 ,u)

f +

.

2

S(r=r0 ,u)

du + 

2

Z

u∗

Z

! −2

u 1

2

(df )

du.

S(r=r0 ,u)

Then, starting from the case n = 0 and arguing by iteration on the largest integer

289

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

below n, one immediately deduces for any real n ≥ 0 Z

n

sup

Z

2

n

(1 + u )f

S(r=r0 ,u)

Z

1 u∗

Z

2

S(r=r0 ,1)

Z

u∗

2

du

S(r=r0 ,u)

!

Z

n−2

(1 + u 1

du

! n

(1 + u )h 1

2

S(r=r0 ,u)

Z

f +

. +

!

Z

(1 + u )f +

1≤u≤u∗

2

u∗

)(df )

2

du.

S(r=r0 ,u)

Now, a simple trace estimate yields Z Z   (1 + un )h2 . (1 + un ) |h|2 + |e3 (h)|2 Cu

S(r=r0 ,u)

so that Z u∗

!

Z

n

2

(1 + u )h 1

u∗

Z

Z

du .

S(r=r0 ,u)

Cu

1

Z

  (1 + un ) |h|2 + |e3 (h)|2 du

(1 + un )(d≤1 h)2 .

. (int) M

We deduce Z

n

sup

Z

u∗

!

Z

n

(1 + u )f +

1≤u≤u∗

S(r=r0 ,u)

Z

2

Z

1

u∗

2

du

S(r=r0 ,u)

(1 + un )(d≤1 h)2 (int) M

S(r=r0 ,1)

Z

(1 + u )f 1

f +

. +2

2

Z

! (1 + un−2 )(df )2

du

S(r=r0 ,u)

which concludes the proof of Lemma 6.12. 6.2.5

Proof of Proposition 6.14

In this section, we infer from the Teukolsky-Starobinsky identity, see Proposition 2.101, a parabolic equation for α. Corollary 6.21. α satisfies on Σ∗ the following equation: 6me ν α + r4 /d?2 /d?1 /d1 /d2 α     1 3 4 2 2 −3 = e3 (r e3 (rq)) + 2ωr e3 (rq) − r Err[T S] − r ρκ − 6am e4 α r3 2       3 2m 2Υ 12m − − r4 ρ + 3 κ + 3mr κ − − e3 α 2 r r r where the vectorfield νe is defined by (6.2.3).

290

CHAPTER 6

Proof. Recall from (2.3.15) that we have    3  e3 (r2 e3 (rq)) + 2ωr2 e3 (rq) = r7 /d?2 /d?1 /d1 /d2 α + ρ κe4 − κe3 α + Err[T S]. 2 This yields  3 4  r ρ κe4 − κe3 α + r4 /d?2 /d?1 /d1 /d2 α 2

=

 1 2 2 e (r e (rq)) + 2ωr e (rq) 3 3 3 r3 −r−3 Err[T S].

Now, we have in view of the definition of νe  3 4  r ρ κe4 − κe3 − 6me ν 2         3 4 3 4 2m 2Υ 12m r ρκ − 6am e4 + − r ρ + 3 κ + 3mr κ − − e3 . = 2 2 r r r We infer 6me ν α + r4 /d?2 /d?1 /d1 /d2 α    1 3 4 2 2 −3 = e (r e (rq)) + 2ωr e (rq) − r Err[T S] − r ρκ − 6am e4 α 3 3 3 r3 2       3 2m 2Υ 12m − − r4 ρ + 3 κ + 3mr κ − − e3 α. 2 r r r This concludes the proof of the corollary. Corollary 6.22. α satisfies on Σ∗ 6me ν α + r4 /d?2 /d?1 /d1 /d2 α

= F

where F satisfies Z max

0≤k≤ksmall +18

Σ∗

u2+2δdec |dk F |2

. 20 .

Proof. In view of Corollary 6.17, α satisfies 6me ν α + r4 /d?2 /d?1 /d1 /d2 α

= F

with F F1

:= e3 (e3 (q)) + F1 ,  1 2 3 2 := e (r e (r)q) + e (r )e (q) + 2ωr e (rq) − r−3 Err[T S] 3 3 3 3 3 r3  3 4 − r ρκ − 6am e4 α 2       3 2m 2Υ 12m − − r4 ρ + 3 κ + 3mr κ − − e3 α. 2 r r r

291

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

Recall also that Err[T S] is given schematically by, see Proposition 2.101,  Err[T S] = r4 d/Γb + rΓb · Γb ) · α + r2 Γb e3 (rq) + (d≤1 Γb )rq

+ r7 d≤2 e3 η · β + r5 d≤3 Γb · Γg . We infer that F1 is given schematically by F1

= r d/Γb + rΓb · Γb ) · α + r−1 Γb e3 (rq) + (d≤1 Γb )rq

+ r4 d≤2 e3 η · β + r2 d≤3 Γb · Γg + r−1 Γb

= r−1 Γb + r2 d≤3 Γb · Γg + r4 d≤3 Γg · β



where we have used: • The fact we are working here with the global frame of Proposition 3.26 which has the property that η ∈ Γg . • The fact that Γb behave better than rΓg . • The fact that q ∈ rΓg . • The fact that α and e3 (q) behaves at least as good as Γb . −1 • The fact that ρ + 2m Γg . r 3 behaves as good as r • The fact that e3 (r) + 1 belongs to rΓb . Now, recall from Lemma 5.1 that the global frame of Proposition 3.26 satisfies in particular2 n 7 1 max sup r 2 +δdec −2δ0 |dk β| + r2 u 2 +δdec −2δ0 |dk Γg | 0≤k≤ksmall +22 M o +ru1+δdec −2δ0 |dk Γb | . . (6.2.4) Together with the schematic form of F1 and the behavior (3.3.4) of r on Σ∗ , and the fact that δ0 can be chosen to satisfy3 8δ0 ≤ δdec , we infer 3

1

3

sup ru 2 + 2 δdec |dk F1 | . u∗2

max

+δdec

0≤k≤ksmall +18 Σ∗

In view of the definition of F , this yields Z max u2+2δdec |dk F |2 . 20 + 0≤k≤ksmall +18

Σ∗

sup(r−1 ) + 2 . 0 . Σ∗

Z max

0≤k≤ksmall +19

Σ∗

u2+2δdec |dk e3 (q)|2 .

Together with Theorem M1, and the fact that δextra > δdec , we infer Z max u2+2δdec |dk F |2 . 20 . 0≤k≤ksmall +18

2 Here

Σ∗

we use (3.4.11) with kloss = 22. from Lemma 5.1 that we have

3 Recall

δ0 =

kloss . klarge − ksmall

Since we have here kloss = 22, and since we have 2ksmall ≤ klarge + 1 and klarge δdec  1, we deduce δ0  δdec and we have indeed 8δ0 ≤ δdec .

292

CHAPTER 6

This concludes the proof of the corollary. We are now ready to prove Proposition 6.14. In view of Corollary 6.22, α satisfies 6me ν α + r4 /d?2 /d?1 /d1 /d2 α

=

F.

6me ν (dk α) + r4 /d?2 /d?1 /d1 /d2 (dk α)

=

Commuting with dk , we infer Fk

where Fk is defined by := −6m[dk , νe]α − 6

Fk

k X j=1

dj (m)dk−j νeα − [dk , r /d?2 ]r /d?1 r /d1 r /d2 α

−r /d?2 [dk , r /d?1 ]r /d1 r /d2 α k +d F.

− r /d?2 r /d?1 [dk , r /d1 ]r /d2 α − r /d?2 r /d?1 r /d1 [dk , r /d2 ]α

Note that we have schematically   [e ν , d/] = O(r−1 ) + rΓb d,

[d, d/] = rΓb d,

[e ν , re4 ] = O(r−1 )d,

[e ν , e3 ] = O(r−1 )d.

Together with the fact that α behaves at least as good as Γb , we infer, schematically, Fk

=

dk F + r−1 d≤k+4 Γb + rd≤k+4 (Γ2b ).

In view of (6.2.4) and the behavior (3.3.4) of r on Σ∗ , we have max

3

1

3

sup ru 2 + 2 δdec |r−1 d≤k+4 Γb + rd≤k+4 (Γ2b )| . u∗2

+δdec

0≤k≤ksmall +18 Σ∗

sup r−1 + 2 Σ∗

. 0 . This yields Z max

0≤k≤ksmall +18

Σ∗

u2+2δdec |Fk |2

.

20 +

max

sup u2+2δdec |dk F |2 .

0≤k≤ksmall +18 Σ∗

Together with the estimate for F of Corollary 6.22, we infer Z max u2+2δdec |Fk |2 . 4 + 20 . 20 . 0≤k≤ksmall +18

Σ∗

This concludes the proof of Proposition 6.14. 6.2.6

Proof of Lemma 6.16

In this section we prove Lemma 6.16, i.e., we derive estimates for the control of the parabolic equation appearing in the statement of Proposition 6.14. The following identity will be useful.

DECAY ESTIMATES FOR α AND α (THEOREMS M2, M3)

293

Lemma 6.23. We have for any reduced scalar f Z  Z Z Z νe f2 = 2 f νe(f ) + (−2af e4 (f ) + κf 2 ) + a (2f e4 (f ) + κf 2 ) S S S  Z S  2 +Err e3 f . S

Proof. Recall from the definition of νe that νe = e3 + ae4 . We infer, in view of Proposition 2.64, Z  2 νe f ZS  Z  2 2 = e3 f + ae4 f S S  Z  Z Z 2 2 = (2f e3 (f ) + κf ) + Err e3 f + a (2f e4 (f ) + κf 2 ) S S S Z Z Z = 2 f νe(f ) + (−2af e4 (f ) + κf 2 ) + a (2f e4 (f ) + κf 2 ) S S  Z S  2 +Err e3 f . S

This concludes the proof of the lemma. We are now ready to prove Lemma 6.16. Recall from Lemma 6.23 that we have Z  Z Z Z νe f2 = 2 f νe(f ) + (−2af e4 (f ) + κf 2 ) + a (2f e4 (f ) + κf 2 ) S S S  Z S  +Err e3 f2 . S

In view of the equation satisfied by f , we infer Z  Z  Z  1 1 2 4 ? ? νe f = − r f /d2 /d1 /d1 /d2 f + hf 3m 3m S S S Z Z + (−2af e4 (f ) + κf 2 ) + a (2f e4 (f ) + κf 2 ) S S  Z  2 +Err e3 f . S

294

CHAPTER 6

Now, from the definition of νe, we have νe(u) = 2/ς. We deduce  Z  Z  un νe un f2 + r4 f /d?2 /d?1 /d1 /d2 f 3m S Z S  Z Z n u n = hf + u (−2af e4 (f ) + κf 2 ) + aun (2f e4 (f ) + κf 2 ) 3m S S S  Z  Z 2 n−1 n 2 2 +u Err e3 f + nu f . ς S S This yields in view of the bootstrap assumptions  Z  Z  un νe un f2 + r4 f /d?2 /d?1 /d1 /d2 f 3m S S   Z Z un 1 −1 . khkL2 (S) kf kL2 (S) + + u un |f ||d≤1 f | + nun−1 f2 3m r S S Z Z un n−1 ≤1 n−1 . khkL2 (S) kf kL2 (S) + u |f ||d f | + nu f2 3m S S where we have used in the last inequality the behavior (3.3.4) of r on Σ∗ . Next, we rely on the Poincar´e inequality of Lemma 6.19 to deduce  Z  Z Z Z Z νe un f 2 + un f 2 . un h2 + 2 un−2 (df )2 + nun−1 f 2. S

S

S

S

S

Integrating in u between 1 and u∗ , and recalling that νe(u) = 2/ς, we infer Z Z Z Z Z un f 2 . f2 + un h2 + 2 un−2 (df )2 + n un−1 f 2 . Σ∗ ∩C1

Σ∗

Σ∗

In particular, we have for n = 0 Z Z 2 f . Σ∗

Σ∗

2

Z

f +

Σ∗ ∩C1

2

h + Σ∗

Σ∗

2

Z

u−2 (df )2 .

Σ∗

Then, starting from the case n = 0 and arguing by iteration on the largest integer below n, one immediately deduces for any real n ≥ 0 Z Z Z Z (1 + un )f 2 . f2 + (1 + un )h2 + 2 (1 + un−2 )(df )2 Σ∗

Σ∗ ∩C1

Σ∗

which concludes the proof of Lemma 6.16.

Σ∗

Chapter Seven Decay Estimates (Theorems M4, M5) In this chapter, we rely on the decay of q, α and α to prove the decay estimates for all the other quantities. More precisely, we rely on the results of Theorems M1, M2 and M3 to prove Theorems M4 and M5.

7.1

PRELIMINARIES TO THE PROOF OF THEOREM M4

In what follows we give a detailed proof of Theorem M4, which, we recall, provides the main decay estimates in (ext) M. The proof makes use of the bootstrap assumptions BA-D, BA-E, and the results of Theorems M1, M2, M3 and Lemmas 3.15, 3.16. In this section, we start with some preliminaries. 7.1.1

Geometric structure of Σ∗

The proof of Theorem M4 depends in a fundamental way on the geometric properties of the GCM hypersuface Σ∗ , the spacelike future boundary of (ext) M introduced in section 3.1.2. For the convenience of the reader, we recall below its main features. 1. The affine parameter s is initialized on Σ∗ such that s = r. 2. There exists a constant c∗ such that Σ∗ := {u + r = c∗ }. 3. Let ν∗ = e3 + a∗ e4 be the unique vectorfield tangent to the hypersurface Σ∗ , perpendicular to the foliation S(u) induced on Σ∗ and normalized by the condition g(ν∗ , e4 ) = −2. The following normalization condition holds true at the south pole SP of every sphere S, 2m a∗ = −1 − . (7.1.1) r SP 4. We have − 23

r ≥ 0

dec u1+δ ∗

on Σ∗ .

(7.1.2)

5. The following GCM conditions hold on Σ∗ : κ=

2 , r

/d?2 /d?1 κ = 0, Z

Φ

Z

ηe = S

S

/d?2 /d?1 µ = 0,

ξeΦ = 0.

(7.1.3)

(7.1.4)

296

CHAPTER 7

Moreover on S∗ = Σ∗ ∩ C∗ , Z βeΦ = 0,

Z

S∗

eθ (κ)eΦ = 0.

(7.1.5)

S∗

6. According to the definition of the Hawking mass, i.e., 1 − GCM assumption for κ we also have   r 2m κ = − 1− . 2 r

2m r

2

= − r4 κκ, and the (7.1.6)

Thus on Σ∗ , e3 (r) =

r r (κ + A) = −Υ + A, 2 2

e4 (r) = 1.

(7.1.7)

7. In view of the definition of ν∗ and that of ς we easily deduce1 the following relation between a∗ and ς on Σ∗ : 2 r a∗ = − + Υ − A. ς 2

(7.1.8)

8. Since on Σ∗ we have r = s we deduce r Ω = e3 (r) = −Υ + A on Σ∗ . 2 7.1.2

(7.1.9)

Main assumptions

We reformulate below the main bootstrap assumptions2 in the form needed in the proof of Theorem M4. Definition 7.1. We make use of the following norms on S = S(u, r) ⊂ kf k∞ (u, r) : = kf k


,

L∞ S(u,r)

kf k∞,k (u, r) :=

k X i=0

kdi f k∞ (u, r),

kf k2 (u, r) := kf k

(ext)

M,


,

L2 S(u,r)

kf k2,k (u, r) :=

k X i=0

kdi f k2 (u, r).

(7.1.10)

To simplify the exposition it also helps to introduce the following schematic notation for the connection coefficients (recall ω, ξ = 0 and ζ = −η), n o n 2 2Υ o Γg = κ ˇ , ϑ, η, ζ, κ ˇ ∪ κ − ,κ + , r o n r n o (7.1.11) ˇ ∪ ω − m , r−1 (ς − 1), r−1 (Ω + Υ) . Γb = ϑ, η, ω ˇ , ξ, A, r−1 ςˇ, r−1 Ω, r2 Remark 7.2. It is important to note that η belongs to Γb rather than Γg as it may have been expected. Note also that A ∈ Γb in view of Proposition 2.64 and the fact 1 Indeed, since ν is tangent to Σ along which u = −r + c , using also (7.1.7), 2 = e (u) = ∗ ∗ ∗ 3 ς ν∗ (u) = −ν∗ (r) = −e3 (r) − a∗ e4 (r) = −a∗ + Υ − r2 A. 2 Based on bootstrap assumptions BA-D, BA-E, Theorems M1, M2, M3, and Lemmas 3.15, 3.16.

297

DECAY ESTIMATES (THEOREMS M4, M5)

n o ˇ ∈ rΓb . We also note that the averaged quantities κ − 2 , κ + 2Υ and that (ˇ ς , Ω) r r n o m −1 −1 ω − r2 , r (ς − 1), r (Ω + Υ) are actually better behaved in view of Lemmas 3.15, 3.16. Ref 1. According to our bootstrap assumptions BA-D, and the pointwise estimates of Proposition 3.19, which themselves follow from BA-E, as well as the control of averages in Lemma 3.15 and the control of the Hawking mass in Lemma 3.16, we have on (ext) M, 1. For 0 ≤ k ≤ ksmall , n o 1 kΓg k∞,k .  min r−2 u− 2 −δdec , r−1 u−1−δdec , (7.1.12)

ke3 Γg k∞,k−1 . r−2 u−1−δdec , kΓb k∞,k . r−1 u−1−δdec .

2. For k ≤ klarge − 5, kΓg k∞,k

.

r−2 ,

kΓb k∞,k . r−1 .

Ref 2. The quantity3 q satisfies on (ext) M, for all 0 ≤ k ≤ ksmall + 20, n o 1 kqk∞,k . 0 min u−1−δextra , r−1 u− 2 −δextra , ke3 qk∞,k−1 . 0 r−1 u−1−δextra . In addition, on the last slice Σ∗ , for all k ≤ ksmall + 20, Z |e3 dk q|2 + |e4 dk q|2 + r−2 |q|2 . 20 (1 + τ )−2−2δdec .

(7.1.13)

(7.1.14)

(7.1.15)

Σ∗ (τ,τ∗ )

According to Theorem M2 we have on (ext) M, for all 0 ≤ k ≤ ksmall + 20, n o 1 kαk∞,k . 0 min r−3 (u + 2r)− 2 −δextra , log(1 + u)r−2 (u + 2r)−1−δextra , n 1 ke3 αk∞,k−1 . 0 min r−4 (u + 2r)− 2 −δextra , (7.1.16) o log(1 + u)r−3 (u + 2r)−1−δextra . According to Theorem M3, the component α verifies the following estimate4 holds on T , for 0 ≤ k ≤ ksmall + 16, sup u1+δdec |dk α| . 0 ,

(7.1.17)

T

3 Recall (see Remark 2.110) that the quantity q we are working with is defined relative to the global frame of Proposition 3.26. 4 In fact, the corresponding estimate in Theorem M3 holds on (int) M, and hence in particular on T since T ⊂ (int) M.

298

CHAPTER 7

and on the last slice Σ∗ for all k ≤ ksmall + 18 Z |dk α|2 . 20 (1 + τ )−2−2δdec .

(7.1.18)

Σ∗ (τ,τ∗ )

Ref 3. In view of the bootstrap assumptions BA-D and the pointwise estimates of Proposition 3.19 for the curvature components, which themselves follow from BA-E, we have in (ext) M, i. For all 0 ≤ k ≤ ksmall , n o 1 kβk∞,k .  min r−3 (u + 2r)− 2 −δdec , r−2 (u + 2r)−1−δdec , n o 1 ke3 βk∞,k−1 .  min r−4 (u + 2r)− 2 −δdec , r−3 (u + 2r)−1−δdec ,

  n o

−3 − 12 −δdec −2 −1−δdec

ρˇ, ρ + 2m .  min r u , r u ,

r3 ∞,k

  (7.1.19)

−3 −1−δdec

e3 ρˇ, ρ + 2m . r u ,

r3 ∞,k−1

2m

µ

. r−3 u−1−δdec ,

ˇ, µ − r3 ∞,k

kβk∞,k . r−2 u−1−δdec . Since K = −ρ − 14 κκ + 14 ϑϑ = r12 − (ρ − ρ) − 14 (κκ − κκ) + l.o.t. we also deduce for all 0 ≤ k ≤ ksmall ,

n o

1

K − 1 .  min r−3 u− 2 −δdec , r−2 u−1−δdec .

2 r ∞,k ii. For all k ≤ klarge − 5, 7

r2+

δB 2



 kαk∞,k + kβk∞,k . ,

r3 kˇ ρk∞,k + r2 kβk∞,k + rkαk∞,k . .

(7.1.20)

Remark 7.3. In view of the control of averages Lemma 3.15 we have in fact better estimates for the scalars, 2 κ− , r

κ+

2Υ , r

ω−

m , r2

ρ+

2m . r3

In particular they can be estimated by  replaced by 0 in Ref 1. 1 Remark 7.4. Note that r(ˇ ρ, ρ + 2m r 3 ), r(K − r 2 ) behave as Γg . For convenience we shall just simply add them to Γg . Similarly (rβ, α) behave as Γb . Thus, our extended Γg , Γb are   n o  2 2Υ 2m Γg = κ ˇ , ϑ, η, ζ, κ ˇ , rρˇ ∪ κ − , κ + ,r ρ + 3 , r r r n o n o ˇ rβ, α ∪ ω − m , r−1 (ς − 1), r−1 (Ω + Υ) . Γb = ϑ, η, ω ˇ , ξ, A, r−1 ςˇ, r−1 Ω, 2 r

299

DECAY ESTIMATES (THEOREMS M4, M5)

Note also that we can write e3 (Γg ) = r−1 dΓb . 7.1.3 7.1.3.1

Basic lemmas Commutation identities

Lemma 7.5. We have, schematically, [ d/, e4 ]ψ = Γg d% ψ + l.o.t., [ d/, e3 ]ψ = rΓb e3 ψ + Γ≤1 b d% ψ + l.o.t.

(7.1.21)

Proof. Follows from Lemma 2.68 and the symbolic notation introduced in (7.1.11), see also Remark 7.4. 7.1.3.2

Interpolation and product estimates

We estimate quadratic error terms with the help of the following lemma. Lemma 7.6. Let kloss = 25. Then, the following interpolation estimates hold true for all 0 ≤ k ≤ ksmall + kloss :

  δdec 1

kΓg k∞,k + r ρˇ, β, α . r−2 u− 2 − 2 , ∞,k (7.1.22) δdec k(Γb , α)k∞,k + rkβk∞,k . r−1 u−1− 2 . Also, the following product estimates hold true for all 0 ≤ k ≤ ksmall + kloss :





kΓg · Γg k∞,k + r ρˇ, β, α · Γg . 0 r−4 u−1−δdec , ∞,k



   3



+ rkΓg · βk∞,k + r ρˇ, β, α · Γb . 0 r−3 u− 2 −δdec

Γg · Γb , α ∞,k ∞,k



 (7.1.23) 9

. 0 r− 2 u−1−δdec ,

β, α · Γb ∞,k





+ rkβ · Γb k∞,k . 0 r−2 u−2−δdec .

Γb , α · Γb ∞,k

Proof. All estimates are easy to prove in the range 0 ≤ k ≤ ksmall . We shall thus assume that ksmall ≤ k ≤ ksmall + kloss . Since kloss < ksmall we have k/2 < ksmall for all k in that range. For simplicity of notation we write L := klarge − 5, S := ksmall . By standard interpolation inequalities, for all S ≤ k ≤ L, kΓg k∞,k

kΓb k∞,k

L−k h 1 i L−S k−S L−k L−S L−S . kΓg k∞,L kΓg k∞,S . r−2 u− 2 −δdec k−S h 1 i L−S 1 . r−2 u− 2 −δdec u 2 +δdec , L−k h i L−S k−S L−k L−S L−S . kΓb k∞,L kΓb k∞,S . r−1 u−1−δdec k−S h i L−S . r−1 u−1−δdec u1+δdec .

300

CHAPTER 7

Now, we may assume that kloss satisfies5 kloss ≤

δdec (klarge − ksmall ). 3

Thus, for ksmall ≤ k ≤ ksmall + kloss , we have h

u

1 2 +δdec

k−S i L−S

h

1+δdec

+ u

k−S i L−S

h

1+δdec

. u

kloss large −5−ksmall

ik

h

1+δdec

. u

i δdec 3

.u

δdec 2

and hence kΓg k∞,k

.

kΓb k∞,k

.

1

r−2 u− 2 −

δdec 2

δ −1− dec 2

r−1 u

, .

Since rρˇ satisfies the same estimates as Γg and rβ and rα satisfy even better estimates, and that α and rβ satisfy the same estimate as Γb , we infer

 

kΓg k∞,k + r ρˇ, β, α

∞,k

1

. r−2 u− 2 −

k(Γb , α)k∞,k + rkβk∞,k . r−1 u−1−

δdec 2

δdec 2

, ,

which is the desired interpolation bound. The first, second and last product estimates follow immediately from the above interpolation bound. Finally, the third product estimate follows from the above interpolation estimate for Γb together with the following interpolation estimate for ksmall ≤ k ≤ ksmall + kloss : k−S

k(β, α)k∞,k

. . .

L−k

L−S L−S k(β, α)k∞,L k(β, α)k∞,S k−S h L−k h 7 δB i L−S  7 δB  i L−S 1  r− 2 − 2 min r− 2 − 2 , r−3 u− 2 −δdec h 7 δB i1−δdec h iδdec 1  r− 2 − 2 r−3 u− 2 −δdec 7

. r− 2 u−

δdec 2

where we have used in the last inequality the fact that δB > 2δdec . 7.1.3.3

Elliptic estimates

We shall often make use of the results of Proposition 2.33 and Lemma 2.38 which we rewrite as follows with respect to the L2 based hk (S) spaces introduced in Definition 2.39. Lemma 7.7. Under the assumptions Ref 1–Ref 3, the following elliptic estimates 5 Recall

that we have  0 < δdec  1,

δdec klarge  1,

ksmall =

 1 klarge + 1. 2

In particular, we have δdec (klarge − ksmall )  1 and hence we may indeed assume that kloss = 25 satisfies the required constraints.

301

DECAY ESTIMATES (THEOREMS M4, M5)

hold true for the Hodge operators /d1 , /d2 , /d?1 , /d?2 , for all k ≤ ksmall + 20. 1. If f ∈ s1 (S), k d/f khk (S) + kf khk (S) . rk /d1 f khk (S) . 2. If f ∈ s2 (S), k d/f khk (S) + kf khk (S) . rk /d2 f khk (S) . 3. If f ∈ s0 (S), k d/f khk (S) . rk /d?1 f khk (S) . 4. If f ∈ s1 (S), kf khk+1 (S)

Z . rk /d?2 f khk (S) + r−2 eΦ f . S

5. If f ∈ s1 (S),

R

f eΦ Φ

S

f − R 2Φ e

e S

hk+1 (S)

7.1.4

. rk /d?2 f khk (S) .

Main equations

The proof of Theorem M4 relies heavily on the null structure and null Bianchi identities derived in section 2.2.4, see Proposition 2.63. We also rely on Proposition 2.73 for equations verified by the check quantities. We rewrite them below in a schematic form. Proposition 7.8 (Transport equations for checked quantities). We have the following transport equations in the e4 direction, e4 κ ˇ + κκ ˇ = Γg · Γg , 1 1 e4 κ ˇ + κˇ κ+ κ ˇ κ = −2 /d1 ζ + 2ˇ ρ + Γg · Γb , 2 2 e4 ω ˇ = ρˇ + Γg · Γb , 3 3 e4 ρˇ + κˇ ρ + ρˇ κ = /d1 β + Γb · α + Γg · β + κ ˇ · ρˇ, 2 2 3 3 e4 µ ˇ + κˇ µ + µˇ κ = r−1 Γg · d/≤1 Γg . 2 2

(7.1.24)

Also, we have in the e3 direction, e3 κ ˇ = r−1 d/≤1 Γb + Γb · d/≤1 Γb ,

e3 ρˇ = r−2 d/≤1 Γb + r−1 Γb · d/≤1 Γb .

(7.1.25)

Proof. The statements follow from the precise formulas of Proposition 2.73 and the symbolic notation in (7.1.11). We also use the convention made in Remark 7.4 according to which we write rρˇ, rµ ˇ ∈ Γg , (rβ, α) ∈ Γb and e3 (Γg ) = r−1 (dΓb ).

302

CHAPTER 7

7.1.5

Equations involving q

Recall that our main quantity q has been introduced in Definition 2.98 with respect to the global frame of Proposition 3.26 (see Remark 2.110). The passage from the geodesic frame (e3 , eθ , e4 ) of (ext) M to the global frame (e03 , e0θ , e04 ) is given by   1 2 1 0 e0θ = eθ + f e3 , e03 = Υ−1 e3 , (7.1.26) e4 = Υ e4 + f eθ + f e3 , 4 2 with a reduced scalar f which was constructed in Proposition 3.20. We recall below the main relevant statements of Proposition 3.20 in connection to the construction of the global frame. Proposition 7.9. Under assumptions Ref 1–2 on (ext) M there exists a frame transformation of the form (7.1.26) verifying the following properties:6 1. Everywhere in (ext) M we have ξ 0 = 0. 2. The transition function f verifies, relative to the background frame (e3 , eθ , e4 ), the estimates7  , for k ≤ ksmall + 20 on 1 ru 2 +δdec −2δ0 + u1+δdec −2δ0  |dk−1 e03 f | . 1+δ −2δ0 for k ≤ ksmall + 20 on (ext) M. ru dec |dk f | .

(ext)

M, (7.1.27)

3. The primed Ricci coefficients and curvature components verify8 (   1 max sup r2 u 2 +δdec −2δ0 + ru1+δdec −2δ0 |dk Γ0g | 0≤k≤ksmall +kloss

(ext) M

+ru1+δdec −2δ0 |dk Γ0b |   2Υ 0 2 0 0 0 0 2 1+δdec −2δ0 k−1 0 0 +r u e3 κ − ,κ + ,ϑ ,ζ ,η ,η d r r   1 3 +δ −2δ 2 1+δ 0 dec + r (u + 2r) 2 + r (u + 2r) dec −2δ0 |dk (α0 , β 0 )|   1 + r3 (2r + u)1+δdec + r4 (2r + u) 2 +δdec −2δ0 |dk−1 e03 (α0 )|   1 + r3 u1+δdec + r4 u 2 +δdec −2δ0 |dk−1 e03 (β 0 )|   1 + r3 u 2 +δdec −2δ0 + r2 ru1+δdec −2δ0 |dk ρˇ0 | )   1+δdec −2δ0 2 k 0 k 0 +u r |d β | + r|d α | . . We have the following analog of Proposition 2.99.

6 We

denote by primes the Ricci and curvature components w.r.t. the primed frame. fact, the estimates hold for ksmall + kloss , see Proposition 3.20, and we choose here kloss = 20. 8 Note that u and r here are the outgoing optical function and area radius of the foliation of (ext) M. 7 In

303

DECAY ESTIMATES (THEOREMS M4, M5)

Proposition 7.10. We have, relative to  3 r4 /d?2 /d?1 ρ + κρϑ + 4

the background frame of  3 κρϑ = q + Err 4

(ext)

M, (7.1.28)

with error term expressed schematically in the form Err

= r2 d/≤2 (Γb · Γg ).

(7.1.29)

Proof. We make use of Proposition 2.99. Recall (see Remark 2.110) that the quantity q we are working with is defined relative to the global frame of Proposition 3.26. We thus write9   3 3 q = r4 ( /d?2 )0 ( /d?1 )0 ρ0 + κ0 ρ0 ϑ0 + κ0 ρ0 ϑ + Err0 , 4 4 Err0

=

r4 e03 η 0 · β 0 + r2 d≤1 Γb · Γg ),

where the primes refer to the global frame in which q was defined. Since in that frame e03 η 0 ∈ r−1 dΓb and β 0 ∈ r−1 Γg we can simplify and write Err0

= r2 d≤1 Γb · Γg ).

We also have in view of Proposition 2.90 ρ0

=

β0

=

α0

=

ρ + f β + O(f 2 α), 1 β + f α, 2 α,

0

=

κ + f ξ,

0

=

κ + /d1 0 (f ) + f (ζ + η) + O(r−1 f 2 ),

0

=

ϑ0

=

ϑ − /d?2 0 (f ) + f (ζ + η) + O(r−1 f 2 ),

κ κ ϑ

ϑ + f ξ.

Note that ( /d?1 )0 ρ =

1 1 −e0θ (ρ) = −eθ ρ − f e3 ρ = /d?1 ρ − f e3 ρ. 2 2

We deduce ( /d?2 )0 ( /d?1 )0 ρ0

= = = =

9 The

( /d?2 )0 ( /d?1 )0 ρ + ( /d?2 )0 ( /d?1 )0 (Γb · Γg ) + l.o.t.   1 ? 0 ? ( /d2 ) /d1 − f e3 ρ + r−2 d/≤2 (Γb · Γg ) 2   1 ? ? /d2 /d1 − f e3 ρ + r−2 d/≤2 (Γb · Γg ) 2 1 /d?2 /d?1 ρ − /d?2 f e3 ρ + r−2 d/≤2 (Γb · Γg ). 2

values of r and r0 differ only by lower order terms which do not affect the result.

304

CHAPTER 7

Similarly, κ0 ρ0 ϑ0 κ0 ρ0 ϑ0


= ρκ ϑ − /d?2 f + r−3 d/≤1 (Γg · Γg ), = κρϑ + r−3 d/≤1 (Γb · Γg ).

We deduce 3 3 ( /d?2 )0 ( /d?1 )0 ρ0 + κ0 ρ0 ϑ0 + κ0 ρ0 ϑ0 4 4

= −

3 3 /d?2 /d?1 ρ + κρϑ + κρϑ 4 4  1 ? 3 /d2 f e3 ρ + κρ + r−2 d/≤2 (Γb · Γg ). 2 2

Note that /d?2 f



 3 e3 ρ + κρ 2

=

/d?2 f



1 /d1 β − ϑα + l.o.t. 2



= r−2 d/≤2 (Γg · Γb ).

Hence 3 3 ( /d?2 )0 ( /d?1 )0 ρ0 + κ0 ρ0 ϑ0 + κ0 ρ0 ϑ0 4 4

=

3 3 /d?2 /d?1 ρ + κρϑ + κρϑ + r−2 d/≤2 (Γb · Γg ). 4 4

This concludes the proof of Proposition 7.10. We shall also need the following analogue of Proposition 2.100. Proposition 7.11. The following identity holds true in (ext) M, with respect to its background frame ( ) 3 3 3 3 e3 (rq) = r5 /d?2 /d?1 /d1 β − ρ /d?2 /d?1 κ − κρ /d?2 ζ − κρα + (2ρ2 − κκρ)ϑ 2 2 2 4 (7.1.30) + Err[e3 (rq)], where Err[e3 (rq)]


= r3 d≤3 Γb · Γg .

(7.1.31)

Proof. We start with the result of Proposition 2.100 which we write in the form (r0 )−5 e03 (r0 q)

Err0 [e03 (r0 q)]

3 3 3 ( /d?2 /d?1 /d1 )0 β 0 − κ0 ρ0 α0 − ρ0 ( /d?2 /d?1 )0 κ0 − κ0 ρ(0 /d?2 )0 ζ 0 2 2 2 3 0 + (2(ρ0 )2 − κ0 κ0 ρ0 )ϑ + (r0 )−5 Err[e03 (r0 q)] 4

= r0 Γb q + r5 d0≤1 e03 η 0 · β 0 + r03 d≤2 Γb · Γg .

=

Since e03 η 0 ∈ r−1 Γb and q ∈ Γb , we deduce
Err0 [e03 (r0 q)] = r3 d≤2 Γb · Γg . Now, in view of Proposition 2.90,   1 ? ? 0 ? ? 0 0 ( /d2 /d1 /d1 ) β = ( /d2 /d1 /d1 ) β + f α = ( /d?2 /d?1 /d1 )0 β + r−2 d/3 (Γb · Γg ). 2

305

DECAY ESTIMATES (THEOREMS M4, M5)

Proceeding in the same manner with all other terms we find 3 3 3 3 ( /d?2 /d?1 /d1 )0 β 0 − κ0 ρ0 α0 − ρ0 ( /d?2 /d?1 )0 κ0 − κ0 ρ(0 /d?2 )0 ζ 0 + (2(ρ0 )2 − κ0 κ0 ρ0 )ϑ0 2 2 2 4 3 3 ? ? 3 3 ? ? ? 2 = /d2 /d1 /d1 β − κρα − ρ /d2 /d1 κ − κρ /d2 ζ + (2ρ − κκρ)ϑ 2 2 2 4 +r−2 d/≤3 (Γb · Γg ) from which the result easily follows. 7.1.6

Additional equations

The following proposition is an immediate corollary of Proposition 2.74. Proposition 7.12. We have, schematically,   1 1 2 /d?1 ω = κ + 2ω η + e3 (ζ) − β − κξ + r−1 Γg + Γb · Γb , 2 2
? 2 /d2 /d2 η = κ −e3 (ζ) + β − e3 (eθ (κ)) + r−2 d/≤1 Γg + r−1 d/≤1 (Γb · Γb ),
2 /d2 /d?2 ξ = κ e3 (ζ) − β − e3 (eθ (κ)) + r−2 d/≤1 Γg + r−1 d/≤1 (Γb · Γb ). Remark 7.13. Note that in fact Γg = {ˇ κ, ϑ, ζ, κ ˇ , rρˇ} and Γb = {ϑ, η, ξ, ω ˇ , rβ, α} in the derivation of this proposition. It is important to note also that the terms denoted schematically by d/(Γb · Γb ) do not contain derivatives of ω ˇ. The following corollary of Proposition 7.12 will be very useful later on. Proposition 7.14. The following identities hold true on Σ∗ .   2 /d?2 /d?1 /d1 /d2 /d?2 η = κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 e3 (eθ (κ)) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.

(7.1.32)

    2 /d?2 /d?1 /d1 /d2 /d?2 ξ = e3 ( /d?2 /d2 + 2K) /d?2 /d?1 κ − κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β (7.1.33) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. Remark 7.15. Here, as in Remark 7.13, we have Γg = {ˇ κ, ϑ, ζ, κ ˇ , rρˇ} and Γb = {ϑ, η, ξ, ω ˇ , rβ, α}. The quadratic terms denoted l.o.t. are lower order both in terms of decay in r, u as well in terms of number of derivatives. They also contain only angular derivatives d/ and not e3 nor e4 . Proof. We make use of Proposition 7.12 . We shall also make use of the conventions mentioned in Remark 7.4, i.e., ρˇ, µ ˇ ∈ r−1 Γg , β ∈ r−1 Γb , α ∈ Γb . We start with
2 /d2 /d?2 η = κ −e3 (ζ) + β − e3 (eθ (κ)) + r−2 d/≤1 Γg + r−1 d/(Γb · Γb ).

306

CHAPTER 7

We apply /d?1 /d1 to derive 2 /d?1 /d1 /d2 /d?2 η


= κ − /d?1 /d1 e3 (ζ) + /d?1 /d1 β − /d?1 /d1 e3 (eθ (κ)) + r−4 d/≤3 Γg +r−3 d/3 (Γb · Γb )


= κ −e3 ( /d?1 /d1 (ζ) + /d?1 /d1 β − /d?1 /d1 e3 (eθ (κ)) − κ[ /d?1 /d1 , e3 ]ζ + r−4 d/≤3 Γg + r−3 d/3 (Γb · Γb ).

Making use of the commutation formula, see Lemma 7.5, and the null structure equations for e3 ζ, e4 ζ, [ /d1 , e3 ]ζ = −ηe3 ζ + r−2 d/ζ + Γb e4 ζ + l.o.t. = r−1 Γb · Γb + r−2 d/Γg + l.o.t., we deduce, schematically, [ /d?1 /d1 , e3 ]ζ

=

/d?1 [ /d1 , e3 ]ζ + [ /d?1 , e3 ] /d1 ζ


= r−1 d/ r−1 Γb · Γb + r−2 d/ζ + l.o.t. + Γb e3 /d1 ζ + r−2 /d1 ζ + l.o.t.
= r−2 d/(Γb · Γb ) + r−3 d/2 ζ + Γb /d1 e3 ζ + Γb e3 ζ + r−2 d/ζ + l.o.t. = r−2 d/(Γb · Γb ) + r−2 Γb d/(dΓb ) + r−1 Γb · Γb · Γb + r−4 d/2 Γg = r−2 d/(Γb d≤1 Γb ) + r−4 d/2 Γg + l.o.t. Hence,   2 /d?1 /d1 /d2 /d?2 η = κ − e3 ( /d?1 /d1 ζ) + /d?1 /d1 β − /d?1 /d1 e3 (eθ (κ)) + r−4 d/≤3 Γg + r−3 d/2 (Γb · d/Γb ).

(7.1.34)

Since µ = − /d1 ζ − ρ + 14 ϑϑ, we deduce 1 ? /d1 (ϑϑ), 4 1 e3 /d?1 µ = −e3 ( /d?1 /d1 ζ) − e3 /d?1 ρ + e3 /d?1 (ϑϑ) 4 /d?1 µ = − /d?1 /d1 ζ − /d?1 ρ +

= −e3 ( /d?1 /d1 ζ) − /d?1 e3 ρ − [ /d?1 , e3 ]ρ +

1 ? 1 /d1 e3 (ϑϑ) + [e3 , /d?1 ](ϑ · ϑ). 4 4

Making use of the equations for e3 ρ = /d1 β − 32 κρ + Γg · Γb and also the equations for10 e4 ρ, e3 ϑ, e3 ϑ, e4 ϑ, e4 ϑ (and writing /d1 β = r−1 d/β = r−2 d/Γb ) Γb e3 ρ + Γb e4 ζ + r−2 d/ρ = r−2 Γb d/Γb + r−3 d/Γg + l.o.t.,
= Γb e3 (ϑ · ϑ) + Γb e4 (ϑ · ϑ) + r−2 d/(ϑ · ϑ) = r−2 d/ Γb · Γg + l.o.t.

[e3 , /d?1 ]ρ = [e3 , /d?1 ](ϑ 10 This

· ϑ)

is to avoid the presence of e3 , e4 derivatives in the error terms.

307

DECAY ESTIMATES (THEOREMS M4, M5)

We deduce, ignoring the lower order terms,  
3 e3 /d?1 µ = −e3 ( /d?1 /d1 ζ) − /d?1 /d1 β − κρ + Γg · Γb + r−2 Γb d/Γb + r−2 d/ Γb Γg 2 +r−3 d/Γg 3 = −e3 ( /d?1 /d1 ζ) − /d?1 /d1 β + κ /d?1 ρ + r−3 d/Γg + r−2 d/≤1 (Γb · Γb ). 2 Hence, e3 ( /d?1 /d1 ζ) = −e3 ( /d?1 µ) − /d?1 /d1 β + r−3 d/Γg + r−2 d/≤2 (Γb · Γb ) + l.o.t. (7.1.35) and thus, back to (7.1.34),  
2 /d?1 /d1 /d2 /d?2 η = κ e3 ( /d?1 µ) + 2 /d?1 /d1 β − /d?1 /d1 e3 eθ (κ) + r−4 d/≤3 Γg + r−3 d/≤3 (Γb · Γb ) + l.o.t. Applying /d?2 and commuting once more with e3 , i.e.,  
2 /d?2 /d?1 /d1 /d2 /d?2 η = κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 e3 eθ (κ)   + κ[ /d?2 , e3 ] /d?1 µ + r−1 d/Γg · e3 ( /d?1 µ) + 2 /d?1 /d1 β

(7.1.36)

(7.1.37)

+ r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ). Note that, in view of (7.1.36), we can write
e3 ( /d?1 µ) = 2κ−1 /d?2 /d?1 /d1 e3 eθ (κ) − 2 /d?1 /d1 β + 2κ−1 /d?1 /d1 /d2 /d?2 η = r−3 d/≤4 Γb + l.o.t.

(7.1.38)

Hence,   r−1 d/Γg · e3 ( /d?1 µ) + 2 /d?1 /d1 β = r−4 d/Γg · d/≤4 Γb . Similarly, Γb · e3 /d?1 µ + Γb e4 /d?1 µ + r−3 d/2 µ + l.o.t.
= r−3 Γb · d/≤4 Γb + Γb /d?1 e4 µ + [e4 , /d?1 ]µ + r−4 d/2 Γg + l.o.t.

[ /d?2 , e3 ] /d?1 µ =

Thus, making use of the equation for e4 µ and combining with the estimate above,   κ[ /d?2 , e3 ] /d?1 µ + r−1 d/Γg · e3 ( /d?1 µ) + 2 /d?1 /d1 β = r−4 Γb · d/≤4 Γb + r−5 d/≤2 Γg . Back to (7.1.37) we deduce  
2 /d?2 /d?1 /d1 /d2 /d?2 η = κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 e3 eθ (κ) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) as desired.

308

CHAPTER 7

To prove the second part we start with the formula for /d2 /d?2 ξ in Corollary 7.12 2 /d2 /d?2 ξ


= κ e3 (ζ) − β − e3 (eθ (κ)) + r−2 d/≤1 Γg + r−1 d/(Γb · Γb ).

Applying /d?1 /d1 and proceeding exactly as before in the derivation of (7.1.34) we derive
2 /d?1 /d1 /d2 /d?2 ξ = −e3 ( /d?1 /d1 eθ (κ)) + κ e3 ( /d?1 /d1 ζ) − /d?1 /d1 β (7.1.39) + r−4 d/≤3 Γg + r−3 d/2 (Γb · dΓb ). Making use of (7.1.35) we deduce, as in (7.1.36),   2 /d?1 /d1 /d2 /d?2 ξ = −e3 ( /d?1 /d1 eθ (κ)) + κ − e3 ( /d?1 µ) − 2 /d?1 /d1 β + r−4 d/3 Γg + r−3 d/≤2 (Γb · dΓb ) + l.o.t.

(7.1.40)

Applying /d?2 and proceeding as in the derivation of (7.1.37), by making use of (7.1.39) and (7.1.38) we obtain   2 /d?2 /d?1 /d1 /d2 /d?2 ξ = −e3 ( /d?2 /d?1 /d1 eθ (κ)) − κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. The identity /d?1 /d1 = /d2 /d?2 + 2K yields, together with the bootstrap assumptions,   2 /d?2 /d?1 /d1 /d2 /d?2 ξ = −e3 (( /d?2 /d2 + 2K) /d?2 eθ (κ)) − κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.   = e3 (( /d?2 /d2 + 2K) /d?2 /d?1 (κ)) − κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. as desired.

7.2

STRUCTURE OF THE PROOF OF THEOREM M4

We rephrase the statement of Theorem M4 as follows. Theorem 7.16. Let M = (int) M ∪ (ext) M be a GCM admissible spacetime.11 Under the basic bootstrap assumptions and the results of Theorems M1–M4 (all encoded in Ref 1–Ref 4) the following estimates12 hold true, for all k ≤ ksmall + 8, everywhere on (ext) M, n o 1 kΓg k∞,k . 0 min r−2 u− 2 −δdec , r−1 u−1−δdec , ke3 Γg k∞,k−1 . 0 r−2 u−1−δdec , kΓb k∞,k . 0 r−1 u−1−δdec ,

11 In

particular the conditions (7.1.1)–(7.1.5) hold on the spacelike boundary Σ∗ . Remark 7.4 for the definition of Γg , Γb used here.

12 See

(7.2.1)

DECAY ESTIMATES (THEOREMS M4, M5)

309

and, n o 1 kβk∞,k . 0 min r−2 (u + 2r)−1−δdec , r−3 (u + 2r)− 2 −δdec , ke3 βk∞,k−1 . 0 r−3 (u + 2r)−1−δdec , n o 1 kˇ ρk∞,k . 0 min r−2 u−1−δdec , r−3 u− 2 −δdec ,

(7.2.2)

ke3 ρˇk∞,k . 0 r−3 u−1−δdec , kˇ µk∞,k . 0 r−3 u−1−δdec ,

kβk∞,k . 0 r−2 u−1−δdec . Moreover, everywhere in

(ext)

M,

kαk∞,k

. 0 r−1 u−1−δdec .

(7.2.3)

Here is a short sketch of the proof of the theorem. 1. Estimates on Σ∗ . To start with, we only have good13 estimates for q, α and α, according to Ref 2. To proceed we make use in an essential way of all the GCM conditions (7.1.3)–(7.1.5) on the spacelike boundary Σ∗ to estimate all the Ricci and curvature coefficients along Σ∗ . We also take full advantage of the dominance 1+δ condition r ≥ −1 on Σ∗ . The main result is stated in Proposition 7.28. 0 u∗ The proof is divided in the following intermediary steps. a) In Proposition 7.22, we derive flux type estimates along Σ∗ for the quantities β and Γb . These estimates take advantage in an essential way of the improved flux estimates for q and α, see (7.1.15) and (7.1.18). This step also makes use of Proposition 7.11 and the identities of Proposition 7.21 for η, ξ. Moreover, as a byproduct of the flux estimates, we obtain the desired estimates on Σ∗ for β and Γb . b) We next estimate the ` = 1 modes of the Ricci and curvature coefficients in Proposition 7.26. Besides the information provided by the estimates for q, α, α and the GCM conditions, an important Ringredient in the proof is the vanishing of the ` = 1 mode of eθ (K), i.e., eθ (K)eΦ = 0. The flux estimates derived in Proposition 7.22 play an essential role in deriving the desired estimate for the ` = 1 mode of β. c) We make use of the previous steps to complete the proof for the remaining desired estimates on Σ∗ in Proposition 7.28. This step also uses, in addition to the GCM conditions, Proposition 7.10 relating q to /d?2 /d?1 ρ, the Codazzi equations and elliptic estimates on 2-surfaces. 2. First estimates in (ext) M. We make use of the propagation equations in e4 and the estimates on Σ∗ to derive some of the desired estimates of Theorem 7.16, more precisely the better estimates in powers of r for the Γg quantities. Note that these estimates decay only like u−1/2−δdec in powers of u. a) We first prove the desired estimates for κ ˇ, µ ˇ by simply integrating the corresponding e4 equations. Note that these estimates are also well behaved in terms of powers of u. This is done in section 7.4.3. 13 That

is, estimates in terms of 0 .

310

CHAPTER 7

b) We derive spacetime estimates for all the ` = 1 modes in Lemma 7.34. This is done by propagating them from the last slice in the e4 direction, combined with Codazzi equations and the vanishing of the ` = 1 mode of eθ (K). c) We provide all the optimal estimates in terms of powers14 of r for the quantities ϑ, ζ, η, κ ˇ , β, ρˇ. This is achieved in Proposition 7.33 with the help of the estimates on the last slice, the propagation equation for these quantities and the estimates for the ` = 1 modes derived in the previous step. 3. Optimal u-decay estimates in (ext) M. We derive all the remaining estimates of ˇ ςˇ. The main remaining difficulty Theorem 7.16 for all but the quantities ξ, ω ˇ , Ω, is to get the top decay in powers of u for ϑ, ζ, η, κ ˇ , β, ρˇ, β. The result is stated in Proposition 7.35. We proceed as follows. a) One would like to start with ϑ by using the equation e4 ϑ + κϑ = −2α. This unfortunately cannot work by integration15 starting from the last slice Σ∗ . Similar problems occur for ζ, β, ρˇ. On the other hand the quantities κ ˇ and ϑ could in principle be propagated using their corresponding e4 equations from Σ∗ , but unfortunately they are strongly coupled with the other quantities for which we don’t yet have information. For example, we have, 1 1 e4 κ ˇ + κˇ κ+ κ ˇ κ = −2 /d1 ζ + 2ˇ ρ + Γ g · Γb , 2 2 and therefore we cannot derive the estimate for κ ˇ , by integration, before estimating /d1 ζ and ρˇ. To circumvent this difficulty we proceed by an indirect method as follows. b) We can derive optimal decay information on various mixed quantities. For example making use of the equation   1 3 e3 α + κ − 4ω α = − /d?2 β − ϑρ + 5ζβ, 2 2 we infer the desired decay in u for the quantity /d?2 β − 32 ϑρ. Other such information can be derived from the Codazzi equations for ϑ, ϑ, the Bianchi identity for β and the identity (7.1.28) of Lemma 7.10. c) We combine the control we have for α, κ ˇ, µ ˇ with the control for the mixed quantities mentioned above with a propagation equation for an intermediary quantity,
Ξ := r2 eθ (κ) + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β . We show in the crucial Lemma 7.36 that Ξ is also a good mixed quantity, i.e., it has optimal decay in u. It is important to note that this estimate does not depend linearly on α for which we only have information on the last slice and T. d) We can combine the control of Ξ with all other available information mentioned above, to derive good estimates, simultaneously, for /d?2 /d?1 κ, /d?2 ζ and 1

estimates also provide weak decay in u, i.e., u− 2 −δdec decay. would work however if instead we would integrate from the interior, but we don’t possess information about optimal u-decay in the interior, for example, on the timelike boundary T of (ext) M. 14 These 15 It

311

DECAY ESTIMATES (THEOREMS M4, M5)

/d?2 β. This is achieved in a sequence of crucial lemma in section 7.5.2. Unfortunately this step is heavily dependent on the estimate of Ref 2 for α and therefore the estimates we derive are only useful on T.  ? e) We also show that we have good estimates for /d2 ζ, /d?1 κ ˇ , β, β, /d?1 ρˇ . To es  timate κ ˇ , ζ, β, β, ρˇ from /d?2 ζ, /d?1 κ ˇ , β, β, /d?1 ρˇ we rely on the elliptic Hodge Lemma 7.7 and the control we have for the ` = 1 modes from Lemma 7.34 derived earlier. We obtain estimates for η, ϑ, ϑ as well. This establishes all the estimates of Proposition 7.35 on T . f) The estimates mentioned above on T can now be propagated by integrating forward the e4 null structure and null Bianchi equations. This ends the proof of Proposition 7.35 in (ext) M.

4. In Proposition 7.43 we derive improved decay estimates for e3 (β, ϑ, ζ, κ ˇ , ρˇ) and ˇ ςˇ in terms of u−1−δdec decay. The estimates for ω estimates for ξ, ω ˇ , Ω, ˇ and ξ are propagated from the last slice using their e4 propagation equations. The estimate for ω ˇ can be easily derived by integrating e4 (ˇ ω ) = ρˇ + Γg · Γb from the last slice Σ∗ . The estimate for ξ follows by integrating e4 (ξ) = −e3 (ζ) + β − κζ + Γb · Γb and making use of the previously derived estimates for e3 ζ, β, ζ. The estimates ˇ ςˇ follow easily from the equations (2.2.19). for Ω,

7.3 7.3.1

DECAY ESTIMATES ON THE LAST SLICE Σ∗ Preliminaries

We shall make use of the following norms on Σ∗ . X kψk∗∞,k (u, r) := kdj∗ ψkL∞ (S(u,r)) , dj∗ = j≤k

Recall that ν∗ = ν

Σ∗

X

d/j1 (ν∗ )j2 .

(7.3.1)

j1 +j2 ≤j

= e3 + a∗ e4 is the tangent vector to Σ∗ and (see (7.1.8)

(7.1.9)), along Σ∗ , a∗

2 r 2 = − + Υ − A = − − Ω. ς 2 ς

(7.3.2)

Since ς − 1 and Ω + Υ belong to rΓb in view of (7.1.11), we deduce a∗ + 1 +

2m ∈ rΓb . r

(7.3.3)

As immediate consequence of the commutation Corollary 7.5 we derive the following. Lemma 7.17. We have, schematically, [ d/, ν∗ ]ψ

= rΓb (ν∗ ψ) + d≤1 Γb · dψ.

(7.3.4)

312

CHAPTER 7

Proof. Indeed, see Lemma 7.5, [ d/, e4 ]ψ = Γg d% ψ,

(7.3.5)

[ d/, e3 ]ψ = rΓb e3 ψ + Γb d% ψ + l.o.t. Hence, since d/a∗ ∈ r d/Γb , [ d/, ν∗ ]ψ

=

[ d/, e3 + a∗ e4 ]ψ = rΓb e3 ψ + Γb d% ψ + a∗ Γg d% ψ + d/a∗ e4 ψ

=

rΓb (ν∗ ψ − a∗ e4 ψ) + a∗ Γg d% ψ + d/a∗ e4 ψ

= =

rΓb ν∗ ψ − a∗ (Γb d% ψ + Γg d% ψ) + d/Γb · dψ rΓb ν∗ ψ + d≤1 Γb · dψ

as desired. To estimate derivatives of the ` = 1 modes on Σ∗ we make use of the following. Lemma 7.18. For every scalar function h we have the formula Z  Z ν∗ h = (ς)−1 ς (ν∗ (h) + (κ + a∗ κ)h) . S

(7.3.6)

S

In particular ν∗ (r)

=

r −1 (ς) ς(κ + a∗ κ). 2

(7.3.7)

Proof. We consider the coordinates u, θ along Σ∗ with ν∗ (θ) = 0. In these coordinates we have ν∗ =

2 ∂u . ς

The lemma follows easily by expressing the volume element of the surfaces S ⊂ Σ∗ with respect to the coordinates u, θ (see also the proof of Proposition 2.64). Lemma 7.19. Given ψ ∈ s1 , we have the formula Z  Z Z
3 ν∗ ψeΦ = (ν∗ ψ)eΦ + κ − 2κ − Ω κ ψeΦ + Err[ψ, ν∗ ](7.3.8) 2 S S S with error term = r4 Γb ν∗ (ψ) + r3 Γb ψ.

Err[ψ, ν∗ ] Proof. We have Z  Φ ν∗ ψe

= ς −1

S

= ς −1

Z ZS S

ς ν∗ (ψeΦ ) + (κ + a∗ κ)ψeΦ





ς ν∗ ψeΦ + e−Φ ν∗ (eΦ ) + κ + a∗ κ ψeΦ .

313

DECAY ESTIMATES (THEOREMS M4, M5)

Recalling that e4 (Φ) = 12 (κ − ϑ), e3 (Φ) = 12 (κ − ϑ) we deduce e−Φ ν∗ (eΦ ) + κ + a∗ κ =

3 1 (κ + a∗ κ) − (ϑ − a∗ ϑ). 2 2

ˇ Hence, writing also ςa∗ = −2 − ςΩ, ς = ς + ςˇ, κ = κ + κ ˇ, κ = κ + κ ˇ , and Ω = Ω + Ω, Z   Z  Z 3 1 −1 Φ −1 Φ ν∗ ψe = ς ς ν∗ ψ + (κ + a∗ κ)ψ e − ς ς(ϑ − a∗ ϑ)ψeΦ 2 2 S S S  Z  3 −1 = ς ς ν∗ ψ + (κ − Ωκ)ψ eΦ 2 S  Z  Z 3 −1 Φ −1 + ς ςˇ ν∗ ψ + (κ − Ωκ) e − 3ς κψeΦ 2 S S Z 1 −1 − ς ς(ϑ − a∗ ϑ)ψeΦ 2  Z  S Z 3 = ν∗ ψ + (κ − Ωκ)ψ eΦ − 3ς −1 κψeΦ 2 S  S Z  3 −1 + (ς ς − 1) ν∗ ψ + (κ − Ωκ)ψ eΦ 2 S  Z  Z 3 1 + ς −1 ςˇ ν∗ ψ + (κ − Ωκ)ψ eΦ − ς −1 ς(ϑ − a∗ ϑ)ψeΦ 2 2 S S Z Z
3 Φ Φ = (ν∗ ψ)e + κ − 2κ − Ω κ ψe + Err[ψ, ν∗ ] 2 S S where, Err[ψ, ν∗ ]

=

r4 Γb ν∗ (ψ) + r3 Γb ψ + r3 Γb ψ + r3 Γg ψ

and the conclusion follows from the fact that Γg behaves at least as good as Γb . Corollary 7.20. Given ψ ∈ s1 and k ≥ 1, the following estimate holds true: Z (ν∗k ψ)eΦ S

Z  k X j
Φ . ψe + d≤k−1 r4 Γb ν∗ (ψ) + r3 Γb ψ .(7.3.9) ν∗ j=0

S

Proof. We prove (7.3.9) by iteration. First, (7.3.9) holds true for k = 1 in view of Lemma 7.19. Also, assuming (7.3.9) for k ≥ 1, we apply it with ψ replaced by ν∗ ψ which implies Z (ν∗k+1 ψ)eΦ S

.

Z  k X j ≤k−1 4
Φ ν∗ ν ψe r Γb ν∗2 (ψ) + r3 Γb ψ . ∗ + d j=0

S

Applying Lemma 7.19 with ψ replaced by ν∗ ψ to all terms in the sum of the lefthand side, we infer (7.3.9) with k replaced by k + 1 which shows that (7.3.9) holds indeed for all k by iteration.

314 7.3.2

CHAPTER 7

Differential identities involving GCM conditions on Σ∗

Recall our GCM conditions on Σ∗ : κ=

2 , r

/d?2 /d?1 µ = 0,

/d?2 /d?1 κ = 0,

Z

ηeΦ = 0,

S

Also, on S∗ , the last cut of Σ∗ , Z βeΦ = 0, S∗

Z

Z

ξeΦ = 0.

(7.3.10)

S

eθ (κ)eΦ = 0.

(7.3.11)

S∗

The goal of this section is to derive identities involving the GCM conditions which will be used later, see Lemma 7.25. Proposition 7.21. The following identities hold true on Σ∗ .   2 /d?2 /d?1 /d1 /d2 /d?2 η = κ ν∗ ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 ν∗ (eθ (κ)) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.,

(7.3.12)

    2 /d?2 /d?1 /d1 /d2 /d?2 ξ = ν∗ ( /d?2 /d2 + 2K) /d?2 /d?1 κ − κ ν∗ ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β (7.3.13) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. The quadratic terms denoted l.o.t. are lower order both in terms of decay in r, u as well as in terms of number of derivatives. In particular, if the GCM conditions (7.3.10) are verified, we deduce /d?2 /d?1 /d1 /d2 /d?2 η = κ /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.,

/d?2 /d?1 /d1 /d2 /d?2 ξ = −κ /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.

(7.3.14)

Proof. The proof is a straightforward application of Proposition 7.14. Indeed according to (7.1.32) we have   2 /d?2 /d?1 /d1 /d2 /d?2 η = κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 e3 (eθ (κ)) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. On the other hand since ν∗ = e3 + a∗ e4 with a∗ = e3 ( /d?2 /d?1 µ)

2 ς∗

− Υ + 2r A, see (7.1.8),

= ν∗ ( /d?2 /d?1 µ) − a∗ e4 ( /d?2 /d?1 µ)

= ν∗ ( /d?2 /d?1 µ) − a∗ ( /d?2 /d?1 e4 µ + [e4 , /d?2 /d?1 ]ˇ µ) .

315

DECAY ESTIMATES (THEOREMS M4, M5)

Also, in the same fashion,16 /d?2 /d?1 /d1 e3 (eθ (κ))

= =

/d?2 /d?1 /d1 [ν∗ (eθ (κ)) − a∗ e4 eθ κ]

/d?2 /d?1 /d1 [(ν∗ (eθ (κ))] − a∗ /d?2 /d?1 /d1 (e4 eθ κ) X +r−3 d/i a∗ d/j (e4 eθ κ) i+j=2

/d?2 /d?1 /d1 [ν∗ (eθ (κ)) − a∗ eθ e4 κ − [e4 , eθ ]κ] X = r−2 d/i Γb d/j (eθ (e4 κ) + [eθ , e4 ]κ) . =

i+j=2

Thus, after using the transport equations for e4 µ, e4 κ and the commutator lemma applied to [e4 , eθ ] we easily deduce   2 /d?2 /d?1 /d1 /d2 /d?2 η = κ ν∗ ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 ν∗ (eθ (κ)) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. which confirms the first identity of the proposition. The second part of the proposition can be derived in the same manner starting with the identity (7.1.33)     2 /d?2 /d?1 /d1 /d2 /d?2 ξ = e3 ( /d?2 /d2 + 2K) /d?2 /d?1 κ − κ e3 ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. This concludes the proof of the proposition. 7.3.3

Control of the flux of some quantities on Σ∗

The goal of this section is to establish the following. Proposition 7.22. The following estimate holds true for all k ≤ ksmall + 18: Z  2 2  r2 d/≤3 dk∗ β + d/≤4 dk∗ Γb . 20 u−2−2δdec . (7.3.15) Σ∗ (u,u∗ )

We also have for k ≤ ksmall + 17 rk d/≤1 βk∗∞,k + k d/≤2 Γb k∗∞,k

. 0 r−1 u−1−δdec .

(7.3.16)

Remark 7.23. The flux estimates (7.3.15) will be used in the proof of Proposition 7.26 on the control of the ` = 1 mode of various quantities. They also improve the bootstrap assumption on the flux estimate for η on Σ∗ which is part of the decay norm (ext) Dk [η]. Proof. Note that (7.3.16) follows immediately from (7.3.15) using the trace theorem and Sobolev. We thus concentrate our attention on deriving (7.3.15). Step 1. We first prove the corresponding estimates for β away from its ` = 1 mode. 16 Note

that in view of (7.3.3), we have / da∗ ∈ rΓb .

316

CHAPTER 7

More precisely we prove: Lemma 7.24. The following estimate holds true for all k ≤ ksmall + 18: Z 2 r4 /d?2 ( d/≤2 dk∗ β) . 20 u−2−2δdec . (7.3.17) Σ∗ (u,u∗ )

Proof. We make use of Proposition 7.11 according to which ( ) 3 3 ? ? 3 3 5 ? ? ? 2 e3 (rq) = r /d2 /d1 /d1 β − κρα − ρ /d2 /d1 κ − κρ /d2 ζ + (2ρ − κκρ)ϑ 2 2 2 4 +

Err[e3 (rq)]

where Err[e3 (rq)]


= r3 d≤3 Γb · Γg .

In view of Lemma 7.6, we have for all k ≤ ksmall + 18, 3

kErr[e3 (rq)]k∞,k (u, r) . 0 u− 2 −δdec .

(7.3.18)

We can also check, making use of the estimates (7.1.13), and Lemma 7.6 for ϑ,   δdec kρ /d?2 /d?1 κ, κρ /d?2 ζ, κκρϑ, ρ2 ϑk∞,k .  r−7 + r−6 u−1− 2 . In view of our assumption for r on Σ∗ we have r ≥ 0 u1+δdec , we thus deduce for all k ≤ ksmall + 18

   

δ 3 −2 −1 −1− dec

e3 (rq) − r5 /d?2 /d?1 /d1 β − 3 κρα 2 .  r + r u + 0 u− 2 −δdec

2 ∞,k 3

. 0 u− 2 −δdec . We infer that r−1 kr4 dk∗ /d?2 /d?1 /d1 βkL2 (S)

. r−1 kr−1 dk∗ e3 (rq)kL2 (S) + r−1 kdk∗ αkL2 (S) 3

+0 r−1 u− 2 −δdec

where dk∗ = ν∗k1 d/k2 denote the tangential derivatives to Σ∗ . Thus integrating on the last slice Σ∗ and making use of the assumptions (7.1.15) and (7.1.18), i.e., Z |e3 dk q|2 + r−2 |q|2 + |dk α|2 . 20 (1 + u)−2−2δdec , k ≤ ksmall + 18, Σ∗ (u,u∗ )

we deduce Z

2 r8 dk∗ ( /d?2 /d?1 /d1 β)

. 20 u−2−2δdec .

Σ∗ (u,u∗ )

Taking into account the commutator Lemma 7.17, as well as the product Lemma

317

DECAY ESTIMATES (THEOREMS M4, M5)

7.6, we deduce, for k ≤ ksmall + 18, Z 2 r8 /d?2 /d?1 /d1 (dk∗ β)

. 20 u−2−2δdec .

(7.3.19)

Σ∗ (u,u∗ )

Since /d?1 /d1 = /d2 /d?2 + 2K, we infer that Z

2 r8 ( /d?2 /d2 + 2K) /d?2 (dk∗ β)

.

20 u−2−2δdec .

Σ∗ (u,u∗ )

In view of the coercivity of /d?2 /d2 + 2K we deduce Z 2 r4 /d?2 ( d/≤2 dk∗ β) . 20 u−2−2δdec , Σ∗ (u,u∗ )

k ≤ ksmall + 18.

This concludes the proof of Lemma 7.24. Step 2. We make use of Lemma 7.24 to prove the desired estimate for ϑ, i.e., Z ≤4 k 2 d/ d∗ ϑ| . 20 u−2−2δdec , k ≤ ksmall + 18. (7.3.20) Σ∗ (u,u∗ )

Proof of (7.3.20). One starts with the Codazzi equation /d2 ϑ

= −2β − /d?1 (κ) − ζκ + Γg · Γb .

Differentiating w.r.t. /d?2 and then taking tangential derivatives d/≤2 dk∗ we derive   d/≤2 dk∗ /d?2 /d2 ϑ = −2 d/≤2 dk∗ /d?2 β − d/≤2 dk∗ /d?2 /d?1 (κ) − d/≤2 dk∗ r−2 d/Γg + r−1 d/ (Γg · Γb ) . Making use of the GCM condition /d?2 /d?1 κ = 0 along Σ∗ and the interpolation estimates of Lemma 7.6, for all k ≤ ksmall + 18, d/≤2 dk∗ /d?2 /d2 ϑ

or, since r ≥

= −2 d/≤2 dk∗ /d?2 β + r−2 d/≤2 dk+1 Γg + r−1 d/≤2 dk+1 (Γg · Γb )   δdec 1 = −2 d/≤2 dk∗ /d?2 β + O r−4 u− 2 − 2

 1+δdec , 0 u

d/≤2 dk∗ /d?2 /d2 ϑ

  3 = −2 d/≤2 dk∗ /d?2 β + O 0 r−3 u− 2 −δdec .

Moreover, /d?2 /d2 d/≤2 dk∗ ϑ

  3 = −2 d/≤2 dk∗ /d?2 β + O 0 r−3 u− 2 −δdec + [ d/≤2 dk∗ , /d?2 /d2 ]ϑ.

Using the commutator estimates of Lemma 7.17 and the interpolation estimates of

318

CHAPTER 7

Lemma 7.6, we derive   3 = −2dk∗ /d?2 β + O 0 r−3 u− 2 −δdec .

/d?2 /d2 dk∗ ϑ

Integrating and using the previously derived estimate for β we deduce Z 2 r4 /d?2 /d2 d/≤2 dk∗ ϑ . 20 u−2−2δdec , k ≤ ksmall + 18. Σ∗ (u,u∗ )

In view of the coercivity of /d?2 /d2 we infer that Z ≤4 k 2 d/ d∗ ϑ . 20 u−2−2δdec , Σ∗ (u,u∗ )

k ≤ ksmall + 18

as desired. Step 3. We next derive a non-sharp, preliminary estimate for the ` = 1 mode of β with the help of the Codazzi equation for ϑ, 2β

= − /d2 ϑ + eθ (κ) − κζ + Γg · Γb = − /d2 ϑ + r−1 d/≤1 Γg + Γg · Γb .

Projecting on the ` = 1 mode, this yields Z 2 βeΦ = r2 d/≤1 Γg + r3 Γg · Γb . S

Differentiating, and using Lemma 7.6, we deduce Z  k δ Φ − 12 − dec ν∗ 2 , βe . u S

k ≤ ksmall + 18.

(7.3.21)

Together with Corollary 7.20, we infer Z δ (ν∗k β)eΦ . u− 12 − dec 2 + r4 |dk (β · Γb )|. S

1+δdec Together with product estimates of Lemma 7.6, and since r ≥ (−1 on Σ∗ , 0 )u we deduce, for k ≤ ksmall + 18, Z (ν∗k β)eΦ . 0 ru− 32 −δdec . (7.3.22) S

We combine the result of Lemma 7.24 with (7.3.22) to deduce Z 2 r2 d/≤3 dk∗ β . 20 u−2−2δdec , k ≤ ksmall + 18. Σ∗ (u,u∗ )

(7.3.23)

319

DECAY ESTIMATES (THEOREMS M4, M5)

Indeed, according to the last elliptic estimate of Lemma 7.7 and (7.3.22), we have Z

2 r2 d/≤3 dk∗ β





. r

4

S

. r4

Z ZS

Z 2 ? ≤2 k 2 /d2 ( d/ d∗ β) + r−2 (ν∗k β)eΦ S

? ≤2 k 2 /d2 ( d/ d∗ β) + 20 u−3−2δdec .

S

Thus, integrating and making use of estimate (7.3.17), we infer Z Z 2 2 r2 d/≤3 dk∗ β . r4 /d?2 ( d/≤2 dk∗ β) + 20 u−2−2δdec . 20 u−2−2δdec Σ(u,u∗ )

Σ(u,u∗ )

which concludes the proof of (7.3.23). Step 4. Next, we establish the estimates for η and ξ. We first estimate /d?2 (η, ξ). Lemma 7.25. We have for k ≤ ksmall + 18 Z   r2 | /d?2 ( d/≤5 dk∗ η)|2 + | /d?2 ( d/≤5 dk∗ ξ)|2

. 20 u−2−2δdec . (7.3.24)

Σ∗ (u,u∗ )

Proof. We prove Lemma 7.25 based on the identities of Proposition 7.21. To derive the desired flux estimate for η we make use of the first part of Proposition 7.21 according to which we have   2 /d?2 /d?1 /d1 /d2 /d?2 η = κ ν∗ ( /d?2 /d?1 µ) + 2 /d?2 /d?1 /d1 β − /d?2 /d?1 /d1 ν∗ (eθ (κ)) + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t. Since, /d?1 /d1 = /d2 /d?2 + 2K, we deduce /d?2 ( /d2 /d?2 + 2K) /d2 /d?2 η

=

i 1h κν∗ ( /d?2 /d?1 µ) − /d?2 /d?1 /d1 ν∗ (eθ (κ)) 2 +κ /d?2 ( /d2 /d?2 + 2K)β + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) +l.o.t.,

or i 1h κν∗ ( /d?2 /d?1 µ) − /d?2 /d?1 /d1 ν∗ (eθ (κ)) 2 + r−5 d/≤4 Γg + r−4 d/≤4 (Γb · Γb ) + l.o.t.

  ( /d?2 /d2 + 2K) /d?2 /d2 /d?2 η − κ /d?2 β =

Taking higher tangential derivatives and using our GCM assumptions on Σ∗ h i h i dk∗ ( /d2 /d?2 + 2K) /d?2 /d2 /d?2 η − κ /d?2 β = dk∗ r−5 d/≤4 Γg + r−4 d/≤3 (Γb · dΓb ) + l.o.t. Making use of the commutation Lemma 7.17 we can rewrite h i r2 ( /d2 /d?2 + 2K) /d?2 /d2 /d?2 (dk∗ η) − κ /d?2 (dk∗ β) h i X = d/≤2 r−3 d/≤2 dj∗ Γg + r−2 d/≤1 dj∗ (Γb · dΓb ) . j≤k

320

CHAPTER 7

Using the ellipticity of the operators ( /d2 /d?2 + 2K) and /d?2 /d2 , Lemma 7.6 and the 1+δdec dominance condition r ≥ (−1 on Σ∗ , we derive, for k ≤ ksmall + 18, 0 )u k /d?2 ( d/≤4 dk∗ η)kL2 (S) .

3

rk /d?2 ( d/≤2 dk∗ β)kL2 (S) + 0 r−1 u− 2 −δdec .

(7.3.25)

Finally, squaring, integrating on Σ∗ and taking into account the flux estimate for β in (7.3.23) we deduce, for k ≤ ksmall + 18, Z Z 2 2 ? ≤4 k 2 r /d2 ( d/ d∗ η) . r4 /d?2 ( d/≤2 dk∗ β) + 0 u−2−2δdec . 0 u−2−2δdec Σ∗ (u,u∗ )

Σ∗ (u,u∗ )

as stated. This completes the proof of Lemma 7.25 for η. The proof for ξ is very similar and left to the reader. Step 5. In this step, we derive the desired estimates for η and ξ, i.e., we show Z   | d/≤5 dk∗ η|2 + | d/≤5 dk∗ ξ|2 . 20 u−2−2δdec , k ≤ ksmall + 18. (7.3.26) Σ∗ (u,u∗ )

To this end, we prove the following estimates for the ` = 1 mode of ξ and η: Z Z (ν∗k η)eΦ + (ν∗k ξ)eΦ . 0 r2 u−2−δdec , k ≤ ksmall + 18. (7.3.27) S

S

Then, (7.3.26) follows from (7.3.27) and Lemma 7.25 using a Poincar´e inequality. To prove (7.3.27), we apply Corollary 7.20 to η and ξ. This yields for k ≥ 1 Z Z (ν∗k ξ)eΦ + (ν∗k η)eΦ S

S

Z  Z   k  X j j Φ Φ . ξe + ν∗ ηe ν∗ j=0

S

S


+r4 dk Γb · Γb . In view of the GCM condition for the ` = 1 mode of η and ξ, we infer Z Z
(ν∗k ξ)eΦ + (ν∗k η)eΦ . r4 dk Γb · Γb . S

S

For k ≤ ksmall + 18, we infer, using the product Lemma 7.6, Z Z (ν∗k ξ)eΦ + (ν∗k η)eΦ . 20 u−2−2δdec S

S

which concludes the proof of (7.3.27), and hence of (7.3.26). ˇ and A. Adding κ times the Step 6. Next, we derive the flux estimates for ω ˇ , ςˇ, Ω first equation to the second equation of Proposition 7.12, we obtain   1 1 ? ? 2κ /d1 ω = −e3 (eθ (κ)) − 2 /d2 /d2 η + κ κ + 2ω η − κ2 ξ 2 2 +r−2 d/≤1 Γg + r−1 d/≤1 (Γb · Γb ).

321

DECAY ESTIMATES (THEOREMS M4, M5)

In view of the GCM condition for κ, the fact that ν∗ = e3 +a∗ e4 , and Raychaudhuri, we have −e3 (eθ (κ))

= a∗ e4 (eθ (κ)) = r−1 d/(Γg · Γg )

and hence r /d?1 ω

1 r2 = − r2 /d2 /d?2 η + κ 2 4



 1 r2 κ + 2ω η − κ2 ξ + d/≤1 Γg + r d/≤1 (Γb · Γb ). 2 8

The flux estimate for ω ˇ follows easily from the above identity, the flux estimates for η and ξ derived in Step 4 and Step 5, the interpolation estimate of Lemma 7.6 for ζ, as well as the dominance property of r on Σ∗ . ˇ follow easily from the equations The flux estimates for ςˇ and Ω ς −1 eθ (ˇ ς) ˇ eθ (Ω)

= =

η − ζ, −ξ − (η − ζ)Ω,

the flux estimates for η and ξ derived in Step 4 and Step 5, the interpolation estimate of Lemma 7.6 for ζ, as well as the dominance property of r on Σ∗ . To estimate A, note first that the flux estimate for A − A follows immediately ˇ It then remains to control from formula (7.1.9) and the above flux estimate for Ω. A. In view of (2.2.22), we have ςA =


ˇ + Ωˇ ˇ −κˇ ς + κ ςΩ ς + ςˇκ ˇ − Ω ςˇκ ˇ − Ωςκ

and hence, taking the average, we infer A =

ˇ + ςˇκ ˇ κ −(ς − 1)A − ςˇA ˇ − Ω ςˇκ ˇ − Ως ˇ.

The flux estimate for A follows then from the product estimates of Lemma 7.6. Step 7. It remains to derive the flux estimate for ω, Ω and ς. Recall (4.2.4): (         m r 2 1 2 2 2m ω− 2 = e3 κ − + κ κ− − 2ω κ − −2 ρ+ 3 r 4 r 2 r r r     1 2 ˇ 1 1ˇ + κ κ− Ω − 2ˇ ωκ ˇ + ϑϑ − 2ζ 2 − Ω − ϑ2 + κ ˇ2 2 r 2 2 ) 1 1 ˇ 4 (ˇ ˇκ . +Ω(e κ) + κˇ κ) − κ ˇκ ˇ + Ωˇ 2 r Using the GCM condition for κ, the fact that ν∗ = e3 + a∗ e4 , Lemma 2.72 for e4 (κ − 2/r), and the identity (2.2.12) for ρ, we infer ω−

m r2

= rΓb · Γg

which together with Lemma 7.6 yields the flux estimate for ω.

322

CHAPTER 7

Next, taking the average of (7.1.9), we have Ω+Υ

=

r A. 2

The flux estimate for Ω follows from the above identity and the flux estimate for A derived in Step 6. Finally, we derive the flux estimate for ς. Recall equation (7.1.8) 2 r = − +Υ− A ς 2

a∗

and the GCM condition for a∗ , see (7.1.1), a∗

SP

= −1 −

2m . r

We deduce 2 ς

=

SP

r 2 − A . 2 SP

Since ς = ς SP + ςˇ SP , we infer ς −1

= ςˇ SP +

1 1 − 4r A

r A 4 SP SP

and the flux estimate for ς follows from the above identity and the flux estimates for ςˇ and A of Step 6. This concludes the proof of Proposition 7.22. 7.3.4

Estimates for some ` = 1 modes on Σ∗

In this section, we control the ` = 1 modes of eθ (κ), eθ (ρ), eθ (µ) and of β. We summarize the results in the following proposition. Proposition 7.26. The following estimates hold true: Z Z Z  k Φ −1 −1−δdec eθ (ρ)eΦ + eθ (µ)eΦ + ν∗ max βe , k≤ksmall +20 . 0 r u S S S Z (7.3.28) eθ (κ)eΦ . 0 u−2−δdec . S

Remark 7.27. We note that the estimates for the ` = 1 modes of eθ (κ) and β are sharp.17 During the proof we shall also need to derive sharp estimates for the ` = 1 modes of ζ and β, see (7.3.30) and (7.3.32). 17 Consistent

in fact with strong peeling.

323

DECAY ESTIMATES (THEOREMS M4, M5)

Proof. We will rely on the following auxiliary bootstrap assumptions: Z Z βeΦ . r−1 u−1−δdec , eθ (κ)eΦ . u−2−δdec . S

(7.3.29)

S

Step 1. We start with proving an intermediary estimate for the ` = 1 mode of ζ. In view of the Codazzi equations and the GCM condition on κ, /d2 ϑ

2 = −2β + (eθ (κ) + ζκ) + Γg · Γg = −2β + ζ + Γg · Γg r

and hence Z

ζeΦ

= −r

S

Z

βeΦ + r4 Γg · Γg .

S

Thus, using the product estimates of Lemma 7.6, Z Z ζeΦ . r βeΦ + 0 u−1−δdec . S

S

In particular, in view of (7.3.29), we infer Z ζeΦ . u−1−δdec .

(7.3.30)

S

Step 2. Next, we establish an intermediary estimate for the ` = 1 mode of β. We start with the Codazzi equation for ϑ, 2β

= − /d2 ϑ + eθ (κ) +

2Υ ζ + Γ g · Γb r

and project on the ` = 1 mode, i.e., Z Z Z Z 2Υ Φ Φ Φ 2 βe = ζe + eθ (κ)e + Γg · Γb eΦ . r S S S S

(7.3.31)

We make use of the estimate (7.3.30) for ζ, the auxiliary estimate (7.3.29) for eθ (κ), 1+δdec the dominance condition r ≥ −1 on Σ∗ , and bootstrap assumptions for Γg 0 u to deduce Z Z Z Z −1 Φ −2 Φ −1 Φ Γg · Γb r βe . r ζe + r eθ (κ)e + S S S S Z −2 −1−δdec −1 −2−δdec −2 − 12 −δdec . r u + r u + r u |Γb | S

.

3

0 u−3−2δdec + 0 u− 2 −2δdec kΓb kL2 (S) .

Thus, we infer Z

u∗

u

Z Z r−1 βeΦ du . 0 u−2−2δdec + 0 S

u∗

u

u−3−4δdec

1/2 Z

u∗

u

kΓb k2L2 (S)

1/2

324

CHAPTER 7

which together with the flux estimate of Proposition 7.22 implies Z Z u∗ r−1 βeΦ du . 0 u−2−2δdec . u

(7.3.32)

S

Step 3. Next, we provide an intermediary estimate for the ` = 1 mode of ρ. We start by differentiating the Gauss equation K = −ρ − 14 κ κ + 14 ϑϑ. Using the GCM condition for κ we derive eθ (ρ)

= −eθ (K) −

1 1 eθ (κ) + eθ (ϑϑ). 2r 4

We make use of the vanishing of the ` = 1 mode of eθ (K) (see Lemma 2.32) to derive Z Z Z 1 1 Φ Φ eθ (ρ)e = − eθ (κ)e + eθ (ϑϑ)eΦ . (7.3.33) 2r S 4 S S Using the auxiliary estimate (7.3.29) for the ` = 1 mode of eθ (κ) Z Z eθ (ρ)eΦ . r−1 u−2−δdec + | d/≤1 (Γg · Γb )|. S

S

1+δdec Making use of r ≥ −1 and the bootstrap assumptions on Γg , we deduce 0 u Z eθ (ρ)eΦ . 0 u−3−2δdec + 0 u− 32 −2δdec k d/≤1 Γb kL2 (S) . S

Integrating in u we derive Z

u∗

u

Z Z eθ (ρ)eΦ du . 0 u−2−δdec + 0 S

u∗

u−3−4δdec

 12 Z

u

Using the flux estimate in Proposition 7.22, we infer Z u∗ Z eθ (ρ)eΦ du . 0 u−2−δdec . u

u∗

u

k d/≤1 Γb k2L2 (S)

 12 .

(7.3.34)

S

Step 4. Next, we control the ` = 1 mode of eθ (κ) on Σ∗ . To this end, we need the

DECAY ESTIMATES (THEOREMS M4, M5)

325

precise identity18 of Proposition 2.74:
3 = −2 /d2 /d?2 ξ + κ e3 (ζ) − β + κ2 ζ − κeθ κ + 6ρξ − 2ωeθ (κ) 2 +Err[ /d2 /d?2 ξ],   1 1 1 Err[ /d2 /d?2 ξ] = 2 /d1 ξ + κ ϑ + 2ηξ − ϑ2 η + 2eθ (ηξ) − eθ (ϑ2 ) 2 2 2   1 1 1 1 +κ ϑζ − ϑξ − ϑeθ κ − ϑϑξ 2 2 2 2     1 2 −ζ 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ + ξ − ϑϑ − 2 /d1 ζ + 2ζ 2 2 −6ηζξ − 6eθ (ζξ). e3 (eθ (κ))

The error term can be written schematically as

Err[ /d2 /d?2 ξ] = r−1 d/≤1 Γb · Γb + r−1 d/≤1 Γg · Γb . Note also that we can write     1 3 1 2 κ κζ − 2ωζ − κeθ κ + ζ κ + 2ω κ + 6ρξ − 2ωeθ (κ) 2 2 2   4Υ2 1 8m 12m = ζ+ 3− eθ (κ) − 3 ξ + r−1 d/≤1 (Γg · Γb ). r2 r r r Also, using the transport equation for e4 (κ) and the GCM condition for κ, we have e4 (eθ (κ))

= eθ e4 κ + [e4 , eθ ]κ   1 1 = eθ − κ κ − 2 /d1 ζ + 2ρ + Γg · Γb + κeθ κ + l.o.t. 2 2 1 ? −1 = − eθ κ + 2 /d1 /d1 ζ + 2eθ (ρ) + r d/(Γg · Γb ) r 1 = − eθ κ + 2( /d2 /d?2 + 2K)ζ + 2eθ (ρ) + r−1 d/(Γg · Γb ). r

We can also write, since ν∗ = e3 + a∗ e4 , e3 (ζ)

18 Note

= ν∗ (ζ) − a∗ e4 (ζ) = ν∗ (ζ) − a∗ (−κζ − β + Γg · Γg )    2m 2 = ν∗ (ζ) − 1 + ζ + β + Γb · Γg . r r

that the schematic form of Proposition 7.12 is not suitable here.

326

CHAPTER 7

Combining, we deduce 2Υ ν∗ (eθ (κ)) = −2 /d2 /d?2 ξ + κν∗ (ζ) + β + E1 + E2 ,   r  4 6m 2 3m 12m E1 = 2 1 − ζ+ 2− eθ (κ) − 3 ξ r r r r r (7.3.35)       2m 2Υ 2m 2m ? −2 1+ /d2 /d2 ζ + 1+ β−2 1+ eθ (ρ), r r r r
E2 = r−1 d/≤1 Γb · Γb + r−1 d/≤1 (Γg · Γb ). We introduce the following notation f := eθ (κ) − κζ.

(7.3.36)

Using the fact that ν∗ = e3 + a∗ e4 , and the transport equation for e3 (κ) and e4 (κ), we have ν∗ (f )

= ν∗ (eθ (κ)) − κν∗ (ζ) − ν∗ (κ)ζ

= ν∗ (eθ (κ)) − κν∗ (ζ) − e3 (κ) + a∗ e4 (κ))ζ   4 4m = ν∗ (eθ (κ)) − κν∗ (ζ) + 2 1 − ζ + r−1 d/≤1 (Γg · Γb ). r r Together with (7.3.35), we deduce, with a similar E2 ,   2Υ 4 4m ? ν∗ (f ) = −2 /d2 /d2 ξ + β+ 2 1− ζ + E1 + E2 . r r r

(7.3.37)

Projecting on the ` = 1 mode and integrating /d2 /d?2 ξ by parts, we derive Z

ν∗ (f )eΦ

S

=

2Υ r

Z S

βeΦ +

4 r2

 1−

4m r

Z S

ζeΦ +

Z

(E1 + E2 )eΦ .

S

In view of Ref 1, we have schematically Z   1 E2 eΦ = O k d/≤1 Γb k2L2 (S) + r−1 u− 2 −δdec k d/≤1 Γb kL2 (S) . S

Also, using the GCM condition for the ` = 1 mode of ξ and integrating /d?2 /d2 ζ by parts, we infer   Z   4 4m Φ 1 − ζ + E 1 e 2 r r S  Z  Z 8 5m 2 3m Φ = 1 − ζe + 2 − eθ (κ)eΦ r2 r r r  S Z  SZ 2Υ 2m 2m Φ + 1+ βe − 2 1 + eθ (ρ)eΦ . r r r S S Hence, in view of the intermediate assumption 7.3.29 for the ` = 1 modes of β and

327

DECAY ESTIMATES (THEOREMS M4, M5)

eθ (κ) as well as estimate (7.3.30) for the ` = 1 mode of ζ, Z     4 4m Φ −2 −1−δdec 1 − ζ + E u + r−1 u−2−2δdec 1 e . r 2 r r S Z + eθ (ρ)eΦ . S

We deduce Z (ν∗ f )eΦ .

r

S

−1

Z Z βeΦ + eθ (ρ)eΦ + k d/≤1 Γb k2 2 L (S) S

S

1

+r−1 u− 2 −δdec k d/≤1 Γb kL2 (S) + r−2 u−1−δdec + r−1 u−2−2δdec , 1+δdec or, making use of the assumption r ≥ −1 , 0 u Z Z Z (ν∗ f )eΦ . r−1 βeΦ + eθ (ρ)eΦ + k d/≤1 Γb k2 2 L (S) S

S

S

+0 u−3−2δdec .

(7.3.38)

On the other hand, according to Lemma 7.19, Z  Z Z
3 ν∗ f eΦ = (ν∗ f )eΦ + κ − 2κ − Ω κ f eΦ + Err[f, ν∗ ] 2 S S S

(7.3.39)

with error term Err[f, ν∗ ]

=

r4 Γb ν∗ (f ) + r3 Γb f.

(7.3.40)

Note that r−1

Z

f eΦ

=

S

r−1

Z

Z

κζeΦ Z Z Z −1 Φ −2 Φ −1 r eθ (κ)e + 2r Υ ζe − r κ ˇ ζeΦ . S

=

eθ (κ)eΦ − r−1

S

S

S

S

Thus in view of our auxiliary assumption (7.3.29) for eθ (κ), estimate (7.3.30) of Step 1, and the dominance condition for r, we deduce Z −1 Φ −1 −2−2δdec r f e + r−2 u−1−δdec . 0 u−3−3δdec . (7.3.41) . r u S

Also, we have in view of the definition of f and (7.3.37) f = r−1 d/≤1 Γg ,

ν∗ (f ) = r−2 d/≤2 Γb .

Together with (7.3.40), we infer Err[f, ν∗ ]

=

r4 Γb r−2 d/≤2 Γb + r2 Γb · d/≤1 Γg .

328

CHAPTER 7

We deduce, together with (7.3.38), (7.3.39) and (7.3.41), Z Z Z  Φ −1 Φ ν∗ eθ (ρ)eΦ + k d/≤4 Γb k2 2 + 0 u−3−2δdec f e . r βe + L (S) S

S

S

where we have also used the control of Γg and Sobolev. Integrating in u, and making use of Proposition 7.22 on the flux estimates for Γb , as well as the estimate (7.3.32) for the ` = 1 mode of β and the estimate (7.3.34) for the ` = 1 mode of eθ (ρ), we infer  Z u∗ Z 0 Φ −2−δdec ν∗ f e . du . 0 u u

S

We deduce Z Φ fe S(u)

Z Z u∗ Z  0 Φ ν∗ . f eΦ + f e du S∗ u S Z Φ . f e + 0 u−2−δdec . S∗

Together with the definition of f and the GCM condition for the ` = 1 mode of eθ (κ) on S∗ , this yields Z Z Z Φ Φ Φ eθ (κ)e . κζe + κζe + 0 u−2−δdec S(u) S(u) S∗ Z Z −1 Φ −1 Φ . r ζe + r ζe + r3 |Γg · Γg | + 0 u−2−δdec . S(u) S∗ Together with the estimate (7.3.30) for the ` = 1 mode of ζ, the control of Γg , and the dominance condition of r on Σ∗ , we obtain Z eθ (κ)eΦ . r−1 u−1−δdec + 0 u−2−δdec . 0 u−2−δdec S

which improves the estimate for the ` = 1 mode of eθ (κ) in (7.3.29) and establishes the desired estimate for the ` = 1 mode of eθ (κ). Step 5. We establish the desired estimate for the ` = 1 mode of eθ (ρ). In view of (7.3.33), we have Z Z Z eθ (ρ)eΦ . r−1 eθ (κ)eΦ + | d/≤1 (Γg · Γb )|. (7.3.42) S

S

S

Using the improved estimate of Step 4 for the ` = 1 mode of eθ (κ) and the control of Γb and Γg , we infer Z eθ (ρ)eΦ . 0 r−1 u−2−δdec + 2 r−1 u− 32 −δdec . 0 r−1 u−1−δdec S

which is the desired estimate for the ` = 1 mode of eθ (ρ).

329

DECAY ESTIMATES (THEOREMS M4, M5)

Step 6. To estimate the ` = 1 mode of µ we differentiate the relation µ = −div ζ − ρ + Γg · Γb by eθ and obtain eθ (µ)

= = =

/d?1 /d1 ζ − eθ (ρ) + r−1 d/(Γg · Γb )

( /d2 /d?2 + 2K)ζ − eθ (ρ) + r−1 d/(Γg · Γb ) 2 /d2 /d?2 ζ + 2 ζ − eθ (ρ) + r−1 d/(Γg · Γb ) + l.o.t. r

Hence, Z

eθ (µ)eΦ

=

S

2r−2

Z S

ζeΦ −

Z S

eθ (ρ)eΦ + r2 d/(Γg · Γb ).

(7.3.43)

Using the estimate (7.3.30) for the ` = 1 mode of ζ and the estimate of Step 5 for the ` = 1 mode of eθ (ρ), we deduce, using also the dominant condition for r on Σ∗ , Z eθ (µ)eΦ . r−2 u−1−δdec + 0 r−1 u−1−δdec . 0 r−1 u−1−δdec S

which is the desired estimate for the ` = 1 mode of eθ (µ). Step 7. It remains to estimate the ` = 1 mode of β. We start with the e3 β equation e3 β + κβ

= − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα.

Also, taking into account
the e4 equation for β and recalling that a∗ satisfies the identity a∗ = − 1 + 2m + rΓb , r ν∗ (β)

= e3 (β) + a∗ e4 β = −κβ − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα + a∗ (−2κβ + /d2 α + ζα)   6m 6 m 2m ? = − /d1 ρ − 3 η + 1− β− 1+ /d2 α + r−1 Γb · d/≤1 Γg . r r r r

Projecting on the ` = 1 mode, and using the GCM condition for the ` = 1 mode of η, Z Z Z Z 6 m ν∗ (β)eΦ = eθ (ρ)eΦ + 1− βeΦ + Γb · d/≤1 Γg . (7.3.44) r r S S S S On the other hand, making use of Lemma 7.19, the auxiliary assumption (7.3.29) 1+δdec for the ` = 1 mode of β and r ≥ −1 , 0 u Z  Z Z
3 ν∗ βeΦ = (ν∗ β)eΦ + κ − 2κ − Ω κ βeΦ + r4 Γb ν∗ (β) + r3 Γb β 2 S S S Z Z 6 = (ν∗ β)eΦ − βeΦ + r2 Γb · d≤1 Γg r S S

330

CHAPTER 7

where we have used the fact that ν∗ β = r−2 Γg . Together with (7.3.44), we deduce Z Z Z  Φ ν∗ eθ (ρ)eΦ + 1 βeΦ + r2 Γb · d≤1 Γg . βe . r S

S

S

Using (7.3.29) and (7.3.42), we infer Z Z  1 Φ Φ −2 −1−δdec ν∗ βe . e (κ)e + r2 |Γb · d≤1 Γg | θ + r u r S ZS + | d/≤1 (Γg · Γb )|. S

Using the control of the ` = 1 mode for eθ (κ) derived in Step 4, our control for Γg and Sobolev, as well as the dominance condition in r on Σ∗ , we infer Z  1 Φ ν∗ βe . 0 r−1 u−2−δdec + r−1 u− 2 −δdec k d/≤2 Γb kL2 (S) . (7.3.45) S

Integrating (7.3.45) in u, and making use of Proposition 7.22 on the flux estimates for Γb , we infer  Z u∗ Z Φ ν∗ βe . 0 r−1 u−1−δdec u

S

+r

−1

Z

u∗

0 −1−2δdec

 12

Z

u u

Σ∗ (u,u∗ )

! 12 2

|Γb |

. 0 r−1 u−1−δdec . In view of the GCM condition for the ` = 1 mode of β on S∗ , we infer on Σ∗ Z βeΦ . 0 r−1 u−1−δdec S

which is the desired estimate for k = 0. Also, coming back to (7.3.45), and using our control for Γb , we have Z  Φ ν∗ βe . 0 r−1 u−1−δdec S

which is our desired estimate for k = 1. It remains to consider the case 2 ≤ k ≤ ksmall + 20. In view of Corollary 7.20, we easily derive the following estimate: Z  Z  Z  Z k ≤k−2 Φ 2 Φ Φ Φ ν∗ ν∗ βe . ν βe + ν βe + βe ∗ ∗ S S S S + d≤k−1 (r4 Γb ν∗ (β) + r3 Γb β) . Since ν∗ β = r−2 Γg , and using our product estimates, as well as the above improved estimates for the ` = 1 mode of β for k = 0 and k = 1, we deduce the following, for

331

DECAY ESTIMATES (THEOREMS M4, M5)

2 ≤ k ≤ ksmall + 20, Z  k Φ ν∗ βe

Z  . ν∗≤k−2 ν∗2 βeΦ + 0 r−1 u−1−δdec .

S

(7.3.46)

S

In view of (7.3.46), we need to estimate ν∗2 β. Recall from above that   6m 6 m 2m ν∗ (β) = − /d?1 ρ − 3 η + 1− β− 1+ /d2 α + r−1 Γb · d/≤1 Γg . r r r r Differentiating again, and relying on the commutation formula of Lemma 7.17, we infer   6m 6 m 2m 2 ? ν∗ (β) = − /d1 (ν∗ ρˇ) − 3 ν∗ η + 1− ν∗ β − 1 + /d2 (ν∗ α) r r r r +r−1 d≤1 (Γb · d/≤1 Γg ). Also, using the Bianchi identity for e3 (ˇ ρ) and e4 (ˇ ρ), as well as ν∗ = e3 + a∗ e4 , we have ν∗ ρˇ =

/d1 β + r−2 d≤1 Γg

and hence ν∗2 (β)

=

  6m 6 m 2m − 3 ν∗ η + 1− ν∗ β − 1 + /d2 (ν∗ α) r r r r

− /d?1 /d1 β

+r−3 d≤2 Γg + r−1 d≤1 (Γb · d/≤1 Γg ). This yields Z ν∗2 (β)eΦ S

= −

Z S ≤2

+d

/d?1 /d1 βeΦ

6m − 3 r

Z

6 m ν∗ ηe + 1− r r S Φ

Z

ν∗ βeΦ

S

Γg + r2 d≤1 (Γb · d/≤1 Γg ).

Since /d?1 /d1 = /d2 /d?2 + 2K, we infer Z Z Z Z 2 6m 6 m ν∗2 (β)eΦ = − 2 βeΦ − 3 ν∗ ηeΦ + 1− ν∗ βeΦ r r r r S S S S +d≤2 Γg + r2 d≤1 (Γb · d/≤1 Γg ). Also, applying Lemma 7.19 to the last two integrals of the RHS, and using again that ν∗ β = r−2 Γg , we obtain Z  Z  Z Z 2 6m 6 m ν∗2 (β)eΦ = − 2 βeΦ − 3 ν∗ ηeΦ + 1− ν∗ βeΦ r S r r r S S S +d≤2 Γg + r2 d≤1 (Γb · d/≤1 Γg ). Together with (7.3.46), the control of Γb and Γg , and the dominance of r on Σ∗ ,

332

CHAPTER 7

and arguing again by iteration for the β term, we deduce Z  Z  Z  k 1 k−2 1 k−1 Φ Φ Φ −1 −1−δdec ν∗ βe . ν βe + ν ηe . ∗ ∗ + 0 r u 2 3 r r S S S Together with the GCM condition for the ` = 1 mode for η and the estimate (7.3.21), we infer for 2 ≤ k ≤ ksmall + 20 Z  k Φ −1 −1−δdec ν∗ βe . 0 r u S

as desired. This completes the proof of Proposition 7.26. 7.3.5

Decay of Ricci and curvature components on Σ∗

Recall that: • we have already derived improved pointwise estimates for α and α, respectively in Theorem M2 and Theorem M3, • we have already derived improved pointwise estimates for β and Γb on Σ∗ , see (7.3.16) in Proposition 7.22, • κ ˇ = 0 on Σ∗ in view of the GCM condition for κ. In the following proposition, we derive improved pointwise estimates on Σ∗ for the remaining quantities, i.e., κ ˇ , ρˇ, ζ, µ ˇ, ϑ and β. Proposition 7.28. The following estimates hold along Σ∗ for all k ≤ ksmall + 18: kˇ κ, rµ ˇk∗∞,k . 0 r−2 u−1−δdec , 1

kϑ, ζ, rρˇk∗∞,k . 0 r−2 u− 2 −δdec , kβk∗∞,k

. 0 r

−3

− 12 −δdec

(2r + u)

(7.3.47) .

Also, for all k ≤ ksmall + 17




ν∗ ϑ, ζ, rρˇ ∗ . 0 r−2 u−1−δdec , ∞,k 1

kν∗ βk∗∞,k . 0 r−4 u− 2 −δdec .

(7.3.48)

Proof. We proceed in several steps. Step 1. In this step we control κ ˇ . First, note from Proposition 2.73 that we have e3 (ˇ κ) = r−1 d/≤1 Γb + r−1 Γg + Γb · Γb ,

e4 (ˇ κ) = r−1 d/≤1 Γg ,

and hence ν∗ (ˇ κ)

= r−1 d/≤1 Γb + r−1 d/≤1 Γg + Γb · Γb .

Together with the improved control of Γb in (7.3.16), the control of Γg , and the dominance in r condition, we infer, for all k ≤ ksmall + 17, kν∗ κ ˇ k∗∞,k

.

0 r−2 u−1−δdec + r−3 . 0 r−2 u−1−δdec .

333

DECAY ESTIMATES (THEOREMS M4, M5)

It then remains to control d/k κ ˇ . Since we have /d?2 /d?1 κ = 0 in view of our GCM condition, we infer, using a Poincar´e inequality, Z r−1 k d/k κ ˇ kL2 (S) . r−2 eθ (κ)eΦ . 0 r−2 u−1−δdec S

where we have used Proposition 7.26 to estimate the ` = 1 mode of κ. Together with the above estimate for ν∗ κ ˇ , we infer, for all k ≤ ksmall + 18, the desired estimate kˇ κk∗∞,k

0 r−2 u−1−δdec .

.

Step 2. Next, we estimate ρˇ. First, note from Proposition 2.73 that we have e3 (ˇ ρ) = r−1 d/≤1 β + r−2 Γg + Γb · Γg ,

e4 (ˇ ρ) = r−2 d/≤1 Γg ,

and hence ν∗ (ˇ ρ)

=

r−1 d/≤1 β + r−2 d/≤1 Γg + Γb · Γg .

Together with the improved control of β in (7.3.16), the control of Γg , and the dominance in r condition, we infer, for all k ≤ ksmall + 17, kν∗ ρˇk∗∞,k

. 0 r−3 u−1−δdec + r−4 . 0 r−3 u−1−δdec .

It then remains to control d/k ρˇ. In view of Proposition 7.10, we have, relative to the background frame of (ext) M,   3 3 r4 /d?2 /d?1 ρ + κρϑ + κρϑ = q + r2 d/≤2 (Γb · Γg ) + l.o.t. 4 4 Using the improved control for q in Ref 2, the improved control for ϑ in (7.3.16), and the product Lemma 7.6, we have, for all k ≤ ksmall + 17, kqk∗∞,k + kϑk∗∞,k + kr2 d/≤2 (Γb · Γg )k∗∞,k

.

1

0 r−1 u− 2 −δdec .

Also, using our condition on r along Σ∗ kϑk∗∞,k

. r−2 . 0 r−1 u−1−δdec .

We deduce, for all k ≤ ksmall + 17, k /d?2 /d?1 ρˇk∗∞,k

1

. 0 r−5 u− 2 −δdec .

We infer, using a Poincar´e inequality, for all k ≤ ksmall + 19, Z 1 1 −1 k −2 Φ r k d/ ρˇkL2 (S) . r eθ (ρ)e + 0 r−3 u− 2 −δdec . 0 r−3 u− 2 −δdec S

where we have used Proposition 7.26 to estimate the ` = 1 mode of ρ. Together with the above estimate for ν∗ κ ˇ , we infer, for all k ≤ ksmall + 18, the desired estimate kˇ ρk∗∞,k

1

. 0 r−3 u− 2 −δdec .

334

CHAPTER 7

Step 3. Next, we control d/k µ ˇ and d/k ζ. Since we have /d?2 /d?1 µ = 0 in view of our GCM condition, we infer, using a Poincar´e inequality, Z r−1 k d/k µ ˇkL2 (S) . r−2 eθ (µ)eΦ . 0 r−3 u−1−δdec S

where we have used Proposition 7.26 to estimate the ` = 1 mode of µ. Also, from the definition of µ = − /d1 ζ − ρ + 14 ϑϑ, we have /d1 ζ

= −ˇ µ − ρˇ + Γg · Γb

and hence, using also the product Lemma 7.6, we infer r−1 k d/k ζkL2 (S)

.

k d/k µ ˇkL2 (S) + k d/k ρˇkL2 (S) + 0 r−2 u−1−δdec .

Together with the above improved estimates for µ ˇ, and the improved estimates for ρˇ of Step 2, we deduce 1

r−1 k d/k ζkL2 (S) . 0 r−2 u− 2 −δdec ,

r−1 k d/k µ ˇkL2 (S) . 0 r−3 u−1−δdec .

Step 4. We conclude in this step the control of ζ and µ ˇ. From the null structure equations, we have e3 (ζ) = r−1 d/≤1 Γb + r−1 Γg + Γb · Γb ,

e4 (ζ) = r−1 d/≤1 Γg ,

and hence ν∗ (ζ)

= r−1 d/≤1 Γb + r−1 d/≤1 Γg + Γb · Γb .

Together with the improved control of Γb in (7.3.16), the control of Γg , and the dominance in r condition, we infer, for all k ≤ ksmall + 17, k d/≤1 ν∗ ζk∗∞,k

. 0 r−2 u−1−δdec + r−3 . 0 r−2 u−1−δdec .

Using µ ˇ = − /d1 ζ − ρˇ + Γg · Γb , we also have for all k ≤ ksmall + 17, in view of the above improved estimate for ν∗ ζ, the commutation formula of Lemma 7.17, and the improved estimate of ρˇ of Step 2, kν∗ ρˇk∗∞,k

.

0 r−3 u−1−δdec .

Together with the improved estimate for d/k ζ and d/k ρˇ of Step 3, we infer, for all k ≤ ksmall + 18, the desired estimates 1

kζk∗∞,k . 0 r−2 u− 2 −δdec ,

kˇ µk∗∞,k . 0 r−3 u−1−δdec .

Step 5. Next, we estimate ϑ. From the null structure equations, we have e3 (ϑ) = r−1 d/≤1 Γb + r−1 Γg + Γb · Γb ,

e4 (ϑ) = r−1 d/≤1 Γg ,

335

DECAY ESTIMATES (THEOREMS M4, M5)

and hence = r−1 d/≤1 Γb + r−1 d/≤1 Γg + Γb · Γb .

ν∗ (ϑ)

Together with the improved control of Γb in (7.3.16), the control of Γg , and the dominance in r condition, we infer, for all k ≤ ksmall + 17, kν∗ ϑk∗∞,k

.

0 r−2 u−1−δdec + r−3 . 0 r−2 u−1−δdec .

It remains to control d/k ϑ. We use Codazzi and the GCM equation for κ, which yields /d2 ϑ

= −2β + (eθ (κ) + ζκ) + Γg · Γg , 2 = −2β + ζ + Γg · Γg . r

Making use of the bootstrap assumption for β and the estimate of Step 3 for ζ,



∗ r−1 d/k /d2 ϑ L2 (S) . β ∞,k + r−1 ζ ∞,k + 0 r−3 u−1−δdec 7

. r− 2 −δdec + 0 r−3 u−1/2−δdec 1+δdec from which we derive, using also the condition r ≥ −1 and a Poincar´e 0 u inequality,

r−1 d/k ϑ L2 (S) . 0 r−2 u−1/2−δdec .

Together with the above estimate for ν∗ ϑ, we infer, for all k ≤ ksmall + 18, kϑk∗∞,k

. 0 r−2 u−1/2−δdec .

Step 6. Finally, we estimate β. From Bianchi, we have e3 β = r−1 d/ρˇ + r−1 β + r−3 Γb + r−1 Γb · Γg ,

e4 (β) = r−1 d/α + r−1 β + r−1 Γg · Γg

and hence ν∗ (β)

= r−1 d/ρˇ + r−1 d/α + r−1 β + r−3 Γb + r−1 Γb · Γg .

Together with the improved control of Γb in (7.3.16), the bootstrap assumptions for α and β, the improved estimate for ρˇ of Step 2, and the dominance in r condition, we infer, for all k ≤ ksmall + 17, kν∗ βk∗∞,k

1

9

1

. 0 r−4 u− 2 −δdec + r− 2 . 0 r−4 u− 2 −δdec .

It remains to control d/k β. We have from Bianchi /d?2 β

= e3 α + r−1 α + r−3 ϑ + Γb · (α, β) + r−1 Γg · Γg .

Hence, using in particular the improved estimate for ϑ of Step 6, and the improved estimate for α and e3 α of Ref 2, we infer

k ?

d/ /d2 βkL2 (S) . 0 r−4 (2r + u)− 12 −δdec .

336

CHAPTER 7

This yields, using a Poincar´e inequality, Z 1 1 r−1 k d/k βkL2 (S) . r−3 βeΦ + 0 r−3 (2r + u)− 2 −δdec . 0 r−3 (2r + u)− 2 −δdec S

where we have used Proposition 7.26 to estimate the ` = 1 mode of β. Together with the above estimate for ν∗ β, we infer, for all k ≤ ksmall + 18, the desired estimate kβk∗∞,k

1

. 0 r−4 u− 2 −δdec .

This concludes the proof of Proposition 7.28.

7.4

CONTROL IN

7.4.1

(ext)

M, PART I

Preliminaries

7.4.1.1

Commutation lemmas

Here and below we write schematically d/ = r /d,

d% = {re4 , d/},

d = (e3 , re4 , r /d),

T=

1 (e3 + Υe4 ) . 2

Lemma 7.29. We have, schematically, [T, e4 ] = r−1 Γb d% ,

ˇ g d% + Γg . [ d/, e4 ] = Γ

(7.4.1)

Also, T(r) =

1 A = r−1 Γb . 2r

Proof. The identity for [ d/, e4 ] has already been discussed in Corollary 7.5. According to Lemma 2.69, we have     m m 2 e4 (m) [T, e4 ] = ω− 2 − κ− + e4 + (η + ζ)eθ . r 2r r r In view of Ref 4 and bootstrap assumptions Ref 2, the factors of e4 and eθ , on the right-hand side, behave at worst like Γb . Thus schematically [T, e4 ] = r−1 Γb d% . 7.4.1.2

Transport lemmas

The following lemma will be used repeatedly in what follows. Lemma 7.30. If f verifies the transport equation p e4 (f ) + κf = F, 2

337

DECAY ESTIMATES (THEOREMS M4, M5)

we have for fixed u and any r0 ≤ r ≤ r∗ , Z r rp kf k∞ (u, r) . r0p kf k∞ (u, rH ) + λp kF k∞ (u, λ)dλ, r0 Z r∗ rp kf k∞ (u, r) . r∗p kf k∞ (u, r∗ ) + λp kF k∞ (u, λ)dλ,

(7.4.2)

r

where r is the area radius at fixed u. Proof. According to Corollary 2.67 we have e4 (rp f ) = rp F . The desired estimates follow easily by integration with respect to the affine parameter s, where we recall that e4 (s) = 1. Proposition 7.31. The following inequalities hold true for all k ≤ klarge − 5, r0 ≤ r ≤ r∗ : Z r∗ p p r kf k∞,k (u, r) . r∗ kf k∞,k (u, r∗ ) + λp kF k∞,k (u, λ)dλ, Zr r (7.4.3) p p r kf k∞,k (u, r) . r0 kf k∞,k (u, r0 ) + λp kF k∞,k (u, λ)dλ. r0

Proof. Commuting the equation e4 (rp f ) = rp F with d/, applying the commutation Lemma 7.29 and our bootstrap assumptions on Γg , we derive
e4 (rp | d/f |) . rp | d/F | + rp | d/F | + r−2 rp | d/f | + re4 (rp f ) . rp (| d/F | + |F |) + r−2 rp (| d/f | + |f |).

Similarly, commuting with T,
e4 T(rp f ) = T(rp F ) − [T, e4 ](rp f ) = rp F − prp−1 T(r)F − r−1 Γb d% rp f. Hence, e4 rp Tf





= rp T(F ) − prp−1 T(r)F − r−1 Γb d% rp f − pe4 rp−1 T(r)f ,

i.e.,
e4 rp |Tf |

.



rp |TF | + |F | + O(r−2 ) rp |F | + | d/f | + |f | .

Similarly, commuting the equation with re4 we derive


e4 rp |rf | . rp |re4 F | + |F | + O(r−2 ) rp |F | + | d/f | + |f | . Integrating the inequalities, e4 (rp | d/f |) .
e4 rp |Tf | .
e4 rp |rf | .

rp (| d/F | + |F |) + r−2 rp (| d/f | + |f |)

rp |TF | + |F | + r−2 rp | d/f | + |f |

rp |re4 F | + |F | + r−2 rp |F | + | d/f | + |f |

and applying Gronwall, we derive the desired estimates in (7.4.3) for k = 1. Repeating the procedure for e dk , any combination of derivatives of the form

338

CHAPTER 7

e dk = Tk1 d/k2 with k1 + k2 = k, estimating the corresponding commutators using our assumptions Ref 1, we deduce for all 0 ≤ k ≤ klarge − 5, e4 (rp |e d≤k f |) . rp |e d≤k F | + r−2 rp |e d≤k f | and the desired estimates follow by integration. 7.4.1.3

Transport equations for ` = 1 modes

To estimate ` = 1 modes we make use of the following. Lemma 7.32. The following equation holds for reduced scalars ψ ∈ s1 ( (ext) M): Z  Z Z Z 1 3 e4 ψeΦ = e4 (ψ)eΦ + κ ψeΦ + (3ˇ κ − ϑ) ψeΦ . (7.4.4) 2 2 S S S S Proof. This is an immediate consequence of Proposition (2.64). Indeed according to it and e4 Φ = 12 (κ − ϑ), Z   Z Z  1 e4 ψeΦ = (e4 (ψeΦ ) + κψeΦ ) = e4 (ψ) + (3κ − ϑ)ψ eΦ 2 S S S  Z  Z 3 1 = e4 (ψ) + κψ eΦ + (3ˇ κ − ϑ) ψeΦ 2 2 S ZS Z Z 3 1 Φ Φ = e4 (ψ)e + κ ψe + (3ˇ κ − ϑ) ψeΦ 2 S S S 2 as desired. 7.4.2

Proposition 7.33

In what follows we prove the stronger estimates in terms of powers of r for the quantities κ ˇ, µ ˇ, ϑ, ζ, κ ˇ , β, ρˇ. More precisely we establish the following. Proposition 7.33. The following estimates hold in (ext) M for all k ≤ ksmall + 20:

κ ˇ k∞,k . 0 r−2 u−1−δdec ,

(7.4.5)

µ ˇk∞,k−2 . 0 r−3 u−1−δdec . Also, for all k ≤ ksmall + 18,

ϑ, ζ, κ ˇ , rρˇ ∞,k



β ∞,k

e3 β ∞,k−1

eθ K ∞,k−1

. 0 r−2 u−1/2−δdec , . 0 r−3 (2r + u)−1/2−δdec , . 0 r−4 u−1/2−δdec , . 0 r−4 u−1/2−δdec .

(7.4.6)

339

DECAY ESTIMATES (THEOREMS M4, M5)

7.4.3

Estimates for κ ˇ, µ ˇ in

(ext)

M

Step 1. We prove the following estimates for κ ˇ in (ext) M.

κ ˇ k∞,k . 0 r−5/2 u−1−δdec , k ≤ ksmall + 20.

(7.4.7)

We make use of the equation  1 2 1 2 1 2 e4 (ˇ κ) + κ κ ˇ = F := − κ ˇ − κ ˇ − ϑ − ϑ2 . 2 2 2 In view of our assumptions Ref 1–2 and Lemma 7.6





F ∞,k (u, λ) . 0 λ−7/2 u−1−δdec . Applying Proposition 7.31, we deduce r2 kˇ κk∞,k (u, r) . r∗2 kˇ κk∞,k (u, r∗ ) + 0 u−1−δdec

Z

r∗

λ2 λ−7/2 dλ

r

. r∗2 kˇ κk∞ (u, r∗ ) + 0 r−1/2 u−1−δdec . In view of the control on the last slice we infer that, everywhere in kˇ κk∞,k (u, r) .

(ext)

M,

0 r−5/2 u−1−δdec .

Step 2. We prove the estimate

µ ˇk∞,k . 0 r−3 u−1−δdec ,

k ≤ ksmall + 18.

(7.4.8)

Recall that we have 3 3 e4 (ˇ µ) + κˇ µ = − µˇ κ + F, 2 2 3 1 F : = − µ ˇκ ˇ+ µ ˇκ ˇ + Err[e4 µ ˇ] − Err[e4 µ ˇ], 2 2   1 2 3 ? 2 Err[e4 µ ˇ] = − κϑ − ϑ /d2 ζ − ϑζ + 2eθ (κ) − 2β + κζ ζ. 8 2 In view of Lemma 7.6 we check kF k∞,k (u, λ) .

0 λ−9/2 u−1−δdec .

Applying Proposition 7.31 and the estimates on the last slice for µ ˇ we deduce Z r∗ r3 ke dk µ ˇk∞,k (u, λ) . r∗3 ke dk µ ˇk∞,k (u, r∗ ) + 0 u−1−δdec λ3 λ−9/2 r

. r∗3 ke dk µ ˇk∞,k (u, r∗ ) + 0 u−1−δdec r−1/2 . 0 u−1−δdec

from which the desired estimate (7.4.8) follows.

340 7.4.4

CHAPTER 7

Estimates for the ` = 1 modes in

(ext)

M

We extend the validity of Lemma 7.26 to the entire region

(ext)

M.

Lemma 7.34. The following estimates hold true on (ext) M for all k ≤ ksmall +19:

Z



βeΦ (u, r) . 0 r−1 u−1−δdec ,

S

∞,k

Z



ζeΦ (u, r) . 0 ru−1−δdec ,

S ∞,k

Z



eθ (ρ)eΦ (u, r) . 0 r−1 u−1−δdec ,

S ∞,k

Z

eθ (κ)eΦ (u, r) . 0 u−1−δdec ,

S

(7.4.9)

∞,k

Z



βeΦ

S

(u, r) . 0 u−1−δdec .

∞,k

Proof. We first note that the estimate for the ` = 1 mode of µ ˇ is an immediate consequence of the estimate (7.4.8). To prove the remaining estimates we proceed in steps as follows. Step 1. Observe that the estimates of Lemma 7.26 remain valid when we replace the norms k k∗∞,k by k k∞,k . To show this it suffices to prove estimates for re4 of all ` = 1 modes. This can easily be achieved with the help of Lemma 7.32 and our e4 transport equations for ζ, ρˇ, µ ˇ, κ ˇ , β. Step 2. We establish the estimate

Z



βeΦ . 0 r−1 u−1−δdec ,

S

∞,k

k ≤ ksmall + 20.

(7.4.10)

In view of (7.4.4) and the Bianchi identity for e4 (β) Z  Z Z Z 3 1 Φ e4 βe = e4 βeΦ + κ βeΦ + (3ˇ κ − ϑ) βeΦ 2 S S S S 2 Z Z Z 3 1 ? Φ Φ = (−2κβ + /d2 α + ζα)e + κ βe + (3ˇ κ − ϑ) βeΦ 2 S S S 2  Z Z  κ 1 Φ = − βe + ζα + (−ˇ κ + ϑ)β eΦ , 2 S 2 S and hence Z e4

Φ

βe S



κ + 2

Z

Φ

βe S

Z  = S

 1 ζα + (−ˇ κ + ϑ)β eΦ . 2

Recall that (α, β)

. r−3 (2r + u)−1/2−δdec .

(7.4.11)

341

DECAY ESTIMATES (THEOREMS M4, M5)

We deduce   Z Φ e4 r βe

. r0 r

−2 −1/2−δdec −3

u

r

(2r + u)

−1/2−δdec

Z

S

S

|eΦ |

. 0 r−1 u−1/2−δdec (2r + u)−1/2−δdec . 0 r−1−δ u−1−δdec , i.e., in view of the estimate on Σ∗ , everywhere on (ext) M, Z βeΦ . r−1 u−1−δdec .

(7.4.12)

S

Commuting with T, d/ and re4 we also easily deduce Z

ek2

βeΦ . r−1 u−1−δdec , ∀ k1 + k2 ≤ ksmall + 20

d (re4 )k1 ∞

S

from which (7.4.10) follows. Step 3. We prove the estimate

Z



ζeΦ . 0 r1/2 u−1−δdec ,

S

∞,k

k ≤ ksmall + 19

(7.4.13)

which is better than the desired estimate in Lemma 7.34. This follows, as for the corresponding estimate on Σ∗ , by projecting the Codazzi equation for ϑ on the ` = 1 mode  Z   Z Z Z Z  r 2Υ Φ Φ Φ Φ Φ 2 βe − eθ (κ)e − ϑζe − κ− ζe . ζe = 2Υ r S S S S S Note that in view of the estimates for κ ˇ in (7.4.7) already established19 we have

Z



eθ (κ)eΦ . 0 r−1/2 u−1−δdec , k ≤ ksmall + 19.

S

∞,k

Also, making use of (7.4.10),

Z



βeΦ . 0 r−1 u−1−δdec ,

S

∞,k

k ≤ ksmall + 19.

Thus, since Υ is bounded away from zero in (ext) M, we easily deduce

Z



ζeΦ . 0 r1/2 u−1−δdec , k ≤ ksmall + 19.

S

∞,k

19 Note that the estimate for κ ˇ is stronger in powers of r than the corresponding bootstrap assumption.

342

CHAPTER 7

Step 4. We prove the estimate

Z



eθ (ρ)eΦ . 0 r−1 u−1−δdec ,

S

∞,k

k ≤ ksmall + 19.

(7.4.14)

We proceed as in Step 4 of the proof of Lemma 7.26. In view of the definition of µ and the identity /d?1 /d1 = /d2 /d?2 + 2K, we write Z Z Z Z 1 eθ (ρ)eΦ = − eθ (µ)eΦ + /d?1 /d1 ζeΦ + eθ (ϑϑ)eΦ 4 S S ZS ZS Z 1 Φ ? Φ = − eθ (µ)e + ( /d2 /d2 + 2K)ζe + eθ (ϑϑ)eΦ 4 S S S  Z Z Z  Z 2 1 1 Φ Φ Φ = − eθ (µ)e + 2 ζe + 2 K − 2 ζe + eθ (ϑϑ)eΦ . r S r 4 S S S Together with the above estimate for the ` = 1 mode of ζ, the estimate (7.4.8) for µ ˇ and the bootstrap assumptions, we infer that

Z



eθ (ρ)eΦ . 0 r−1 u−1−δdec .

S

∞,k

Step 5. We prove the estimate

Z



eθ (κ)eΦ . 0 u−1−δdec ,

S

∞,k

k ≤ ksmall + 19.

(7.4.15)

As in the corresponding estimate on the last slice we make use of the remarkable identity for the ` = 1 mode of eθ (K), i.e., Z Z Z 1 1 Φ Φ eθ (ρ)e + eθ (κκ)e − eθ (ϑϑ)eΦ = 0. 4 S 4 S S We infer Z eθ (κ)e

Z

Z Z r r Φ = −2r eθ (ρ)e − κeθ (κ)e + eθ (ϑϑ)eΦ 2 S 2 S S  Z  r 2 − κ− eθ (κ)eΦ . 2 S r

Φ

S

Φ

The estimate (7.4.15) follows easily from the above estimate for the ` = 1 mode of eθ (ρ), the estimate for κ ˇ in (7.4.7) and the bootstrap assumptions. Step 6. We prove the estimate

Z



βeΦ . 0 u−1−δdec ,

S

∞,k

k ≤ ksmall + 19.

Projecting the Codazzi for ϑ on the ` = 1 mode, we have Z Z Z Z −2 βeΦ + eθ (κ)eΦ − κζeΦ + ϑζeΦ = S

S

S

S

(7.4.16)

0

343

DECAY ESTIMATES (THEOREMS M4, M5)

and hence Z βeΦ

=

S

1 2

Z

Υ eθ (κ)e + r S Φ

Z

1 ζe − 2 S Φ

Z  S

2Υ κ+ r



1 ζe + 2 Φ

Z

ϑζeΦ .

S

The desired estimate follows easily in view of the above estimates for the ` = 1 mode of eθ (κ), the ` = 1 mode of ζ and the bootstrap assumptions. 7.4.5

Completion of the proof of Proposition 7.33

We prove the second part of Proposition 7.33, i.e., we prove for all k ≤ ksmall + 18

ϑ, ζ, κ ˇ , rρˇ ∞,k . 0 r−2 u−1/2−δdec ,



β . 0 r−3 (2r + u)−1/2−δdec , ∞,k (7.4.17)

e3 β . 0 r−4 u−1/2−δdec , ∞,k−1

eθ K . 0 r−4 u−1/2−δdec . ∞,k−1 We also prove the stronger estimate for β

βk∞,k . 0 log(1 + u)r−3 (2r + u)−1/2−δextra .

(7.4.18)

Proof. We proceed in steps as follows. Step 1. We derive the estimate



ϑ . 0 r−2 u−1/2−δdec , ∞,k

∀ k ≤ ksmall + 19,

(7.4.19)

with the help of the equation e4 ϑ + κϑ = F := −2α − κ ˇ ϑ and the corresponding estimate on the last slice. Note that



α . 0 r−3−δ (2r + u)−1/2−δdec ∞,k where δ > 0 is a small constant, δ < δextra − δdec . Thus, using also the product estimates of Lemma 7.6, we easily check that kF k∞,k

. 0 r−3−δ u−1/2−δdec + 0 r−7/2 u−1−δdec ,

k ≤ ksmall + 20.

Making use of Proposition 7.31 we deduce, for all k ≤ ksmall + 19, Z r∗ r2 kdk ϑk∞,k (u, r) . r∗2 kdk ϑk∞,k (u, r∗ ) + 0 u−1/2−δdec λ−1−δ dλ. r

Thus, in view of the results on the last slice Σ∗ , we deduce kdk ϑk∞ (u, r) . Step 2. We derive the estimate



β . 0 r−3 (2r + u)−1/2−δdec , ∞,k

r−2 u−1/2−δdec .

∀ k ≤ ksmall + 19.

(7.4.20)

344

CHAPTER 7

We proceed exactly as in the estimates
for β on the last slice Σ∗ by making use of the Bianchi identity e3 α + 12 κ − 4ω α = − /d?2 β − 32 ϑρ + 5ζβ, from which we deduce



k /d?2 βk∞,k−1 . e3 α + r−1 α + r−3 ϑk∞,k−1 + 0 r−5 u−1−δdec . ∞,k−1

∞,k−1

Thus, in view of the above estimate for ϑ and Ref 2 for α, . 0 r−4 (2r + u)−1/2−δdec + 0 r−5 u−1/2−δdec .

k /d?2 βk∞,k−1

On the other hand we have, according to (7.4.10),

Z



βeΦ . 0 r−1 u−1−δdec , k ≤ ksmall + 20.

S

∞,k

Estimate (7.4.20) follows then easily, according to part 4 of the elliptic Hodge Lemma 7.7. We can prove a stronger estimate for β. Indeed we have, in view of Ref 2. |α| . log(1 + u)r−3 (2r + u)−1/2−δextra ,

|e3 α| . r−4 (2r + u)−1/2−δextra .
Hence, using the equation e3 α + 12 κ − 4ω α = − /d?2 β − 32 ϑρ + 5ζβ,

?

/d2 βk∞,k

.

0 log(1 + u)r−4 (2r + u)−1/2−δextra + 0 r−5 u−1/2−δdec .

According to Lemma 7.7 kβkhk+1 (S)

.

rk /d?2

βkhk (S) + r

−2

Z eΦ β S

and thus, in view of the estimate (7.4.10) for the ` = 1 mode of β, kβkhk+1 (S)

. 0 log(1 + u)r−2 (2r + u)−1/2−δextra + 0 r−3 u−1−δdec . 0 log(1 + u)r−2 (2r + u)−1/2−δextra .

The estimates for the T and e4 derivatives are derived in the same manner and hence

βk∞,k . 0 log(1 + u)r−3 (2r + u)−1/2−δextra , ∀ k ≤ ksmall + 19. (7.4.21) This improvement is needed in the next step. Step 3. We derive the estimate



ζ . 0 r−2 u−1/2−δdec , ∞,k

∀ k ≤ ksmall + 19.

(7.4.22)

For this we make use of the transport equation for ζ, e4 ζ + κζ = F := −β + Γg · Γg , and the improved estimate for β in the previous step. Thus, making use of the

345

DECAY ESTIMATES (THEOREMS M4, M5)

product Lemma 7.6,





F . β ∞,k + 0 r−7/2 u−1−δdec ∞,k . 0 log(1 + u)r−3 (2r + u)−1/2−δextra + 0 r−7/2 u−1−δdec . 0 r−3−δ u−1/2−δdec + 0 r−7/2 u−1−δdec . Making use of Proposition 7.31 we deduce 2

k

r∗2 kdk ζk∞,k (u, r∗ )

r kd ζk∞,k (u, r) .

+ 0 u

−1/2−δdec

Z

r∗

λ−1−δ dλ.

r

Thus, in view of the estimates on the last slice, r2 kdk ζk∞ (u, r) .

0 u−1/2−δdec ,

k ≤ ksmall + 19

as desired. Step 4. We derive the estimate



ρˇ . 0 r−3 u−1/2−δdec , ∞,k

∀ k ≤ ksmall + 18.

(7.4.23)

We make use of the definition of µ from which we infer that µ ˇ =

− /d1 ζ − ρˇ + Γg · Γb .

Hence, in view of the product lemma and the estimates already derived, for all k ≤ ksmall + 18,







ρˇ . r−1 ζ ∞,k+1 + µ ˇ ∞,k + 0 r−3 u−1−δdec ∞,k .

0 r−3 u−1/2−δdec

as desired. Step 5. We derive the estimate



κ ˇ ∞,k . 0 r−2 u−1/2−δdec ,

∀ k ≤ ksmall + 18.

(7.4.24)

We make use of the equation 1 1 e4 κ ˇ + κˇ κ = F := −2 /d1 ζ − κ ˇ κ + 2ˇ ρ + Γ g · Γb . 2 2 In view of the previously derived estimates,



F . 0 r−3 u−1/2−δdec , ∞,k

k ≤ ksmall + 18.

Making use of Proposition 7.31 we deduce, for all k ≤ ksmall + 18, Z r∗ rkdk κ ˇ k∞,k (u, r) . r∗ kdk κ ˇ k∞,k (u, r∗ ) + 0 u−1/2−δdec λ−2 dλ k

. r∗ kd κ ˇ k∞,k (u, r∗ ) + 0 r

r −1 −1/2−δdec

u

.

346

CHAPTER 7

Thus, in view of the estimates on the last slice, rkdk κ ˇ k∞ (u, r) .

0 (r∗ )−1 u−1/2−δdec + 0 r−1 u−1/2−δdec

from which the desired estimate easily follows. Step 6. We derive the estimate

e3 β . 0 r−4 u−1/2−δdec , ∞,k

∀ k ≤ ksmall + 17,

(7.4.25)

making use of the equation e3 β + (κ − 2ω)β = − /d?1 ρ + 3ζρ + Γg β + Γb α and the estimates derived above for β, /d?1 ρ, ζ. Hence,





e3 β . r−1 β ∞,k + /d?1 ρ ∞,k + r−3 ζ ∞,k + 0 r−4 u−1−δdec ∞,k . 0 r−4 u−1/2−δdec . Step 7. As a corollary of the above estimates (see also Ref 4) we also derive, in (ext) M,

K − K . 0 r−3 u−1/2−δdec , k ≤ ksmall + 18, ∞,k−1



(7.4.26)

K − 1 . 0 r−3 u−1/2−δdec , k ≤ ksmall + 18.

2 r ∞,k−1 In view of the definition of K we have,   1 1 1 eθ (K) = −eθ ρˇ − κˇ κ − κˇ κ + ϑϑ . 4 4 4 Thus, in view of the above estimates, for all k ≤ ksmall + 17,

eθ K . 0 r−4 u−1/2−δdec ∞,k from which the desired estimate easily follows.

7.5

CONTROL IN

(ext)

M, PART II

We derive the crucial decay estimates which imply, in particular, decay of order ˇ (except ξ, ω ˇ which will be treated sepau−1−δdec for all quantities in Γ and R ˇ, Ω rately) in the interior. More precisely we prove the following: Proposition 7.35. The following estimates hold in (ext) M, for all k ≤ ksmall + 8,

ϑ, ζ, η, κ (7.5.1) ˇ , ϑ, rβ, rρˇ, rβ, αk∞,k . 0 r−1 u−1−δdec . To prove the proposition we make use of the fact that we already have good decay estimates in terms of powers of u for κ ˇ, µ ˇ. We also derive below decay estimates for various renormalized quantities.

347

DECAY ESTIMATES (THEOREMS M4, M5)

7.5.1

Estimate for η

We start with the following simple estimate for η in terms of ζ: . kζk∞,k + 0 r−1 u−1−δdec ,

kηk∞,k

k ≤ ksmall + 17.

(7.5.2)

This can be derived by propagation from the last slice with the help of the equation 1 e4 (η − ζ) + κ(η − ζ) 2

1 = − ϑ(η − ζ) = Γg · Γb . 2

Note that . 0 r−3 u−1−δdec .

kΓg · Γb k∞,k

Thus making use of Proposition 7.31 we deduce rkη − ζk∞,k (u, r) . r∗ kη − ζk∞,k (u, r∗ ) +

Z

r∗

r

λkΓg · Γb k∞,k (u, λ)

. r∗ kη − ζk∞,k (u, r∗ ) + 0 u−1−δdec with r∗ the value of r on C(u) ∩ Σ∗ . On the last slice we have derived the estimates, recorded in Proposition 7.22 and Proposition 7.33, kηk∗∞,k kζk∗∞,k

. 0 r−1 u−1−δdec , . 0 r−2 u−1/2−δdec .

In view of the dominance condition on r on Σ∗ we deduce kη − ζk∗∞,k (u, r) . 0 r−1 u−1−δdec and therefore, also, r∗ kη − ζk∞,k (u, r∗ ) . 0 u−1−δdec . Therefore, rkηk∞,k (u, r) .

rkζk∞,k (u, r) + 0 u−1−δdec

as desired. 7.5.2

Crucial lemmas

We start with the following lemma. Lemma 7.36. The s1 (M) reduced tensor Ξ: verifies in

(ext)

= r2 eθ (κ) + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β

M the estimate



Ξ . 0 u−1−δdec , ∞,k

∀ k ≤ ksmall + 13.




(7.5.3)

(7.5.4)

348

CHAPTER 7

Proof. To calculate e4 Ξ we make use of the equations 1 1 e4 (κ) + κκ = −2 /d1 ζ + 2ρ − ϑϑ + 2ζ 2 , 2 2 e4 ζ + κζ = −β − ϑζ, e4 β + 2κβ

=

/d2 α + ζα.

Since we already have an estimate for µ ˇ we re-express ρ = −µ − /d1 ζ + 14 ϑϑ and derive 1 e4 (κ) + κκ = 2

−2µ − 4 /d1 ζ + 2ζ 2 .

Commuting with /d?1 and making use of [ /d?1 , e4 ] = 12 (κ + ϑ) /d?1 we derive 1 e4 ( /d?1 κ) + κ /d?1 κ + κ /d?1 κ = − /d?1 µ ˇ − 4 /d?1 /d1 ζ + 2 /d?1 (ζ 2 ) + ϑ /d?1 κ. (7.5.5) 2 Hence, since e4 (r) = 2r κ, e4 (r2 /d?1 κ)

1 = r2 (κ − κ) /d?1 κ − r2 κ /d?1 κ ˇ − 4r2 /d?1 /d1 ζ − r2 /d?1 µ ˇ 2 +r2 ( /d?1 (ζ 2 ) + ϑ /d?1 κ) 1 = − r2 κ /d?1 κ ˇ − 4r2 /d?1 /d1 ζ + Err1 2

where Err1 :

=


1 − r2 κ /d?1 κ ˇ − r2 /d?1 µ ˇ + r2 κ ˇ /d?1 κ + /d?1 (ζ 2 ) + ϑ /d?1 κ . 2

In view of the estimate already established for κ ˇ, µ ˇ and the product Lemma 7.6 we check

Err1 k∞,k . 0 r−2 u−1−δdec , k ≤ ksmall + 17. To simplify notation we introduce the following. Definition 7.37. We say that a quantity ψ ∈ sk (M) is r−p Gooda provided that it verifies the estimate, everywhere in (ext) M,



ψ . 0 r−p u−1−δdec , ∀k ≤ ksmall + a. (7.5.6) ∞,k Using this notation we write e4 (r2 /d?1 κ)

= −4r2 /d?1 /d1 ζ + r−2 Good17 .

(7.5.7)

Using the same notation, the transport equation for ζ can be written in the form e4 ζ + κζ = −β − ϑζ − κ ˇ ζ = −β + r−7/2 Good20 .

349

DECAY ESTIMATES (THEOREMS M4, M5)

Commuting with (r /d?1 )(r /d1 ) (making use of Lemma 7.29) we derive e4 (r2 /d?1 /d1 ζ) + κ(r2 /d?1 /d1 ζ) = −r2 /d?1 /d1 β + r−7/2 Good18 . Since e4 (r) = 12 rκ we deduce e4 (r3 /d?1 /d1 ζ)

1 = − κr3 /d?1 /d1 ζ − r3 /d?1 /d1 β + r−5/2 Good18 . 2

(7.5.8)

Similarly the transport equation for β takes the form e4 β + 2κβ

=

/d2 α + ζα − 2ˇ κβ = /d2 α + r−9/2 Good20

and e4 (r2 /d?1 /d1 β) + 2κr2 /d?1 /d1 β = r2 /d?1 /d1 /d2 α + r−9/2 Good18 . As before, since e4 (r) = 12 rκ, we deduce e4 (r4 /d?1 /d1 β) = −κr4 /d?1 /d1 β + r4 /d?1 /d1 /d2 α + r−5/2 Good18 .

(7.5.9)

Combining (7.5.7)–(7.5.9) we deduce h
i e4 Ξ = e4 r2 − /d?1 κ + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β   1 = 4r2 /d?1 /d1 ζ + 4 − κr3 /d?1 /d1 ζ − r3 /d?1 /d1 β 2
4 ? − 2 −κr /d1 /d1 β + r4 /d?1 /d1 /d2 α + r−2 Good17     2 2 = −2 κ − r3 /d?1 /d1 ζ + 2r4 κ − /d?1 /d1 β − 2r4 /d?1 /d1 /d2 α + r−2 Good17 . r r Making use of Ref 4 estimates for κ −

2 r

and the estimates for α in Ref 2, i.e.,

r4 | /d?1 /d1 /d2 α| . 0 r−1 (2r + u)−1−δextra . 0 r−1−δ u−1−δextra +δ ,

0 < δ < δextra ,

i.e., r4 /d?1 /d1 /d2 α

=

r−1−δ Good13 ,

we thus deduce e4 Ξ

= r−1−δ Good13 .

We deduce kΞk∞,k (u, r) . kΞk∞,k (u, r∗ ) + 0 u−1−δdec

Z

r∗

λ−1−δ dλ,

r

∀k ≤ ksmall + 13.

In view of the estimates on the last slice it is easy to check that kΞk∞,k (u, r∗ ) . 0 u−1−δdec ,

∀k ≤ ksmall + 13.

350

CHAPTER 7

Indeed, on the last slice, k /d?1 κk∞,k

.

0 r−3 u−1/2−δdec ,

.

0 r−4 u−1/2−δdec ,

k /d?1 /d1 βk∞,k

.

0 r−5 u−1/2−δdec .

k /d?1 /d1 ζk∞,k

Hence, since r  u on Σ∗ , kΞk∞,k (u, r∗ ) . 0 r−1 u−1/2−δdec . 0 u−1−δdec . Thus everywhere on

(ext)

kΞk∞,k

M, .

0 u−1−δdec ,

∀k ≤ ksmall + 13

as desired. In the following lemma, we make use of the control we have already established for q, α, α, κ ˇ, µ ˇ in (ext) M to derive two nontrivial relations between angular derivatives of ζ, κ ˇ and β. Remark 7.38. According to Theorem M3 we only have good estimates for α along T and on the last slice Σ∗ . To keep track of this fact we denote by r−p Gooda (α) those r−p Gooda terms which depend linearly on α and their derivatives. Lemma 7.39. Let A, B be the operators A := /d?2 /d2 − 3ρ, B = /d?2 /d2 + 2K. The following identities hold true: 3 AB /d?2 ζ − κ ρ /d?2 eθ (κ) ∈ r−6 Good14 (α) 4
2 9 2 ? A B /d2 β + κ ρ /d?2 eθ (κ) ∈ r−9 Good9 (α) 8

(7.5.10)

Proof. In view of the improved control for α in Theorem M2, α in Theorem M3, and q in Theorem M1, the bootstrap assumptions and product lemma, and the control we have already derived for κ ˇ and µ ˇ in (ext) M, we obtain /d2 ϑ + 2β − κζ ∈ r−3 Good20 , −3

/d2 ϑ + 2β − eθ (κ) + κζ ∈ r Good20 , 3 /d?2 β + ρϑ ∈ r−3 Good15 , 2 3 /d?2 β + ρϑ ∈ r−2 Good15 (α), 2 3 3 /d?2 /d?1 ρ + κ ρϑ + κ ρϑ ∈ r−4 Good18 , 4 4

Codazzi and control of κ ˇ, Codazzi, Bianchi and control of α,

(7.5.11)

Bianchi and control of α, (7.1.28) and control of q,

where we used Codazzi for the two first inequalities, Bianchi for the third and fourth inequalities, the definition of µ for the fifth one, and the identity relating q and /d?2 /d?1 ρ for the last one. Combining the first statement with the third and the second with the fourth we

351

DECAY ESTIMATES (THEOREMS M4, M5)

infer that ( /d?2 /d2 − 3ρ)ϑ − κ /d?2 ζ

( /d?2 /d2 − 3ρ)ϑ − /d?2 eθ (κ) + κ /d?2 ζ

∈ ∈

r−3 Good14 , r−2 Good14 (α),

or, setting A := /d?2 /d2 − 3ρ, Aϑ − κ /d?2 ζ ∈ r−3 Good14 ,

Aϑ − /d?2 eθ (κ) + κ /d?2 ζ ∈ r−2 Good14 (α).

(7.5.12)

From the fifth equations we deduce   3 3 ? ? A /d2 /d1 ρ + κ ρϑ + κ ρϑ ∈ r−6 Good18 , 4 4 i.e., 3 3 A /d?2 /d?1 ρ + κ ρ Aϑ + κ ρ Aϑ ∈ r−6 Good18 . 4 4 Making use of (7.5.12) we deduce
3
3 A /d?2 /d?1 ρ + κ ρ κ /d?2 ζ + κ ρ /d?2 eθ (κ) − κ /d?2 ζ ∈ r−6 Good14 (α). 4 4 Hence, simplifying, 3 A /d?2 /d?1 ρ + κ ρ /d?2 eθ (κ) ∈ r−6 Good14 (α). 4

(7.5.13)

Next, in view of the identity /d?1 /d1 = /d2 /d?2 + 2K, ( /d?2 /d2 + 2K) /d?2 ζ

=

/d?2 /d2 /d?2 ζ + 2K /d?2 ζ

=

/d?2 ( /d2 /d?2 ζ + 2Kζ) − 2 /d?2 Kζ

=

/d?2 /d?1 /d1 ζ + r−9/2 Good19 .

Recalling the definition of µ = − /d1 ζ − ρ + 14 ϑϑ and the product lemma we write /d?1 /d1 ζ

= − /d?1 µ − /d?1 ρ +

1 ? /d1 (ϑϑ) = − /d?1 µ − /d?1 ρ + r−4 Good19 . 4

In view of the estimates for µ ˇ we have already established we deduce /d?1 /d1 ζ = − /d?1 ρ + r−4 Good17 . Thus, ( /d?2 /d2 + 2K) /d?2 ζ

=

− /d?2 /d?1 ρ + r−5 Good16 .

(7.5.14)

352

CHAPTER 7

Therefore, making use of (7.5.13), A( /d?2 /d2 + 2K) /d?2 ζ

= −A /d?2 /d?1 ρ + r−6 Good14 3 = κ ρ /d?2 eθ (κ) + r−6 Good14 (α), 4

i.e., 3 A( /d?2 /d2 + 2K) /d?2 ζ − κ ρ /d?2 eθ (κ) = r−6 Good14 (α) 4

(7.5.15)

as desired. To prove the second statement of the lemma we write, using (7.5.12), ( /d?2 /d2 + 2K)Aϑ

=

κ ( /d?2 /d2 + 2K) /d?2 ζ + r−5 Good12 (α).

Hence applying A and making use of (7.5.15), A( /d?2 /d2 + 2K)Aϑ

= =

κA( /d?2 /d2 + 2K) /d?2 ζ + r−7 Good10 (α) 3 2 ? κ ρ /d2 eθ (κ) + r−7 Good10 (α). 4

Finally, making use of the relation /d?2 β + 32 ρ ϑ ∈ r−3 Good15 , we have A2 ( /d?2 /d2 + 2K) /d?2 β

= A( /d?2 /d2 + 2K)A /d?2 β + r−9 Good9 (α) 3 = − ρA( /d?2 /d2 + 2K)Aϑ + r−9 Good9 (α) 2   3 3 2 ? = − ρ κ ρ /d2 eθ (κ) + r−8 Good10 (α) + r−9 Good9 2 4 9 2 2 ? = − κ ρ /d2 eθ (κ) + r−9 Good9 (α) 8

as desired. This concludes the proof of the lemma. Corollary 7.40. The s1 (M) tensor eθ (κ) = − /d?1 κ ˇ verifies the following fifth order elliptic equation in (ext) M: A2 /d?2 (eθ κ) −

12m ? 36m2 ? A /d2 eθ (κ) + /d2 eθ (κ) ∈ r−7 Good8 (α). 3 r r6

(7.5.16)

Proof. According to Lemma 7.39 3 AB /d?2 ζ − κ ρ /d?2 eθ (κ) ∈ r−6 Good14 (α), 4
2 9 2 ? A B /d2 β + κ ρ /d?2 eθ (κ) ∈ r−9 Good9 (α), 8 we have 3 κ ρ A /d?2 eθ (κ) + r−8 Good12 (α), 4
2 9 A2 B /d?2 β = − κ ρ /d?2 eθ (κ) + r−9 Good9 (α). 8 A2 B /d?2 ζ =

(7.5.17)

353

DECAY ESTIMATES (THEOREMS M4, M5)

In view of Lemma 7.36 we have, on

(ext)

M,

eθ (κ) + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β ∈ r−2 Good13 .

(7.5.18)

Thus,   A2 /d?2 eθ (κ) + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β ∈ r−7 Good8 . Making use of /d?2 /d?1 /d1

=

  /d?2 /d2 /d?2 + 2K = ( /d?2 /d2 + 2K) /d?2 − eθ (K),

we deduce A2 /d?2 (eθ κ)

=

−4rA2 /d?2 /d?1 /d1 ζ + 2r2 A2 /d?2 /d?1 /d1 β + r−7 Good8

= −4rA2 ( /d?2 /d2 + 2K) /d?2 ζ + 2r2 A2 ( /d?2 /d2 + 2K) /d?2 β + r−7 Good8 = −4rA2 B /d?2 ζ + 2r2 A2 B /d?2 β + r−7 Good8 . Thus, in view of the lemma, A2 /d?2 (eθ κ)


= −3r κ ρA /d?2 eθ (κ) + r−8 Good12 
2 9 2 − r κ ρ /d?2 eθ (κ) + r−9 Good9 (α) + r−7 Good8 . 4

We deduce 9 2 A2 /d?2 eθ κ + 3r (κ ρ) A /d?2 eθ (κ) + r2 (κ ρ) /d?2 eθ (κ) ∈ r−7 Good8 (α). 4 Finally, A2 /d?2 eθ κ −

12m ? 36m2 ? A /d2 eθ (κ) + /d2 eθ (κ) ∈ r−7 Good8 (α) 3 r r6

as desired. Lemma 7.41. We have the following Poincar´e inequality on (ext) M for f ∈ s2 (M) with A = ( /d?2 /d2 − 3ρ):   Z Z Z 12m 36m2 1 9 2 f A2 − 3 A + f ≥ ( / d f ) + f 2. 2 6 2 4 r r 4r r S S S Proof. Recall that we have the following Poincar´e inequality for /d2 : Z Z ( /d2 f )2 ≥ 4 Kf 2 . S

S

354

CHAPTER 7

Since K − r−2 . r−2 , Z Z Z f Af = f ( /d?2 /d2 − 3ρ)f ≥ (4K − 3ρ)f 2 S S S  Z 4 6m −2 + + O(r ) f 2. = r2 r3 S Since A is positive self-adjoint,  Z Z Z 4 6m 2 1/2 1/2 −2 fA f = (A f )A(A f ) = + 3 + O(r ) |A1/2 f |2 r2 r S S S  Z 4 6m −2 = + 3 + O(r ) f Af. r2 r S This yields    Z Z 12m 4 6m 12m 2 −2 f A f − 3 Af = + 3 − 3 + O(r ) f Af r r2 r r S S  Z 4 6m = − 3 + O(r−2 ) f Af, r2 r S and therefore,   Z 12m 36m2 f A2 − 3 A + r r6 S

Z Z 4 6m 36m2 −2 − + O(r ) f Af + f2 6 r2 r3 r S S    Z 4 3m −2 = 1− + O(r ) f Af r2 2r S Z 36m2 + 6 f 2. r S 

≥

Note that for r > 2m we have 1−

3m 1 > . 2r 4

We deduce, for sufficiently small , everywhere in (ext) M,   Z  Z Z 36m2 12m 1 36m2 2 2 f A − 3 A+ > f Af + f . r r6 r2 r4 S S S −3 Now, since ρ + 2m , 3 r . 0 r Z

Z f Af

S

Hence, Z

f ( /d?2 /d2

= S

36m2 f Af + r4 S

Z

2

f > S

− 3ρ)f =

Z  S

Z  S

2

| /d2 f | +



  6m −3 2 + O(r 0 ) |f | . r3

6m 36m2
2  | /d2 f | + + |f | > r3 r6 2

Z S

| /d2 f |2

355

DECAY ESTIMATES (THEOREMS M4, M5)

or, since

R S

( /d2 f )2 ≥ 4

R

1 S r2

R S

|f |2 + O(0 r−3 )

  12m 36m2 2 f A − 3 A+ ≥ r r6 S

Z

R S

|f |2 . We deduce

1 4r2

Z

9 ( /d2 f ) + 4 r S 2

Z

f2

S

as desired. This concludes the proof of the lemma. Applying the lemma to f = /d?2 eθ κ in (7.5.16), i.e., A2 /d?2 (eθ κ) −

12m ? 36m2 ? A / d e (κ) + /d2 eθ (κ) ∈ r−7 Good8 (α) 2 θ r3 r6

or, in any region where kαk2,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 16,

we have

12m 36m2 ?

2 ?

/ d e (κ)

A /d2 (eθ κ) − 3 A /d?2 eθ (κ) +

2 θ r r6 2,k −6 −1−δdec . 0 r u , k ≤ ksmall + 8. We deduce, by L2 -elliptic estimates, . 0 r−2 u−1−δdec ,

k /d?2 eθ κ ˇ k2,k

k ≤ ksmall + 12.

(7.5.19)

Since we control the ` = 1 mode of eθ κ ˇ we infer that keθ κ ˇ k2,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 13,

i.e., kˇ κk2,k

. 0 u−1−δdec ,

k ≤ ksmall + 14.

Therefore, using the Sobolev embedding, kˇ κk∞,k

.

0 r−1 u−1−δdec

k ≤ ksmall + 12.

This proves the following: Proposition 7.42. In any region of

(ext)

M where

kαk2,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 16,

kˇ κk∞,k

.

0 r−1 u−1−δdec ,

k ≤ ksmall + 12.

we also have

7.5.3

(7.5.20)

Proof of Proposition 7.35, Part I

We first prove Proposition 7.35 in the region where the estimate kαk2,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 16,

(7.5.21)

356

CHAPTER 7

holds true. Step 1. We prove the estimates kζk∞,k . 0 r−1 u−1−δdec ,

kβk∞,k . 0 r

−2 −1−δdec

u

k ≤ ksmall + 15,

(7.5.22)

k ≤ ksmall + 12.

,

According to (7.5.17) A2 B /d?2 ζ

=

A2 B /d?2 β

=

3 κ ρ A /d?2 eθ (κ) + r−8 Good12 (α), 4
2 9 − κ ρ /d?2 eθ (κ) + r−9 Good9 (α). 8

In view of (7.5.19) we deduce, in L2 norms, kA2 B /d?2 ζk2,k

kA

2

B /d?2 βk2,k

r−4 kA /d?2 eθ κk2,k + 0 r−7 u−1−δdec ,

.

. r

−8

k /d?2 eθ κk2,k

+ 0 r

−8 −1−δdec

u

,

k ≤ ksmall + 12,

k ≤ ksmall + 9.

Thus, in view of the estimates for κ ˇ derived above, . 0 r−7 u−1−δdec ,

kA2 B /d?2 ζk2,k

kA

2

B /d?2 βk2,k

. 0 r

−8 −1−δdec

u

,

k ≤ ksmall + 12, k ≤ ksmall + 9.

Thus, by elliptic estimates, k /d?2 ζk2,k

k /d?2 βk2,k

. 0 r−1 u−1−δdec , . 0 r

−2 −1−δdec

u

,

k ≤ ksmall + 16, k ≤ ksmall + 13.

In view of the estimates for the ` = 1 modes of ζ, β we deduce kζk2,k . 0 u−1−δdec ,

kβk2,k . 0 r

−1 −1−δdec

u

k ≤ ksmall + 17, ,

k ≤ ksmall + 14.

Passing to L∞ norms we derive kζk∞,k . 0 r−1 u−1−δdec ,

kβk∞,k . 0 r

−2 −1−δdec

u

,

k ≤ ksmall + 15,

k ≤ ksmall + 13.

(7.5.23)

Step 2. We prove the estimate kηk∞,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 15.

This follows immediately from the estimate from ζ and the previously derived estimate (7.5.2). Indeed, kηk∞,k

.

kζk∞,k + 0 r−1 u−1−δdec . 0 r−1 u−1−δdec .

Step 3. We derive the estimate kϑk∞,k

. r−1 u−1−δdec ,

k ≤ ksmall + 11.

(7.5.24)

357

DECAY ESTIMATES (THEOREMS M4, M5)

This follows easily in view of the equation (see (7.5.11)) /d2 ϑ + 2β − κζ

∈ r−3 Good20

from which, in view of Step 1, . 0 r−1 u−1−δdec ,

k /d2 ϑk2,k

k ≤ ksmall + 12.

The desired estimate follows by elliptic estimates and Sobolev. Step 4. We derive the intermediate estimate for ϑ, kϑk∞,k

.

0 u−1−δdec ,

k ≤ ksmall + 12.

(7.5.25)

To show this we combine the equations (see (7.5.11)) /d2 ϑ + 2β − eθ (κ) + κζ ∈ r−3 Good20 , 3 /d?2 β + ρϑ ∈ r−2 Good15 , 2 to deduce /d?2 /d2 ϑ − 3ρϑ

=

/d?2 eθ κ + κ /d?2 ζ + r−2 Good15 ,

and hence, . 0 r−1 u−1−δdec ,

kAϑk2,k

k ≤ ksmall + 12.

Thus, kϑk2,k

. 0 ru−1−δdec ,

k ≤ ksmall + 14

kϑk∞,k

. 0 u−1−δdec ,

k ≤ ksmall + 12

and hence,

as desired. Step 5. We derive the estimate kˇ ρk∞,k

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 14.

(7.5.26)

From 3 3 /d?2 /d?1 ρ + κ ρϑ + κ ρϑ 4 4

∈ r−4 Good20 ,

we deduce k /d?2 /d?1 ρk2,k

. r−4 (kθk2,k + kθk2,k ) + 0 r−3 u−1−δdec , . 0 r−3 u−1−δdec ,

k ≤ ksmall + 14.

k ≤ ksmall + 14

358

CHAPTER 7

Since we control the ` = 1 mode of /d?1 ρ (see Lemma 7.34) we infer that kˇ ρk2,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 16,

i.e., kˇ ρk∞,k

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 14

as desired. Step 6. We derive the estimate



β . 0 r−2 u−1−δdec , ∞,k with the help of the identity ( e3 (rq)

= r

5

/d?2 /d?1 /d1 β

∀ k ≤ ksmall + 9

(7.5.27)

3 3 3 3 − κρα − ρ /d?2 /d?1 κ − κρ /d?2 ζ + (2ρ2 − κκρ)ϑ 2 2 2 4

)

+Err[e3 (rq)], Err[e3 (rq)]

= r4 (e3 Γb ) · d/≤1 β + rΓb · q + r2 d/3 (Γg · Γb ),

of Proposition 7.11. In view of (7.3.18) we have kErr[e3 (rq)]k∞,k (u, r) .

0 u−1−δdec ,

k ≤ ksmall + 16.

We can now make use of the estimates for κ ˇ , , ζ, ϑ, ϑ already derived and the Ref 2 estimate for e3 (q) and α to deduce, for all k ≤ ksmall + 10, kρ /d?2 /d?1 κk∞,k

. 0 r−6 u−1−δdec ,

2

. 0 r−6 u−1−δdec ,

kκρ /d?2 ζk∞,k

. 0 r−6 u−1−δdec ,

kκκϑk∞,k

. 0 r−5 u−1−δdec , . 0 r−5 u−1−δdec ,

kρ ϑk∞,k

kκραk∞,k ke3 (rq)k∞,k

. 0 u−1−δdec .

Therefore, k /d?2 /d?1 /d1 βk∞,k

. 0 r−5 u−1−δdec ,

k ≤ ksmall + 10,

k /d?2 /d?1 /d1 βk2,k

.

0 r−4 u−1−δdec ,

k ≤ ksmall + 10.

i.e.,

Making use of the identity /d?1 /d1 = /d2 /d?2 + 2K, we deduce

?

( /d2 /d2 + K) /d?2 β 2,k

. 0 r−4 u−1−δdec .

359

DECAY ESTIMATES (THEOREMS M4, M5)

Since /d?2 /d2 + K is coercive we deduce

?

/d2 β . 0 r−2 u−1−δdec , 2,k

∀ k ≤ ksmall + 10.

Since we control the ` = 1 mode of β (see Lemma 7.34 ) according to Lemma 7.26,



β 2,k

0 r−1 u−1−δdec ,

.

∀ k ≤ ksmall + 11.

Hence,



β ∞,k

0 r−2 u−1−δdec ,

.

∀ k ≤ ksmall + 9.

(7.5.28)

Step 7. Using the above estimate for β we can improve the estimate for ϑ derived in Step 4. We show, in the region where the estimate (7.5.21) for α holds, kϑk∞,k

.

0 r−1 u−1−δdec ,

k ≤ ksmall + 9.

(7.5.29)

Indeed in view of the Codazzi equation ∈ r−3 Good20 ,

/d2 ϑ + 2β − eθ (κ) + κζ we infer that, for all k ≤ ksmall + 11, k /d2 ϑk2,k

. .

kβk2,k + r−1 kˇ κk2,k+1 + r−1 kζk2,k + 0 r−2 u−1−δdec 0 r−2 u−1−δdec .

Thus, for all k ≤ ksmall + 12, kϑk2,k

. 0 r−1 u−1−δdec

and hence, kϑk∞,k

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 10.

(7.5.30)

This ends the proof of Proposition 7.35 in the region for which the desired estimate (7.5.21) for α holds true. Since (7.5.21) for α holds true on T in view of20 Theorem M3, this ends the proof of Proposition 7.35 on T . 7.5.4

Proof of Proposition 7.35, Part II

We extend the validity of Proposition 7.35 to all of (ext) M propagating the estimates derived in the first part on T . We also recall that we have good decay estimates for κ ˇ and µ ˇ everywhere on (ext) M. Step 1. We first derive estimates for ϑ in Mext making use of the transport 20 Recall that r is bounded on T and that T ⊂ on T in view of Theorem M3. Then, since we have indeed true for (ext) α on T .

(int) M

(ext) α

so that (7.5.21) holds true for (int) α = ((ext) Υ)2 (int) α on T , (7.5.21) holds

360

CHAPTER 7

equation e4 (ϑ) + κϑ

= −2α − (κ − κ)ϑ = −2α + Γg · Γg .

Making use of Proposition 7.31 we derive, for all r ≥ r0 = rT , Z r r2 kϑk∞,k (u, r) . r02 kϑk∞,k (u, r0 ) + λ2 kαk∞,k (u, λ)dλ + 0 u−1−δdec . r0

We now make use of the estimate kαk∞,k (u, r) . 0 r−2 u−1−δdec ,

k ≤ ksmall + 20

and kϑk∞,k (u, r0 ) . 0 u−1−δdec derived above in (7.5.24), to derive r2 kϑk∞,k (u, r) . 0 u−1−δdec + 0 ru−1−δdec . Therefore, everywhere on

(ext)

M,

kϑk∞,k (u, r) . 0 r−1 u−1−δdec .

(7.5.31)

Step 2. Next, we estimate β from the equation e4 β + 2κβ

=

/d2 α − (κ − κ)β + Γg · α = /d2 α + Γg · (α, β)

to deduce in the same manner r4 kβk∞,k (u, r) . r02 kβk∞,k (u, r0 ) +

Z

r

r0

λ4 k /d2 αk∞,k (u, λ)dλ + 0 ru−1−δdec .

Thus, in view of the estimates for α in (7.5.23) and the estimates for α in Ref 2, i.e., for 0 ≤ k ≤ ksmall + 20, 1

kαk∞,k . 0 min{r−2 log(1 + u)(u + 2r)−1−δextra , r−3 (u + 2r)− 2 −δextra }. Rr Thus we have with I(u, r) := r0 λ4 k /d2 αk∞,k (u, λ)dλ 

Z

r

I(u, r) . 0 min log(1 + u)

−1−δextra

λ(u + 2λ) r0

Z

r

dλ,

(u + 2λ)

−1/2−δextra

r0

If r ≤ 2u we have Z r λ(u + 2λ)−1−δextra dλ . r2 u−1−δextra . r2 (u + 2r)−1−δextra r0

and r−4 I(u, r) .

0 r−2 log(1 + u)(u + 2r)−1−δextra .

 dλ .

361

DECAY ESTIMATES (THEOREMS M4, M5)

If r ≥ 2u we have Z

r

(u + 2λ)−1/2−δextra

.

(u + 2r)1/2+δextra

r0

and r−4 I(u, r) . r−4 (u + 2r)1/2+δextra . r−2 (u + 2r)−1−δextra . We deduce kβk∞,k

. r−4 kβk∞,k (u, r0 ) + 0 r−2 log(1 + u)(u + 2r)−1−δextra .

Thus, in view of (7.5.23), kβk∞,k

. 0 r−2 log(1 + u)(u + 2r)−1−δextra .

(7.5.32)

Step 3. We now estimate ζ using the equation = −β + Γg · Γg .

e4 (ζ) + κζ

This can be done exactly as in Step 1 making use of the estimates already derived for β and the estimate (7.5.23) for ζ along T . We thus derive kζk∞,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall+15 .

Step 4. We estimate ρˇ using equation ρˇ = − /d1 ζ − µ ˇ + Γg · Γb , the previous estimate for ζ and µ ˇ in kˇ ρk∞,k

(ext)

M. We deduce

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 14.

(7.5.33)

Step 5. We estimate κ ˇ using the equation 1 1 e4 κ ˇ + κˇ κ+ κ ˇ κ = −2 /d1 ζ + 2ˇ ρ + Γg · Γb . 2 2 Making use of the estimates in (ext) M for κ ˇ , ζ and ρˇ as well as the estimates for κ ˇ on T in Proposition 7.42 we derive, everywhere on (ext) M, kˇ κk∞,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 12.

(7.5.34)

Alternatively, we could make use
of the estimate for the auxiliary quantity Ξ = r2 eθ (κ) + 4r /d?1 /d1 ζ − 2r2 /d?1 /d1 β in Lemma 7.36, which holds everywhere on (ext) M, and the above estimates for ζ, β. Step 6. We estimate β everywhere on

(ext)

M with the help of the equation

e4 β + κβ = − /d1 ρ − 3ζρ − ϑβ − (κ − κ)β together with the estimate (7.5.28) for β on T and the above derived estimates for

362

CHAPTER 7

ρˇ, ζ in

(ext)

M to infer that



β . 0 r−2 u−1−δdec , ∞,k

Step 7. We estimate ϑ everywhere on equation for ϑ in (7.5.11),

∀ k ≤ ksmall + 9.

(ext)

(7.5.35)

M by making use of the Codazzi

∈ r−3 Good20 .

/d2 ϑ + 2β − eθ (κ) + κζ

Using the estimates already derived above, we infer that, for all k ≤ ksmall + 11, k /d2 ϑk2,k

. .

Hence, everywhere in

kβk2,k + r−1 kˇ κk2,k+1 + r−1 kζk2,k + 0 r−2 u−1−δdec 0 r−2 u−1−δdec . (ext)

M,

kϑk2,k

. 0 r−1 u−1−δdec ,

for all k ≤ ksmall + 12,

kϑk∞,k

.

0 r−2 u−1−δdec ,

for all k ≤ ksmall + 10.

and therefore,

Step 8. We estimate α everywhere on

(ext)

M by making use of the equation

1 3 1 e4 α + κα = − /d?2 β − ϑρ − 5ζβ − (κ − κ)α 2 2 2 as well as the estimate (7.5.21) for α on T and the above estimates in all for β and ϑ. Proceeding as before we derive kαk∞,k

.

0 r−1 u−1−δdec ,

for all k ≤ ksmall + 8.

(ext)

M

(7.5.36)

This concludes the proof of Proposition 7.35.

7.6

CONCLUSION OF THE PROOF OF THEOREM M4

So far we have established the following estimates, for all k ≤ ksmall + 8,

κ ˇ , rµ ˇk∞,k . 0 r−2 u−1−δdec ,

ϑ, ζ, κ ˇ , rρˇ ∞,k . 0 r−2 u−1/2−δdec ,

ϑ, ζ, η, κ ˇ , ϑ, rβ, rρˇ, rβ, αk∞,k . 0 r−1 u−1−δdec ,

β, re3 β . 0 r−3 (2r + u)−1/2−δdec .

(7.6.1)

∞,k

It only remains to derive improved decay estimates for e3 (β, ϑ, ζ, κ ˇ , ρˇ) and the ˇ as well as ς + 1 and Ω + Υ in terms of u−1−δdec decay. More estimates for ξ, ω ˇ , ςˇ, Ω precisely it remains to prove the following. Proposition 7.43. The following estimates hold true on

(ext)

M for all integers k

363

DECAY ESTIMATES (THEOREMS M4, M5)

such that k ≤ ksmall + 7:

e3 (ϑ, ζ , κ ˇ ), re3 β, re3 ρˇ ∞,k

ξ, ω ˇ ∞,k

ˇ ς + 1, Ω + Υ

ςˇ, Ω, ∞,k

. 0 r−2 u−1−δdec , . 0 r−1 u−1−δdec , . 0 u−1−δdec .

Proof. We proceed in steps as follows. Step 1. We make use of the equation e3 ϑ = − 12 κ ϑ + 2ωϑ − 2 /d?2 η − 12 κ ϑ + 2η 2 and the previously derived estimates to derive

e3 ϑ . 0 r−2 u−1−δdec , k ≤ ksmall + 9. (7.6.2) ∞,k Step 2. We make use of the equation e3 β + (κ − 2ω)β = − /d?1 ρ + 3ηρ + Γg β + Γb α and the previously derived estimates for β, ρˇ, β to derive

e3 β ∞,k

0 r−3 u−1−δdec ,

.

k ≤ ksmall + 9.

Step 3. To estimate e3 ζ in the next step we actually need a stronger estimate for e3 β than the one derived above. At the same time we derive an improved estimate for β. We show in fact, for some 0 < δ,



β . 0 r−2−δ u−1−δdec , k ≤ ksmall + 10, ∞,k

(7.6.3)

e3 β . 0 r−3−δ u−1−δdec , k ≤ ksmall + 10. ∞,k−1 This makes use of the equation e4 β + 2κβ

=

/d2 α + Γg · α = F := /d2 α + Γg · α − 2ˇ κβ

and the estimates for α in Ref 2. Thus, for some 0 < δ < δextra − δdec , kF k∞,k

. log(1 + u)r−3 (2r + u)−1−δextra + 0 r−4 u−1−δdec . 0 u−1−δdec r−3−δ .

Integrating from T , where r = rT = r0 . 1, we deduce with the help of Proposition 7.31 Z r 4 4 r kβk∞,k (u, r) . r0 kβk∞,k (u, r0 ) + λ4 kF k∞,k (u, λ)dλ r0 Z r . kβk∞,k (u, r0 ) + 0 λ1−δ dλ. r0

Based on the previously derived estimate for β we have kβk∞,k (u, rH ) . 0 u−1−δdec . Hence, kβk∞,k (u, r) . 0 r−4 u−1−δdec + 0 r−4 r2−δ u−1−δdec . 0 r−2−δ u−1−δdec as desired. To prove the second estimate in (7.6.3) we commute the transport equation for

364

CHAPTER 7

β with T and make use of the corresponding estimate for Tα (which follows from Ref 2) kTαk∞,k

. 0 log(1 + u)r−4 (2r + u)−1−δextra . 0 u−1−δdec r−4−δ

as well as the fact that we control Tβ on T , i.e., kTβk∞,k−1 (u, r0 ) . 0 u−1−δdec . Step 4. We make use of the equation e4 ζ + κζ = −β + Γg · Γg to derive ke3 ζkk,∞

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 9.

(7.6.4)

Indeed commuting the equation with T we derive e4 Tζ + κTζ = F := −Tβ + [T, e4 ]ζ + ζTκ + T(Γg · Γg ). It is easy to check, in view of the commutation Lemma 7.29, kF k∞,k−1

.

kTβk∞,k−1 + 0 r−4 u−1−δdec .

Thus, in view of the estimate for e3 ζ derived in Step 3 and the estimate for e4 ζ we infer that kF k∞,k−1

. 0 r−3−δ u−1−δdec .

Integrating from T , and relying also on the previously derived estimate for ζ, i.e., kζkk,∞ . 0 r−1 u−1−δdec , we infer Z r r2 kTζk∞,k−1 . r0 2 kTζk∞,k−1 (u, r0 ) + 0 u−1−δdec λ−1−δ dλ r0

. kTζk∞,k−1 (u, r0 ) + 0 r

−δ −1−δdec

u

. 0 u−1−δdec .

Hence kTζk∞,k−1

. 0 r−2 u−1−δdec

from which the desired estimate easily follows. Step 5. We make use of the equation e4 (ˇ ω ) = ρˇ+Γg ·Γb and the previously derived estimates for ρˇ as well as the estimates of ω ˇ on the last slice (see Proposition 7.28) to derive the estimate kˇ ω k∞,k

. 0 r−1 u−1−δdec ,

k ≤ ksmall + 9.

(7.6.5)

Indeed, ke4 ω ˇ k∞,k

. kˇ ρk∞,k + 0 r−3 u−1−δdec . 0 r−2 u−1−δdec .

Thus, applying Proposition 7.31, integrating from Σ∗ and using the previously

365

DECAY ESTIMATES (THEOREMS M4, M5)

derived estimate for ω ˇ on Σ∗ , kˇ ω k∞,k (u, r∗ ) + 0 u−1−δdec

kˇ ω k∞,k (u, r) .

Z

r∗

λ−2 dλ

r

. 0 r−1 u−1−δdec as desired. Step 6. We derive the estimate kξk∞,k

0 r−1 u−1−δdec ,

.

k ≤ ksmall + 9

(7.6.6)

by making use of the transport equation e4 (ξ) = F := −e3 (ζ)+β− 12 κ(ζ +η)+Γb ·Γb . In view of the previously derived estimates for e3 ζ, β, ζ, η we derive kF k∞,k

.

0 r−2 u−1−δdec .

Integrating from Σ∗ and making use of the estimate for ξ on Σ∗ (see Proposition 7.28) we derive kξk∞,k (u, r) . kξk∞,k (u, r∗ ) + 0 r−1 u−1−δdec . 0 r−1 u−1−δdec . Step 7. We derive the estimate ˇ ∞,k kΩk

0 u−1−δdec ,

.

k ≤ ksmall + 8.

(7.6.7)

This follows immediately from the equation eθ (Ω) = −ξ − (η − ζ)Ω, see (2.2.19), and the previous estimate for ξ. Note that Ω has been estimated in Lemma 3.15. Step 8. We derive the estimate . 0 u−1−δdec ,

kς − 1k∞,k

k ≤ ksmall + 8.

(7.6.8)

The estimate follows from the propagation equation e4 (ς) = 0 and the estimate for ς − 1 on the last slice Σ∗ . Step 9. We derive the estimate ke3 ρˇk∞,k

.

0 r−3 u−1−δdec ,

k ≤ ksmall + 8

(7.6.9)

with the help of the equation (see Proposition 7.8) e3 ρˇ = r−2 d/≤1 Γb + r−1 Γb · Γb ˇ ςˇ. and the previously derived estimates for β, κ ˇ , ρˇ, Ω, Step 10. We derive the estimate ke3 κ ˇ k∞,k

. 0 r−2 u−1−δdec ,

k ≤ ksmall + 8

(7.6.10)

366

CHAPTER 7

using the equation (see Proposition 7.8) e3 κ ˇ = r−1 d/≤1 Γb + Γb · Γb ˇ ςˇ. This ends the proof of Propoand the previously derived estimates for κ ˇ , ξ, ω ˇ , Ω, sition 7.43 and Theorem M4.

7.7

PROOF OF THEOREM M5

Recall from Theorem M3 that we have obtained the following estimate for in (int) M: sup u1+δdec |dk α| . 0 .

max

0≤k≤ksmall +16

(int)

α

(7.7.1)

(int) M

Step 1. We consider the control of the other curvature components, as well as the Ricci components on T . Recall that the (u, (int) s) foliation is initialized on T as follows: • u and

(int)

s are defined on T by (int)

u = u and

(ext)

s=

s on T .

In particular, on the hypersurface T , the 2-spheres S(u, (int) s) coincide with the 2-sphere S(u, (ext) s). • In view of the above initialization, and the fact that T = {r = rT }, we infer that (int)

• The null frame ( (int) e3 , (int)

e4 =

(ext)

(ext)

r=

(int)

λ (ext) e4 ,

e4 ,

(int)

(int)

r = rT , (int)

m=

(ext)

eθ ) is defined on T by

e3 = ( (ext) λ)−1 (ext) e3 ,

where (ext)

m.

λ=1−

(int)

eθ =

(ext)

eθ on T

2 (ext) m . (ext) r

In particular, we deduce the following identities for the curvature components and Ricci coefficients on T . Lemma 7.44. We have on T (int)

(int)

(ext) κ

ς

= −

Ω

= λ − λ2

+

(ext)

A

(ext) κ

λ−1 (ext) ς,

(ext) κ (ext) κ

+

(ext) A

−λ

(ext) κ (ext) κ

where λ=

(ext)

λ=1−

2 (ext) m . (ext) r

+

(ext) A

(ext)

Ω,

367

DECAY ESTIMATES (THEOREMS M4, M5)

Moreover, we have on T (int)

α = λ2 (ext) α,

(int)

α = λ−2 (ext) α,

(int)

ξ = 0,

(int)

(int)

κ = λ (ext) κ,

(int)

(int)

ω = 0, (int)

β = λ (ext) β,

ζ=

(ext)

ϑ = λ (ext) ϑ,

(int)

(ext)

ρ=

(int)

ρ,

ζ,

(int)

η = − (ext) ζ,

(int)

κ = λ−1 (ext) κ,

β = λ−1 (ext) β,

(int)

ϑ = λ−1 (ext) ϑ,

and (int)

(int)

ξ

ω

(int)

λ2 (ext) κ

=

(ext) κ

(ext) A

λ (ext) κ

=

η

+

(ext) κ (ext)

=

+

ζ−

( (ext) ζ − (ext)

(ext) A

+

η),

ω,

(ext) κ (ext) κ

(ext)

(ext)

(ext) A

ξ.

Proof. The following vectorfield is tangent to T : νT

:=

(ext)

e3 −

(ext) κ

+

(ext)

A (ext)

(ext) κ

e4 ,

which can also be written as νT

=

λ (int) e3 −

(ext) κ

+

(ext)

A

(ext) κ

λ−1 (int) e4 .

Since νT is tangent to T , and in view of the definition of u and ately infer

(int)

s, we immedi-

νT (u) = νT (u) and νT ( (int) s) = νT ( (ext) s) on T and hence, using the identities (ext)

e4 (u) =

(int)

e3 (u) = 0,

(ext)

e4 ( (ext) s) = 1,

(int)

e3 ( (int) s) = −1,

we deduce on T − −λ −

(ext) κ

+

(ext)

(ext) κ

(ext) κ

+

(ext)

(ext) κ

A

A

λ−1 (int) e4 (u)

=

(ext)

e3 (u),

λ−1 (int) e4 ( (int) s)

=

(ext)

e3 ( (ext) s) −

(ext) κ

+

(ext)

(ext) κ

A

.

368

CHAPTER 7

(ext)

ς,

(int)

(ext) κ

+

(ext)

In view of the definition of (int)

(int)

ς

= −

Ω

= λ − λ2

(ext)

ς, A

(ext) κ

Ω and

+

(ext) A

−λ

e4 = λ (ext) e4 ,

(int)

e3 = λ−1

(ext)

(ext) κ (ext) κ

(int)

Next, we consider the Ricci coefficients of (int)

α = λ2 (ext) α,

(int)

α = λ−2 (ext) α,

(int)

β = λ (ext) β,

(int)

e3 ,

(int)

(ext)

(ext) A

+

Ω.

M on T . From

the fact that λ is constant on T , and the fact that on T (int)

Ω, this yields

λ−1 (ext) ς,

(ext) κ (ext) κ

(int)

(ext)

(ext)

eθ =

eθ is tangent to T , we infer

(ext)

ρ,

(int)

ϑ = λ (ext) ϑ,

(int)

ρ=

eθ on T ,

β = λ−1 (ext) β,

and (int)

ζ=

(ext)

(int)

ϑ = λ−1 (ext) ϑ.

ζ,

(int)

Also, since the foliation of (int)

κ = λ (ext) κ,

(int)

ω = 0,

It remains to find identities for (int) ξ, T and νT tangent to T , we have on T DνT

(int)

e4 = λDνT

κ = λ−1 (ext) κ,

M is ingoing geodesic, we have

(int)

ξ = 0,

(int)

(ext)

e4 ,

(int)

(int)

DνT

η = − (int) ζ. (int)

ω and

(int)

η. Since λ is constant on

e3 = λ−1 DνT

(ext)

e3

and hence g(DνT

(int)

g(DνT

(int)

g(DνT

(int)

e4 ,

(int)

e4 ,

(int)

e3 ,

(int)

eθ ) e3 ) eθ )

= λg(DνT = g(DνT = λ

−1

(ext)

(ext)

g(DνT

(ext)

e4 ,

e4 , (ext)

eθ ),

(ext)

e3 ,

e3 ), (ext)

eθ ).

We deduce 2λ

(int)

η−2

−4λ (int) ω − 4 2λ (int) ξ − 2

(ext) κ

+

(ext)

A

(ext) κ (ext) κ

+

(ext)

A

(ext) κ (ext) κ

+

(ext)

(ext) κ

λ

−1 (int)

ξ

λ−1 (int) ω

A

λ−1 (int) η

= λ 2

(ext)

η−2

= −4 (ext) ω − 4

(ext) κ

(ext)

+

(ext) κ

(ext) κ

+

(ext)

A (ext)

(ext) κ

= λ−1 2 (ext) ξ −2

(ext) κ

+

(ext)

(ext) κ

A (ext)

A (ext)

! η ,

! ξ

ω,

,

369

DECAY ESTIMATES (THEOREMS M4, M5)

and thus (int)

(int)

ξ

=

ω

(int)

=

η

=

λ2 (ext) κ (ext) κ

+

(ext) A

( (ext) ζ −

λ (ext) κ (ext) κ (ext)

+

ζ−

(ext)

(ext) A

+

η),

ω,

(ext) κ (ext) κ

(ext)

(ext)

(ext) A

ξ.

This concludes the proof of the lemma. Remark 7.45. Since the 2-spheres S(u, (int) s) coincide on T with the 2-spheres S(u, (ext) s), the above lemma immediately yields κ ˇ = λ (ext) κ ˇ , (int) κ ˇ = λ−1 (ext) κ ˇ, 1 1 (int) µ ˇ = − (ext) µ ˇ − 2 (ext) ρˇ + (ext) ϑ (ext) ϑ − (ext) ϑ (ext) ϑ, 2 2 ! (ext) (ext) ω ω (int) (ext) ω ˇ=λ κ − , (ext) κ + (ext) A (ext) κ + (ext) A (int)

ρˇ =

(ext)

ρˇ,

(int)

and (int)

ˇς = −

(int) ˇ

Ω

=

1



λ (ext) κ

−λ2 (ext) κ

A) (ext) ς − ( (ext) κ +

1 (ext) κ

+

(ext)

−λ (ext) κ

(ext)

( (ext) κ +

(ext) κ

+

(ext) A

Ω (ext) A

−

−

+

+

,

(ext) A

!

(ext) Ω (ext) κ



!

1 (ext) κ

(ext) A) (ext) ς

(ext) A

.

Together with the estimates on T for the outgoing geodesic foliation of (ext) M derived in Theorem M4, we infer the control of tangential derivatives to T , i.e., (eθ , TT ) derivatives. Recovering the transversal derivative thanks to the transport equations in the direction e3 , we infer for the ingoing geodesic foliation of (int) M on T



max sup u1+δdec dk (int) α, (int) β, (int) ρˇ, (int) β, (int) µ ˇ , (int) κ ˇ, 0≤k≤ksmall +8 T

(int)

κ ˇ , (int) ϑ,  (int) ˇ ξ, (int) ω ˇ , (int)ˇς , (int) Ω

2 ϑ,

(int)

ζ,

(int)

η,

(int)

L (S)

. 0 .

Step 2. Relying on the estimates of the ingoing geodesic foliation of (int) M on T derived in Step 1, we propagate these estimates to (int) M thanks to transport equations in the e3 direction given by the null structure equations and Bianchi identities. Recalling that α has already been estimated in Theorem M3, see (7.7.1), quantities are recovered in the following order:

370

CHAPTER 7

1. We recover κ ˇ , with a control of ksmall + 8 derivatives, from e3 κ ˇ + κκ ˇ =

Err[e3 κ ˇ ].

2. We recover ϑ, with a control of ksmall + 8 derivatives, from = −2α.

e3 (ϑ) + κ ϑ

3. We recover β, with a control of ksmall + 8 derivatives, from e3 β + 2κ β

/d2 α − ζα.

=

4. We recover ζ, with a control of ksmall + 8 derivatives, from = β − ϑζ.

e3 (ζ) + κζ

5. We recover η, with a control of ksmall + 8 derivatives, from 1 e3 (η + ζ) + κ(η + ζ) 2

1 = − ϑ(η + ζ). 2

6. We recover µ ˇ , with a control of ksmall + 8 derivatives, from 3 3 e3 µ ˇ + κˇ µ + µˇ κ = 2 2

Err[e3 µ ˇ ].

7. We recover ρˇ, with a control of ksmall + 7 derivatives, from 3 3 e3 ρˇ + κˇ ρ + ρˇ κ = 2 2

/d1 β + Err[e3 ρˇ].

8. We recover κ ˇ , with a control of ksmall + 7 derivatives, from 1 1 e3 κ ˇ + κˇ κ+ κ ˇκ = 2 2

2 /d1 ζ + 2ˇ ρ + Err[e3 κ ˇ ].

9. We recover ϑ, with a control of ksmall + 7 derivatives, from 1 e3 ϑ + κ ϑ 2

1 = −2 /d?2 ζ − κ ϑ + 2ζ 2 . 2

10. We recover β, with a control of ksmall + 6 derivatives, from e3 β + κβ

= eθ (ρ) + 3ζρ − ϑβ.

11. We recover α, with a control of ksmall + 5 derivatives, from 1 e3 α + κα 2

=

3 − /d?2 β − ϑρ + 5ζβ. 2

12. We recover ω ˇ , with a control of ksmall + 7 derivatives, from e3 ω ˇ

=

ρˇ + Err[e3 ω ˇ ].

371

DECAY ESTIMATES (THEOREMS M4, M5)

ˇ with a control of ksmall + 7 derivatives, from 13. We recover Ω, ˇ e3 (Ω)

ˇ = −2ˇ ω+κ ˇ Ω.

14. We recover ξ, with a control of ksmall + 6 derivatives, from e3 (ξ)

1 1 = e4 (ζ) + β + κ(ζ − η) + ϑ(ζ − η). 2 2

15. We recover ς, with a control of ksmall + 8 derivatives, from e3 (ς − 1) = 0. As the estimates are significantly simpler to derive21 and in the same spirit as the corresponding ones in Theorem M4, we leave the details to the reader. This concludes the proof of Theorem M5.

21 Note

that r is bounded on

(int) M

and that all quantities behave the same in

(int) M.

Chapter Eight Initialization and Extension (Theorems M6, M7, M8) In this chapter, we prove M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies.

8.1

PROOF OF THEOREM M6

Step 1. Let r0 such that −2

r0

:= d0 0 3 ,

(8.1.1)

where the constant d0 satisfies 1 ≤ d0 ≤ 2 2 and will be suitably chosen in Step 3. Also, let δ0 > 0 sufficiently small. Consider ◦

the unique sphere S of the initial data layer on C(1+δ0 ,L0 ) with area radius r0 . Then, denoting S(uL0 , (ext) sL0 ) the spheres of the outgoing portion of the initial data layer, we have ◦

◦ ◦

S = S(u, s),

◦

◦

|s − r0 | . 0 .

u = 1 + δ0 ,

Relying on the control of the initial data layer given by (3.3.5), i.e., 5

Iklarge +5 ≤ 03 , we then invoke Theorem GCMS-II of section 3.7.4 with the choices ◦

◦

δ =  = 0 ,

smax = klarge + 5, ◦

to produce a unique GCM sphere S∗ , which is a deformation of S, satisfying 2 κS∗ = S∗ , /d?2S∗ /d?1S∗ κS∗ = /d?2S∗ /d?1S∗ µS∗ = 0 r Z Z S∗ Φ β e = 0, eSθ ∗ (κS∗ )eΦ = 0. S∗

on S∗ ,

S∗

Remark 8.1. In order to apply Theorem GCMS-II to the above setting, one needs to check that the initial data foliation layer satisfies the assumptions of the theorem,

373

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and in particular ◦

1

|d≤smax Γb | . () 3 r0−2 , Z Z ◦ ◦ Φ Φ r0 βe . δ, r0 eθ (κ)e . δ, S

Z ◦ Φ r0 eθ (κ)e . δ.

S

◦

S

◦

Now, in view of the above choice for smax , δ,  and r0 , this follows from 5

r|d≤klarge +5 Γb | . 0 ,

r3 |β| + r2 | d/κ ˇ | + r2 | d/κ ˇ | . 03

and hence from 5

Iklarge +5 ≤ 03 which is (3.3.5). Step 2. Starting from S∗ constructed in Step 1, and relying on the control of the initial data layer, we then invoke Theorem GCMH of section 3.7.4 to produce a smooth spacelike hypersurface Σ∗ included in the initial data layer, passing through the sphere S∗ , and a scalar function u defined on Σ∗ such that • The following GCM conditions hold: 2 κ= , r

/d?2 /d?1 κ

=

/d?2 /d?1 µ

Z = 0,

Φ

Z

ηe = S

ξeΦ = 0 on Σ∗ .

S

• We have, for some constant cΣ∗ , u + r = cΣ∗ ,

along

Σ∗ .

• The following normalization condition holds true at the south pole SP of every sphere S, 2m a = −1 − r SP where a is such that we have ν = e3 + ae4 , with ν the unique vectorfield tangent to the hypersurface Σ∗ , normal to S, and normalized by g(ν, e4 ) = −2. Furthermore, we have1 max

  sup r |dk f | + |dk f | + |dk log(λ)|

k≤klarge +4 Σ∗

1 We

. 0 ,

(8.1.2)

have in fact max

  sup kdk f kL2 (S) + kdk f kL2 (S) + kdk log(λ)kL2 (S)

k≤klarge +6 Σ∗

.

0 ,

and then use the Sobolev embedding on the 2-spheres S foliating Σ∗ to deduce (8.1.2).

374

CHAPTER 8

and   sup |m − m0 | + |r − r0 |

. 0 ,

(8.1.3)

Σ∗

where (f, f , λ) are the transition functions from the frame of the initial data layer to the frame of Σ∗ . Step 3. Provided δ0 > 0 has been chosen sufficiently small, the spacelike hypersurface Σ∗ of Step 2 intersects the curve of the south poles of the spheres foliating the outgoing cone C(1,L0 ) of the initial data layer. We then call S1 the unique sphere of Σ∗ such that its south pole coincides with the south pole of a sphere of C(1,L0 ) , and ◦

we calibrate u such that u = 1 on S1 . We then can compare u = 1 + δ0 to u(S∗ ) and obtain |u(S∗ ) − 1 − δ0 | . 0 δ0 , so that 1 ≤ u ≤ u(S∗ )

on Σ∗ where 1 < u(S∗ ) < 1 + 2δ0 .

Together with the estimate (8.1.3), and in view of the choice (8.1.1) for r0 , we have inf r Σ∗

= =

−2

r(S∗ ) = r0 + O(0 ) = d0 0 3 + O(0 )   5  −2 0 3 (u(S∗ ))1+δdec d0 + O(δ0 ) + O 03 . 1 2

Thus, we may choose the constant d0 in the range

≤ d0 ≤ 2 such that

−2

inf r = 0 3 (u(S∗ ))1+δdec Σ∗

so that the dominant condition (3.3.4) for r is satisfied. Step 4. In view of Step 1 to Step 3, Σ∗ satisfies all the required properties for the future spacelike boundary of a GCM admissible spacetime, see item 3 of Definition 3.2. We now control the outgoing geodesic foliation initialized on Σ∗ and covering the region we denote by (ext) M, which is included in the initial data layer. Let (f, f , λ) the transition functions from the frame of the outgoing part of the initial data layer to the frame of (ext) M. Since both frames are outgoing geodesic, we may apply Corollary 2.93 which yields for (f , f, log(λ)) the following transport equations: λ−1 e04 (rf ) −1 0 λ e4 (log(λ))   λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω

= E10 (f, Γ), = E20 (f, Γ), = E30 (f, f , λ, Γ),

375

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

where E10 (f, Γ) E20 (f, Γ) E30 (f, f , λ, Γ)

r r = − κ ˇ f − ϑf + l.o.t., 2 2 1 2 1 = f ζ − f ω − ηf − f 2 κ + l.o.t., 2  4  r 2 2 = − κ ˇf + r κ ˇ− κ− e0θ (log(λ)) 2 r   r +r2 /d01 (f ) + λ−1 ϑ0 e0θ (log(λ)) − κ ˇ Ωf + rE3 (f, f , Γ) 2 −2r2 e0θ (E2 (f, Γ)) + rΩE1 (f, Γ),

and where E1 , E2 and E3 are given in Lemma 2.92. Integrating these transport equations from Σ∗ , using the control (8.1.2) of (f, f , λ) on Σ∗ , and together with the assumption (3.3.5) for the Ricci coefficients of the foliation of the initial data layer, we obtain   sup r |d≤klarge +4 (f, log(λ))| + |d≤klarge +3 f | . 0 . (8.1.4) (ext) M(r≥2m

0 (1+δH ))

Then, let T = {r = 2m0 (1 + δH )}, i.e., we choose rT = 2m0 (1 + δH ). We initialize the ingoing geodesic foliation of (int) M on T using the outgoing geodesic foliation of (ext) M as in item 4 of Definition 3.2. Using the control of (f, f , λ) induced on T by (8.1.4), and using the analog of Corollary 2.93 in the e3 direction for ingoing foliations, we obtain similarly   sup |d≤klarge +3 (f , log(λ))| + |d≤klarge +2 f | . 0 . (8.1.5) (int) M

Then, in view of (8.1.4), (8.1.5), and the assumption (3.3.5) for the Ricci coefficients and curvature components of the foliation of the initial data layer, and using the transformation formulas of Proposition 2.90, we deduce (  7  max sup r 2 +δB (|dk α| + |dk β|) + r3 |dk ρˇ| + r2 |dk β| + r|dk α| k≤klarge

(ext) M

+ sup r2 (|dk κ ˇ | + |dk ϑ| + |dk ζ| + |dk κ ˇ |) (ext) M ) + sup r(|dk η| + |dk ϑ| + |dk ω ˇ | + |dk ξ|)

. 0 ,

(ext) M

and max

k≤klarge

sup (int) M



ˇ + |dk Γ| ˇ |dk R|



In particular, we infer that (En)

(Dec)

Nklarge + Nksmall which concludes the proof of Theorem M6.

. 0

.

0 .

376 8.2

CHAPTER 8

PROOF OF THEOREM M7

In view of the assumptions of Theorem M7, we are given a GCM admissible spacetime M = M(u∗ ) ∈ ℵ(u∗ ) verifying the following improved bounds, for a universal constant C > 0, (Dec)

Nksmall +5 (M) ≤ C0

(8.2.1)

provided by Theorems M1–M5. We then proceed as follows. Step 1. We extend M by a local existence argument, to a strictly larger spacetime M(extend) , with a naturally extended foliation and the following slightly increased bounds (Dec)

Nksmall +5 (M(extend) ) ≤ 2C0 , but which may not verify our admissibility criteria. Step 2. We then invoke Theorem GCMH of section 3.7.4 to extend the hypersur(extend) face Σ∗ in M(extend) \ M as a smooth spacelike hypersurface Σ∗ , together (extend) with a scalar function u , satisfying the same GCM conditions as Σ∗ . Step 3. We consider the outgoing geodesic foliation (u(extend) , s(extend) ) initialized (extend) (extend) on Σ∗ to the future of Σ∗ in M(extend) . Note in particular that we have (extend) from the definition of Σ∗ and Σ∗ u(extend) + s(extend) = cΣ∗ . (extend)

We define the following spacetime region to the future of Σ∗ : n o e := R u∗ ≤ u(extend) ≤ u∗ + δext , cΣ∗ ≤ u(extend) + s(extend) ≤ cΣ∗ + ∆ext , where ∆ext :=

d0 r∗ δext , u∗

r∗ := r(S∗ ),

S∗ := Σ∗ ∩ C∗ ,

e ⊂ M(extend) , and with d0 a with δext > 0 chosen sufficiently small so that R constant satisfying 1 ≤ d0 ≤ 1 2 which will be suitably chosen in Step 11 below. From now on, for convenience, we drop the index (extend) and simply denote u(extend) and s(extend) by u and s. (extend)

Step 4. Since we have on Σ∗ the GCM conditions /d?2 /d?1 κ = /d?2 /d?1 µ = 0, and Φ ? since e generates the kernel of /d2 , we infer R R e (κ)eΦ Φ e (µ)eΦ Φ (extend) ? ? SR θ SR θ /d1 κ = − e , / d µ = − e , on Σ∗ . 1 2Φ 2Φ e e S S

377

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Thus, introducing the following two scalar functions R R e (κ)eΦ e (µ)eΦ SR θ SR θ C(u) := − , M (u) := − , e2Φ e2Φ S S

(extend)

on Σ∗

,

(extend)

we rewrite the GCM conditions on Σ∗ 2 , r

κ=

as follows: Z Z /d?1 µ = M (u)eΦ , ηeΦ = ξeΦ = 0.

/d?1 κ = C(u)eΦ ,

S

S (extend)

Propagating these GCM quantities in the e4 direction from Σ∗ , and propagating the scalar functions C and M by e4 (r4 C) = 0 and e4 (r5 M ) = 0 so that we e we obtain for all k ≤ ksmall + 4 have2 C = C(u, s) and M = M (u, s) in R,    

2 2 k 3 k−1 ? Φ 4 k−1 ? Φ sup r d κ − +r d /d1 κ − Ce +r d /d1 µ − M e r e R 0 . ∆ext r and sup r

−2

Z   Z ξeΦ + ηeΦ . S

e R

S

0 ∆ext . r (extend)

Next, in view of (4.1.2) and the fact that ν = e3 + ae4 , we have on Σ∗ Z Z   0 0 20 Φ Φ ν ν βe . , e (κ)e θ ru1+δdec . ru1+δdec + u2+2δdec . S S −2

In particular, since r(S∗ ) = 0 3 (u(S∗ ))1+δdec in view of (3.3.4) and u(S∗ ) = u∗ , we −2

(extend)

infer r ∼ 0 3 u1+δdec on Σ∗ (u∗ ≤ u ≤ u∗ + δext ) and hence Z  Z  0 (extend) Φ Φ ν ν βe + e (κ)e (u∗ ≤ u ≤ u∗ + δext ). θ . ru1+δdec on Σ∗ S

S

We integrate from S∗ where we have Z Z Φ βe = S∗

eθ (κ)eΦ = 0

S∗

and obtain sup (extend)

Σ∗

(u∗ ≤u≤u∗ +δext )

Z   Z Φ Φ r βe + r eθ (κ)e . S

S

(extend)

We now integrate in the e4 direction from Σ∗

of

(u∗ ≤ u ≤ u∗ + δext ) where we

e and M = r−5 M f, with C e and M f given by the restriction precisely, we have C = r−4 C (extend) 5 e e f f and r M to Σ∗ so that C = C(u) and M = M (u). Note also that r = r(u, s).

2 More

r4 C

0 δext . u∗

378

CHAPTER 8

have the above estimate as well as eθ (κ) = 0. We obtain Z Z   Z sup r βeΦ + r eθ (κ)eΦ + r eθ (κ)eΦ . S

e R∩{u≥u ∗}

S

S

.

0 0 δext + ∆ext u∗ r 0 ∆ext . r

Also, recall that ν = e3 + a∗ e4 denotes the unique tangent vectorfield to Σ∗ which is orthogonal to eθ and normalized by g(ν, e4 ) = −2. Then, one has, since u + r is constant on Σ∗ and s = r on Σ∗ , 0 = ν(u + s) = e3 (u) + ae4 (u) + e3 (s) + ae4 (s) =

2 +Ω+a ς

and hence a

2 = − − Ω on Σ∗ . ς

Together with the GCM condition on a, we infer   2 2m +Ω = 1+ on Σ∗ . ς r SP As above, propagating forward in e4 , we infer     2 2m sup +Ω − 1+ ς r SP e R (extend)

Finally, arguing as we did above on Σ∗ −2 e ∩ {u ≥ u∗ } and hence r ∼ 0 3 u1+δdec on R sup e R∩{u≥u ∗}

r2 |Γb | .

0 ∆ext . r

.

(u∗ ≤ u ≤ u∗ + δext ), we have

 r  1 0 . 03 . 1+δ dec u e R∩{u≥u ∗} sup

Step 5. We fix the following sphere of the (u(extend) , s(extend) ) foliation in the e ∩ {u ≥ u∗ }: region R ◦

◦ ◦

S := S(u, s),

◦

u := u∗ +

δext , 2

◦

s := r∗ +

3d0 r∗ δext . 4u∗

(8.2.2)

Define ◦

δ :=

0 d0 0 δext ∆ext = , r u∗

◦

 := 0 ,

◦

and the small spacetime neighborhood of S n ◦ ◦ ◦ R(, δ) := |u − u| ≤ δR , ◦

o ◦ |s − s| ≤ δR ,

◦ ◦
− 1 2

δR = δ 

.

◦ e In view of the estimates in Step 4, we are in position to Note that R(, δ) ⊂ R.

379

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

apply Theorem GCMS-II of section 3.7.4, with smax = ksmall + 4, which yields the ◦ ◦ ◦ existence of a unique sphere Se∗ , which is a deformation of S, is included in R(, δ), and is such that the following GCM conditions hold on it: Z Z 2 e/d? e/d? κ e/d? e/d? µ e Φ= e = e = 0, κ e = , βe eeθ (e κ)eΦ = 0, 2 1 2 1 re e∗ e∗ S S where the tilde refer to the quantities and tangential operators on Se∗ . Step 6. Starting from Se∗ constructed in Step 5, and in view of the estimates in Step 4, we may apply Theorem GCMH of section 3.7.4, with smax = ksmall + 4, e∗ which yields the existence of a smooth small piece of spacelike hypersurface Σ e starting from S∗ towards the initial data layer, together with a scalar function u e e ∗ , whose level surfaces are topological spheres denoted by S, e so that defined on Σ e ∗: • The following GCM conditions are verified on Σ Z Z 2 Φ e/d? e/d? κ e/d? e/d? µ e e = e = 0, κ e = , η e e = ξeΦ = 0, 2 1 2 1 re e e S S e ∗. where the tilde refer to the quantities and tangential operators on Σ • We have, for some constant cΣ e∗ , u e + re = cΣ e∗ ,

e ∗. along Σ

• The following normalization condition holds true at the south pole SP of every e sphere S, e a

SP

= −1 −

2m e re

where e a is such that we have νe = ee3 + e aee4 , e ∗ , normal to S, e and with νe the unique vectorfield tangent to the hypersurface Σ normalized by g(e ν , ee4 ) = −2. e∗ • The transition functions (f, f , λ) from the frame of M(extend) to the frame of Σ k(f, f , log(λ))khksmall +5

◦

. δ.

e ∗ has been constructed in Step 6 in a Step 7. The spacelike GCM hypersurface Σ e small neighborhood of S∗ . We now focus on proving that it in fact extends all the way to the initial data layer. To this end, we denote by u1 with ◦

1 ≤ u1 < u, the minimal value of u such that

380

CHAPTER 8

• We have ◦ e ∗ ∩ Cu 6= ∅ for any u1 ≤ u ≤ u. Σ

(8.2.3)

• There exists a large constant D ≥ 1 such that we have for any sphere Se of e ∗ (u ≥ u1 ) Σ k(f, f , log(λ))khk

◦

≤ Du∗ δ.

(S) small +5 e

(8.2.4)

e ∗ (u ≥ u1 ) • For the same large constant D ≥ 1 as above, we have along Σ |ψ(s)|

◦

≤ Du∗ δ,

(8.2.5)

where the function ψ(s) is such that the curve   ◦ u = −s + cΣ e ∗ + ψ(s), s, θ = 0 with ψ(s) = 0,

(8.2.6)

e ∗ and the constant c e is fixed coincides with the south poles of the sphere Se of Σ Σ∗ ◦

by the condition ψ(s) = 0. ◦

The fact that ψ(s) = 0 together with the bounds of Step 6 implies that (8.2.3), ◦ ◦ (8.2.4), (8.2.5) hold for u1 < u with u1 close enough to u. By a continuity argument based on reapplying Theorem GCMH, it suffices to show that we may improve the bounds (8.2.4), (8.2.5) independently of the value of u1 . Step 8. We now focus on improving the bounds (8.2.4), (8.2.5). We first prove e ∗ (u ≥ u1 ) is included in R. e Indeed, (8.2.4), (8.2.5) imply that Σ sup e ∗ (u≥u1 ) Σ

|u + s − cΣ e∗ | .

sup e ∗ (u≥u1 ) Σ



|ψ| + r|f | + r|f |



◦

. .

Du∗ δ Du∗ 0 ∆ext r 2

.

03 D0 ∆ext

.

0 ∆ext . ◦

◦ On the other hand, by construction, ψ(s) = 0 and the south pole of S and Se∗ coincide, so that we have

cΣ e∗

= =

δext 3d0 r∗ ◦ ◦ u + s = u∗ + r∗ + + δext 2 4u∗   3 2u∗ cΣ∗ + 1+ ∆ext 4 3d0 r∗

381

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and hence 3 sup u + s − cΣ∗ − ∆ext . 4 e ∗ (u≥u1 ) Σ



 u∗ + 0 ∆ext 2d0 r∗

2

03 ∆ext .

. e we infer In view of the definition of R, e ∗ (u ≥ u1 ) ⊂ R e Σ

(8.2.7)

as claimed. e ∗ (u ≥ u1 ) ⊂ R, e the bounds of Step 4 apply, and hence we have Step 9. Since Σ     ◦ 2 2m 0 sup +Ω − 1+ . r ∆ext . δ, ς r SP e R and for all k ≤ ksmall + 4     ◦ 2 0 2 k 2 k−2 2 ? ? 3 k−2 2 ? ? sup r d κ − (r /d2 /d1 κ)| + r |d (r /d2 /d1 µ)| . ∆ext . δ, + r |d r r e R as well as sup e R∩{u≥u ∗}

Z Z   Z r βeΦ + r eθ (κ)eΦ + r eθ (κ)eΦ . S

S

S

◦ 0 ∆ext . δ. r

Together with the a priori estimates of Chapter 9 on the GCM construction, this yields   2m e 2 0 |ψ (s)| . 1 + + Ω+ + |λ − 1| re ς SP m e m 0 . − + |λ − 1| + ∆ext . re r r In view of (8.2.4), we have ◦

|e r − r| + |m e − m| . sup r(|f | + |f |) . Du∗ δ e S

and we infer Du∗ ◦ ◦ δ+δ r  ◦ 2 . 1 + 03 D δ

|ψ 0 (s)| .

◦

. δ.

(8.2.8)

382

CHAPTER 8 ◦

◦

Integrating from s where ψ(s) = 0, we infer ◦ ◦

|ψ(s)| . |s − s|δ ◦

. u∗ δ which improves (8.2.5) for D ≥ 1 large enough. Similarly, we obtain Z   Z ◦ −2 Φ Φ k(f, f , log(λ))khk . r f e + f e + δ e ( S) small +5 S

S

and Z  Z  Φ Φ ν˜ f e + ν˜ fe S

◦

1 . r δ+ r 2

S

Z   Z f eΦ + f eΦ . S

S

In view of (8.2.4), we infer Z  Z  ◦ ◦ Φ Φ ν˜ f e + ν˜ f e . r2 δ + rDu∗ δ S

S

and integrating from Se∗ , we infer Z   Z −2 Φ Φ r . fe + fe S

.

D(u∗ )2 ◦ δ r   ◦ 2 1 + 03 D u∗ δ

.

u∗ δ.

S

◦

u∗ δ +

◦

This yields ◦

k(f, f , log(λ))khk

e (S) small +5

. u∗ δ

e∗ which improves (8.2.4) for D ≥ 1 large enough. We thus conclude that u1 = 1, Σ e e extends all the way to the initial data layer, Σ∗ ⊂ R, and we have the bounds k(f, f , log(λ))khk

◦ (S) small +5 e

. u∗ δ,

◦

|ψ(s)| . u∗ δ.

◦

e∗ In view of the definition of δ, we infer in particular for any sphere Se of Σ k(f, f , log(λ))khk

small +5

e (S)

. 0 δext ,

|ψ(s)| . 0 δext .

(8.2.9)

e ∗ extends all the way to the initial data layer, this allows us to Step 10. As Σ e ∗ by fixing the value u calibrate u ˜ along Σ e = 1 as in (3.1.5): e ∗ ∩ {e Se1 = Σ u = 1} is such that Se1 ∩ C(1,L0 ) ∩ SP 6= ∅,

(8.2.10)

e ∗ such that its south pole intersects the south pole i.e., Se1 is the unique sphere of Σ

383

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

of one of the spheres of the outgoing null cone C(1,L0 ) of the initial data layer. Now that u e is calibrated, we define u ˜∗ := u ˜(Se∗ ).

(8.2.11)

For the proof of Theorem M7, we need in particular to prove that u ˜∗ > u∗ . First, e ∗ , we have note that, since u ˜ + r˜ is constant along Σ n o e∗ = u Σ e + re = 1 + re(Se1 ) . (8.2.12) e ∗ , and in view of (8.2.12), (8.2.2), (8.2.6), we infer Since Se∗ ⊂ Σ   ◦ e∗ ) − u∗ + δext = u e∗ ) − u(S) u e ( S e ( S 2   ◦ = 1 + re(Se1 ) − re(Se∗ ) − −s(S) + cΣ e∗ . Next, note from s = r on Σ∗ ,

r e4 (r − s) = 2



2 κ− r



that we have sup |r − s| . e R

Together with (8.2.8), this yields   δext e u e(S∗ ) − u∗ + 2 .

0 ∆ext . 0 δext . r

(8.2.13)

1 + re(Se1 ) − cΣ e ∗ + 0 δext .

Since cΣ e ∗ in (8.2.6) is a constant, we have in particular e e e cΣ e ∗ = u(S1 ) + r(S1 ) − ψ(s(S1 )) and thus   δext e e1 ) − u(Se1 ) − r(Se1 ) + ψ(s(Se1 )) + 0 δext u e ( S ) − u + . + r e ( S 1 ∗ ∗ 2 e . 1 − u(Se1 ) + e r(S1 ) − r(Se1 ) + ψ(s(Se1 )) + 0 δext . In view of (8.2.9) and (8.2.8), we infer   e∗ ) − u∗ + δext u e ( S 2

. 1 − u(Se1 ) + 0 δext .

Also, since (recall in particular (3.1.5)) u = 1 on S1 ∩ SP,

0 eL 4 (u) = O

  0

r2

,

384

CHAPTER 8

and since the south pole of S1 coincides with the one of the corresponding sphere of CL0 ,1 , we infer sup e L ,1 ∩SP R∩C 0

|u − 1| . ∆ext

0 . 0 δext . r2

This yields   e∗ ) − u∗ + δext u e ( S 2

. 0 δext .

(8.2.14)

In particular, we deduce, for 0 small enough, u e(Se∗ ) > u∗

(8.2.15)

as desired. Step 11. We would like to check that the dominant condition (3.3.4) for r holds e ∗ , i.e., we need to prove that there exists a choice of constant d0 satisfying on Σ 1 ≤ d0 ≤ 1 such that 2 2

re(Se∗ )

=

− 0 3 (e u(Se∗ ))1+δdec .

To this end, note that we have in view of (8.2.8), (8.2.13) and (8.2.14) 2

− re(Se∗ ) − 0 3 (e u(Se∗ ))1+δdec  1+δdec ◦ δext − 23 = s(S) + O (0 δext ) − 0 u∗ + + O (0 δext ) 2 ◦ 1 + δdec − 23 −2 = s(S) − 0 3 (u∗ )1+δdec − 0 (u∗ )δdec δext 2   δext − 23 δdec +0 (u∗ ) δext O + 0 + O (0 δext ) . u∗

Together with (8.2.2), we infer 2

− re(Se∗ ) − 0 3 (e u(Se∗ ))1+δdec 3d0 r∗ 1 + δdec − 23 −2 = r∗ + δext − 0 3 (u∗ )1+δdec − 0 (u∗ )δdec δext 4u∗ 2   δext − 23 δdec +0 (u∗ ) δext O + 0 + O (0 δext ) u∗   3d0 r∗ 1 + δdec − 23 δext − 23 1+δdec 1+δdec = r∗ − 0 (u∗ ) + − 0 (u∗ ) 4 2 u∗   δext −2 +0 3 (u∗ )δdec δext O + 0 + O (0 δext ) . u∗

Since we have by the condition (3.3.4) of r on Σ∗ −2

dec r∗ = 0 3 u1+δ , ∗

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

385

we deduce 2

= =

− re(Se∗ ) − 0 3 (e u(Se∗ ))1+δdec     3d0 1 + δdec r∗ δext δext −2 − + 0 3 (u∗ )δdec δext O + 0 + O (0 δext ) 4 2 u∗ u∗    3r∗ δext 2 + 2δdec δext d0 − + O 0 + . 4u∗ 3 u∗

Thus, we may choose the contant d0 such that

1 2

≤ d0 ≤ 1 and

2

re(Se∗ )

− = 0 3 (e u(Se∗ ))1+δdec

as desired. e ∗ obtained so far: Step 12. We summarize the properties of Σ e ∗ is a spacelike hypersurface included in the spacetime region R. e • Σ e ∗ and its level sets are topological 2-spheres • The scalar function u e is defined on Σ e denoted by S. e ∗: • The following GCM conditions hold on Σ Z Z 2 Φ e/d? e/d? κ e/d? e/d? µ e e = e = 0, κ e = , η e e = ξeΦ = 0. 2 1 2 1 re e e S S e ∗: • In addition, the following GCM conditions hold on the sphere Se∗ of Σ Z Z e Φ= βe eeθ (e κ)eΦ = 0. e∗ S

e∗ S

• We have, for some constant cΣ e∗ , u e + re = cΣ e∗ ,

e ∗. along Σ

• The following normalization condition holds true at the south pole SP of every e sphere S: e a

SP

= −1 −

2m e re

where e a is such that we have νe = ee3 + e aee4 , e ∗ , normal to S, e and with νe the unique vectorfield tangent to the hypersurface Σ normalized by g(e ν , ee4 ) = −2. e ∗ , i.e., we have • The dominant condition (3.3.4) for r holds on Σ re(Se∗ )

−2

= 0 3 (e u(Se∗ ))1+δdec .

386

CHAPTER 8

e ∗ by fixing the value u • u ˜ is calibrated along Σ e = 1: e ∗ ∩ {e Se1 = Σ u = 1} is such that Se1 ∩ C(1,L0 ) ∩ SP 6= ∅,

(8.2.16)

e ∗ such that its south pole intersects the south i.e., Se1 is the unique sphere of Σ pole of one of the sphere of the outgoing null cone C(1,L0 ) of the initial data layer. e ∗ satisfies all the required properties for the future spacelike boundary of a Thus Σ GCM admissible spacetime, see item 3 of Definition 3.2. Furthermore, we have on e∗ Σ u e(Se∗ ) > u∗ ,

(8.2.17)

and (f, f , λ) satisfy in view of (8.2.9) and Corollary 9.53 sup kd≤ksmall +5 (f, f , log(λ))kL2 (S) e . 0 δext . e∗ Σ

e we find Together with the Sobolev embedding on the spheres S, sup re |d≤ksmall +3 (f, f , log(λ))| . 0 δext . e∗ Σ

Possibly reducing the size of δext > 0, we deduce 1

sup re u e 2 +δdec |d≤ksmall +3 (f, f , log(λ))| . 0 .

(8.2.18)

e∗ Σ

e ∗ . We Step 13. We now control the outgoing geodesic foliation initialized on Σ f the region covered by this outgoing geodesic foliation. Let denote by (ext) M (e4 , e3 , eθ ) of (ext) M be extended to the spacetime M(extend) , and satisfy, as discussed in Step 1 to Step 3, (Dec)

Nksmall +5 (M(extend) ) .

0 .

(8.2.19)

Let (f, f , λ) the transition functions from the null frame (e4 , e3 , eθ ) to the null f Since both frames are outgoing geodesic, we may apply frame (e e4 , ee3 , eeθ ) of (ext) M. Corollary 2.93 which yields for (f , f, log(λ)) the following transport equations: λ−1 e04 (rf )

= E10 (f, Γ),

λ−1 e04 (log(λ)) = E20 (f, Γ),   λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω = E30 (f, f , λ, Γ),

387

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

where E10 (f, Γ) E20 (f, Γ) E30 (f, f , λ, Γ)

r r = − κ ˇ f − ϑf + l.o.t., 2 2 1 2 1 = f ζ − f ω − ηf − f 2 κ + l.o.t., 2  4  r 2 2 = − κ ˇf + r κ ˇ− κ− e0θ (log(λ)) 2 r   r +r2 /d01 (f ) + λ−1 ϑ0 e0θ (log(λ)) − κ ˇ Ωf + rE3 (f, f , Γ) 2 −2r2 e0θ (E2 (f, Γ)) + rΩE1 (f, Γ),

and where E1 , E2 and E3 are given in Lemma 2.92. Integrating these transport e ∗ , using the control (8.2.18) of (f, f , λ) on Σ e ∗ , and together with equations from Σ the control (8.2.19) for the Ricci coefficients of the foliation of M(extend) , we obtain  1  sup re u e 2 +δdec + u e1+δdec   (ext) M f

r e≥2m0 (1+

δH 2

)



× |d≤ksmall +3 (f, log(λ))| + |d≤ksmall +2 f |



Then, for any rT in the interval     δH 3δH 2m0 1 + ≤ rT ≤ 2m0 1 + , 2 2

.

0 .

(8.2.20)

(8.2.21)

f T ] on re = rT using the outwe initialize the ingoing geodesic foliation of (int) M[r f as in item 4 of Definition 3.2. Using the control going geodesic foliation of (ext) M of (f, f , λ) induced on re = rT by (8.2.20), and using the analog of Corollary 2.93 in the e3 direction for ingoing foliations, we obtain similarly, for any rT in the interval (8.2.21),   sup u e1+δdec |d≤ksmall +2 (f , log(λ))| + |d≤ksmall +1 f | . 0 . (8.2.22) (int) M[r f

T

]

Let now, for any rT in the interval (8.2.21), M[rT ]

:=

(ext)

f r ≥ rT ) ∪ M(e

(int)

f T ]. M[r

Then, in view of (8.2.20), (8.2.22), and (8.2.19), and using the transformation formulas of Proposition 2.90, we deduce (Dec)

Nksmall (M[rT ]) .

0

which concludes the proof of Theorem M7.

8.3

PROOF OF THEOREM M8

So far, we have only improved our bootstrap assumptions on decay estimates. We ˇ now improve our bootstrap assumptions on energies and weighted energies for R

388

CHAPTER 8

ˇ relying on an iterative procedure which recovers derivatives one by one.3 and Γ Let Im0 ,δH the interval of R defined by      δH 3δH , 2m0 1 + . (8.3.1) Im0 ,δH := 2m0 1 + 2 2 Remark 8.2. Recall that the results of Theorems M0–M7 hold for any rT ∈ Im0 ,δH , see Remark 3.30. More precisely, • they hold on (ext) M(r ≥ 2m0 (1 + δ2H )), and hence on (ext) M(r ≥ rT ) for any rT ∈ Im0 ,δH , • they hold on (int) M[rT ] for any rT ∈ Im0 ,δH , where (int) M[rT ] is initialized on T = {r = rT } using (ext) M(r ≥ rT ) as in section 3.1.2. It is at this stage that we need to make a specific choice of rT in the context of a Lebesgue point argument. More precisely, we choose rT such that we have Z Z ≤klarge ˇ 2 ˇ 2. |d R| = inf |d≤klarge R| (8.3.2) r0 ∈Im0 ,δH

{r=rT }

{r=r0 }

Remark 8.3. In case the above infimum is achieved for several values of r, we choose rT to be the largest of such values, so that rT is uniquely defined. Note also that the infimum could a priori be infinite, and will only be shown to be finite — and more precisely O(0 ) — at the end of the proof of Theorem M8, see section 8.3.4. This could be made rigorous in the context of a continuity argument. In view of the definition of rT , and since T = {r = rT }, we have ! Z Z Z 1 ˇ2 ≤ ˇ 2 dr0 |d≤klarge R| |d≤klarge R| 2m0 δH Im0 ,δ T {r=r0 } H

and hence4 Z T

≤klarge

|d

ˇ2 R|

Z . (ext) M

 r∈Im0 ,δH

ˇ 2.  |d≤klarge R|

(8.3.3)

From now on, we may thus assume that the spacetime M satisfies 3 See also [33] for a related strategy to recover higher order derivatives from the control of lower order ones. 1 4 We use the coarea formula, dM = √ d{r = r0 }dr0 , and the fact that, for r ∈ Im0 ,δH , g(Dr,Dr)

2 )≥ g(Dr, Dr) = −e3 (r)e4 (r) = Υ + O() ≥ δ2H + O( + δH see the convention for . made at the end of section 3.3.1.

δH . 4

−1 Note that . here depends on δH ,

389

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

• the conclusions of Theorem M0, i.e., ( "   7 max sup r 2 +δB |dk (ext) α| + |dk (ext) β| 0≤k≤klarge

C1

# +r

9 2 +δB

|d

k−1

e3 (

(ext)

α)|

"

  k (ext) 2m0 + sup r d ρ + 3 + r2 |dk (ext) β| r C1 #) 3

+r|dk (ext) α|

.

0

(8.3.4)

and " max

k (int)

sup |d

0≤k≤klarge C 1

k (int)

α| + |d

  2m0 β| + dk (int) ρ + 3 r # +|dk (int) β| + |dk (int) α|

(8.3.5)

. 0 ,

• the conclusions of Theorem M7, i.e., (Dec)

Nksmall

. 0 ,

(8.3.6) (Dec)

see section 3.2.3 for the definition of the combined norm on decay Nk • the estimate Z Z ˇ2 . ˇ 2.   |d≤klarge R| |d≤klarge R| (ext) M

T

,

(8.3.7)

r∈Im0 ,δH

The goal of this section is to prove Theorem M8, i.e., to prove that the following bound holds on M for the weighted energies: (En)

Nklarge . 0 , (En)

see section 3.2.3 for the definition of the combined norm on weighted energies Nk 8.3.1

.

Main norms

We recall below our norms for measuring weighted energies for curvature components and Ricci coefficients, see sections 3.2.1 and 3.2.2. Let r0 ≥ 4m0 . Then, we

390

CHAPTER 8

have for 

(ext)

(ext)

M

0 ˇ R≥r 0 [R]

Z

2

=



sup 0≤u≤u∗

Z



+

r4+δB α2 + r4 β 2



Cu (r≥r0 )

r4+δB (α2 + β 2 ) + r4 (ˇ ρ)2 + r2 β 2 + α2

ZΣ∗



+ (ext) M(r≥r



r3+δB (α2 + β 2 ) + r3−δB (ˇ ρ)2 + r1−δB β 2

0)

 +r−1−δB α2 ,



(ext)

0 ˇ R≤r 0 [R]

(ext)



(ext)



Z

2

(ext) M(r≤r

ˇ R0 [R]

1−

=

(ext)

=

0)

0 ˇ R≥4m [R] + 0

(ext)

2  2 Z (ext) ≤k ˇ ˇ Rk [R] = R0 [d R] +

 (ext) M(r≤4m

0)

3m r

2

ˇ 2, |R|

0 ˇ R≤4m [R], 0

ˇ2 |d≤k−1 NR|

 ˇ2 , +|d≤k−1 R|

for k ≥ 1,

and 

(ext)

0 G≥r k

  2 ˇ Γ

"

Z

  r2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2 + (d≤k κ ˇ )2

= Σ∗

# +

≤k

(d

≤k

2

ϑ) + (d

+

sup λ≥4m0

+ λ

2−δB

(ext)

  2 0 ˇ G≤r Γ k (ext)

(d

Z

Also, we have for

(ext) M(≤4m

(int)



ω ˇ ) + (d

,

0)

  0 ˇ G≤4m Γ + k

(ext)

  0 ˇ G≥4m Γ . k

M

(int)

2

ξ)

  λ2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2

≤k
2 ˇ , d Γ

(ext)

≤k

κ ˇ )2 + (d≤k ϑ)2 + (d≤k η)2 + (d≤k ω ˇ )2 #!

=

  ˇ = Gk Γ

2

{r=λ} ≤k

+ λ−δB (d≤k ξ)2 

η) + (d "

Z

≤k

2

2 ˇ Rk [R]

Z = (int) M

ˇ 2, |d≤k R|

391

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and 

(int)

2 ˇ Gk [Γ]

Z = (int) M

ˇ 2. |d≤k Γ|

Finally, we recall the following Morawetz type norms, see section 5.1.4. For δ > 0, we have Bδ [ψ](τ1 , τ2 ) Z

 2   3m 1 2 2 = |Rψ| + r |ψ| + 1 − |∇ / ψ| + 2 |T ψ| r r (trap) M(τ ,τ ) 1 2 Z
 + rδ−3 |dψ|2 + |ψ|2 (trap)

−2

2

2

M(τ1 ,τ2 )

where the scalar function τ andthe spacetime region (trap) M have been introduced in section 5.1.1, and where (trap) M denotes the complement of (trap) M. Also, we have   Z 1 1 2 2 2 2 2 −2 2 Eδ [ψ](τ ) = (NΣ , e3 ) |e4 ψ| + (NΣ , e4 ) |e3 ψ| + |∇ / ψ| + r |ψ| 2 2 Σ(τ ) Z   + rδ |e4 ψ|2 + r−2 |ψ|2 . Σ≥4m0 (τ )

Here Σ(τ ) denotes the level set of τ , see section 5.1.1, NΣ denotes a choice for the normal to Σ, and recall that we have ( NΣ = e 3 on (int) Σ, NΣ = NΣ = e 4 on (ext) Σ, with

(int)

Σ and

(ext)

Σ defined in section 5.1.1, and

(NΣ , e3 ) ≤ −1 and (NΣ , e4 ) ≤ −1

on

(trap)

Σ.

Moreover, we have Z Fδ [ψ](τ1 , τ2 )



= A(τ1 ,τ2 )

−1 δH |e4 Ψ|2 + δH |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2

Z + Σ∗ (τ1 ,τ2 )




 |e3 Ψ|2 + rδ |e4 ψ|2 + |∇ / ψ|2 + r−2 |ψ|2

with A(τ1 , τ2 ) = A ∩ M(τ1 , τ2 ) and Σ∗ (τ1 , τ2 ) = Σ∗ ∩ M(τ1 , τ2 ). 8.3.2

Control of the global frame

Some quantities will be controlled based on the wave equation they satisfy, and will thus need to be defined w.r.t. a global frame, i.e., a smooth frame on M. To this end, we will rely on the global frame of section 3.5.2. We recall below the main properties of that global frame. From Definition 3.22, the region where the frame of (int) M and a conformal

392

CHAPTER 8

renormalization of the frame of (ext) M are matched is given by     3 (ext) Match := M ∩ (int) r ≤ 2m0 1 + δH 2     1 ∪ (int) M ∩ (int) r ≥ 2m0 1 + δH , 2 where (int) r denotes the area radius of the ingoing geodesic foliation of (int) M and its extension to (ext) M. The following proposition concerning the global frame is an immediate consequence of Proposition 3.23 and the decay estimates (8.3.6). Proposition 8.4. Assume (8.3.6). Then, there exists a global null frame defined on (int) M ∪ (ext) M and denoted by ((glo) e4 , (glo) e3 , (glo) eθ ) such that a) In

(ext)

M \ Match, we have

((glo) e4 , (glo) e3 , (glo) eθ ) = b) In

(int)



(ext)

 Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ .

M \ Match, we have ((glo) e4 , (glo) e3 , (glo) eθ ) =



(int)

 e4 , (int) e3 , (int) eθ .

c) In the matching region, we have ˇ (glo) R) ˇ . 0 , u1+δdec dk ((glo) Γ, 0≤k≤ksmall −2 Match∩ (int) M ˇ (glo) R) ˇ . 0 , max sup u1+δdec dk ((glo) Γ, max

sup

0≤k≤ksmall −2 Match∩ (ext) M

where

(glo)

ˇ and R

(glo) ˇ

Γ are given by   2m (glo) ˇ R = α, β, ρ + 3 , β, α , r   m 2Υ 2 (glo) ˇ Γ = ξ, ω + 2 , κ − , ϑ, ζ, η, η, κ + , ϑ, ω, ξ . r r r

d) Furthermore, we may also choose the global frame such that, in addition, one of the following two possibilities hold: i. We have on all

(ext)

M

((glo) e4 , (glo) e3 , (glo) eθ ) = ii. We have on all

(int)



(ext)

 Υ (ext) e4 , (ext) Υ−1(ext) e3 , (ext) eθ .

M

((glo) e4 , (glo) e3 , (glo) eθ ) =



(int)

 e4 , (int) e3 , (int) eθ .

393

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

8.3.3

Iterative procedure

Recall our norms for measuring energies for curvature components and Ricci coˇ (ext) Rk [R] ˇ and (int) Gk [Γ], ˇ efficients which are given respectively by (int) Rk [R], (ext) ˇ Gk [Γ], see sections 3.2.1 and 3.2.2. Recall also our combined weighted energy norm (En)

=

Nk

(ext)

ˇ + Rk [R]

(ext)

ˇ + Gk [Γ]

(int)

ˇ + Rk [R]

(int)

ˇ Gk [Γ].

We also introduce the following norm controlling on the matching region the Ricci coefficients and curvature components of the global frame of Proposition 8.4: (match) Nk

Z := Match

1 ≤k (glo) ˇ (glo) ˇ 2 2 Γ, R) . d (

(8.3.8)

To initiate the iterative procedure, we rely on the following lemma. Lemma 8.5. We have (En)

(match)

Nksmall + Nksmall −2

0 .

.

(8.3.9)

Proof. The estimate (8.3.6) and Proposition 8.4 imply in particular (ext)

≥4m0 ˇ R\ ksmall [R] +

(ext)

0 ˇ R≤4m ksmall [R] +

ˇ + + (int) Rksmall [R]

(int)

(ext)

ˇ Gksmall [Γ]

ˇ + N (match) Gksmall [Γ] ksmall −2

. 0

(8.3.10)

where the first term of the right-hand side is defined by Z  2 ≥4m0 ˇ (ext) \ Rk [R] := sup r4 |d≤k β|2 0≤u≤u∗ Cu (r≥4m0 ) Z   + r4 |d≤k ρˇ|2 + r2 |d≤k β|2 + |d≤k α|2 ZΣ∗  + r3−δB |d≤k ρˇ|2 + r1−δB |d≤k β|2 (ext) M(r≥4m

+r

−1−δB

0)

≤k

|d

 α|2 . (En)

In view of the definition of the combined weighted energy norm Nk " Z (En)

(match)

Nksmall + Nksmall −2

. 0 +

sup 1≤u≤u∗

Z + Σ∗

Cu (r≥4m0 )

, we infer

r4+δB |d≤ksmall α|2

r4+δB (|d≤ksmall α|2 + |d≤ksmall β|2 )

(8.3.11) # 12

Z +

r (ext) M(r≥4m ) 0

3+δB

≤ksmall

(|d

2

≤ksmall

α| + |d

2

β| )

.

Note that the terms on the RHS of the above estimate can not be estimated directly by (8.3.6) since δdec < δB .

394

CHAPTER 8

Next we claim the estimate Z Z 4+δB ≤ksmall 2 sup r |d α| + r4+δB (|d≤ksmall α|2 + |d≤ksmall β|2 ) 1≤u≤u∗ Cu (r≥4m0 ) Σ∗ Z + r3+δB (|d≤ksmall α|2 + |d≤ksmall β|2 ) (ext) M(r≥4m ) 0

.



(ext)

0 ˇ R≤4m ksmall [R]

2

+



(ext)

2 (En) ˇ Gksmall [Γ] + 20 + 20 (Nksmall )2 .

(8.3.12)

The proof of (8.3.12) relies on rp -weighted estimates for the Bianchi pair (α, β) and is postponed to section 8.7.3. Then (8.3.9) follows immediately from (8.3.10), (8.3.11) and (8.3.12) for 0 > 0 small enough. Next, for J such that ksmall − 2 ≤ J ≤ klarge − 1, consider the iteration assumption (En)

NJ

(match)

+ NJ

. B [J],

(8.3.13)

where B [J] :=

J X

`(J)

j=ksmall −2

 Z  B :=

(0 )`(j) B 1−`(j) + 0

B,

`(j) := 2ksmall −2−j , (8.3.14)

 12 (ext) M

 r∈Im0 ,δH

ˇ 2 .  |d≤klarge R|

Lemma 8.6. The following estimate holds true for B [J] as defined above: 1

1

B [J] + B 2 (B [J]) 2 + 0 B . B [J + 1].

(8.3.15)

Proof. We clearly have B [J] + 0 B . B [J + 1].

(8.3.16)

Also, we have, using `(j) = 2`(j + 1), BB [J] .

J X

`(J)

(0 )`(j) B 2−`(j) + 0

j=ksmall −2 J+1 X

.

2`(J+1)

(0 )2`(j) B 2−2`(j) + 0

j=ksmall −1

 . 

J+1 X

B2 2

(0 )`(j) B 1−`(j) +

j=ksmall −2

=

B2

`(J+1)  0 B

(B [J + 1])2

which concludes the proof of the lemma. In view of (8.3.9), (8.3.13) holds for J = ksmall − 2. The propositions below will

395

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

allow us to prove Theorem M8 in the next section. Proposition 8.7. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Consider the global frame constructed in Proposition 8.4. In that frame, let ρ˜ := r2 ρ + 2mr−1 .

(8.3.17)

Then, under the iteration assumption (8.3.13), we have  2 (En) (match) sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ ) + FδJ [˜ ρ](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 .

τ ∈[1,τ∗ ]

Proposition 8.8. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Consider the global frame constructed in Proposition 8.4. In that frame, under the iteration assumption (8.3.13), we have sup EδJ [α + Υ2 α](τ ) + BδJ [α + Υ2 α](1, τ∗ ) + FδJ [α + Υ2 α](1, τ∗ )

τ ∈[1,τ∗ ]

.

 2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 .

Proposition 8.9. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Consider the global frame constructed in Proposition 8.4. In that frame, under the iteration assumption (8.3.13), we have i  2 h (En) (match) J B−2 ρˇ, α, α, β, β (1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 . Proposition 8.10. Let J such that ksmall −2 ≤ J ≤ klarge −1. Under the iteration assumption (8.3.13), we have for r0 ≥ 4m0 (int)

ˇ + RJ+1 [R]

(ext)

ˇ ≤ RJ+1 [R]

0 ˇ R≥r J+1 [R]     (En) (match) +O r010 B [J] + 0 NJ+1 + NJ+1

(ext)

and (ext)

0 ˇ R≥r J+1 [R] .

   (En) (match) 10 0 ˇ r0−δB (ext) G≥r [ Γ] + r  [J] +  N + N . B 0 0 J+1 J+1 J+1

Proposition 8.11. Let J such that ksmall −2 ≤ J ≤ klarge −1. Under the iteration assumption (8.3.13), we have   (ext) ˇ + (int) RJ+1 [R] ˇ + (ext) RJ+1 [R] ˇ . B [J] + 0 N(En) + N (match) . GJ+1 [Γ] J+1 J+1 Proposition 8.12. Let J such that ksmall −2 ≤ J ≤ klarge −1. Under the iteration assumption (8.3.13), we have (int)

ˇ GJ+1 [Γ]

 12   Z (En) (match) ˇ 2 . B [J] + 0 NJ+1 + NJ+1 + |dJ+1 ((ext) R)| . T

Proposition 8.13. Let J such that ksmall −2 ≤ J ≤ klarge −1. Under the iteration

396

CHAPTER 8

assumption (8.3.13), we have (match)

Z

(En)

NJ+1

. NJ+1 +

T

ˇ 2 |dJ+1 ((ext) R)|

 12 .

The proofs of Propositions 8.7, 8.8, 8.9, 8.10, 8.11, 8.12 and 8.13 are postponed respectively to sections 8.4, 8.5, 8.6, 8.7, 8.8, 8.9 and 8.10. 8.3.4

End of the proof of Theorem M8

To prove Theorem M8, we rely on Propositions 8.11, 8.12 and 8.13. Note that among these propositions, only the last two involve the dangerous boundary term
1 R ˇ 2 2 . We proceed as follows. |dJ+1 ((ext) R)| T Step 1. As mentioned earlier, the estimate (8.3.9) trivially implies the iteration assumption (8.3.13) with J = ksmall − 2. We assume that the iteration assumption (8.3.13) holds for any fixed J such that ksmall − 2 ≤ J ≤ klarge − 2. In view of Proposition 8.12, we have (int)

 21   Z J+1 (ext) ˇ 2 ˇ . B [J] + 0 N(En) + N (match) + GJ+1 [Γ] |d ( R)| .(8.3.18) J+1 J+1 T

We need to deal with the last term in the RHS of (8.3.18). Relying on a trace theorem in the spacetime region (ext) M(r ∈ Im0 ,δH ), as well as the fact that J + 2 ≤ klarge , we obtain Z T

ˇ 2 |dJ+1 ((ext) R)|

 12 .

 Z 

 41 (ext) M

 r∈Im0 ,δH

ˇ 2  ( (ext) RJ+1 [R]) ˇ 12  |dklarge R|

ˇ + (ext) RJ+1 [R].

(8.3.19)

Proposition 8.11, (8.3.18) and (8.3.19) yield, for 0 > 0 small enough so that we can absorb some of the terms to the left, (En) NJ+1

.

 Z  B [J] +

 14 (ext) M

 r∈Im0 ,δH

ˇ 2  |dklarge R|

   12 (En) (match) (match) × B [J] + 0 NJ+1 + NJ+1 + 0 NJ+1 ,

397

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and using also Proposition 8.13, (match) NJ+1

.

.

(En) NJ+1

Z

J+1 (ext)

+ T

 Z B [J] + 

|d

(

ˇ 2 R)|

 12  14

(ext) M

 r∈Im0 ,δH

ˇ 2  |dklarge R|

   12 (En) (match) (match) × B [J] + 0 NJ+1 + NJ+1 + 0 NJ+1 . For 0 > 0 small enough, we infer, by absorbing the appropriate terms to the left, (En)

.

(match)

NJ+1 + NJ+1  Z  B [J] +

 14

(ext) M

.

 r∈Im0 ,δH

ˇ 2  |dklarge R|

   12 (En) (match) × B [J] + 0 NJ+1 + NJ+1   14 Z  1 ˇ 2  B [J] 2   |dklarge R| B [J] +  (ext) M

 Z +

r∈Im0 ,δH

 14 (ext) M

 r∈Im0 ,δH

ˇ 2  |dklarge R|

   12 (En) (match) 0 NJ+1 + NJ+1

and hence (En)

(match)

NJ+1 + NJ+1

.

 Z B [J] +   Z  +0

 14 (ext) M

 r∈Im0 ,δH

ˇ 2  |dklarge R|



B [J]

 12

 12 (ext) M

 r∈Im0 ,δH

ˇ 2 .  |dklarge R|

In view of Lemma 8.6, we deduce (En)

(match)

NJ+1 + NJ+1

. B [J + 1]

which is (8.3.13) for J + 1 derivatives. We deduce that the estimate (8.3.13) holds for all J ≤ klarge − 1, and hence (En)

(match)

Nklarge −1 + Nklarge −1

. B [klarge − 1].

(8.3.20)

398

CHAPTER 8

Step 2. Next, Proposition 8.11 implies in view of (8.3.20) (ext)

ˇ + Gklarge [Γ]

(int)

ˇ + Rklarge [R]

(ext)

ˇ Rklarge [R]

. B [klarge − 1] (8.3.21)   (En) (match) +0 Nklarge + Nklarge .

In particular, we have  Z 

 12 (ext) M

 r∈Im0 ,δH

ˇ 2  |d≤klarge R|

≤

(ext)

ˇ Rklarge [R]

  (En) (match) . B [klarge − 1] + 0 Nklarge + Nklarge . In view of the definition of B [klarge − 1], we infer for 0 > 0 small enough  Z 

 12 (ext) M

 r∈Im0 ,δH

  ˇ 2  . 0 + 0 N(En) + N (match)  |d≤klarge R| klarge klarge

and hence   (En) (match) B [klarge − 1] . 0 + 0 Nklarge + Nklarge which yields, together with (8.3.21), ˇ + (int) Rk ˇ + Gklarge [Γ] [R] large   (En) (match) 0 + 0 Nklarge + Nklarge . (ext)

.

(ext)

ˇ Rklarge [R] (8.3.22)

Step 3. Next, Proposition 8.12 implies in view of (8.3.22) (int)

ˇ Gklarge [Γ]

. 0 +



(En) 0 Nklarge

+

(match) Nklarge



Z + T

klarge (ext)

|d

(

ˇ 2 R)|

 12

and hence, for 0 > 0 small enough, using again (8.3.22), (En)

Nklarge

(match)

. 0 + 0 Nklarge

Z + T

ˇ 2 |dklarge ((ext) R)|

 12 .

Together with Proposition 8.13, we infer for 0 > 0 small enough (En) Nklarge

+

(match) Nklarge

Z . 0 + T

J+1 (ext)

|d

(

ˇ 2 R)|

 12 .

Step 4. It remains to estimate the last term of the RHS of the previous inequality.

399

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Now, in view of (8.3.7) and (8.3.22), we have Z T

ˇ 2 |dklarge ((ext) R)|

 Z 

 12 .

 12 

(ext) M

r∈Im0 ,δH

ˇ 2  |d≤klarge R|

ˇ Rklarge [R]

.

(ext)

.

0 + 0 Nklarge

(En)

so that we finally obtain, for 0 > 0 small enough, (En)

.

Nklarge

0 .

This concludes the proof of Theorem M8.

8.4

PROOF OF PROPOSITION 8.7

8.4.1

A wave equation for ρ˜

Proposition 8.14. The following wave equations hold true. 1. The curvature component ρ verifies the identity g ρ = κe4 ρ + κe3 ρ +

 3 κ κ + 2ρ ρ + Err[g ρ], 2

where Err[g ρ]

  3 1 := ρ − ϑ ϑ + 2(ξ ξ + η η) 2 2    3 1 + κ − 2ω ϑ α − ζ β − 2(η β + ξ β) 2 2 1 − ϑ /d?2 β + (ζ − η)e3 β − ηe3 (Φ)β − ξ(e4 β + e4 (Φ)β) − ββ 2   1 −e3 − ϑ α + ζ β + 2(η β + ξ β) 2   − /d?1 (κ)β + 2 /d?1 (ω)β + 3η /d?1 (ρ) − /d1 − ϑβ + ξα − 2ηeθ ρ.

2. The small curvature quantity ρ˜ := r

2



2m ρ+ 3 r



verifies the wave equation 8m g (˜ ρ) + 3 ρ˜ = r


  g (r) − 2r − 2m 3m 4Υ r2 −6m − κκ + 2 r2 r r 3m − (Aκ + Aκ) + Err[g ρ˜], r

400

CHAPTER 8

where ! 6m 3 2 3 4 e3 (r) 4 e4 (r) Err[g ρ˜] := − AA + 2 ρ˜ + A + A ρ˜ r r 2 3 r 3 r    8m  3 8m 2 + κκ − 3 + 2 g (r2 ) + 3 ρ˜ 2 r 3r r 2 2 −Ae3 (˜ ρ) − Ae4 (˜ ρ) + Ae3 (m) + Ae4 (m) r   r 1 2 a +4D (m)Da + g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ], r r and where we recall that A=

2 e4 (r) − κ, r

A=

2 e3 (r) − κ. r

Proof. See section B.1. 8.4.2

Control of g (r)

Lemma 8.15. Let r the function on M associated to the global frame constructed in Proposition 8.4, see Definition 4.14. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Under the iteration assumption (8.3.13), we have 2 2 2m d g (r) − − 2 r r (int) M∪ (ext) M(r≤4m ) 0    2 Z 2 2m + sup dJ g (r) − − 2 r r r0 ≥4m0 {r=r0 }  2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 

Z

J





and 2 (ext) m d e4 g ( r) − (ext) − (ext) 2 r ( r) (trap) M  2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 . Z

.



J





(ext)

2

Proof. Recall that, according to Definition 4.14, r is defined on as follows: • on

• on

(ext)

(int)

M \ Match, we have (glo)

r=

(ext)

r,

(glo)

r=

(int)

r,

M \ Match, we have

2

(ext)

M∪

(int)

M

401

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

• on the matching region, we have (glo)

r

(1 − ψm0 ,δH ( (int) r)) (int) r + ψm0 ,δH ( (int) r) (ext) r,

=

where the matching region of Proposition 8.4 is given by     3 (ext) Match := M ∩ (int) r ≤ 2m0 1 + δH 2     1 ∪ (int) M ∩ (int) r ≥ 2m0 1 + δH , 2 and where ψm0 ,δH is given by r − 2m0 1 + 12 δH 2m0 δH

ψm0 ,δH (r) = ψ


!

    1 3 on 2m0 1 + δH ≤ r ≤ 2m0 1 + δH 2 2

with ψ : R → R a smooth cut-off function such that 0 ≤ ψ ≤ 1, ψ = 0 on (−∞, 0] and ψ = 1 on [1, +∞). We have on (ext) M   1 (ext) (ext) (ext) g ( r) = −e3 e4 ( r) + 4 /( r) + 2ω − κ e4 ( (ext) r) 2 1 − κe3 ( (ext) r) + 2ηeθ ( (ext) r). 2 Here, (e4 , e3 , eθ ) denotes the frame of puted w.r.t. this frame, so we have (ext)

e4 ( (ext) r) =

r

2

(ext)

M and the Ricci coefficients are com-

(ext)

e3 ( (ext) r) =

κ,

2

r

(κ + A),

eθ ( (ext) r) = 0

and hence g ( (ext) r)

 (ext)

= =

1 (ext) r κ− κ (κ + A) 2 2 2 2   (ext) (ext) (ext) r e3 ( (ext) r) 1 r r − e3 (κ) − κ + 2ω − κ κ− κκ 2 2 2 2 4 −e3

(ext)

−

(ext)

=

−

r

2 (ext)

−

r

4

4

r

r

   1 κ + 2ω − κ 2

(ext)

r

κA   1 (ext) r 1 e3 (κ) − κ (κ + A) + 2ω − κ 2 2 2 (ext)

κκ −

4

r

κA.

(ext)

2

r

κ

402

CHAPTER 8

Now, we have e3 (κ)

= e3 (κ) + Err[e3 κ] 1 1 = − κκ + 2ωκ + 2ρ + 2 /d1 η − ϑϑ + 2η 2 + Err[e3 κ] 2 2 1 1 = − κκ + 2ωκ + 2ρ − ϑϑ + 2η 2 + Err[e3 κ] 2 2

and hence (ext)

g (

(ext)

  1 1 2 r) = − − κκ + 2ωκ + 2ρ − ϑϑ + 2η + Err[e3 κ] 2 2 2   (ext) (ext) (ext) 1 r 1 r r − κ (κ + A) + 2ω − κ κ− κκ − 2 2 2 2 4 r

(ext)

4

r

κA.

Together with (8.3.4) and the iteration assumption (8.3.13), we easily infer5 2 2 (ext) m d g ( r) − (ext) − (ext) 2 r ( r) (ext) M(r≤4m ) 0    2 Z (ext) 2 2 m J (ext) + sup d g ( r) − (ext) − (ext) 2 r ( r) r0 ≥4m0 {r=r0 }  2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 . (8.4.1) 

Z

J



(ext)



2

Also, using again (8.3.4) and the iteration assumption (8.3.13), we have 2 (ext) m d e4 g ( r) − (ext) − (ext) 2 r ( r) (trap) M  2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 , Z

.



J



(ext)



2

2

(8.4.2)

where we have used the null structure equations for e4 (κ), e4 (κ), e4 (ω), e4 (ϑ), e4 (ϑ), e4 (η), the equations for e4 (Ω),e4 (ς), e4 (r), and the Bianchi identity for e4 (ρ). Remark 8.16. Note that we have used in the last estimate the following observations to avoid a potential loss of one derivative:   1 e4 (κ) = −2 /d1 ζ + · · · = 2 ρ + µ − ϑϑ + · · · , 4 e4 (ρ) = /d1 β + · · · = · · · , e4 (Err[e3 κ]) = 2e4 (ς −1 ςˇ/d1 η) + · · · = 2ς −1 ςˇ/d1 e4 η + · · · = −2ς −1 eθ (ς)e4 η + · · · . Note also that there is no term involving dJ ρ (without average) as such a term appears only in the null structure equations for e4 (κ), as well as e4 (ω), and vanishes 5 Recall

in particular that ρ is under control in view of Lemma 3.15.

403

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

due to the cancellation   1 e4 2ω − κ = 2

1 2e4 (ω) − e4 (κ) 2 1 2ρ + · · · − (−2 /d1 ζ + 2ρ) + · · · 2 2µ + · · · .

= =

This is important as such a term would otherwise violate (8.4.2) at r = 3m. Remark 8.17. Recall that the global null frame constructed in Proposition 8.4 • coincides with the null frame of (int) M in the region (int) M \ Match, • coincides with a conformal renormalization of the null frame of (ext) M in the region (ext) M \ Match. Thus, J + 1 derivatives of its Ricci coefficients and curvature components are controlled (match)

• by NJ+1

in Match,

(En)

• by NJ+1 in M \ Match, (En)

(match)

and hence by NJ+1 + NJ+1 (En) NJ+1

(match) NJ+1

+ (8.4.1), (8.4.2).

on the right-hand side of numerous estimates, see for example

Arguing similarly for

(int)



Z

r, we obtain the following analog of (8.4.1):

dJ



(int) M

.

on M. This explains the occurrence of the term



g ( (int) r) − (En)



2 (int) r

(match)

(B [J])2 + 20 NJ+1 + NJ+1

2

−

2 (int) m ( (int) r)2

2

.

(8.4.3)

Then, since • on

(ext)

M \ Match, we have g (r) = g ( (ext) r),

• on

(int)

m=

(ext)

m=

(int)

m,

M \ Match, we have g (r) = g ( (int) r),

m,

we immediately infer from (8.4.1), (8.4.2) and (8.4.3) Z 

 (int) M∪ (ext) M(r≤4m

 0)

\Match

dJ



 g (r) −

  2 2 2m + sup dJ g (r) − − 2 r r r0 ≥4m0 {r=r0 }  2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 Z



2 2m − 2 r r

2

404

CHAPTER 8

and Z



(trap) M

  2 2 2m dJ e4 g (r) − − 2 r r

 2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1

which are the desired estimates outside of the matching region. Note that we have used the fact that (trap) M ∩ Match = ∅. It remains to derive the desired estimates in the matching region. To this end, we need to estimate (ext) r − (int) r and (int) m − (ext) m in the matching region. Step 7 or the proof of Lemma 4.16 in section 4.6.2 yields6 Z   2 (En) (match) 2 dJ+1 (ext) r − (int) r, (ext) m − (int) m . (NJ )2 + (NJ ) . (int) M

We infer, in view of the iteration assumption (8.3.13), Z   2 dJ+1 (ext) r − (int) r, (ext) m − (int) m . (B [J])2 .

(8.4.4)

(int) M

Then, since we have on the matching region r

=

m

=

g (r)

=

(1 − ψm0 ,δH ( (int) r)) (int) r + ψm0 ,δH ( (int) r) (ext) r,

(1 − ψm0 ,δH ( (int) r)) (int) m + ψm0 ,δH ( (int) r) (ext) m,

(1 − ψm0 ,δH ( (int) r))g ( (int) r) + ψm0 ,δH ( (int) r)g ( (ext) r) 0 +2ψm ( (int) r)Dα ( (int) r)Dα ( (ext) r − 0 ,δH

+(

(ext)

r−

(int)

(int)

r)

r)g (ψm0 ,δH ),

we deduce there  g (r) − =

2 2m − 2 r r



   2 2 (int) m (1 − ψm0 ,δH ( (int) r)) g ( (int) r) − (int) − (int) 2 r ( r)    (ext) 2 2 m +ψm0 ,δH ( (int) r) g ( (ext) r) − (ext) − (ext) 2 r ( r)   2 2 2 (int) m 2m (int) +(1 − ψm0 ,δH ( r)) (int) − − (int) 2 + 2 r r r ( r)   (ext) 2 2 2 m 2m +ψm0 ,δH ( (int) r) (ext) − − (ext) 2 + 2 r r r ( r) 0 +2ψm ( (int) r)Dα ( (int) r)Dα ( (ext) r − 0 ,δH

+(

(ext)

r−

(int)

(int)

r)

r)g (ψm0 ,δH )

6 The proof of Lemma 4.16 in section 4.6.2 is done in the particular case J = k large − 1 but extends immediately to the case ksmall − 2 ≤ J ≤ klarge − 1.

405

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and thus, in view of (8.4.1), (8.4.3) and (8.4.4), we have on the matching region 

Z

J

d Match



 g (r) −

2 2m − 2 r r

2

 2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1

as desired. This concludes the proof of the lemma. Corollary 8.18. Let N0 the RHS of the wave equation for ρ˜ provided by Proposition 8.14, i.e.,
  g (r) − 2r − 2m 3m 4Υ 3m r2 N0 = −6m − κκ + 2 − (Aκ + Aκ) + Err[g ρ˜]. 2 r r r r Then, N0 − Err[g ρ˜] satisfies Z (int) M∪ (ext) M(r≤4m ) 0

Z + sup r0 ≥4m0

{r=r0 }


2 dJ (N0 − Err[g ρ˜])


2 dJ (N0 − Err[g ρ˜])

 2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 and   dJ e4 N0 − Err[g ρ˜] = where aJ satisfies Z (trap) M

dJ e4 aJ


2

−

12mκ J d ρ + aJ on r

(trap)

M

 2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 .

Proof. The first estimate is an immediate consequence of Lemma 8.15, (8.3.4) and the iteration assumption (8.3.13). Concerning the second estimate, note that the term dJ ρ is due to the null structure equations for e4 (κ), i.e., e4 (κ)

= −2 /d1 ζ + 2ρ + · · ·

=

4ρ + · · · .

Then, the estimate for aJ follows from Lemma 8.15, (8.3.4) and the iteration assumption (8.3.13). 8.4.3

End of the proof of Proposition 8.7

In view of Proposition 8.14, ρ˜ satisfies (0 + V0 )˜ ρ = N0 ,

V0 =

8m , r3

406

CHAPTER 8

where N0

g (r) − 2r − := −6m r2

2m r2


−

3m r

 κκ +

4Υ r2

 −

3m (Aκ + Aκ) + Err[g ρ˜]. r

We may thus apply the estimate (10.5.2) of Theorem 10.67 with φ = ρ˜ and s = J to obtain for any ksmall ≤ J ≤ klarge − 1 sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ ) + FδJ [˜ ρ](1, τ∗ )

τ ∈[1,τ∗ ]

. EδJ [˜ ρ](1) + sup EδJ−1 [˜ ρ](τ ) + BδJ−1 [˜ ρ](1, τ∗ ) + FδJ−1 [˜ ρ](1, τ∗ ) τ ∈[1,τ∗ ]

 2 Z 1 (d≤J ρ˜)2 ≤ksmall 2 +δdec +DJ [Γ] sup rutrap |d ρ˜| + r3 M Σ(τ∗ ) Z Z + r1+δ |d≤J N0 |2 + T (dJ ρ˜)dJ N0 , (trap) M

M

where DJ [Γ] is defined by Z DJ [Γ] :=

ˇ 2 (d≤J Γ)

(int) M∪ (ext) M(r≤4m

0)

Z + sup

≤J

r0

r0 ≥4m0

{r=r0 }

|d

2

Γg | +

r0−1

!

Z

≤J

{r=r0 }

|d

Γb |

2

.

Next we use the iteration assumption (8.3.13) which yields in particular DJ [Γ] . (B [J])2 . Also, we have   2m ρ˜ = r2 ρ − 3 + r2 ρˇ r and hence, using again the iteration assumption (8.3.13), as well as the control on averages provided by Lemma 3.15, we infer Z (d≤J ρ˜)2 sup EδJ−1 [˜ ρ](τ ) + BδJ−1 [˜ ρ](1, τ∗ ) + FδJ−1 [˜ ρ](1, τ∗ ) + . (B [J])2 . 3 r τ ∈[1,τ∗ ] Σ(τ∗ ) Together with the control of d≤ksmall ρ˜ provided by the decay estimate (8.3.6), we infer from the above estimates sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ ) + FδJ [˜ ρ](1, τ∗ )

τ ∈[1,τ∗ ]

.

EδJ [˜ ρ](1)

2

Z

+ (B [J]) +

r M

1+δ

≤J

|d

Z N0 | + (trap) 2

T (d ρ˜)d N0 . J

M

Next, using the form of N0 , as well as Corollary 8.18, we derive Z  2 (En) (match) r1+δ |d≤J N0 |2 . (B [J])2 + 20 NJ+1 + NJ+1 . M

J

407

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Also, decomposing T as a combination of R and e4 , integrating e4 by parts, using again the form of N0 , as well as Corollary 8.18, we have Z J J T (d ρ ˜ )d N 0 (trap) M Z Z . R(dJ ρ˜)dJ (N0 − Err[g ρ˜]) + e4 (N0 − Err[g ρ˜])dJ N0 (trap) M (trap) M Z + |T (dJ ρ˜)||dJ Err[g ρ˜]| (trap) M Z J J . d ρ˜e4 (d (N0 − Err[g ρ˜])) (trap) M



+ B [J] + Z



(En) 0 NJ+1

+

(match) NJ+1



(match) NJ+1



! 12 sup τ ∈[1,τ∗ ]

EδJ [˜ ρ](τ )

+

BδJ [˜ ρ](1, τ∗ )

+

BδJ [˜ ρ](1, τ∗ )

(dJ ρ˜)2

. (trap) M



+ B [J] +



(En) 0 NJ+1

+

! 12 sup τ ∈[1,τ∗ ]

EδJ [˜ ρ](τ )

.

In view of the above, we infer sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ ) + FδJ [˜ ρ](1, τ∗ )

τ ∈[1,τ∗ ]

Z . (trap) M

 2 (En) (match) (dJ ρ˜)2 + (B [J])2 + 20 NJ+1 + NJ+1 .

Next, note that we have R(r − 3m)

= =

1 3 (e4 (r) − Υe3 (r)) − (e4 (m) − Υe3 (m)) 2 2 1 (trap) Υ + O(0 ) ≥ on M, 6

and hence, using also integration by parts, Z Z J 2 (d ρ˜) . R(r − 3m)(dJ ρ˜)2 (trap) M (trap) M Z 3m J J . 1 − r |d ρ˜||Rd ρ| (trap) M ! 12 . B [J]

sup τ ∈[1,τ∗ ]

EδJ [˜ ρ](τ )

+

BδJ [˜ ρ](1, τ∗ )

.

We deduce  2 (En) (match) sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ ) + FδJ [˜ ρ](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1

τ ∈[1,τ∗ ]

as desired. This concludes the proof of Proposition 8.7.

408 8.5 8.5.1

CHAPTER 8

PROOF OF PROPOSITION 8.8 A wave equation for α + Υ2 α

Lemma 8.19. We have 2 (α + Υ2 α)

=

     2 3m  16m 1− e3 (α) − Υe4 (α) + − 2 + 3 α r r r   h i 2Υ 2m 8m2 − 2 1− − 2 α + Err 2 (α + Υ2 α) r r r 4 r

where h i Err 2 (α + Υ2 α)   4m 8m 8m2 α 2 α = Υ V + 2 Υg (r) − 3 ΥD (r)Dα (r) + 4 D (r)Dα (r) α r r r     m 2Υ + 4 ω+ 2 +2 κ− e3 (α) − 4ωe4 (α) r r    m m e4 (m) −4Υ Υ ω + 2 + 2 (e4 (r) − 1) − e3 (α) r r r     2 (e3 (r) + 1) e3 (m) 2 2 + 4Υ ω + 2Υ κ + − 4mΥ + 4Υ e4 (α) r r2 r (      8m 2m 1 4Υ + −4ρ − 3 + 2 ω κ − 3 + κ κ + 2 − 4e4 (ω) − 8ωω r r 2 r ) ( 8Υ α 4Υ 8m −10κ ω α + D (m)Dα (r) − g (m) − 3 Dα (r)Dα (m) 2 r r r ) (    m 8m α 8m 2m 2 + D (m)Dα α+Υ −4ρ − 3 − 4 e3 (ω) − 3 r r r r )     2m 1 4Υ −10 κ ω − 3 + κ κ + 2 − 8ωω + 2κω α r 2 r    4m 2 2m + 2 Υ g (r) − − 2 α r r r    8m 3m − 3 1− − e4 (r)e3 (r) − Υ + (eθ (r))2 α + 4Υeθ (Υ)eθ (α) r r +Err[g α] + Υ2 Err[g α]. Proof. Recall from Proposition 2.107 that the curvature components α and α verify the following Teukolsky equations: 2 α = −4ωe4 (α) + (4ω + 2κ)e3 (α) + V α + Err[g α], 1 V = −4ρ − 4e4 (ω) − 8ωω + 2ω κ − 10κ ω + κ κ, 2

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

409

where

=

Err(g α) 1 3 1 ϑe3 (α) + ϑ2 ρ + eθ (Φ)ϑβ − κ(ζ + 4η)β − (ζ + η)e4 (β) − ξe3 (β) 2 4 2 +eθ (Φ)(2ζ + η)α + β 2 + e4 (Φ)ηβ + e3 (Φ)ξβ − (ζ + 4η)e4 (β)

−(e4 (ζ) + 4e4 (η))β − 2(κ + ω)(ζ + 4η)β + 2eθ (κ + ω)β − eθ ((2ζ + η)α) 3 1 −3ξeθ (ρ) + 2ηeθ (α) + ϑ /d1 β + 3ρ(η + η + 2ζ)ξ + /d1 ηα + κϑα − 2ωϑα 2 4 1 3 1 − ϑϑα + ξξα + η 2 α + ϑζβ + 3ϑ(ηβ + ξβ) − ϑ(ζ + 4η)β, 2 2 2 and 2 α = −4ωe3 (α) + (4ω + 2κ)e4 (α) + V α + Err[g α], 1 V = −4ρ − 4e3 (ω) − 8ωω + 2ωκ − 10κ ω + κ κ, 2 where Err(g α) 1 3 1 = ϑe4 (α) + ϑ2 ρ + eθ (Φ)ϑ β − κ(−ζ + 4η)β − (−ζ + η)e3 (β) − ξe4 (β) 2 4 2 2 +eθ (Φ)(−2ζ + η)α + β + e3 (Φ)ηβ + e4 (Φ)ξ β − (−ζ + 4η)e3 (β)

−(−e3 (ζ) + 4e3 (η))β − 2(κ + ω)(−ζ + 4η)β + 2eθ (κ + ω)β − eθ ((−2ζ + η)α) 3 1 −3ξeθ (ρ) + 2ηeθ (α) + ϑ /d1 β + 3ρ(η + η − 2ζ)ξ + /d1 ηα + κϑ α − 2ωϑ α 2 4 1 3 1 − ϑϑα + ξξα + η 2 α − ϑζβ + 3ϑ(ηβ + ξβ) − ϑ(−ζ + 4η)β. 2 2 2

We infer from the above wave equations 2 (α + Υ2 α) = 2 (α) + Υ2 2 (α) + 2Dµ (Υ2 )Dµ (α) + 0 (Υ2 )α = −4ωe4 (α) + (4ω + 2κ)e3 (α)   +Υ2 − 4ωe3 (α) + (4ω + 2κ)e4 (α) − 2Υe3 (Υ)e4 (α) − 2Υe4 (Υ)e3 (α)   +V α + Υ2 V + 0 (Υ2 ) α + 4Υeθ (Υ)eθ (α) + Err[g α] + Υ2 Err[g α]

410

CHAPTER 8

and hence

=

2 (α + Υ2 α)    4 3m  1− e3 (α) − Υe4 (α) r r   4m 8m 8m2 +V α + Υ2 V + 2 Υg (r) − 3 ΥDα (r)Dα (r) + 4 Dα (r)Dα (r) α r r r      m 2Υ + 4 ω+ 2 +2 κ− e3 (α) − 4ωe4 (α) r r    m m e4 (m) −4Υ Υ ω + 2 + 2 (e4 (r) − 1) − e3 (α) r r r     2 (e3 (r) + 1) e3 (m) + 4Υ2 ω + 2Υ2 κ + − 4mΥ + 4Υ e4 (α) r r2 r 8Υ α 4Υ 8m D (m)Dα (r) − g (m) − 3 Dα (r)Dα (m) 2 r r r ! m 8m α + D (m)Dα α + 4Υeθ (Υ)eθ (α) + Err[g α] + Υ2 Err[g α]. r r +

Next, we have in view of the formula for V 1 = −4ρ − 4e4 (ω) − 8ωω + 2ω κ − 10κ ω + κ κ 2       2 16m 8m 2m 1 4Υ = − 2 + 3 + −4ρ − 3 + 2 ω κ − 3 + κκ + 2 r r r r 2 r −4e4 (ω) − 8ωω − 10κ ω.

V

Also, we have in view of the formula for V V

1 = −4ρ − 4e3 (ω) − 8ωω + 2ωκ − 10κ ω + κ κ 2      2 8m 2m 2m = − 2 + −4ρ − 3 − 4 e3 (ω) + 3 − 10 κ ω − 3 r r r r   1 4Υ + κ κ + 2 − 8ωω + 2κω. 2 r

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Moreover, we have 4m 8m 8m2 Υg (r) − 3 ΥDα (r)Dα (r) + 4 Dα (r)Dα (r) 2 r r r    4mΥ 2 2m 4m 2 2m = − 2 + 2 Υ g (r) − − 2 r2 r r r r r   8m 3m − 3 1− (−e4 (r)e3 (r) + (eθ (r))2 ) r r    16m2 Υ 4m 2 2m = + 2 Υ g (r) − − 2 r4 r r r    8m 3m − 3 1− − e4 (r)e3 (r) − Υ + (eθ (r))2 r r and hence 4m 8m 8m2 Υ2 V + 2 Υg (r) − 3 ΥDα (r)Dα (r) + 4 Dα (r)Dα (r) r r  r  2Υ 2m 8m2 = − 2 1− − 2 r r r (      8m 2m 2m +Υ2 −4ρ − 3 − 4 e3 (ω) + 3 − 10 κ ω − 3 r r r )      1 4Υ 4m 2 2m + κ κ + 2 − 8ωω + 2κω + 2 Υ g (r) − − 2 2 r r r r    8m 3m − 3 1− − e4 (r)e3 (r) − Υ + (eθ (r))2 . r r We deduce 2

2 (α + Υ α)

     2 3m  16m = 1− e3 (α) − Υe4 (α) + − 2 + 3 α r r r   h i 2Υ 2m 8m2 − 2 1− − 2 α + Err 2 (α + Υ2 α) r r r 4 r

411

412

CHAPTER 8

where h i Err 2 (α + Υ2 α)   4m 8m 8m2 = Υ2 V + 2 Υg (r) − 3 ΥDα (r)Dα (r) + 4 Dα (r)Dα (r) α r r r      m 2Υ + 4 ω+ 2 +2 κ− e3 (α) − 4ωe4 (α) r r    m m e4 (m) −4Υ Υ ω + 2 + 2 (e4 (r) − 1) − e3 (α) r r r     2 (e3 (r) + 1) e3 (m) + 4Υ2 ω + 2Υ2 κ + − 4mΥ + 4Υ e4 (α) r r2 r (      8m 2m 1 4Υ + −4ρ − 3 + 2 ω κ − 3 + κ κ + 2 − 4e4 (ω) − 8ωω r r 2 r ) ( 8Υ α 4Υ 8m −10κ ω α + D (m)Dα (r) − g (m) − 3 Dα (r)Dα (m) r2 r r ) (    m 8m α 8m 2m 2 + D (m)Dα α+Υ −4ρ − 3 − 4 e3 (ω) − 3 r r r r )     2m 1 4Υ −10 κ ω − 3 + κ κ + 2 − 8ωω + 2κω α r 2 r    4m 2 2m + 2 Υ g (r) − − 2 α r r r    8m 3m − 3 1− − e4 (r)e3 (r) − Υ + (eθ (r))2 α + 4Υeθ (Υ)eθ (α) r r +Err[g α] + Υ2 Err[g α] as desired. This concludes the proof of the lemma. Lemma 8.20. We have (     1 ρ˜ 3 3m 2Υ 6m(e4 (r) − Υ) ? −1 + κ˜ ρ − κ − + e3 (α) = − κα − /d2 /d1 e4 2 2 3 2 r 2r r r r4 ) 2e4 (m) 1 3 − + ϑα − ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (ζ + 4η)β r3 2 2 and (     1 ρ˜ 3 3m 2 6m(e3 (r) + 1) ? −1 e4 (α) = − κα − /d2 /d1 e3 + κ˜ ρ − κ + + 2 r2 2r2 r3 r r4 ) 2e3 (m) 1 3 − + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β. r3 2 2

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

413

Proof. Recall that we have 2 (α + Υ2 α)

=

     2 3m  16m 1− e3 (α) − Υe4 (α) + − 2 + 3 α r r r   h i 2Υ 2m 8m2 − 2 1− − 2 α + Err 2 (α + Υ2 α) . r r r 4 r

We first express e3 (α) − Υe4 (α) in terms of ρ˜, where we recall that ρ˜ = r2 ρ + Using Bianchi, we have e3 (α)

=

/d1 β

= = =

2m r .

1 3 − κα − /d?2 β + 4ωα − ϑρ + (ζ + 4η)β, 2 2 3 1 e4 (ρ) + κρ + ϑα − ζβ − 2(ηβ + ξβ) 2  2 ρ˜ 2m 3 1 e4 − 3 + κρ + ϑα − ζβ − 2(ηβ + ξβ) r2 r 2 2     ρ˜ 3 3m 2Υ 6m(e4 (r) − Υ) 2e4 (m) e4 + κ˜ ρ − κ − + − r2 2r2 r3 r r4 r3 1 + ϑα − ζβ − 2(ηβ + ξβ) 2

and hence (     1 ρ˜ 3 3m 2Υ 6m(e4 (r) − Υ) ? −1 e3 (α) = − κα − /d2 /d1 e4 + κ˜ ρ − κ − + 2 r2 2r2 r3 r r4 ) 2e4 (m) 1 3 − + ϑα − ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (ζ + 4η)β. r3 2 2 Similarly, we have e4 (α)

=

/d1 β

= = =

1 3 − κα − /d?2 β + 4ωα − ϑρ + (−ζ + 4η)β, 2 2 3 1 e3 (ρ) + κρ + ϑα + ζβ − 2(ηβ + ξβ) 2  2 ρ˜ 2m 3 1 e3 − 3 + κρ + ϑα + ζβ − 2(ηβ + ξβ) 2 r r 2 2     ρ˜ 3 3m 2 6m(e3 (r) + 1) 2e3 (m) e3 + κ˜ ρ − κ + + − 2 2 3 r 2r r r r4 r3 1 + ϑα + ζβ − 2(ηβ + ξβ) 2

and hence e4 (α)

(     1 ρ˜ 3 3m 2 6m(e3 (r) + 1) ? −1 = − κα − /d2 /d1 e3 + κ˜ ρ − κ + + 2 2 3 2 r 2r r r r4 ) 2e3 (m) 1 3 − + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β. r3 2 2

414

CHAPTER 8

This concludes the proof of the lemma. Corollary 8.21. We have   2 2m 2 (α + Υ2 α) − 2 1 + (α + Υ2 α) r r       8 3m ρ˜ 6 3m = − 1− /d?2 /d−1 R − 1 − (ϑ − Υϑ)ρ 1 r r r2 r r       4 3m 3 3m 2Υ 6m(e4 (r) − Υ) − 1− /d?2 /d−1 κ˜ ρ − κ − + 1 r r 2r2 r3 r r4       4Υ 3m 3 3m 2 6m(e3 (r) + 1) + 1− /d?2 /d−1 κ˜ ρ − κ + + + Err1 , 1 r r 2r2 r3 r r4 where Err1

"   3m ρ˜ := 1− 4ωα + (ζ + 4η)β − Υ(−ζ + 4η)β + [Υ, /d?2 /d−1 ]e 3 1 r r2   2e4 (m) 1 − /d?2 /d−1 − + ϑα − ζβ − 2(ηβ + ξβ) 1 r3 2  # 2e (m) 1 3 −1 +Υ /d?2 /d1 − + ϑα + ζβ − 2(ηβ + ξβ) r3 2       4Υ 3m 1 2Υ m + 1− κ− −4 ω+ 2 α r r 2 r r    h i 2 3m 2 − 1− κ+ α + Err 2 (α + Υ2 α) . r r r 4 r



Proof. Recall from Lemma 8.19 that we have      2 4 3m  16m 2 2 (α + Υ α) = 1− e3 (α) − Υe4 (α) + − 2 + 3 α r r r r   h i 2Υ 2m 8m2 − 2 1− − 2 α + Err 2 (α + Υ2 α) . r r r

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

415

In view of Lemma 8.20, we have e3 (α) − Υe4 (α)

=

    ρ˜ 1 1 3 −2 /d?2 /d−1 R − κα + Υ κ − 4ω α − (ϑ − Υϑ)ρ 1 r2 2 2 2     3 3m 2Υ 6m(e4 (r) − Υ) − /d?2 /d−1 κ˜ ρ − κ − + 1 2r2 r3 r r4     3 3m 2 6m(e3 (r) + 1) +Υ /d?2 /d−1 κ˜ ρ − κ + + 1 2r2 r3 r r4   ρ˜ +4ωα + (ζ + 4η)β − Υ(−ζ + 4η)β + [Υ, /d?2 /d−1 ]e 3 1 r2   2e4 (m) 1 − /d?2 /d−1 − + ϑα − ζβ − 2(ηβ + ξβ) 1 r3 2   2e (m) 1 3 +Υ /d?2 /d−1 − + ϑα + ζβ − 2(ηβ + ξβ) . 1 r3 2

We infer 2 (α + Υ2 α)         8 3m ρ˜ 2 3m 2 16m ? −1 = − 1− /d2 /d1 R − 1− κα + − 2 + 3 α r r r2 r r r r      4Υ 3m 1 2Υ 2m 8m2 + 1− κ − 4ω α − 2 1 − − 2 α r r 2 r r r   6 3m − 1− (ϑ − Υϑ)ρ r r       4 3m 3 3m 2Υ 6m(e4 (r) − Υ) ? −1 − 1− /d2 /d1 κ˜ ρ− 3 κ− + r r 2r2 r r r4       4Υ 3m 3 3m 2 6m(e3 (r) + 1) ? −1 + 1− /d2 /d1 κ˜ ρ− 3 κ+ + r r 2r2 r r r4 "     4 3m ρ˜ + 1− 4ωα + (ζ + 4η)β − Υ(−ζ + 4η)β + [Υ, /d?2 /d−1 ]e 3 1 r r r2   2e4 (m) 1 − /d?2 /d−1 − + ϑα − ζβ − 2(ηβ + ξβ) 1 r3 2  # h i 2e3 (m) 1 ? −1 2 +Υ /d2 /d1 − + ϑα + ζβ − 2(ηβ + ξβ) + Err  (α + Υ α) . 2 r3 2 Since we have      2 3m 4Υ 3m 1 − 1− κα + 1− κ − 4ω α r r r r 2         4 3m 4Υ 3m 2m 2 3m 2 = 1− α+ 2 1− 1+ α− 1− κ+ α r2 r r r r r r r       4Υ 3m 1 2Υ m + 1− κ− −4 ω+ 2 α, r r 2 r r

416

CHAPTER 8

this yields

=

2 (α + Υ2 α)         8 3m ρ˜ 2 2m 2Υ 4m2 − 1− /d?2 /d−1 R + 1 + α + 1 − α 1 r r r2 r2 r r2 r2   6 3m − 1− (ϑ − Υϑ)ρ r r       4 3m 3 3m 2Υ 6m(e4 (r) − Υ) − 1− /d?2 /d−1 κ˜ ρ − κ − + 1 r r 2r2 r3 r r4       4Υ 3m 3 3m 2 6m(e3 (r) + 1) + 1− /d?2 /d−1 κ˜ ρ − κ + + + Err1 , 1 r r 2r2 r3 r r4

where Err1

=

"   3m ρ˜ 1− 4ωα + (ζ + 4η)β − Υ(−ζ + 4η)β + [Υ, /d?2 /d−1 ]e 3 1 r r2   2e4 (m) 1 − /d?2 /d−1 − + ϑα − ζβ − 2(ηβ + ξβ) 1 r3 2  # 2e (m) 1 3 −1 +Υ /d?2 /d1 − + ϑα + ζβ − 2(ηβ + ξβ) r3 2       4Υ 3m 1 2Υ m + 1− κ− −4 ω+ 2 α r r 2 r r    h i 2 3m 2 1− κ+ α + Err 2 (α + Υ2 α) . − r r r 4 r



Now, since we have     2 2m 2Υ 4m2 1+ α+ 2 1− 2 α r2 r r r

=

2 r2



2m 1+ r



(α + Υ2 α),

we infer

=

  2 2m 2 (α + Υ2 α) − 2 1 + (α + Υ2 α) r r       8 3m ρ˜ 6 3m − 1− /d?2 /d−1 R − 1 − (ϑ − Υϑ)ρ 1 r r r2 r r       4 3m 3 3m 2Υ 6m(e4 (r) − Υ) − 1− /d?2 /d−1 κ˜ ρ − κ − + 1 r r 2r2 r3 r r4       4Υ 3m 3 3m 2 6m(e3 (r) + 1) + 1− /d?2 /d−1 κ˜ ρ − κ + + + Err1 , 1 r r 2r2 r3 r r4

as desired. This concludes the proof of the corollary.

417

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

8.5.2

End of the proof of Proposition 8.8

In view of Corollary 8.21, α + Υ2 α satisfies 2

(2 + V2 )(α + Υ α)

  2m 1+ , r

2 V2 = − 2 r

= N2 ,

where N2

      8 3m ρ˜ 6 3m ? −1 := − 1− /d2 /d1 R − 1− (ϑ − Υϑ)ρ r r r2 r r       4 3m 3 3m 2Υ 6m(e4 (r) − Υ) ? −1 − 1− /d2 /d1 κ˜ ρ− 3 κ− + r r 2r2 r r r4       4Υ 3m 3 3m 2 6m(e3 (r) + 1) ? −1 + 1− /d2 /d1 κ˜ ρ− 3 κ+ + r r 2r2 r r r4 +Err1 .

We may thus apply the estimate (10.5.1) of Theorem 10.67 with ψ = α + Υ2 α and s = J to obtain for any ksmall ≤ J ≤ klarge − 1 sup EδJ [α + Υ2 α](τ ) + BδJ [α + Υ2 α](1, τ∗ ) + FδJ [α + Υ2 α](1, τ∗ )

τ ∈[1,τ∗ ]

. EδJ [α + Υ2 α](1) + sup EδJ−1 [α + Υ2 α](τ ) + BδJ−1 [α + Υ2 α](1, τ∗ ) τ ∈[1,τ∗ ]

 2 1 2 +δdec +FδJ−1 [α + Υ2 α](1, τ∗ ) + DJ [Γ] sup rutrap |d≤ksmall (α + Υ2 α)| M Z Z 1+δ ≤J 2 J 2 J + r |d N2 | + T (d (α + Υ α))d N2 , (trap) M

M

where DJ [Γ] is defined by Z DJ [Γ] :=

ˇ 2 (d≤J Γ)

(int) M∪ (ext) M(r≤4m

0)

Z + sup r0 ≥4m0

≤J

r0 {r=r0 }

|d

2

Γg | +

r0−1

Z {r=r0 }

! ≤J

|d

Γb |

2

.

Next we use the iteration assumption (8.3.13) which yields in particular DJ [Γ] . (B [J])2 and sup EδJ−1 [α + Υ2 α](τ ) + BδJ−1 [α + Υ2 α](1, τ∗ )

τ ∈[1,τ∗ ]

+FδJ−1 [α + Υ2 α](1, τ∗ ) . (B [J])2 . Together with the control of d≤ksmall (α + Υ2 α) provided by the decay estimate

418

CHAPTER 8

(8.3.6), we infer from the above estimates sup EδJ [α + Υ2 α](τ ) + BδJ [α + Υ2 α](1, τ∗ ) + FδJ [α + Υ2 α](1, τ∗ )

τ ∈[1,τ∗ ]

.

Z EδJ [α + Υ2 α](1) + (B [J])2 + r1+δ |d≤J N2 |2 M Z J 2 J + T (d (α + Υ α))d N2 . (trap) M

Next, using the form of N2 , as well as the control of ρ˜ provided by Proposition 8.7, we derive Z  2 (En) (match) r1+δ |d≤J N2 |2 . (B [J])2 + 20 NJ+1 + NJ+1 M

and Z J 2 J (trap) T (d (α + Υ α))d N2 M Z   ˇ 1 − 3m |T (dJ (α + Υ2 α))| |R(dJ ρ˜)| + |dJ ρ˜| + |dJ Γ| r (trap) M Z + |T (dJ (α + Υ2 α))||Err1 |

.

(trap) M

   (En) (match) B [J] + 0 NJ+1 + NJ+1

.

! 12 sup EδJ [˜ ρ](τ ) + BδJ [˜ ρ](1, τ∗ )

.

τ ∈[1,τ∗ ]

In view of the above, we infer sup EδJ [α + Υ2 α](τ ) + BδJ [α + Υ2 α](1, τ∗ ) + FδJ [α + Υ2 α](1, τ∗ )

τ ∈[1,τ∗ ]

.

 2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1

as desired. This concludes the proof of Proposition 8.8.

8.6 8.6.1

PROOF OF PROPOSITION 8.9 Control of α and Υ2 α

We initiate the proof of Proposition 8.9 by deriving a suitable control for α and Υ2 α. Recall from Lemma 8.20 that we have (     1 ρ˜ 3 3m 2 6m(e3 (r) + 1) ? −1 e4 (α) = − κα − /d2 /d1 e3 + κ˜ κ + ρ − + 2 r2 2r2 r3 r r4 ) 2e3 (m) 1 3 − + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β. 3 r 2 2

419

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

We infer e4 (α − Υ2 α)

= e4 (α + Υ2 α) − 2e4 (Υ2 α)

= e4 (α + Υ2 α) − 2Υ2 e4 (α) − 2e4 (Υ2 )α (     ρ˜ 3 3m 2 2 2 ? −1 = e4 (α + Υ α) + 2Υ /d2 /d1 e3 + κ˜ ρ − κ + r2 2r2 r3 r ) 6m(e3 (r) + 1) 2e3 (m) 1 + − + ϑα + ζβ − 2(ηβ + ξβ) + Υ2 κα − 8Υ2 ωα r4 r3 2 8mΥe4 (r) 8Υe4 (m) α+ α. r2 r

+3Υ2 ϑρ − 2Υ2 (−ζ + 4η)β −

Also, recall from Lemma 8.20 that we have (     1 ρ˜ 3 3m 2Υ 6m(e4 (r) − Υ) ? −1 e3 (α) = − κα − /d2 /d1 e4 + κ˜ ρ − κ − + 2 r2 2r2 r3 r r4 ) 2e4 (m) 1 3 − + ϑα − ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (ζ + 4η)β. r3 2 2 We infer e3 (α − Υ2 α)

= −e3 (α + Υ2 α) + 2e3 (α) 2

= −e3 (α + Υ α) −

2 /d?2 /d−1 1

( e4



ρ˜ r2

 +

3m 3 κ˜ ρ− 3 2r2 r

 κ−

6m(e4 (r) − Υ) 2e4 (m) 1 + − + ϑα − ζβ − 2(ηβ + ξβ) r4 r3 2

2Υ r



) − κα + 8ωα

−3ϑρ + 2(ζ + 4η)β. In view of the above identities for e4 (α − Υ2 α) and e3 (α − Υ2 α), and using the control for ρ˜ provided by Proposition 8.7 as well as the control for α+Υ2 α provided by Proposition 8.8, and the iteration assumption (8.3.13), we obtain BδJ−1 [e3 (α − Υ2 α)](1, τ∗ ) + BδJ−1 [re4 (α − Υ2 α)](1, τ∗ )  2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 .

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CHAPTER 8

Also, using the Bianchi identity for /d2 α and /d1 β, we have   /d1 /d2 α = /d1 e4 β + 2(κ + ω)β − (2ζ + η)α − 3ξρ   = e4 ( /d1 β) + [ /d1 , e4 ]β + /d1 2(κ + ω)β − (2ζ + η)α − 3ξρ   3 1 = e4 e4 ρ + ρ + ϑα − ζβ − 2(ηβ + ξβ) + [ /d1 , e4 ]β 2 2   + /d1 2(κ + ω)β − (2ζ + η)α − 3ξρ "     ρ˜ 3 3m 2Υ 6m(e4 (r) − Υ) = e4 e4 + 2 κ˜ ρ− 3 κ− + r2 2r r r r4 # 2e4 (m) 1 − + ϑα − ζβ − 2(ηβ + ξβ) + [ /d1 , e4 ]β r3 2   + /d1 2(κ + ω)β − (2ζ + η)α − 3ξρ . Using the control for ρ˜ provided by Proposition 8.7 as well as the iteration assumption (8.3.13), we obtain BδJ−2 [r2 /d1 /d2 α](1, τ∗ ) .

2  (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 .

Using the control for α + Υ2 α provided by Proposition 8.8, we infer BδJ−2 [r2 /d1 /d2 (α − Υ2 α)](1, τ∗ ) . .

BδJ−2 [r2 /d1 /d2 α](1, τ∗ ) +BδJ−2 [r2 /d1 /d2 (α + Υ2 α)](1, τ∗ )  2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 .

Using a Poincar´e inequality for /d1 and for /d2 , we deduce BδJ−2 [ d/2 (α − Υ2 α)](1, τ∗ ) .

 2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1 .

Together with the above estimate for e3 (α − Υ2 α) and re4 (α − Υ2 α), we deduce  2 (En) (match) BδJ [α − Υ2 α](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 . Together with the control for α+Υ2 α provided by Proposition 8.8, we finally obtain  2 (En) (match) BδJ [α](1, τ∗ ) + BδJ [Υ2 α](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 . 8.6.2

(8.6.1)

Control of α

(8.6.1) provides in particular the control of Υ2 α. In this section, we infer a suitable control for α using the wave equation satisfied by α and the redshift vectorfield.

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

421

Let Y(0) the vectorfield given by     5 5 Y(0) := 1 + (r − 2m) + Υ e3 + 1 + (r − 2m) e4 , 4m 4m where Y(0) has been introduced in Proposition 10.29 in connection with the redshift vectorfield. Lemma 8.22. We have 2 α

=

r2 1 +

4m e2
Y(0) α + N − 2m) + Υ

5 4m (r

e2 is given by where N e2 N


!" 5 m 1 + 4m (r − 2m) 1
1+ − κα 5 2 r 1 + 4m (r − 2m) + Υ (     ρ˜ 3 3m 2 6m(e3 (r) + 1) 2e3 (m) − /d?2 /d−1 e + κ˜ ρ − κ + + − 3 1 r2 2r2 r3 r r4 r3 ) # 1 3 + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β 2 2      m 2 +V α − 4 ω + 2 e3 (α) + 4ω + 2 κ + e4 (α) + Err[g α]. r r

4 := − r

Proof. Recall from Proposition 2.107 that α verifies the following Teukolsky equation: 2 α = −4ωe3 (α) + (4ω + 2κ)e4 (α) + V α + Err[g α], 1 V = −4ρ − 4e3 (ω) − 8ωω + 2ωκ − 10κ ω + κ κ, 2 where Err(g α) 1 3 1 = ϑe4 (α) + ϑ2 ρ + eθ (Φ)ϑ β − κ(−ζ + 4η)β − (−ζ + η)e3 (β) − ξe4 (β) 2 4 2 +eθ (Φ)(−2ζ + η)α + β 2 + e3 (Φ)ηβ + e4 (Φ)ξ β − (−ζ + 4η)e3 (β)

−(−e3 (ζ) + 4e3 (η))β − 2(κ + ω)(−ζ + 4η)β + 2eθ (κ + ω)β − eθ ((−2ζ + η)α) 3 1 −3ξeθ (ρ) + 2ηeθ (α) + ϑ /d1 β + 3ρ(η + η − 2ζ)ξ + /d1 ηα + κϑ α − 2ωϑ α 2 4 1 3 1 2 − ϑϑα + ξξα + η α − ϑζβ + 3ϑ(ηβ + ξβ) − ϑ(−ζ + 4η)β. 2 2 2

We deduce 2 α

=

 4m 4 m e3 (α) − e4 (α) + V α − 4 ω + 2 e3 (α) 2 r r r  2 + 4ω + 2 κ + e4 (α) + Err[g α]. r

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CHAPTER 8

In view of the definition of Y(0) , we infer 2 α
! 5 m 1 + 4m (r − 2m) 4m 4
Y(0) α −
e4 (α) = 1+ 5 5 r r2 1 + 4m (r − 2m) + Υ r 1 + 4m (r − 2m) + Υ     m 2 +V α − 4 ω + 2 e3 (α) + 4ω + 2 κ + e4 (α) + Err[g α]. r r Next, recall from Lemma 8.20 that we have (     1 ρ˜ 3 3m 2 6m(e3 (r) + 1) ? −1 e4 (α) = − κα − /d2 /d1 e3 + 2 κ˜ ρ− 3 κ+ + 2 2 r 2r r r r4 ) 2e3 (m) 1 3 − + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β. r3 2 2 We infer 2 α

=

r2

1+

4m e2
Y(0) α + N − 2m) + Υ

5 4m (r

e2 is given by where N e2 N

=


!" 5 m 1 + 4m (r − 2m) 1
1+ − κα 5 2 r 1 + 4m (r − 2m) + Υ (     ρ˜ 3 3m 2 6m(e3 (r) + 1) 2e3 (m) ? −1 + 2 κ˜ + − − /d2 /d1 e3 ρ− 3 κ+ 2 r 2r r r r4 r3 ) # 1 3 + ϑα + ζβ − 2(ηβ + ξβ) + 4ωα − ϑρ + (−ζ + 4η)β 2 2      m 2 +V α − 4 ω + 2 e3 (α) + 4ω + 2 κ + e4 (α) + Err[g α]. r r 4 − r

This concludes the proof of the lemma. e2 , in the RHS of the wave equation for α introduced in Lemma Lemma 8.23. N 8.22, satisfies Z  2 e2 |2 . (B [J])2 + 20 N(En) + N (match) . |dJ N J+1 J+1 (int) M

e2 , see Lemma Proof. The proof of the lemma follows immediately from the form of N 8.22, as well as the control for ρ˜ provided by Proposition 8.7, (8.3.6), and the iteration assumption (8.3.13). In view of Lemma 8.22, we may apply Proposition 10.69 with ψ = α,

f2 (r, m) =

r2 1 +

4m
. − 2m) + Υ

5 4m (r

423

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

We infer Z

J+1

(d

α)

2

Z

EδJ [α](τ

.

(int) M(1,τ ) ∗

(dJ+1 α)2

= 1) + (ext) M

(1,τ∗ ) r≤ 5 m0 2

2

 +DJ [Γ] 

sup (int) M(1,τ )∪ (ext) M ∗ r≤ 5 m0 2

Z



+

r|d≤ksmall α|

 e2 )2 . (d≤s α)2 + (d≤J+1 N

(int) M(1,τ )∪ (ext) M ∗ r≤ 5 m0 2

Next we use the iteration assumption (8.3.13) which yields in particular DJ [Γ] . (B [J])2 together with the control of d≤ksmall α provided by the decay estimate (8.3.6), as e2 provided by Lemma 8.23 well as the iteration assumption and the control for N to deduce Z  2 (En) (match) (dJ+1 α)2 . (B [J])2 + 20 NJ+1 + NJ+1 (int) M(1,τ ) ∗ Z + (dJ+1 α)2 . (ext) M (1,τ∗ ) r≤ 5 m0 2

2 Note that Υ2 & δH > 0 on Z (ext) M

r≤ 5 m0 2

(ext)

M and hence Z (dJ+1 α)2 . (ext) M

(1,τ∗ )

(dJ+1 (Υ2 α))2 r≤ 5 m0 2

(1,τ∗ )

which together with the control of Υ2 α provided by (8.6.1) yields Z  2 (En) (match) (dJ+1 α)2 . (B [J])2 + 20 NJ+1 + NJ+1 (ext) M (1,τ∗ ) r≤ 5 m0 2

and hence Z

(dJ+1 α)2 (int) M(1,τ ) ∗

 2 (En) (match) . (B [J])2 + 20 NJ+1 + NJ+1 .

Since BδJ [α](1, τ∗ )

Z . (int) M(1,τ ) ∗

(d≤J+1 α)2 + BδJ [Υ2 α](1, τ∗ ),

using again (8.6.1), we finally obtain  2 (En) (match) BδJ [α](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 .

(8.6.2)

424

CHAPTER 8

8.6.3

End of the proof of Proposition 8.9

We have   ρ˜ 2m − ρ − . r2 r3

ρˇ =

Together with the control for ρ˜ provided by Proposition 8.7, as well as the control on averages provided by Lemma 3.15, we infer  2 (En) (match) BδJ [ˇ ρ](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 . Together with the control for α provided by (8.6.1) and the control for α provided by (8.6.2), we infer  2 (En) (match) BδJ [α, ρˇ, α](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 . Together with the Bianchi identities for e4 (β), e3 (β), /d1 β, e4 (β), e3 (β), /d1 β, as well as the iteration assumption (8.3.13), we infer J Bδ−2 [β, β](1, τ∗ ) .

 2 (En) (match) (B [J])2 + 20 NJ+1 + NJ+1

and hence  2 (En) (match) J B−2 [α, β, ρˇ, β, α](1, τ∗ ) . (B [J])2 + 20 NJ+1 + NJ+1 as desired. This concludes the proof of Proposition 8.9.

8.7

PROOF OF PROPOSITION 8.10

J First, note that, by definition of the norms B−2 , we have for any r0 ≥ 4m0 (int)

ˇ + RJ+1 [R]

(ext)

0 ˇ R≤r J+1 [R] .

(int)

ˇ and RJ+1 [R]

(ext)

ˇ RJ+1 [R],

J r10 B−2 [α, β, ρˇ, β, α](1, τ∗ ).

Together with Proposition 8.9, this implies (int)

ˇ + RJ+1 [R]

(ext)

0 ˇ R≤r J+1 [R] .

  2  (En) (match) r010 (B [J])2 + 20 NJ+1 + NJ+1 .

Since we have (int)

ˇ + RJ+1 [R]

(ext)

ˇ RJ+1 [R]

=

(int)

ˇ + RJ+1 [R]

(ext)

0 ˇ R≤r J+1 [R] +

(ext)

0 ˇ R≥r J+1 [R],

we deduce for any r0 ≥ 4m0 ˇ RJ+1 [R]     (En) (match) (ext) ≥r0 ˇ RJ+1 [R] + O r010 B [J] + 0 NJ+1 + NJ+1 . (int)

≤

ˇ + RJ+1 [R]

(ext)

425

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Thus, to prove Proposition 8.10, it suffices to establish the following inequality:    (En) (match) (ext) ≥r0 ˇ 10 0 ˇ RJ+1 [R] . r0−δB (ext) G≥r B [J] + 0 NJ+1 + NJ+1 . k [Γ] + r0 This will follow from rp -weighted estimates for the curvature components. 8.7.1

r-weighted divergence identities for Bianchi pairs

Lemma 8.24. Let k ≥ 1, let a(1) and a(2) real numbers. We consider the following equations. • If ψ(1) , h(1) ∈ sk , ψ(2) , h(2) ∈ sk−1 , let (ψ(1) , ψ(2) ) such that ( e3 (ψ(1) ) + a(1) κψ(1) = − /d?k ψ(2) + h(1) , e4 (ψ(2) ) + a(2) κψ(2)

=

/dk ψ(1) + h(2) .

• If ψ(1) , h(1) ∈ sk−1 , ψ(2) , h(2) ∈ sk , let (ψ(1) , ψ(2) ) such that ( e3 (ψ(1) ) + a(1) κψ(1) = /dk ψ(2) + h(1) , e4 (ψ(2) ) + a(2) κψ(2)

= − /d?k ψ(1) + h(2) .

(8.7.1)

(8.7.2)

Then, the pair (ψ(1) , ψ(2) ) satisfies for any real number b     1   2 2 2 Div rb ψ(1) e3 + Div rb ψ(2) e4 − rb κ − 4a(1) + b + 2 ψ(1) 2   1 2 + rb κ 4a(2) − b − 2 ψ(2) 2 2 2 = 2rb /d1 (ψ(1) ψ(2) ) − 2rb ωψ(1) − 2rb ωψ(2) + 2rb ψ(1) h(1) + 2rb ψ(2) h(2)    r r  2 2 +brb−1 e3 (r) − κ ψ(1) + brb−1 e4 (r) − κ ψ(2) . (8.7.3) 2 2 Remark 8.25. Note that the Bianchi identities can be written as systems of equations of the type (8.7.1), (8.7.2). In particular • • • •

the the the the

Bianchi Bianchi Bianchi Bianchi

pair pair pair pair

(α, β) satisfies (β, ρ) satisfies (ρ, β) satisfies (β, α) satisfies

(8.7.1) with k = 2, a(1) = 12 , a(2) = 2, (8.7.1) with k = 1, a(1) = 1, a(2) = 32 , (8.7.2) with k = 1, a(1) = 32 , a(2) = 1, (8.7.2) with k = 2, a(1) = 2, a(2) = 12 .

Proof of Lemma 8.24. The proof being identical for (8.7.1) and (8.7.2), it suffices to prove it in the case where (ψ(1) , ψ(2) ) satisfies (8.7.1). We compute Dγ eγ4

1 1 = − g(D4 e4 , e3 ) − g(D3 e4 , e4 ) + g(Dθ e4 , eθ ) + g(Dϕ e4 , eϕ ) 2 2 = κ − 2ω

426

CHAPTER 8

and Dγ eγ3

1 1 = − g(D4 e3 , e3 ) − g(D3 e3 , e4 ) + g(Dθ e3 , eθ ) + g(Dϕ e3 , eϕ ) 2 2 = κ − 2ω.

We infer in view of (8.7.1)   2 Dγ rb ψ(1) eγ3 = = =

2 2 2rb ψ(1) e3 (ψ(1) ) + brb−1 e3 (r)ψ(1) + rb ψ(1) Dγ eγ3   2 2 2rb ψ(1) − a(1) κψ(1) − /d?k ψ(2) + h(1) + brb−1 e3 (r)ψ(1) + rb ψ(1) (κ − 2ω)    b r  2 2 −2rb ψ(1) /d?k ψ(2) + rb − 2a(1) + + 1 κψ(1) + brb−1 e3 (r) − κ ψ(1) 2 2 2 −2ωrb ψ(1) + 2rb ψ(1) h(1)

and   2 Dγ rb ψ(2) eγ4 2 2 2rb ψ(2) e4 (ψ(2) ) + brb−1 e4 (r)ψ(2) + rb ψ(2) Dγ eγ4   2 2 = 2rb ψ(2) − a(2) κψ(2) + /dk ψ(1) + h(2) + brb−1 e4 (r)ψ(2) + rb ψ(2) (κ − 2ω)     b r 2 2 = 2rb ψ(2) /dk ψ(1) + rb − 2a(2) + + 1 κψ(2) + brb−1 e4 (r) − κ ψ(2) 2 2 2 −2rb ωψ(2) + 2rb ψ(2) h(2) .

=

We sum the two identities     2 2 Dγ rb ψ(1) eγ3 + Dγ rb ψ(2) eγ4   b 2 = −2rb ψ(1) /d?k ψ(2) + 2rb ψ(2) /dk ψ(1) + rb − 2a(1) + + 1 κψ(1) 2    b r  2 2 +rb − 2a(2) + + 1 κψ(2) + brb−1 e3 (r) − κ ψ(1) 2  2  r b−1 2 b 2 b 2 +br e4 (r) − κ ψ(2) − 2r ωψ(1) − 2r ωψ(2) 2 +2rb ψ(2) h(2) + 2rb ψ(1) h(1) and hence       b 2 2 2 Dγ rb ψ(1) eγ3 + Dγ rb ψ(2) eγ4 − rb κ − 2a(1) + + 1 ψ(1) 2   b 2 +rb κ 2a(2) − − 1 ψ(2) 2   r  2 r  2 = 2rb /d1 (ψ(1) ψ(2) ) + brb−1 e3 (r) − κ ψ(1) + brb−1 e4 (r) − κ ψ(2) 2 2 2 2 −2rb ωψ(1) − 2rb ωψ(2) + 2rb ψ(1) h(1) + 2rb ψ(2) h(2) . This concludes the proof of Lemma 8.24.

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

427

To obtain rp -weighted estimates for higher order derivatives of the curvature components, we will need several lemmas. Lemma 8.26. Let k ≥ 1 and s ≥ 1 two integers. Let ψ(1) ∈ sk and ψ(2) ∈ sk−1 . Then, we have   − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1) = /d1 d/s ψ(1) d/s ψ(2) + E[ d/, s, k, ψ(1) , ψ(2) ] where |E[ d/, s, k, ψ(1) , ψ(2) ]| . r| d/s ψ(1) |

s−1 X j=0

+r| d/s ψ(2) |

| d/s−1−j (ψ(2) )|| d/j (K)|

s−1 X j=0

| d/s−1−j (ψ(1) )|| d/j (K)|.

Proof. Recall our definition d/s for higher angular derivatives. Given f a k-reduced scalar and s a positive integer we define ( p r2p 4 / k f, if s = 2p, s d/ f = p 2p+1 r /dk 4 / k f, if s = 2p + 1. We start with the case s = 2p, i.e., s is even. Since ψ(1) ∈ sk and ψ(2) ∈ sk−1 , we have − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1)   p p p p = r4p − 4 / k ψ(1) 4 / k /d?k ψ(2) + 4 / k−1 ψ(2) 4 / k−1 /dk ψ(1) . Next, recall the commutation formulas − /dk 4 /k + 4 / k−1 /dk

− /d?k 4 / k−1

+

=

4 / k /d?k

=

−(2k − 1)K /dk − keθ (K),

(2k − 1)K /d?k + (k − 1)eθ (K).

We infer p

4 / k−1 /dk

=

=

p

p X

p

p X

/dk 4 /k + /dk 4 /k +

j=1

j=1

  p−j j−1 4 / k−1 4 / k−1 /dk − /dk 4 /k 4 /k   p−j j−1 4 / k−1 − (2k − 1)K /dk − keθ (K) 4 /k

and p

4 / k /d?k

=

=

p

p X

p

p X

/d?k 4 / k−1 + /d?k 4 / k−1 +

j=1

j=1

p−j



 j−1 4 / k /d?k − /d?k 4 / k−1 4 / k−1

p−j



 j−1 (2k − 1)K /d?k + (k − 1)eθ (K) 4 / k−1 .

4 /k 4 /k

428

CHAPTER 8

This yields − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1) ( =

r4p −

p

p

−4 / k ψ(1) /d?k 4 / k−1 ψ(2)

p X

p

p−j

4 / k ψ(1) 4 /k



 j−1 (2k − 1)K /d?k + (k − 1)eθ (K) 4 / k−1 ψ(2)

j=1 p p +4 / k−1 ψ(2) /dk 4 / k ψ(1) p  X p p−j + 4 / k−1 ψ(2) 4 / k−1 j=1

=

 j−1 − (2k − 1)K /dk − keθ (K) 4 / k ψ(1)

)

  /d1 d/s ψ(1) d/s ψ(2) − +

p X j=1 p X j=1

  d/s ψ(1) d/s−2j (2k − 1)r2 K /d?k + (k − 1)r2 eθ (K) d/2j−2 ψ(2)   d/s ψ(2) d/s−2j − (2k − 1)r2 K /dk − kr2 eθ (K) d/2j−2 ψ(1) .

Hence, we infer − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1)

=

  /d1 d/s ψ(1) d/s ψ(2) + E[ d/, s, k, ψ(1) , ψ(2) ]

where |E[ d/, s, k, ψ(1) , ψ(2) ]| .

r2 | d/s ψ(1) |

s−1 X j=0

+r2 | d/s ψ(2) |

| d/s−1−j (ψ(2) )|| d/j (K)|

s−1 X j=0

| d/s−1−j (ψ(1) )|| d/j (K)|.

Next, we deal with the case s = 2p + 1, i.e., s odd. Since ψ(1) ∈ sk and ψ(2) ∈ sk−1 , we have − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1)   p p p p = r4p+2 − /dk 4 / k ψ(1) /dk 4 / k /d?k ψ(2) + /dk−1 4 / k−1 ψ(2) /dk−1 4 / k−1 /dk ψ(1) .

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

429

In view of the case s = 2p above, we infer − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1) ( = r4p+2 −

p X j=1

p

p

− /dk 4 / k ψ(1) /dk /d?k 4 / k−1 ψ(2) p

p−j

/dk 4 / k ψ(1) /dk 4 /k



 j−1 (2k − 1)K /d?k + (k − 1)eθ (K) 4 / k−1 ψ(2)

p

p

+ /dk−1 4 / k−1 ψ(2) /dk−1 /dk 4 / k ψ(1) ) p   X p p−j j−1 + /dk−1 4 / k−1 ψ(2) /dk−1 4 / k−1 − (2k − 1)K /dk − keθ (K) 4 / k ψ(1) . j=1

Next, recall the commutation formula /dk /d?k − /d?k−1 /dk−1

= −2(k − 1)K.

We infer

=

− d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1) (   p p 4p+2 r − /dk 4 / k ψ(1) /d?k−1 /dk−1 − 2(k − 1)K 4 / k−1 ψ(2) −

p X j=1

p

p−j

/dk 4 / k ψ(1) /dk 4 /k p



 j−1 (2k − 1)K /d?k + (k − 1)eθ (K) 4 / k−1 ψ(2) p

+ /dk−1 4 / k−1 ψ(2) /dk−1 /dk 4 / k ψ(1) ) p   X p p−j j−1 + /dk−1 4 / k−1 ψ(2) /dk−1 4 / k−1 − (2k − 1)K /dk − keθ (K) 4 / k ψ(1) j=1

=

  /d1 d/s ψ(1) d/s ψ(2) +2(k − 1)r2 K d/s ψ(1) d/s−1 ψ(2) p   X − d/s ψ(1) d/s−2j (2k − 1)r2 K /d?k + (k − 1)r2 eθ (K) d/2j−2 ψ(2) +

j=1 p X j=1

  d/s ψ(2) d/s−2j − (2k − 1)r2 K /dk − kr2 eθ (K) d/2j−2 ψ(1) .

Hence, we obtain − d/s ψ(1) d/s /d?k ψ(2) + d/s ψ(2) d/s /dk ψ(1)

=

  /d1 d/s ψ(1) d/s ψ(2) + E[ d/, s, k, ψ(1) , ψ(2) ]

430

CHAPTER 8

where |E[ d/, s, k, ψ(1) , ψ(2) ]| . r2 | d/s ψ(1) |

s−1 X j=0

+r2 | d/s ψ(2) |

| d/s−1−j (ψ(2) )|| d/j (K)|

s−1 X j=0

| d/s−1−j (ψ(1) )|| d/j (K)|.

This concludes the proof of the lemma. Corollary 8.27. Let k ≥ 1, let a(1) and a(2) real numbers and let 0 ≤ s ≤ klarge . Consider the outgoing geodesic foliation of (ext) M. We consider the following equations. • If ψ(1) ∈ sk , ψ(2) ∈ sk−1 , let (ψ(1) , ψ(2,s) ) such that ( e3 ( d/s ψ(1) ) + a(1) κ d/s ψ(1) = − d/s /d?k ψ(2) + h(1,s) , e4 ( d/s ψ(2) ) + a(2) κ d/s ψ(2)

=

d/s /dk ψ(1) + h(2,s) .

• If ψ(1) ∈ sk−1 , ψ(2) , h(2) ∈ sk , let (ψ(1) , ψ(2) ) such that ( e3 ( d/s ψ(1) ) + a(1) κ d/s ψ(1) = d/s /dk ψ(2) + h(1,s) , e4 ( d/s ψ(2) ) + a(2) κ d/s ψ(2)

=

− d/s /d?k ψ(1) + h(2,s) .

Then, the pair (ψ(1) , ψ(2) ) satisfies for any real number b     Div rb ( d/s ψ(1) )2 e3 + Div rb ( d/s ψ(2) )2 e4     1 1 − rb κ − 4a(1) + b + 2 ( d/s ψ(1) )2 + rb κ 4a(2) − b − 2 ( d/s ψ(2) )2 2  2  b s s b = 2r /d1 d/ ψ(1) d/ ψ(2) + 2r E[ d/, s, k, ψ(1) , ψ(2) ] − 2rb ω( d/s ψ(1) )2  r  +2rb d/s ψ(1) h(1,s) + 2rb d/s ψ(2) h(2,s) + brb−1 e3 (r) − κ ( d/s ψ(1) )2 2  r  +brb−1 e4 (r) − κ ( d/s ψ(2) )2 2 where E[ d/, s, k, ψ(1) , ψ(2) ] has been introduced in Lemma 8.26. Proof. The proof follows immediately from combining Lemma 8.24 and Lemma 8.26. Lemma 8.28. Let j, k, l three integers. Consider a Bianchi (ψ(1) , ψ(2) ) satisfying

431

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

(8.7.1) or (8.7.2). Then, the pair (ψ(1) , ψ(2) ) satisfies for any real number b     Div rb ( d/j (re4 )k Tl ψ(1) )2 e3 + Div rb ( d/j (re4 )k Tl ψ(2) )2 e4   1 − rb κ − 4a(1) + 2k + b + 2 ( d/j (re4 )k Tl ψ(1) )2 2   1 + rb κ 4a(2) − 2k − b − 2 ( d/j (re4 )k Tl ψ(2) )2 2   = 2rb /d1 d/j (re4 )k Tl ψ(1) d/j (re4 )k Tl ψ(2) +2rb E[ d/, j, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ] − 2rb ω( d/j (re4 )k Tl ψ(1) )2 +2rb d/j (re4 )k Tl ψ(1) h(1),j,k,l + 2rb d/j (re4 )k Tl ψ(2) h(2),j,k,l  r  +brb−1 e3 (r) − κ ( d/j (re4 )k Tl ψ(1) )2 2   r b−1 +br e4 (r) − κ ( d/j (re4 )k Tl ψ(2) )2 2

where E[ d/, s, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ] has been introduced in Lemma 8.26, and where h(1),j,k,l and h(2),j,k,l are given, schematically, by h(1),j,k,l

=

d/≤j+k+l (h(1) ) + kr−1 d/j+1 (re4 )k−1 Tl ψ(2) 

+rdj+k+l Γg ψ(1) , ψ(2) + O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2)

and h(2),j,k,l

=


 d/≤j+k+l (h(2) ) + kr−1 d/j+1 (re4 )k−1 Tl ψ(1) + rdj+k+l Γg ψ(1) , ψ(2)
+O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2) .

Proof. We have the following simple schematic consequences of the commutator identities: [T, e4 ], [T, e3 ] = r−1 Γb d,

[T, /dk ] = −ηe3 + Γg d,

[ d/, e4 ] = Γg d + Γg , [ d/, e3 ] = −rηe3 + rΓg d, r r [re4 , e4 ] = − κe4 + Γg d, [re4 , e3 ] = − κe4 + Γb d, 2 2 [re4 , d/k ] = r−1 d/ + Γg d + Γg . Then, differentiating with d/j (re4 )k Tl the equations ( e3 (ψ(1) ) + a(1) κψ(1) = − /d?k ψ(2) + h(1) , e4 (ψ(2) ) + a(2) κψ(2)

=

/dk ψ(1) + h(2) ,

and using the above commutator identities we infer 
e3 ( d/j (re4 )k Tl ψ(1) ) + a(1) − k2 κ d/j (re4 )k Tl ψ(1)    
 e4 ( d/j (re4 )k Tl ψ(2) ) + a(2) − k2 κ d/j (re4 )k Tl ψ(2)   

=

− d/j /d?k ((re4 )k Tl ψ(2) ) +h(1),j,k,l ,

=

d/j /dk ((re4 )k Tl ψ(1) ) +h(2),j,k,l ,

432

CHAPTER 8

where h(1),j,k,l

=

d/j (re4 )k Tl (h(1) ) + kr−1 d/j+1 (re4 )k−1 Tl ψ(2) + jrηdj+k+l−1 e3 ψ(1) 

+rdj+k+l Γg ψ(1) , ψ(2) + O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2)

and h(2),j,k,l

=

d/j (re4 )k Tl (h(2) ) + kr−1 d/j+1 (re4 )k−1 Tl ψ(1) 

+rdj+k+l Γg ψ(1) , ψ(2) + O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2) .

Also, using the equation e3 (ψ(1) )

=

−a(1) κψ(1) − /d?k ψ(2) + h(1) ,

we obtain jrηdj+k+l−1 e3 ψ(1)



= rdj+k+l Γg ψ(1) , ψ(2) + O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2) +rηdk+j+l−1 (h(1) )

and hence, h(1),j,k,l

=

d/≤j+k+l (h(1) ) + kr−1 d/j+1 (re4 )k−1 Tl ψ(2) 

+rdj+k+l Γg ψ(1) , ψ(2) + O(r−1 )d≤j+k+l−1 ψ(1) , ψ(2) .

We have thus obtained the desired form for h(1),j,k,l and h(2),j,k,l . The divergence identity now follows from the equations 
e3 ( d/j (re4 )k Tl ψ(1) ) + a(1) − k2 κ d/j (re4 )k Tl ψ(1) = − d/j /d?k ((re4 )k Tl ψ(2) )     +h(1),j,k,l ,
j k j k l k l  e ( d/ (re4 ) T ψ(2) ) + a(2) − 2 κ d/ (re4 ) T ψ(2) = d/j /dk ((re4 )k Tl ψ(1) )    4 +h(2),j,k,l , together with Corollary 8.27. This concludes the proof of the lemma. Corollary 8.29. Let r0 ≥ 4m0 and 1 ≤ u0 ≤ u∗ . We introduce the spacetime region Ru0 =

(ext)

M ∩ {r ≥ 4m0 } ∩ {1 ≤ u ≤ u0 }.

Let j, k, l three integers. Assume that the frame of (ext) M satisfies   r r  sup e3 (r) − κ + r |ω| + e4 (r) − κ . 0 . 2 2 (ext) M Consider a pair (ψ(1) , ψ(2) ) satisfying (8.7.1) or (8.7.2). Then, (ψ(1) , ψ(2) ) satisfies, for any real number b, a) If −4a(1) + 2k + b + 2 > 0 and 4a(2) − 2k − b − 2 > 0,

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

then, we have Z

rb ( d/j (re4 )k Tl ψ(1) )2

Cu0 (r≥r0 )

Z

  rb ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2

+ Σ∗ (≤u0 )

Z

  rb−1 ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2

+ Ru0 (r≥r0 )

Z .

(ext) M( r0 2

  rb−1 ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2 ≤r≤r0 )

Z

  rb+1 (h(1),j,k,l )2 + (h(2),j,k,l )2

+ Ru0 (r≥r0 )

Z

rb E[ d/, j, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ].

+ Ru0 (r≥r0 )

b) If −4a(1) + 2k + b + 2 ≤ 0 and 4a(2) − 2k − b − 2 > 0, then, we have Z

rb ( d/j (re4 )k Tl ψ(1) )2

Cu0 (r≥r0 )

Z

  rb ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2

+ Σ∗ (≤u0 )

Z

rb−1 ( d/j (re4 )k Tl ψ(2) )2

+ Ru0 (r≥r0 )

Z .

(ext) M( r0 2

  rb−1 ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2 ≤r≤r0 )

Z +

  rb+1 (h(1),j,k,l )2 + (h(2),j,k,l )2

Ru0 (r≥r0 )

Z +

rb−1 ( d/j (re4 )k Tl ψ(1) )2

Ru0 (r≥r0 )

Z +

rb E[ d/, j, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ].

Ru0 (r≥r0 )

c) If 4a(2) − 2k − b − 2 = 0,

433

434

CHAPTER 8

then, we have Z

rb ( d/j (re4 )k Tl ψ(1) )2

Cu0 (r≥r0 )

Z + Z .

  rb ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2

Σ∗ (≤u0 )

(ext) M( r0 2

  rb−1 ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2 ≤r≤r0 )

Z +

r

b+1−δB

Z

2

Z

rb+1+δB (h(2),j,k,l )2

(h(1),j,k,l ) +

Ru0 (r≥r0 )

Ru0 (r≥r0 )

rb−1+δB ( d/j (re4 )k Tl ψ(1) )2

+ Ru0 (r≥r0 )

Z

rb−1−δB ( d/j (re4 )k Tl ψ(2) )2

+ Ru0 (r≥r0 )

Z

rb E[ d/, j, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ].

+ Ru0 (r≥r0 )

d) If −4a(1) + 2k + b + 2 > 0, then, we have Z

rb ( d/j (re4 )k Tl ψ(1) )2

Cu0 (r≥r0 )

Z

  rb ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2

+ Σ∗ (≤u0 )

Z

rb−1 ( d/j (re4 )k Tl ψ(1) )2

+ Ru0 (r≥r0 )

Z .

(ext) M( r0 2

  rb−1 ( d/j (re4 )k Tl ψ(1) )2 + ( d/j (re4 )k Tl ψ(2) )2 ≤r≤r0 )

Z + Ru0 (r≥r0 )

Z +

rb+1 (h(1),j,k,l )2 +

Z

rb+1 (h(2),j,k,l )2

Ru0 (r≥r0 )

rb−1 ( d/j (re4 )k Tl ψ(2) )2

Ru0 (r≥r0 )

Z +

rb E[ d/, j, k, (re4 )k Tl ψ(1) , (re4 )k Tl ψ(2) ].

Ru0 (r≥r0 )

Proof. We multiply the pair (ψ(1) , ψ(2) ) by a smooth cut-off function in r supported in r ≥ r20 and identically one for r ≥ r0 . We obtain again a solution to (8.7.1) or (8.7.2) up to error terms that are supported in the region r20 ≤ r ≤ r0 . We then integrate the divergence identities of Lemma 8.28 on the region Ru0 and the corollary follows.

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

8.7.2

435

End of the proof of Proposition 8.10

Let r0 ≥ 4m0 . Recall that, to prove Proposition 8.10, it suffices to establish the following inequality:    (En) (match) (ext) ≥r0 ˇ 10 0 ˇ RJ+1 [R] . r0−δB (ext) G≥r [ Γ] + r  [J] +  N + N . B 0 0 J+1 J+1 k To this end, we will rely on the rp -weighted estimates derived in Corollary 8.29 applied to the Bianchi pairs, where we recall Remark 8.25. Remark 8.30. For the Bianchi pair (β, ρ), we replace the Bianchi identities for e4 (ρ) by its analog for e4 (ˇ ρ), i.e., 3 e4 ρˇ + κˇ ρ = 2

3 /d1 β − ρˇ κ + Err[e4 ρˇ], 2

while for the Bianchi pair (ρ, β), we replace the Bianchi identities for e3 (ρ) by its analog for e3 (ˇ ρ), i.e., 3 e3 ρˇ + κˇ ρ = 2


3 3 3 ˇ + ς −1 Ωˇ /d1 β − ρˇ κ − κ ρς −1 ςˇ + κ ρ Ω ς + Err[e3 ρˇ], 2 2 2

see Proposition 2.73 for the derivation of these equations. Let j, k, l three integers such that j + k + l = J + 1. To derive rp -weighted curvature estimates for d/j (re4 )k Tl derivatives in the region r ≥ r0 , we proceed as follows. Step 1. We start with the case k = 0, i.e., we derive rp -weighted curvature estimates for d/j Tl derivatives with j + l = J + 1. First, we apply Corollary 8.29 • • • •

to to to to

the the the the

Bianchi Bianchi Bianchi Bianchi

pair pair pair pair

(α, β) with the choice b = 4 + δB , (β, ρ) with the choice b = 4 − δB , (ρ, β) with the choice b = 2 − δB , (β, α) with the choice b = −δB .

The above choices are such that we are in case (a) of Corollary 8.29 for the Bianchi pairs (α, β) and (β, ρ), and in case (b) of Corollary 8.29 for the last two

436

CHAPTER 8

Bianchi pairs. In particular, we obtain ( Z  X sup r4+δB ( d/j Tl α)2 + r4−δB ( d/j Tl β)2 1≤u≤u∗

j+l=J+1

Cu (r≥r0 )

 Z +r2−δB ( d/j Tl ρˇ)2 + r−δB ( d/j Tl β)2 +

  r4+δB ( d/j Tl α)2 + ( d/j Tl β)2 Σ∗  4−δB j l 2 2−δB j l 2 −δB +r ( d/ T ρˇ) + r ( d/ T β) + r ( d/j Tl α)2 Z    + r3+δB ( d/j Tl α)2 + ( d/j Tl β)2 + r3−δB ( d/j Tl ρˇ)2 (ext) M(r≥r

+r

.

1−δB

j



0)

l

2

( d/ T β) + r

−1−δB

j

l

2

( d/ T α)



)

( ˇ 2 Z (dJ+1 R) + r−1+δB ( d/j Tl ϑ)2 5 r (ext) M( r0 ≤r≤r ) (ext) M(r≥r ) 0 0 2     −1−δB j l 2 +r ( d/ T η) + ( d/j Tl κ ˇ )2 + r−3−δB ( d/j Tl κ ˇ )2 + ( d/j Tl ζ)2 )   −5−δB j l 2 j l 2 j l 2 j lˇ 2 +r ( d/ T ξ) + ( d/ T ϑ) + ( d/ T ςˇ) + ( d/ T Ω) r08+δB

Z

(En)

+(B [J])2 + 20 (NJ+1 )2 . Using Proposition 8.9 to bound the first term on the right-hand side, and using 0 ˇ also the definition of the norm (ext) G≥r k [Γ], we infer that ( Z  X sup r4+δB ( d/j Tl α)2 + r4−δB ( d/j Tl β)2 j+l=J+1

1≤u≤u∗

Cu (r≥r0 )

j

l

+r

Z

+∞

. r0

−δB

2

1−δB

( d/ T ρˇ) + r

(ext) M(r≥r

3−δB

2

j

l

r1+δB

l

2

( d/ T β)

+

0)

( d/ T ρˇ) + r dr

j



Z

   r4+δB ( d/j Tl α)2 + ( d/j Tl β)2 Σ∗  4−δB j l 2 2−δB j l 2 −δB +r ( d/ T ρˇ) + r ( d/ T β) + r ( d/j Tl α)2 Z    + r3+δB ( d/j Tl α)2 + ( d/j Tl β)2 +r

2−δB



j

l

2

( d/ T β) + r

−1−δB

j

l

( d/ T α)

2



)

  2  (En) (match) 10 2 2 0 ˇ 2 ( (ext) G≥r [ Γ]) + r ( [J]) +  N + N B 0 0 J+1 J+1 J+1

437

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and hence (

Z

X



sup

j+l=J+1

1≤u≤u∗

r4+δB ( d/j Tl α)2 + r4−δB ( d/j Tl β)2

Cu (r≥r0 )

 Z +r2−δB ( d/j Tl ρˇ)2 + r−δB ( d/j Tl β)2 +

  r4+δB ( d/j Tl α)2 + ( d/j Tl β)2 Σ∗  2 −δB 4−δB j l 2 2−δB j l +r ( d/ T ρˇ) + r ( d/ T β) + r ( d/j Tl α)2 Z    + r3+δB ( d/j Tl α)2 + ( d/j Tl β)2 (ext) M(r≥r

+r

3−δB

j



0)

l

2

( d/ T ρˇ) + r

1−δB

j

l

2

( d/ T β) + r

−1−δB

j

l

2

( d/ T α)



)

  2  (En) (match) 10 0 ˇ 2 . r0−δB ( (ext) G≥r (B [J])2 + 20 NJ+1 + NJ+1 . J+1 [Γ]) + r0

(8.7.4)

Step 2. We derive additional rp -weighted curvature estimates for d/j Tl derivatives with j + l = J + 1. To this end, we apply Corollary 8.29 • to the Bianchi pair (β, ρ) with the choice b = 4, • to the Bianchi pair (ρ, β) with the choice b = 2, • to the Bianchi pair (β, α) with the choice b = 0. All the above choices are such that we are in case (c) of Corollary 8.29. In particular, we obtain ( Z   X sup r4 ( d/j Tl β)2 + r2 ( d/j Tl ρˇ)2 + ( d/j Tl β)2 1≤u≤u∗

j+l=J+1

Z

Cu (r≥r0 )

    r4 ( d/j Tl β)2 + ( d/j Tl ρˇ)2 + r2 ( d/j Tl β)2 + ( d/j Tl α)2

+

)

Σ∗

.

r08

Z (ext) M( r0 2

≤r≤r0 )

ˇ 2 (dJ+1 R) + r5

(Z

X

 (ext) M(r≥r

j+l=J+1

+r3−δB ( d/j Tl ρˇ)2 + r1−δB ( d/j Tl β)2 + r−1−δB ( d/j Tl α)2

r3+δB ( d/j Tl β)2

0)



)

(

Z

r−1−δB ( d/j Tl η)2 + r−1+δB ( d/j Tl κ ˇ )2

+ (ext) M(r≥r

0)

  +r−3+δB ( d/j Tl κ ˇ )2 + ( d/j Tl ζ)2 +r

−5+δB



j

l

2

j

l

2

j

l

2

j

lˇ 2

( d/ T ξ) + ( d/ T ϑ) + ( d/ T ςˇ) + ( d/ T Ω)



)

(En)

+(B [J])2 + 20 (NJ+1 )2 . Using Proposition 8.9 to bound the first term on the right-hand side, and using

438

CHAPTER 8

(ext)

also the definition of the norm ( Z X sup 1≤u≤u∗

j+l=J+1

Z



0 ˇ G≥r k [Γ], we infer that

r4 ( d/j Tl β)2 + r2 ( d/j Tl ρˇ)2 + ( d/j Tl β)2

Cu (r≥r0 )

    r4 ( d/j Tl β)2 + ( d/j Tl ρˇ)2 + r2 ( d/j Tl β)2 + ( d/j Tl α)2

+



)

Σ∗

Z .

+∞



dr

r1+δB (Z X

r0

+

+r

 (ext) M(r≥r

j+l=J+1 1−δB

  2  (En) (match) 10 2 2 0 ˇ 2 ( (ext) G≥r [ Γ]) + r ( [J]) +  N + N B 0 0 J+1 J+1 J+1

j

l

2

( d/ T β) + r

r3+δB ( d/j Tl β)2 + r3−δB ( d/j Tl ρˇ)2

0)

−1−δB

j

l

( d/ T α)

2



)

and hence (

Z

X

1≤u≤u∗

j+l=J+1

Z



sup

r4 ( d/j Tl β)2 + r2 ( d/j Tl ρˇ)2 + ( d/j Tl β)2

Cu (r≥r0 )

    r4 ( d/j Tl β)2 + ( d/j Tl ρˇ)2 + r2 ( d/j Tl β)2 + ( d/j Tl α)2

+



)

Σ∗

.

  2  (En) (match) 10 2 2 0 ˇ 2 r0−δB ( (ext) G≥r [ Γ]) + r ( [J]) +  N + N B 0 0 J+1 J+1 J+1 (Z  X + r3+δB ( d/j Tl β)2 + r3−δB ( d/j Tl ρˇ)2 (ext) M(r≥r

j+l=J+1

+r

1−δB

j

l

2

( d/ T β) + r

0)

−1−δB

j

l

( d/ T α)

2



) .

Together with (8.7.4), we deduce ( Z  X sup r4+δB ( d/j Tl α)2 + r4 ( d/j Tl β)2 + r2 ( d/j Tl ρˇ)2 j+l=J+1

1≤u≤u∗



Z

Cu (r≥r0 )

  r4+δB ( d/j Tl α)2 + ( d/j Tl β)2 + r4 ( d/j Tl ρˇ)2 Σ∗  Z    2 j l 2 +r ( d/ T β) + ( d/j Tl α)2 + r3+δB ( d/j Tl α)2 + ( d/j Tl β)2 +( d/j Tl β)2 +



(ext) M(r≥r

+r

3−δB

j

l

2

( d/ T ρˇ) + r

1−δB

j

l

2

( d/ T β) + r

0)

−1−δB

j

l

2

( d/ T α)



)

  2  (En) (match) 10 2 2 0 ˇ 2 . r0−δB ( (ext) G≥r [ Γ]) + r ( [J]) +  N + N . B 0 0 J+1 J+1 J+1

(8.7.5)

Step 3. We now argue by iteration on k. For 0 ≤ k ≤ J, we consider the following

439

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

iteration assumption: ( X j+l=J+1−k

Z



sup

1≤u≤u∗

r4+δB ( d/j (re4 )k Tl α)2 + r4 ( d/j (re4 )k Tl β)2

Cu (r≥r0 )

 +r2 ( d/j (re4 )k Tl ρˇ)2 + ( d/j (re4 )k Tl β)2 Z    + r4+δB ( d/j (re4 )k Tl α)2 + ( d/j (re4 )k Tl β)2 + r4 ( d/j (re4 )k Tl ρˇ)2 Σ∗  2 +r ( d/j (re4 )k Tl β)2 + ( d/j (re4 )k Tl α)2 Z    + r3+δB ( d/j (re4 )k Tl α)2 + ( d/j (re4 )k Tl β)2 (ext) M(r≥r

+r .

3−δB

0)

j

k

l

2

( d/ (re4 ) T ρˇ) + r

1−δB

j

k

l

2

( d/ (re4 ) T β) + r

−1−δB

j

k

l

2

( d/ (re4 ) T α)

  2  (En) (match) 10 0 ˇ 2 r0−δB ( (ext) G≥r (B [J])2 + 20 NJ+1 + NJ+1 . J+1 [Γ]) + r0



)

(8.7.6)

(8.7.6) holds true for k = 0 in view of (8.7.5). We now assume that (8.7.6) holds true for k such that 0 ≤ k ≤ J, and our goal is to prove that it also holds for k + 1. First, note that the Bianchi identities for e4 (β), e4 (ˇ ρ), e4 (β) and e4 (α), together with (8.7.6), yield ( Z  X sup r4 ( d/j (re4 )k+1 Tl β)2 1≤u≤u∗

j+l=J+1−(k+1)

Cu (r≥r0 )

 Z +r2 ( d/j (re4 )k+1 Tl ρˇ)2 + ( d/j (re4 )k+1 Tl β)2 +



r4+δB ( d/j (re4 )k+1 Tl β)2 Σ∗  4 j k+1 l 2 2 j k+1 l 2 +r ( d/ (re4 ) T ρˇ) + r ( d/ (re4 ) T β) + ( d/j (re4 )k+1 Tl α)2 Z  + r3+δB ( d/j (re4 )k+1 Tl β)2 + r3−δB ( d/j (re4 )k+1 Tl ρˇ)2 (ext) M(r≥r

+r

1−δB

j

0)

k+1

( d/ (re4 )

l

2

T β) + r

−1−δB

j

( d/ (re4 )

k+1

l

2

T α)

) 

  2  (En) (match) 10 0 ˇ 2 . r0−δB ( (ext) G≥r (B [J])2 + 20 NJ+1 + NJ+1 . J+1 [Γ]) + r0

(8.7.7)

We still need to estimate d/j (re4 )k+1 Tl α. To this end, we apply Corollary 8.29 to the Bianchi pair (α, β) with the choice b = 4 + δB . Since k + 1 ≥ 1, we are in

440

CHAPTER 8

case (d) of Corollary 8.29. In particular, we obtain, arguing similarly as above, ( Z X sup r4+δB ( d/j (re4 )k+1 Tl α)2 1≤u≤u∗

j+l=J+1−(k+1)

Z +

r

4+δB

j

k+1

( d/ (re4 )

Cu (r≥r0 ) l

2

T α) +

r (ext) M(r≥r

Σ∗

(Z

X

.

)

Z

j

( d/ (re4 )

k+1

l

2

T α)

0)

) r3+δB ( d/j (re4 )k+1 Tl β)2 (ext) M(r≥r

j+l=J+1−(k+1)

3+δB

0)

  2  (En) (match) 10 2 2 0 ˇ 2 +r0−δB ( (ext) G≥r [ Γ]) + r ( [J]) +  N + N . B 0 0 J+1 J+1 J+1 Together with (8.7.7), this implies (8.7.6) for k + 1. Hence, by iteration, (8.7.6) holds for any 0 ≤ k ≤ J + 1. This implies ( Z   X sup r4+δB (dk α)2 + r4 (dk β)2 + r2 (dk ρˇ)2 + (dk β)2 1≤u≤u∗

k≤J+1

Z



+

Cu (r≥r0 )



  r4+δB (dk α)2 + (dk β)2 + r4 (dk ρˇ)2 + r2 (dk β)2 + (dk α)2

ZΣ∗



+ (ext) M(r≥r

  r3+δB (dk α)2 + (dk β)2 + r3−δB (dk ρˇ)2 + r1−δB (dk β)2

0)

+r−1−δB (dk α)2



)

  2  (En) (match) ≥r0 ˇ 2 . r0−δB ( (ext) GJ+1 [Γ]) + r010 (B [J])2 + 20 NJ+1 + NJ+1 . Hence, we have obtained (ext)

≥r0 ˇ RJ+1 [R]

   (En) (match) 10 0 ˇ . r0−δB (ext) G≥r B [J] + 0 NJ+1 + NJ+1 J+1 [Γ] + r0

which concludes the proof of Proposition 8.10. 8.7.3

Proof of (8.3.12)

To prove (8.3.12), we argue as in the proof of Proposition 8.10. Let j, k, l three integers such that j + k + l ≤ ksmall . To derive rp -weighted curvature estimates for d/j (re4 )k Tl derivatives of (α, β) in the region r ≥ 4m0 , we proceed as follows. Step 1. We start with the case k = 0, i.e., we derive rp -weighted curvature estimates for d/j Tl derivatives of (α, β) with j+l ≤ ksmall . First, we apply Corollary 8.29 to the Bianchi pair (α, β) with the choice b = 4 + δB . This choice is such that

441

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

we are in case (a) of Corollary 8.29. In particular, we obtain ( Z Z  X sup r4+δB ( d/j Tl α)2 + r4+δB ( d/j Tl α)2 1≤u≤u∗

j+l≤ksmall j

l

2

+( d/ T β)



Cu (r≥4m0 )

Z +

r (ext) M(r≥4m

Z .

(ext) M( 7m0 2

Σ∗

≤r≤4m0 )

3+δB

  ( d/j Tl α)2 + ( d/j Tl β)2

)

0)

ˇ 2 (dJ+1 R) + r5

(

Z

)

r−1+δB ( d/j Tl ϑ)2 (ext) M(r≥4m

+ 20

0)

(En)

+20 (Nksmall )2 . We infer that (

Z

X

sup 1≤u≤u∗

j+l≤ksmall j

l

r

+( d/ T β)

2



4+δB

j

l

Z

2

Cu (r≥4m0 )

Σ∗

Z +

 r4+δB ( d/j Tl α)2

( d/ T α) +

r



3+δB

j

l

2

j

l

( d/ T α) + ( d/ T β)

2



)

(ext) M(r≥4m ) 0

.



(ext)

0 ˇ R≤4m ksmall [R]

2



+

(ext)

2 (En) ˇ Gksmall [Γ] + 20 + 20 (Nksmall )2 .

(8.7.8)

Step 2. We now argue by iteration on k. For 0 ≤ k ≤ ksmall − 1, we consider the following iteration assumption: ( Z X sup r4+δB ( d/j (re4 )k Tl α)2 j+l≤ksmall −k

Z

1≤u≤u∗

Cu (r≥4m0 )



r4+δB ( d/j (re4 )k Tl α)2 + ( d/j (re4 )k Tl β)2

+



Σ∗

Z +

r

3+δB



j

k

l

2

j

k

l

2

( d/ (re4 ) T α) + ( d/ (re4 ) T β)

) 

(ext) M(r≥4m ) 0

.



(ext)

0 ˇ R≤4m ksmall [R]

2

+



(ext)

2 (En) ˇ Gksmall [Γ] + 20 + 20 (Nksmall )2 .

(8.7.9)

(8.7.9) holds true for k = 0 in view of (8.7.8). We now assume that (8.7.9) holds true for k such that 0 ≤ k ≤ ksmall − 1, and our goal is to prove that it also holds for k + 1. First, note that the Bianchi identity for e4 (β), together with (8.7.9), yields (Z X r4+δB ( d/j (re4 )k+1 Tl β)2 Σ∗

j+l≤ksmall −(k+1)

)

Z +

r

3+δB

j

k+1

( d/ (re4 )

l

T β)

2

(ext) M(r≥4m ) 0

.



(ext)

0 ˇ R≤4m ksmall [R]

2

+



(ext)

2 (En) ˇ Gksmall [Γ] + 20 + 20 (Nksmall )2 . (8.7.10)

442

CHAPTER 8

We still need to estimate d/j (re4 )k+1 Tl α. To this end, we apply Corollary 8.29 to the Bianchi pair (α, β) with the choice b = 4 + δB . Since k + 1 ≥ 1, we are in case (c) of Corollary 8.29. In particular, we obtain, arguing similarly as above, ( Z X sup r4+δB ( d/j (re4 )k+1 Tl α)2 1≤u≤u∗

j+l≤ksmall −(k+1)

Z +

r

4+δB

j

k+1

( d/ (re4 )

Cu (r≥4m0 ) l

T α) +

(Z

X 

(ext)

3+δB

j

( d/ (re4 )

k+1

l

2

T α)

0)

) r3+δB ( d/j (re4 )k+1 Tl β)2 (ext) M(r≥4m ) 0

j+l≤ksmall −(k+1)

+

r (ext) M(r≥4m

Σ∗

.

)

Z

2

2  2 (En) (ext) 0 ˇ ˇ R≤4m [ R] + G [ Γ] + 20 + 20 (Nksmall )2 . k small ksmall

Together with (8.7.10), this implies (8.7.9) for k + 1. Hence, by iteration, (8.7.9) holds for any 0 ≤ k ≤ ksmall . Now, (8.7.9) for any 0 ≤ k ≤ ksmall is equivalent to (8.3.12) which is the desired estimate.

8.8

PROOF OF PROPOSITION 8.11

To prove Proposition 8.11, we rely on the following three propositions. Proposition 8.31. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Then, we have (Σ∗ )

ˇ + GJ+1 [Γ]

(Σ∗ )

ˇ G0J+1 [Γ]

(Σ∗ )

.

ˇ + RJ+1 [R]

(Σ∗ )

ˇ GJ [Γ],

where we have introduced the notations Z "   (Σ∗ ) ˇ := Gk [Γ] r2 (d≤k ϑ)2 + (d≤k κ ˇ )2 + (d≤k ζ)2 + (d≤k κ ˇ )2 + (d≤k ϑ)2 Σ∗

# ≤k

+(d

(Σ∗ )

ˇ G0k [Γ]

Z

2

≤k

η) + (d

2

≤k

ω ˇ ) + (d

2

ξ)

,

h i r2 (dk+1 d/κ ˇ )2 + (dk+1 κ ˇ )2 + (d≤k+1 µ ˇ)2 + (dk+1 κ ˇ )2 + (dk+1 ζ)2 ,

:= Σ∗

and (Σ∗ )

ˇ Rk [R]

Z :=




r4+δB (d≤k α)2 + (d≤k β)2 + r4 (d≤k ρˇ)2 + r2 (d≤k β)2 Σ∗  +(d≤k α)2 .

Proposition 8.32. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Then, we have (ext)

0 ˇ G≥4m J+1 [Γ] +

(ext)

0

0 ˇ G≥4m J+1 [Γ] .

ˇ + (Σ∗ ) G0J+1 [Γ] ˇ GJ+1 [Γ] ˇ + (ext) GJ [Γ], ˇ + (ext) RJ+1 [R] (Σ∗ )

443

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

where we have introduced the notation " Z     2 (ext) ≥4m0 0 ˇ Gk [Γ] := sup λ6 dk /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ λ≥4m0 2

{r=λ}

k+1

+λ (d



2 κ ˇ )2 + λ6 dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ

+λ4 (d≤k µ ˇ)2 + λ2 dk eθ (κ) − 4β



2

+ λ2 dk e3 (ζ) + β



2

#! .

Proposition 8.33. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Then, we have (ext)

0 ˇ G≤4m J+1 [Γ] +

(ext)

0

0 ˇ G≤4m J+1 [Γ] .

0

(ext) ≥4m0 ˇ 0 ˇ G≥4m GJ+1 [Γ] J+1 [Γ] + ˇ + (ext) GJ [Γ], ˇ + (ext) RJ+1 [R] (ext)

where we have introduced the notation 0 0 ˇ G≤4m [Γ] k Z sup

(ext)

:=

rT ≤λ≤4m0

"

{r=λ}

    2 λ6 dk /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ



2 +λ2 (dk+1 κ ˇ )2 + λ6 dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ 4

≤k

+λ (d

2

µ ˇ) + λ

2

k

d

eθ (κ) − 4β



2

2

+λ

k−1

d

N e3 (ζ) + β



2

#! .

The proof of Proposition 8.31 is postponed to section 8.8.1, the proof of Proposition 8.32 is postponed to section 8.8.4, and the proof of Proposition 8.33 is postponed to section 8.8.5. The proof of the two latter propositions will rely in particular on basic weighted estimates for transport equations along e4 in (ext) M derived in section 8.8.2, as well as several renormalized identities derived in section 8.8.3. We now conclude the proof of Proposition 8.11. In view of Propositions 8.31, 8.32 and 8.33, we have, for J such that ksmall − 2 ≤ J ≤ klarge − 1, (ext)

ˇ . GJ+1 [Γ]

(ext)

ˇ + RJ+1 [R]

(ext)

ˇ GJ [Γ],

where we have used the fact that (Σ∗ )

ˇ ≤ RJ+1 [R]

(ext)

ˇ RJ+1 [R],

(Σ∗ )

ˇ ≤ GJ [Γ]

(ext)

ˇ GJ [Γ].

In view of the iteration assumption (8.3.13), we infer (ext)

ˇ GJ+1 [Γ]

.

(ext)

ˇ + B [J]. RJ+1 [R]

Since the estimates in Proposition 8.32 are integrated from Σ∗ , we obtain similarly, for any r0 ≥ 4m0 , (ext)

0 ˇ G≥r J+1 [Γ]

.

(ext)

0 ˇ R≥r J+1 [R] + B [J].

444

CHAPTER 8

On the other hand, we have in view of Proposition 8.10, for any r0 ≥ 4m0 ,    (En) (match) (ext) ≥r0 ˇ 10 0 ˇ RJ+1 [R] . r0−δB (ext) G≥r B [J] + 0 NJ+1 + NJ+1 J+1 [Γ] + r0 and (int)

ˇ + RJ+1 [R]

(ext)

0 ˇ R≥r J+1 [R]     (En) (match) +O r010 B [J] + 0 NJ+1 + NJ+1 .

ˇ ≤ RJ+1 [R]

(ext)

Choosing r0 ≥ 4m0 large enough, we infer from the above estimates   (ext) ˇ + (int) RJ+1 [R] ˇ + (ext) RJ+1 [R] ˇ . B [J] + 0 N(En) + N (match) . GJ+1 [Γ] J+1 J+1 This concludes the proof of Proposition 8.11. 8.8.1

Proof of Proposition 8.31

Step 1. We control κ on Σ∗ . Recall the GCM conditions κ = 2/r on Σ∗ . Since νΣ∗ and eθ are tangent, we infer   2 ( d/, νΣ∗ )k κ − = 0. r Together with Raychaudhuri, we infer Z max

k≤J+2

.



(Σ∗ )

r

2



k

d

Σ∗

ˇ + RJ+1 [R]



2 κ− r

(Σ∗ )

2 +r

4


2 d eθ (κ)

!

k

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ , GJ [Γ]

where we have used the fact that e3 is in the span of e4 and νΣ∗ . Note that we have used Codazzi for ϑ to control the term dJ+1 e4 (eθ (κ)). Step 2. We control the ` = 1 modes on Σ∗ . In view of the GCM conditions for κ, and projecting the Codazzi for ϑ on the ` = 1 mode, we infer on Σ∗ Z Z Z r Φ Φ ζe = r βe + ϑζeΦ . 2 S S S Since the vectorfield ν is tangent to Σ∗ , we infer Z  Z Z r ν J+2 ζeΦ = r ν J+2 βeΦ + ν J+2 (ϑζ)eΦ + l.o.t. 2 S S S Z Z Z r r = r ν J+2 βeΦ + ζν J+2 (ϑ)eΦ + ϑν J+2 (ζ)eΦ + l.o.t. 2 S 2 S S where l.o.t. denote, here and below, terms that • either are linear and contain at most J + 1 derivatives of curvature components and J derivatives of Ricci coefficients,

445

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

• or are quadratic and contain at most J + 1 derivatives of Ricci coefficients and curvature components. Using Bianchi identities and the null structure equations, we deduce Z  ν J+2 ζeΦ S Z Z Z r r J+1 = r ν ( /d2 α, /d?1 ρ − 3ρη)eΦ + ζν J+1 /d?2 ηeΦ + ϑν J+1 /d?1 ωeΦ + l.o.t. 2 S 2 S S Z Z Z r J+1 J ? Φ J+1 Φ = r ( /d2 ν α, ν /d1 /d1 (β, β))e − 3ρ ν ηe + ζ /d?2 ν J+1 ηeΦ 2 S S S Z r + ϑ /d?1 ν J+1 ωeΦ + l.o.t. 2 S Z Z Z r r = r ( /d2 ν J+1 α, /d?1 /d1 ν J (β, β))eΦ + ζ /d?2 ν J+1 ηeΦ + ϑ /d?1 ν J+1 ωeΦ 2 S 2 S S +l.o.t., where we have used,R in the last equality, a cancellation due to the fact that ν is tangent to Σ∗ and S ηeΦ = 0 on Σ∗ . Using the identity /d?1 /d1 = /d2 /d?2 + 2K, integration by parts for all terms, and the fact that /d?2 (eΦ ) = 0 so that the top order linear term vanishes, we infer Z  ν J+2 ζeΦ = l.o.t. S

with the above convention for the lower order terms. Also, relying on the null equation for e4 (ζ), i.e., e4 (ζ)

= −κζ − β − ϑζ,

we obtain, with more ease since this estimate is at one lower level of derivatives, Z  J+2 Φ (re4 , ν) ζe = l.o.t. S

We infer  Z 2  Z u∗ −2 k Φ ˇ + max r d ζe . (Σ∗ ) RJ+1 [R] k≤J+2

1

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

S

Next, we have, in view of the definition of µ and the identity /d?1 /d1 = /d2 /d?2 +2K, Z Z Z Z 1 eθ (µ)eΦ = /d?1 /d1 ζeΦ − eθ (ρ)eΦ + eθ (ϑϑ)eΦ 4 S S Z Z S Z S 1 Φ Φ = 2 Kζe − eθ (ρ)e + eθ (ϑϑ)eΦ 4 S S S  Z Z Z  Z 2 2 1 Φ Φ Φ = ζe − eθ (ρ)e + K − 2 ζe + eθ (ϑϑ)eΦ . r2 S r 4 S S S To estimate the RHS, we use in particular the following.

446

CHAPTER 8

• For the second term, in view of Bianchi, we have, schematically, (e3 , re4 )J+2 eθ (ρ) 3 (e3 , re4 )J+1 /d?1 /d1 (rβ, β) − (e3 , re4 )J+1 /d?1 (rκρ, κρ) 2
+(e3 , re4 )J+1 /d?1 rϑα, (rζ, ξ)β, (ζ, η)β, ϑα + l.o.t. 3 = (e3 , re4 )J+1 /d2 /d?2 (rβ, β) + ρ(e3 , re4 )J+1 (reθ (κ), eθ (κ)) 2 +rϑ(e3 , re4 )J+1 /d?1 α + (rζ, ξ)(e3 , re4 )J+1 /d?1 β + (ζ, η)(e3 , re4 )J+1 /d?1 β =

+ϑ(e3 , re4 )J+1 /d?1 α + rα(e3 , re4 )J+1 /d?1 ϑ + β(e3 , re4 )J+1 /d?1 (rζ, ξ)
+β(e3 , re4 )J+1 /d?1 (ζ, η) + α(e3 , re4 )J+1 /d?1 ϑ + l.o.t. =

/d2 (e3 , re4 )J+1 /d?2 (rβ, β) + [(e3 , re4 )J+1 , /d2 ] /d?2 (rβ, β) 3 3 + ρ(e3 , re4 )J+1 (reθ (κ), eθ (κ)) − ρˇ /d?1 (e3 , re4 )J+1 (rˇ κ, κ ˇ) 2 2 +rϑ /d?1 (e3 , re4 )J+1 α + (rζ, ξ) /d?1 (e3 , re4 )J+1 β + (ζ, η) /d?1 (e3 , re4 )J+1 β +ϑ /d?1 (e3 , re4 )J+1 α + rα /d?1 (e3 , re4 )J+1 ϑ + β /d?1 (e3 , re4 )J+1 (rζ, ξ)
+β /d?1 (e3 , re4 )J+1 (ζ, η) + α /d?1 (e3 , re4 )J+1 ϑ + l.o.t.

• For the third term   2 (e3 , re4 ) K− 2 ζ r     2 2 J+2 ζ(e3 , re4 ) K − 2 + K − 2 (e3 , re4 )J+2 ζ + l.o.t. r r     2 J+1 −1 ζ(e3 , re4 ) /d1 (rβ, β, η, r ξ) + K − 2 (e3 , re4 )J+1 eθ (ω) + l.o.t. r     2 J+1 −1 ζ(e3 , re4 ) /d1 rβ, β, η, r ξ + K − 2 (e3 , re4 )J+1 eθ (ω) + l.o.t. r     2 J+1 −1 ζ /d1 (e3 , re4 ) rβ, β, η, r ξ + K − 2 /d?1 (e3 , re4 )J+1 ω ˇ + l.o.t. r J+2

= = = =



• For the fourth term (e3 , re4 )J+2 eθ (ϑϑ)

=

(e3 , re4 )J+1 eθ (ϑ /d?2 (ξ, rζ)) + (e3 , re4 )J+1 eθ (ϑ /d?2 η) + l.o.t.

=

ϑ /d?1 /d?2 (e3 , re4 )J+1 (ξ, rζ) + ϑ /d?1 /d?2 (e3 , re4 )J+1 η + l.o.t.

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

447

We infer J+2

Z

(e3 , re4 ) =

Φ



eθ (µ)e S Z  Z 2 J+2 Φ (e3 , re4 ) ζe + /d2 (e3 , re4 )J+1 /d?2 (rβ, β)eΦ r2 S S Z Z 3 J+1 ? Φ + [(e3 , re4 ) , /d2 ] /d2 (rβ, β)e + ρ (e3 , re4 )J+1 (reθ (κ), eθ (κ))eΦ 2 S S Z 3 − ρˇ /d?1 (e3 , re4 )J+1 (rˇ κ, κ ˇ )eΦ 2 S Z  + rϑ /d?1 (e3 , re4 )J+1 α + (rζ, ξ) /d?1 (e3 , re4 )J+1 β + (ζ, η) /d?1 (e3 , re4 )J+1 β S Z   +ϑ /d?1 (e3 , re4 )J+1 α eΦ + rα /d?1 (e3 , re4 )J+1 ϑ + β /d?1 (e3 , re4 )J+1 (rζ, ξ) S  +β /d?1 (e3 , re4 )J+1 (ζ, η) + α /d?1 (e3 , re4 )J+1 ϑ eΦ  Z Z    2 J+1 −1 Φ + ζ /d1 (e3 , re4 ) rβ, β, ζ, r ξ e + K − 2 /d?1 (e3 , re4 )J+1 ω ˇ eΦ r S S Z Z + ϑ /d?1 /d?2 (e3 , re4 )J+1 (ξ, rζ)eΦ + ϑ /d?1 /d?2 (e3 , re4 )J+1 ζeΦ + l.o.t. S

S

and after integrations by parts and the fact that /dk (F eΦ ) = /dk+1 (F )eΦ ,

/d?k (F eΦ ) = /d?k−1 (F )eΦ ,

we obtain  eθ (µ)eΦ S Z  Z   2 J+2 Φ (e , re ) ζe + /d?1 [(e3 , re4 )J+1 , /d2 ] (rβ, β)eΦ 3 4 2 r S S Z Z 3 3 J+1 + ρ (e3 , re4 ) (reθ (κ), eθ (κ))eΦ + /d2 ρ(e3 , re4 )J+1 (rˇ κ, κ ˇ )eΦ 2 S 2 S Z  + r /d2 ϑ(e3 , re4 )J+1 α + /d2 (rζ, ξ)(e3 , re4 )J+1 β + /d2 (ζ, η)(e3 , re4 )J+1 β S Z   + /d2 ϑ(e3 , re4 )J+1 α eΦ + r /d2 α(e3 , re4 )J+1 ϑ + /d2 β(e3 , re4 )J+1 (rζ, ξ) S  + /d2 β(e3 , re4 )J+1 (ζ, η) + /d2 α(e3 , re4 )J+1 ϑ eΦ   Z Z   2 + /d1 ζ(e3 , re4 )J+1 rβ, β, ζ, r−1 ξ eΦ + /d2 K − 2 (e3 , re4 )J+1 ω ˇ eΦ r S ZS Z J+1 Φ + /d3 /d2 ϑ(e3 , re4 ) (ξ, rζ)e + /d3 /d2 ϑ(e3 , re4 )J+1 ζeΦ + l.o.t.

(e3 , re4 )J+2 =

S

Z

S

Together with the above estimate for the ` = 1 mode of ζ and the estimate of Step

448

CHAPTER 8

1 for κ, we infer Z

u∗

max

k≤J+2

 Z 2 k Φ r d eθ (µ)e 2

1

S

2 ˇ + ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ RJ+1 [R] GJ [Γ]  Z 2 Z u∗ + max r−4 dk eθ (κ)eΦ . 

.

(Σ∗ )

(Σ∗ )

k≤J+1

1

S

In view of the dominant condition (3.3.4) for r on Σ∗ , we infer Z

u∗

max

k≤J+2

 Z 2 k Φ r d eθ (µ)e 2

1

S

2 ˇ + ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ RJ+1 [R] GJ [Γ] 2 Z u∗  Z +0 max dk eθ (κ)eΦ . 

.

(Σ∗ )

(Σ∗ )

k≤J+1

1

S

Next, in view of the remarkable identity for the ` = 1 mode of eθ (K), we have Z Z Z 1 1 eθ (κκ)eΦ + eθ (ϑϑ)eΦ = 0 − eθ (ρ)eΦ − 4 S 4 S S and hence Z eθ (κ)eΦ

Z

r = −2r eθ (ρ)e − 2 ZS r + eθ (ϑϑ)eΦ . 2 S

S

Φ

 Z  Z 2 r κ− eθ (κ)eΦ − κeθ (κ)eΦ r 2 S S

Arguing as for the estimate of the ` = 1 mode of eθ (µ), and using the smallness of 0 , we infer Z

u∗

max

k≤J+2

.



(Σ∗ )

 Z 2 Z k Φ r d eθ (µ)e + max

u∗

2

1

k≤J+2

S

ˇ + RJ+1 [R]

(Σ∗ )

We have thus obtained " Z u∗

max

k≤J+2

r



k

Z

2

ζe

d

1

Φ

+r

2



.

k

Z eθ (µ)e

d

S k

Z

+ d

eθ (κ)e

Φ

2 #

S



(Σ∗ )

ˇ + RJ+1 [R]

(Σ∗ )

eθ (κ)e

Φ

S

S



.

Z

k

d

1 2

ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ GJ [Γ]

−2



2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Φ

2

2

449

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Step 3. Recall the GCM conditions /d?2 /d?1 κ = /d?2 /d?1 µ = 0 on Σ∗ . This yields on Σ∗ R R e (µ)eΦ Φ e (κ)eΦ Φ SR θ SR θ eθ (µ) = e , e (κ) = e . θ e2Φ e2Φ S S Together with Step 2, we infer Z 
2
2  max r4 ( d/, νΣ∗ )k µ ˇ + r2 ( d/, νΣ∗ )k κ ˇ k≤J+2

.



(Σ∗ )

Σ∗

ˇ + RJ+1 [R]

(Σ∗ )

ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ GJ [Γ]

2

.

Then, in view of the null structure equations for e4 (ˇ µ) and e4 (ˇ κ), e4 (ˇ µ) e4 (ˇ κ)

3 = − κˇ µ− 2 1 = − κˇ κ− 2

3 µˇ κ + Err[e4 µ ˇ] 2 1 κˇ κ + 2ˇ µ + 4ˇ ρ + Err[e4 κ ˇ ], 2

we infer, together with the control of κ ˇ provided by Step 1, Z 
2
2  max r4 dk µ ˇ + r2 dk κ ˇ k≤J+2

.



(Σ∗ )

Σ∗

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Step 4. Recall that we have /d1 ζ

1 = −ˇ µ − ρˇ + ϑϑ. 4

Differentiating, and using the Bianchi identities for e4 (ˇ ρ) and e3 (ˇ ρ), and the null structure equations for e4 (ϑ), e3 (ϑ), e4 (ϑ) and e3 (ϑ), we infer /d1 dk ζ

  1   = −dk µ ˇ − dk−1 d/ ρˇ, β, r−1 β + dk−1 d/(ϑϑ), r−1 ϑ d/η, ϑ d/ζ, r−1 ϑ d/ξ 4 +l.o.t.   1   = −dk µ ˇ − d/dk−1 ρˇ, β, r−1 β + d/ ϑdk−1 (ϑ, r−1 η) 4  1  k−1 −1 + d/ ϑd (ϑ, ζ, r ξ) + l.o.t. 4

We infer, since /d1 is invertible in view of the corresponding Poincar´e inequality, dk ζ

  1  = −r d/−1 dk µ ˇ − dk−1 ρˇ, β, r−1 β + ϑdk−1 (ϑ, r−1 η) 4  1  k−1 −1 + ϑd (ϑ, ζ, r ξ) + l.o.t. 4

Together with the estimate for µ ˇ of Step 3, this yields Z  2 (Σ∗ ) ˇ + (Σ∗ ) GJ [Γ] ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . max r2 (dk ζ)2 . RJ+1 [R] k≤J+2

Σ∗

450

CHAPTER 8

Step 5. Recall from the GCM condition that we have on Σ∗ Z ηeΦ = 0. S

Together with the transport equation e4 (η − ζ)

=

1 1 − κ(η − ζ) − ϑ(η − ζ), 2 2

we infer in view of the estimates for ζ of Step 4, Z max

k≤J+1

u∗

r

−4



k

Z

Φ

ηe

d

1

2 .



(Σ∗ )

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

S

Next, recall from Proposition 2.74 that η verifies  
1 2 /d2 /d?2 η = κ −e3 (ζ) + β − e3 (eθ (κ)) − κ κζ − 2ωζ + 6ρη − κeθ κ 2 1 − κeθ (κ) + 2ωeθ (κ) + 2eθ (ρ) + Err[ /d2 /d?2 η], 2     1 1 1 1 Err[ /d2 /d?2 η] = 2 /d1 η − κϑ + 2η 2 η + 2eθ (η 2 ) − κ ϑζ − ϑξ − ϑeθ (κ) 2 2 2 2   1 1 1 3 − 2 /d1 η − ϑϑ + 2η 2 ζ − eθ (ϑ ϑ) − ϑ2 ξ − ϑϑη. 2 2 2 2 Together with the estimates for κ of Step 1, the estimates for κ of Step 3, and the estimates for ζ of Step 4, r2 /d2 (η 2 ) − r2 eθ (η 2 ) + r2 /d1 (ζη) 2 !!2

Z

dk r2 /d2 /d?2 η − r2 eθ (ρ) −

max

k≤J+1

Σ∗

Z .

max

k≤J+1

Σ∗

1 + r2 eθ (ϑ ϑ) 4  ˇ + r−2 |dk η|2 + (Σ∗ ) RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

In view of the dominant condition (3.3.4) for r on Σ∗ , we infer r2 /d2 (η 2 ) − r2 eθ (η 2 ) + r2 /d1 (ζη) 2 !!2

Z

dk r2 /d2 /d?2 η − r2 eθ (ρ) −

max

k≤J+1

.

Σ∗

Z

2 3

0 max

k≤J+1

Σ∗

1 + r2 eθ (ϑ ϑ) 4  ˇ + |dk η|2 + (Σ∗ ) RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

451

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

This yields Z

r2 /d2 /d?2 dk η + r /d?2 [dk , r /d2 ]η + r /d2 [dk , r /d?2 ]η

max

k≤J+1

Σ∗

!2 r2 1 2 k k 2 2 k 2 2 k −r eθ (d ρ) − /d2 d (η ) − r eθ d (η ) + r /d1 d (ζη) + r eθ d (ϑ ϑ) 2 4 Z  2 2 ˇ + (Σ∗ ) GJ [Γ] ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . . 03 max |dk η|2 + (Σ∗ ) RJ+1 [R] 2

k

k≤J+1

Σ∗

We deduce, using a Poincar´e inequality for /d2 , Z Z 2
2 max r /d?2 dk η . 03 max |dk η|2 k≤J+1

k≤J+1

Σ∗

+



(Σ∗ )

Σ∗

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Together with a Poincar´e inequality for r /d?2 and the above control of the ` = 1 mode of η, we infer Z Z 2
2 max dk η . 03 max |dk η|2 k≤J+1

k≤J+1

Σ∗

+



(Σ∗ )

Σ∗

ˇ + RJ+1 [R]

and hence, for 0 small enough, Z 
2 (Σ∗ ) ˇ + max dk η . RJ+1 [R] k≤J+1

(Σ∗ )

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ , GJ [Γ]

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Σ∗

Step 6. Recall from the GCM condition that we have on Σ∗ Z ξeΦ = 0. S

Together with the transport equation e4 (ξ)

= −e3 (ζ) + β − κζ − ζϑ,

we infer in view of the estimates for ζ of Step 4, the estimates for β, and the bootstrap assumptions Z max

k≤J+1

1

u∗

 Z 2  ˇ + r−4 dk ξeΦ . (Σ∗ ) RJ+1 [R] S

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

452

CHAPTER 8

Next, from Proposition 2.74, we have
3 = −e3 (eθ (κ)) + κ e3 (ζ) − β + κ2 ζ − κeθ κ + 6ρξ − 2ωeθ (κ) 2 +Err[ /d2 /d?2 ξ],   1 1 1 Err[ /d2 /d?2 ξ] = 2 /d1 ξ + κ ϑ + 2ηξ − ϑ2 η + 2eθ (ηξ) − eθ (ϑ2 ) 2 2 2   1 1 1 1 + κ ϑζ − ϑξ − ϑeθ κ − ϑϑξ 2 2 2 2   1 2 −ζ 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ 2   2 + ξ − ϑϑ − 2 /d1 ζ + 2ζ − 6ηζξ − 6eθ (ζξ). 2 /d2 /d?2 ξ

Together with the estimates for κ of Step 1, the estimates for κ of Step 3, and the estimates for ζ of Step 4, Z

1 1 dk r2 /d2 /d?2 ξ + eθ (e3 (ˇ κ)) − η /d1 ξ − eθ (ηξ) + eθ (ϑ2 ) + /d2 (ζξ) 2 4 !!2

max

k≤J+1

Σ∗

+3eθ (ζξ) Z .

r−2 |dk ξ|2 +

max

k≤J+1

Σ∗



(Σ∗ )

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

In view of the dominant condition (3.3.4) for r on Σ∗ , we infer Z

1 1 dk r2 /d2 /d?2 ξ + eθ (e3 (ˇ κ)) − η /d1 ξ − eθ (ηξ) + eθ (ϑ2 ) + /d2 (ζξ) 2 4 !!2

max

k≤J+1

Σ∗

+3eθ (ζξ) Z

2 3

. 0 max

k≤J+1

Σ∗

|dk ξ|2 +



(Σ∗ )

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

This yields Z

1 r2 /d2 /d?2 dk ξ + r /d?2 [dk , r /d2 ]ξ + r /d2 [dk , r /d?2 ]ξ + eθ (dk e3 (ˇ κ)) 2 Σ∗ !2 1 k k k 2 k k − /d1 (ηd ξ) − eθ d (ηξ) + eθ d (ϑ ) + /d2 d (ζξ) + 3eθ d (ζξ) 4 Z  2 2 ˇ + (Σ∗ ) GJ [Γ] ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . . 03 max |dk ξ|2 + (Σ∗ ) RJ+1 [R] max

k≤J+1

k≤J+1

Σ∗

453

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

We deduce, using a Poincar´e inequality for /d2 and the estimates for κ of Step 3, Z Z 2
2 max r /d?2 dk ξ . 03 max |dk ξ|2 k≤J+1

k≤J+1

Σ∗

+



(Σ∗ )

Σ∗

ˇ + RJ+1 [R]

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Together with a Poincar´e inequality for r /d?2 and the above control of the ` = 1 mode of ξ, we infer Z Z 2
2 max dk ξ . 03 max |dk ξ|2 k≤J+1

k≤J+1

Σ∗

Σ∗

 ˇ + + (Σ∗ ) RJ+1 [R] and hence, for 0 small enough, Z 
2 (Σ∗ ) ˇ + max dk ξ . RJ+1 [R] k≤J+1

(Σ∗ )

(Σ∗ )

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ , GJ [Γ]

2 ˇ + 0 (Σ∗ ) GJ+1 [Γ] ˇ . GJ [Γ]

Σ∗

Step 7. Using the Codazzi for ϑ and ϑ, the transport equation for ϑ and ϑ in the e4 and e3 direction, the control of κ ˇ of Step 1, the control of κ ˇ of Step 3, the control of ζ of Step 4, the control of η of Step 5, the control of ξ of Step 6, and a Poincar´e inequality for /d2 , we infer Z    2 2 ˇ + (Σ∗ ) GJ [Γ] ˇ +  3 (Σ∗ ) GJ+1 [Γ] ˇ . max r2 (dk ϑ)2 + (dk ϑ)2 . (Σ∗ ) RJ+1 [R] 0 k≤J+1

Σ∗

Step 8. Recall from Proposition 2.74 that ω verifies   1 1 1 ? 2 /d1 ω = − κξ + κ + 2ω + ϑ η + e3 (ζ) − β 2 2 2 1 1 1 + κζ − 2ωζ + ϑζ − ϑξ. 2 2 2 Together with a Poincar´e inequality for /d?1 , the control of ξ from Step 6, the control of η from Step 5, and the control of ζ from Step 4, we infer Z  2 2 (Σ∗ ) ˇ + (Σ∗ ) GJ [Γ] ˇ +  3 (Σ∗ ) GJ+1 [Γ] ˇ . max |dk ω ˇ |2 . RJ+1 [R] 0 k≤J+1

Σ∗

Finally, gathering the estimates of Step 1 to Step 8, we infer (Σ∗ )

ˇ + GJ+1 [Γ]

(Σ∗ )

ˇ G0J+1 [Γ]

.

(Σ∗ )

ˇ + RJ+1 [R]

(Σ∗ )

2

ˇ + 3 GJ [Γ] 0

(Σ∗ )

and hence, for 0 small enough, (Σ∗ )

ˇ + GJ+1 [Γ]

(Σ∗ )

ˇ G0J+1 [Γ]

.

(Σ∗ )

ˇ + RJ+1 [R]

as desired. This concludes the proof of Proposition 8.31.

(Σ∗ )

ˇ GJ [Γ]

ˇ GJ+1 [Γ]

454

CHAPTER 8

8.8.2

Weighted estimates for transport equations along e4 in (ext)

Lemma 8.34. Let the following transport equation in

(ext)

M

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Also, let δB > 0. Then, f satisfies ! Z Z Z 2a−2 2 2a−2 2 sup r0 f . r f + r2a−1+δB h2 . r0 ≥4m0

{r=r0 }

(ext) M(≥4m ) 0

Σ∗

Proof. Multiply by f to obtain 1 a e4 (f 2 ) + κf 2 = hf. 2 2 Next, integrate over Su,r to obtain ! Z Z 1 1 2 e4 f = (e4 (f 2 ) + κf 2 ) 2 Su,r Su,r 2 Z Z a−1 2 = − κf + hf 2 Su,r Su,r Z Z Z a−1 a−1 = − κ f2 − κ ˇf 2 + hf 2 2 Su,r Su,r Su,r and hence 1 e4 2

!

Z f Su,r

2

a−1 + κ 2

a−1 f =− 2 Su,r

Z

2

Z

2

Z

κ ˇf +

hf.

Su,r

Su,r

Also, we multiply by r2a−2 which yields ! Z Z Z 1 a − 1 2a−2 2a−2 2 e4 r f =− r κ ˇ f 2 + r2a−2 hf 2 2 Su,r Su,r Su,r where we used the fact that 2e4 (r) = rκ. This yields ! Z Z Z 1 2a−1+δB 2a−2 2 2a−3−δB 2 −e4 r f ≤r f + r h2 4 Su,r Su,r Su,r and hence −e4

e

−1 −δB −δB r 2a−2

!

Z

r

f Su,r

2

−1 −δB

. e−δB

r

r2a−1+δB

Z

h2

Su,r

where we used the fact that 2e4 (r) = rκ = 2 + O(0 ). Integrating between r = r0

455

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and r = r∗ (u), where r∗ (u) is such that Su,r∗ (u) ⊂ Σ∗ , we infer Z

r02a−2

f 2 . r∗ (u)2a−2

Z

Su,r0

f2 +

Su,r∗ (u)

Z

r∗ (u)

r2a−1+δB

Z

r0

h2 .

(8.8.1)

Su,r

Remark 8.35. Note that we have the following consequences of the coarea formula r √ 2 r ς −κ − A √ dΣ∗ = ς − Υ + A dµu,Σ∗ du, d{r = r0 } = dµu,r0 du, ς 2 κ where we used in particular that Σ∗ = {u + r = cΣ∗ }. Also, we have in dM =

(ext)

M

4ς 2 dµu,r dudr. r2 κ2

(ext)

M, using in particular the dominant condition of r on Σ∗ , r   2  2m0 3 dΣ∗ = 1 + O 0 dµu,Σ∗ du, d{r = r0 } = 1 − (1 + O(0 )) dµu,r0 du, r0 We infer, in

and dM = (1 + O(0 ))dµu,r dudr. Integrating (8.8.1) in u ∈ [1, u∗ ], and relying on Remark 8.35 we deduce for r0 ≥ 4m0 Z Z Z 1 r02a−2 f2 . r2a−2 f 2 + r2a− 2 h2 {r=r0 }

(ext) M(r≥4m ) 0

Σ∗

as desired. This concludes the proof of the lemma. Corollary 8.36. Let the following transport equation in

(ext)

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Also, let δB > 0. Then, f satisfies for 5 ≤ k ≤ klarge + 1 ! Z sup r0 ≥4m0

Z

r02a−2

(dk f )2

{r=r0 }

r2a−2 (d≤k f )2 + sup

.

r0 ≥4m0

Σ∗

Z

r02a−2

!

Z

(d≤k−1 f )2

{r=r0 }

r2a−1+δB (d≤k h)2

+ (ext) M(≥4m ) 0

!2 +

sup (ext) M(≥4m

0)


ra |d≤k−5 f |



(ext)

0 ˇ G≥4m k−1 [Γ] +

(ext)

0 G≥4m [ˇ κ] k

2

.

456

CHAPTER 8

Proof. We first differentiate the equation for f with ( d/, T)l and obtain a e4 (( d/, T)l f ) + κ( d/, T)l f 2

=

hl ,

hl

:=

a ( d/, T)l h − [( d/, T)l , e4 ]f − [( d/, T)l , κ]f. 2

In view of Lemma 8.34, we deduce ! Z r02a−2

sup r0 ≥4m0

l

(( d/, T) f )

2

Z r

.

{r=r0 }

2a−2

l

2

Z

r2a−1+δB h2l .

(d f ) + (ext) M(≥4m

Σ∗

0)

Now, we have the following schematic commutation formulas [T, e4 ] = r−1 Γb d.

[ d/, e4 ] = Γg d + Γg ,

Together with the definition of hl and for 5 ≤ l ≤ klarge + 1, we deduce Z r2a−1+δB h2l (ext) M(≥4m ) 0 Z Z . r2a−1+δB (dl h)2 + 20 r2a−5+δB (d≤l f )2 (ext) M(≥4m

 +

(ext) M

0)


ra |d≤l−5 f |

sup (ext) M

2 

(ext)

ˇ + Gl−1 [Γ]

(ext)

2 Gl [ˇ κ]

and hence r02a−2

sup r0 ≥rT

Z . Σ∗

+20

2a−2

r Z

!

Z

l

l

(( d/, T) f )

2

{r=r0 }

Z

2

r2a−1+δB (dl h)2

(d f ) + (ext) M

r2a−5+δB (d≤l f )2 (ext) M(≥4m ) 0

!2 +

a

sup (ext) M(≥4m

0)

≤l−5

r |d




f|

(ext)

0 ˇ G≥4m l−1 [Γ] +

(ext)

2 0 G≥4m [ˇ κ ] l

or sup r0 ≥4m0

Z r

.

r02a−2

2a−2

l

!

Z

l

(( d/, T) f ) {r=r0 }

Z

2

r2a−1+δB (dl h)2

(d f ) + (ext) M(≥4m ) 0

Σ∗

+20

2

sup r0 ≥4m0

r02a−2

Z

! ≤l

(d f ) {r=r0 }

!2 +

sup (ext) M(≥4m

0)

2


ra |d≤l−5 f |



(ext)

0 ˇ G≥4m l−1 [Γ] +

(ext)

2 0 G≥4m [ˇ κ ] . l

457

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Together with the first equation which yields a re4 (( d/, T)l f ) + rκ( d/, T)l f 2

=

rhl ,

and hence   a (re4 )j (( d/, T)l f ) + (re4 )j−1 rκ( d/, T)l f = 2

(re4 )j−1 (rhl ),

we infer, for 0 > 0 small enough, and for 5 ≤ k ≤ klarge + 1, ! Z r02a−2

sup r0 ≥4m0

Z r

.

2a−2

≤k

(d

(dk f )2

{r=r0 }

Z

2

r2a−1+δB (d≤k h)2

f) + (ext) M(≥4m ) 0

Σ∗

r02a−2

+ sup r0 ≥4m0

!

Z

(d≤k−1 h)2

r0 ≥4m0

!2 +


ra |d≤k−5 f |

sup (ext) M(≥4m

+ sup

{r=r0 }

0)



(ext)

r02a−2

0 ˇ G≥4m k−1 [Γ] +

!

Z

(d≤k−1 f )2

{r=r0 }

(ext)

2 0 G≥4m [ˇ κ] . k

Using a trace estimate, we infer r02a−2

sup r0 ≥4m0

Z

!

Z

(dk f )2

{r=r0 }

r2a−2 (d≤k f )2 +

.

Z

r2a−1+δB (d≤k h)2 (ext) M(≥4m ) 0

Σ∗

+ sup r0 ≥4m0

r02a−2

!

Z

≤k−1

(d

f)

{r=r0 }

!2 +

a

sup (ext) M(≥4m

0)

2

≤k−5

r |d


f|



(ext)

≥4m0 ˇ Gk−1 [Γ] +

(ext)

2 0 G≥4m [ˇ κ] k

as desired. This concludes the proof of the corollary. Lemma 8.37. Let the following transport equation in

(ext)

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Let b > 2a − 2. Then, f satisfies ! Z Z sup r0b f2 + rb−1 f 2 r0 ≥4m0

Z . Σ∗

rb f 2 +

(ext) M(≥4m

{r=r0 }

Z

rb+1 h2 . (ext) M(≥4m

0)

0)

458

CHAPTER 8

Proof. Recall from Lemma 8.34 the following identity: ! Z Z Z Z 1 a−1 a−1 2 e4 f + κ f2 = − κ ˇf 2 + hf. 2 2 2 Su,r Su,r Su,r Su,r We multiply by rb which yields !   Z Z Z Z 1 1 b a−1 b b 2 b 2 2 b e4 r f + a−1− κ r f =− r κ ˇf + r hf 2 2 2 2 Su,r Su,r Su,r Su,r where we used the fact that 2e4 (r) = rκ. We choose b > 2a − 2 and integrate between r = r0 and r = r∗ (u), where r∗ (u) is such that Su,r∗ (u) ⊂ Σ∗ , which yields Z

Z

rb f 2 +

Su,r0

r∗

Z

r0

rb−1 f 2 .

Z

Su,r

rb f 2 +

Z

Su,r∗

r∗

Z

r0

rb+1 h2 .

Su,r

Then, integrating in u in u ∈ [1, u∗ ], and relying on Remark 8.35 we deduce for r0 ≥ 4m0 , Z Z Z Z b 2 b−1 2 b 2 r0 f + r f . r f + rb+1 h2 . (ext) M∩{r≥r

{r=r0 }

0}

(ext) M(≥4m

Σ∗

0)

This concludes the proof of the lemma. (ext)

Corollary 8.38. Let the following transport equation in

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Let b > 2a − 2. Then, f satisfies for 5 ≤ l ≤ klarge + 1 ! Z Z r0b

sup r0 ≥4m0

Z

b

≤k

r (d

.

(dk f )2

rb−1 (dk f )2

+ (ext) M(≥4m ) 0

{r=r0 } 2

f ) + sup r0 ≥4m0

Σ∗

r0b

!

Z

≤k−1

(d

+

sup (ext) M(≥4m

0)

≤k−5

r |d

Z


f|



(ext)

rb−1 (d≤k h)2

+ (ext) M(≥4m

{r=r0 }

!2 b

f)

2

0 ˇ G≥4m k−1 [Γ] +

(ext)

0)

2 0 G≥4m [ˇ κ ] . k

Proof. The proof is based on Lemma 8.37. It is similar to the one of Corollary 8.36 and left to the reader. Lemma 8.39. Let the following transport equation in

(ext)

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies Z Z Z sup f2 . f2 + h2 . rT ≤r0 ≤4m0

{r=r0 }

{r=4m0 }

(ext) M(≤4m ) 0

459

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Proof. Let b > 2a − 2. Recall from Lemma 8.37 the following identity: !   Z Z Z Z 1 1 b a−1 b b 2 e4 r f + a−1− κ rb f 2 = − r κ ˇ f 2 + rb hf. 2 2 2 2 Su,r Su,r Su,r Su,r Choosing b = 2a, we obtain ! Z Z Z Z 1 1 a−1 b b 2 e4 r f − κ rb f 2 = − r κ ˇ f 2 + rb hf. 2 2 Su,r 2 Su,r Su,r Su,r Next, let 1 ≤ u ≤ u∗ and rT ≤ r0 ≤ 4m0 . We now integrate in r0 ≤ r ≤ 4m0 and along Cu in (ext) M. Since r is bounded on (ext) M(r ≤ 4m0 ) from above and below, we obtain, for 0 > 0 small enough, Z Z Z 4m0 Z 2 2 f . f + h2 . Su,r0

Su,4m0

We may now integrate in u to deduce Z u∗ Z Z u∗ Z f2 . 1

Su,r0

1

f2 +

u∗

1

Z

{r=r0 }

Su,r

Z

Su,4m0

Relying on Remark 8.35 we deduce Z sup f2 . rT ≤r0 ≤4m0

rT

f2 +

{r=4m0 }

Z

4m0

Z

rT

h2 .

Su,r

Z

h2 (ext) M(≤4m

0)

as desired. This concludes the proof of the lemma. Corollary 8.40. Let the following transport equation in

(ext)

M

a e4 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies for 5 ≤ l ≤ klarge + 1 Z sup (dk f )2 rT ≤r0 ≤4m0 {r=r0 } Z Z Z ≤k 2 ≤k−1 2 . (d f ) + sup (d f) + (d≤k h)2 rT ≤r0 ≤4m0

{r=4m0 }

!2 +

sup (ext) M(r≤4m ) 0

|d≤k−5 f |




(ext) M(≤4m

{r=r0 }



(ext)

0 ˇ G≤4m k−1 [Γ] +

(ext)

0)

0 G≤4m [ˇ κ] k

2

.

Proof. The proof is based on Lemma 8.39. It is similar to the one of Corollary 8.36 and left to the reader.

460 8.8.3

CHAPTER 8

Several identities

The goal of this section is to prove the identities below that will be used to avoid losing derivatives when controlling the weighted energies of the Ricci coefficients. Lemma 8.41. We have     e4 /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ     −1 ? +2κ /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 + / d / d ρ ˇ 2 2 1   1 ? 1 1 = − − ϑ /d2 + ζe4 (Φ) − β /d?1 κ − ϑ /d1 /d?1 κ + /d?1 (κ + ϑ) /d?1 κ + ( /d?1 κ)2 2 2 2     −1 ? −1 ? ? −1 −2κϑ /d4 /d3 /d2 + /d2 /d1 ρˇ + (κϑ + 2α) /d4 /d3 /d2 + /d?2 /d−1 ˇ 1 ρ h  i +ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 , e4 ρ     3 3 −1 ? −1 ? +ϑ /d4 /d3 /d2 + /d2 /d1 κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2 1 1 − ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) − ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 + /d3 /d2 (−2β − /d?1 κ + κζ − ϑζ) /d?3 ϑ + (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ, 2 2     e4 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 2κ eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ 3 1 = − µeθ (κ) − ϑeθ (µ) − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β 2h 2 i
? ? −ϑ /d2 /d2 ( /d1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ       1 1 ? −1 2 + /d?3 ϑ /d?2 /d−1 ρ ˇ − e ϑ / d / d −ˇ µ + ϑϑ − e ϑ κϑ + ζ θ 2 1 θ 1 4 8   3 3 −2ζ /d1 /d?1 κ − 2eθ (κ) /d?2 ζ − 2(κζ + β + ϑζ)ˇ ρ − 2ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2  
3 +2β /d?2 ζ + eθ κζ 2 + 2κ ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ , 2 and e4 (eθ (κ) − 4β) + κ(eθ (κ) − 4β) 1 1 = 2eθ (µ) + 12ρζ − κeθ (κ) + 4ϑβ − ϑeθ (κ) − eθ (ϑϑ) + 2eθ (ζ 2 ) 2 2   1 = 2 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 12ρζ − κeθ (κ) − 2ϑ /d2 /d?2 ( /d?1 /d1 )−1 β 2 1 +2ζ ρˇ + 4ϑβ − ϑ(eθ (κ) − 4β) − 2ϑβ − eθ (ϑϑ) + 2eθ (ζ 2 ). 2 Proof. Recall Raychaudhuri 1 e4 (κ) + κ2 2

1 = − ϑ2 . 2

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

We commute with /d1 /d?1 which yields   1 1 e4 ( /d1 /d?1 κ) + 2κ /d1 /d?1 κ = − − ϑ /d?2 + ζe4 (Φ) − β /d?1 κ − ϑ /d1 /d?1 κ 2 2 1 ? 1 + /d1 (κ + ϑ) /d?1 κ + ( /d?1 κ)2 − /d1 /d?1 (ϑ2 ). 2 2 We have in view of Codazzi for ϑ

= = = =

=

1 /d1 /d?1 (ϑ2 ) 2 /d1 (ϑ /d?1 ϑ) 1 /d1 (ϑ /d?3 ϑ − ϑ /d2 ϑ) 2  1  ? −1 /d1 ϑ /d3 /d2 (−2β − /d?1 κ + κζ − ϑζ) − ϑ(−2β − /d?1 κ + κζ − ϑζ) 2 1 1 ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) + ϑ /d?2 (−2β − /d?1 κ + κζ − ϑζ) 2 (−2β − / 2 2 1 ? −1 1 − /d3 /d2 (−2β − /d?1 κ + κζ − ϑζ) /d?3 ϑ − (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ 2 2 1 −ϑ /d4 /d?3 /d−1 d?2 β + ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) 2 β − ϑ/ 2 (− / 2 1 1 + ϑ /d?2 (− /d?1 κ + κζ − ϑζ) − /d?3 /d−1 d?1 κ + κζ − ϑζ) /d?3 ϑ 2 (−2β − / 2 2 1 − (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2

Together with Bianchi for e4 (ˇ ρ), we infer

=

1 /d1 /d?1 (ϑ2 ) 2     3 3 −1 ? −1 ? −ϑ /d4 /d3 /d2 + /d2 /d1 e4 (ˇ ρ) + κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2 1 1 + ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) + ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 − /d3 /d2 (−2β − /d?1 κ + κζ − ϑζ) /d?3 ϑ − (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ 2    h 2  i

= −ϑe4 /d4 /d?3 /d−1 d?2 /d−1 ˇ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ 2 + / 1 , e4 ρ     3 3 −ϑ /d4 /d?3 /d−1 d?2 /d−1 κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 + / 1 2 2 1 1 + ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) + ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 ? ? − /d3 /d2 (−2β − /d1 κ + κζ − ϑζ) /d3 ϑ − (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2 2

461

462

CHAPTER 8

In view of the null structure equation for e4 (ϑ), we infer 1 /d1 /d?1 (ϑ2 ) 2  

    = −e4 ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ − (κϑ + 2α) /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ 2 + / 1 ρ h  i −ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 , e4 ρ     3 3 −1 ? −ϑ /d4 /d?3 /d−1 + / d / d κˇ ρ + ρˇ κ − Err[e ρ ˇ ] 2 4 2 1 2 2 1 1 + ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) + ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 ? ? − /d3 /d2 (−2β − /d1 κ + κζ − ϑζ) /d3 ϑ − (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2 2 This yields     e4 /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ     +2κ /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ   1 ? 1 1 = − − ϑ /d2 + ζe4 (Φ) − β /d?1 κ − ϑ /d1 /d?1 κ + /d?1 (κ + ϑ) /d?1 κ + ( /d?1 κ)2 2 2 2     −1 ? −1 ? ? −1 −2κϑ /d4 /d3 /d2 + /d2 /d1 ρˇ + (κϑ + 2α) /d4 /d3 /d2 + /d?2 /d−1 ˇ 1 ρ h  i +ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 , e4 ρ     3 3 −1 ? −1 ? +ϑ /d4 /d3 /d2 + /d2 /d1 κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2 1 1 − ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) − ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 + /d3 /d2 (−2β − /d?1 κ + κζ − ϑζ) /d?3 ϑ + (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2 2 Next, recall that we have 3 e4 µ + κµ = 2

−ϑ /d?2 ζ

 −ϑ

1 κϑ + ζ 2 8



  3 + 2eθ (κ) − 2β + κζ ζ. 2

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

463

We commute with eθ which yields

=

=

e4 (eθ (µ)) + 2κeθ (µ)    3 1 1 − µeθ (κ) − ϑeθ (µ) − eθ ϑ /d?2 /d−1 −ˇ µ − ρ ˇ + ϑϑ 1 2 2 4       1 3 −eθ ϑ κϑ + ζ 2 + eθ 2eθ (κ) − 2β + κζ ζ 8 2 3 1 − µeθ (κ) − ϑeθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 /d?1 ρ + /d?3 ϑ /d?2 /d−1 ˇ 1 ρ 2  2     1 1 −eθ ϑ /d?2 /d−1 −ˇ µ + ϑϑ − eθ ϑ κϑ + ζ 2 1 4 8
3 −2ζ /d1 /d?1 κ − 2eθ (κ) /d?2 ζ − 2ζ /d1 β + 2β /d?2 ζ + eθ κζ 2 . 2

ρ), we have Now, using the Bianchi identities for e4 (β) and e4 (ˇ ϑ /d2 /d?2 ( /d?1 /d1 )−1 /d?1 ρ
= −ϑ /d2 /d?2 ( /d?1 /d1 )−1 e4 β + κβ + 3ρζ + ϑβ   = −e4 ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + e4 (ϑ) /d2 /d?2 ( /d?1 /d1 )−1 β h i
−ϑ /d2 /d?2 ( /d?1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ   = −e4 ϑ /d2 /d?2 ( /d?1 /d1 )−1 β − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β h i
−ϑ /d2 /d?2 ( /d?1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ and ζ /d1 β

= = =

  3 3 ζ e4 (ˇ ρ) + κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2   3 3 e4 (ζ ρˇ) − e4 (ζ)ˇ ρ+ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2   3 3 e4 (ζ ρˇ) + (κζ + β + ϑζ)ˇ ρ+ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] . 2 2

We infer   e4 eθ (µ)) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 2κeθ (µ) =

3 1 − µeθ (κ) − ϑeθ (µ) − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β 2h 2 i
? ? −ϑ /d2 /d2 ( /d1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ       1 1 ? −1 2 + /d?3 ϑ /d?2 /d−1 ρ ˇ − e ϑ / d / d −ˇ µ + ϑϑ − e ϑ κϑ + ζ θ 2 1 θ 1 4 8   3 3 −2ζ /d1 /d?1 κ − 2eθ (κ) /d?2 ζ − 2(κζ + β + ϑζ)ˇ ρ − 2ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2
3 +2β /d?2 ζ + eθ κζ 2 2

464

CHAPTER 8

and hence     e4 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 2κ eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ 3 1 = − µeθ (κ) − ϑeθ (µ) − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β 2h 2 i
−ϑ /d2 /d?2 ( /d?1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ       1 1 ? ? −1 ? −1 2 + /d3 ϑ /d2 /d1 ρˇ − eθ ϑ /d2 /d1 −ˇ µ + ϑϑ − eθ ϑ κϑ + ζ 4 8   3 3 ? ? −2ζ /d1 /d1 κ − 2eθ (κ) /d2 ζ − 2(κζ + β + ϑζ)ˇ ρ − 2ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2  
3 +2β /d?2 ζ + eθ κζ 2 + 2κ ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ . 2 Finally, recall that we have 1 e4 (κ) + κκ = 2 =

1 −2 /d1 ζ + 2ρ − ϑϑ + 2ζ 2 2 2µ + 4ρ − ϑϑ + 2ζ 2 .

We commute with eθ which yields e4 (eθ (κ)) + κeθ (κ)

=

1 1 2eθ (µ) + 4eθ (ρ) − κeθ (κ) − ϑeθ (κ) − eθ (ϑϑ) + 2eθ (ζ 2 ). 2 2

Together with Bianchi for e4 (β), we infer e4 (eθ (κ) − 4β) + κ(eθ (κ) − 4β) 1 1 = 2eθ (µ) + 12ρζ − κeθ (κ) + 4ϑβ − ϑeθ (κ) − eθ (ϑϑ) + 2eθ (ζ 2 ) 2 2   1 ? ? −1 = 2 eθ (µ) + ϑ /d2 /d2 ( /d1 /d1 ) β + 2ζ ρˇ + 12ρζ − κeθ (κ) − 2ϑ /d2 /d?2 ( /d?1 /d1 )−1 β 2 1 +2ζ ρˇ + 4ϑβ − ϑ(eθ (κ) − 4β) − 2ϑβ − eθ (ϑϑ) + 2eθ (ζ 2 ). 2 This concludes the proof of the lemma. 8.8.4

Proof of Proposition 8.32

We introduce the following notation which will constantly appear on the RHS of the equalities below: ˇ R] ˇ N ≥4m0 [J, Γ,

:=

ˇ + (Σ∗ ) G0J+1 [Γ] ˇ + (ext) RJ+1 [R] ˇ + GJ+1 [Γ]   0 (ext) ≥4m0 ˇ 0 ˇ +0 (ext) G≥4m GJ+1 [Γ] . J+1 [Γ] + (Σ∗ )

Step 1. Recall that e4 (ϑ) + κϑ

= −2α.

(ext)

ˇ GJ [Γ] (8.8.2)

465

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

In view of Corollary 8.36 with a = 2, we have for any r0 ≥ 4m0 Z  2 ˇ R] ˇ . max sup r02 (dk ϑ)2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

{r=r0 }

Step 2. Next, recall that 1 1 2 − ϑ2 + ϑ − κ ˇ2 . 4 4

e4 (ˇ κ) + κ κ ˇ =

In view of Corollary 8.36 with a = 2, we have for any r0 ≥ 4m0 Z  2 2 ˇ R] ˇ max sup r0 (dk κ ˇ )2 . N ≥4m0 [J, Γ, k≤J+2 r0 ≥4m0

{r=r0 }

where we have used the null structure equations for e4 (ϑ), e3 (ϑ) and /d2 ϑ to avoid a loss of one derivative for the RHS. Step 3. Next, recall that = −β − ϑζ.

e4 (ζ) + κζ

In view of Corollary 8.36 with a = 2, we have for any r0 ≥ 4m0 Z  2 ˇ R] ˇ . max sup r02 (dk ζ)2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

{r=r0 }

Step 4. Next, recall that 3 e4 (ˇ µ) + κˇ µ = 2

3 − µˇ κ + Err[e4 µ ˇ]. 2

In view of Corollary 8.36 with a = 3, commuting with d/ and T, we have for any r0 ≥ 4m0 Z  2 4 ˇ R] ˇ max sup r0 (dk µ ˇ)2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

{r=r0 }

where we used the estimates for κ ˇ on

(ext)

M derived in Step 2.

Step 5. Next, recall that 1 κ = e4 (ˇ κ) + κˇ 2

1 − κˇ κ + 2ˇ ρ − 2 /d1 ζ + Err[e4 κ ˇ ]. 2

In view of Corollary 8.38 with a = 1 and b = 2 − δB which satisfy the constraint b > 2a − 2, we have for any r0 ≥ 4m0 ! Z Z max

sup

k≤J+1 r0 ≥4m0

.



r02−δB

ˇ R] ˇ N ≥4m0 [J, Γ,

2

(dk κ ˇ )2

{r=r0 }

r1−δB (dk κ ˇ )2

+ (ext) M(r≥4m

0)

466

CHAPTER 8

where we used the estimates for κ ˇ and µ ˇ on and Step 4.

(ext)

M derived respectively in Step 2

Step 6. Next, recall that 1 e4 (ϑ) + κϑ 2

= =

1 2 /d?2 ζ − κϑ + 2ζ 2 2  1 1 ? −1 2 /d2 /d1 −µ − ρ + ϑϑ − κϑ + 2ζ 2 . 4 2

In view of Corollary 8.36 with a = 1, we have for any r0 ≥ 4m0 Z  2 ˇ R] ˇ max sup (dk ϑ)2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

{r=r0 }

where we used the estimates for ϑ and µ ˇ on and Step 4.

(ext)

M derived respectively in Step 1

Step 7. Next, recall that e4 (ˇ ω)

= ρˇ + 3ζ 2 − 3ζ 2 − κ ˇω ˇ.

In view of Corollary 8.38 with a = 0 and b = 0 which satisfy the required constraint b > 2a − 2, we have for any r0 ≥ 4m0 ! Z Z  2 k 2 ˇ R] ˇ . max sup (d ω ˇ) + r−1 (dk ω ˇ )2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

(ext) M(≥4m

{r=r0 }

0)

Step 8. In order to estimate ξ in Step 9, we derive an estimate for e3 (ζ)+β. Recall that we have e4 (ζ) + κζ

=

−β − ϑζ.

Commuting with e3 , we infer e4 (e3 (ζ)) + [e3 , e4 ]ζ + κe3 (ζ) + e3 (κ)ζ

= −e3 (β) − ϑζ.

In view of the null structure equation for e3 (κ), the Bianchi identity for e3 (β) and the commutator identity for [e3 , e4 ], we infer   e4 (e3 (ζ)) + 2ωe4 + 4ζeθ ζ + κe3 (ζ)   1 1 2 + − κκ + 2ωκ + 2 /d1 ζ + 2ρ − ϑϑ + 2ζ ζ 2 2 = (κ − 2ω)β + /d?1 ρ − 3ζρ + ϑβ − ξα − ϑζ. Together with the null structure equation for e4 (ζ), the Bianchi identity for e4 (β)

467

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

to get rid of the term /d?1 ρ, and the definition of µ, we infer   1 1 e4 (e3 (ζ)) + 2ω (−κζ − β − ϑζ) + 4ζ /d?1 /d−1 µ ˇ + ρ ˇ − ϑϑ + ϑϑ 1 4 4   1 +κe3 (ζ) + − κκ + 2ωκ − 2µ + 2ζ 2 ζ 2 = (κ − 2ω)β − e4 (β) − κβ − 3ζρ − ϑβ − 3ζρ + ϑβ − ξα − ϑζ and hence e4 (e3 (ζ) + β) + κ(e3 (ζ) + β)   1 κβ + κκ + 2µ − 6ρ ζ 2

=

  1 1 −ϑβ + ϑβ − ξα − ϑζ + 2ωϑζ − 4ζ /d?1 /d−1 µ ˇ + ρ ˇ − ϑϑ + ϑϑ − 2ζ 3 . 1 4 4 In view of Corollary 8.36 with a = 2, we have for any r0 ≥ 4m0 Z  2 2 ˇ R] ˇ max sup r0 (dk (e3 (ζ) + β))2 . N ≥4m0 [J, Γ, k≤J+1 r0 ≥4m0

{r=r0 }

where we used the estimates for ζ derived in Step 3. Step 9. Next, recall that we have e4 (ξ)

= =

−e3 (ζ) + β − κζ − ζϑ

−(e3 (ζ) + β) + 2β − κζ − ζϑ.

In view of Corollary 8.38 with a = 0 and b = −δB which satisfy the constraint b > 2a − 2, we have for any r0 ≥ 4m0 ! Z Z −δB k 2 max sup r0 (d ξ) + r−1−δB (dk ξ)2 k≤J+1 r0 ≥4m0

.



{r=r0 }

(ext) M(r≥4m

0)

2 ˇ R] ˇ N ≥4m0 [J, Γ,

where we used the estimates for ζ and e3 (ζ) + β on Step 3 and Step 8.

(ext)

M derived respectively in

468

CHAPTER 8

Step 10. Recall that we have     e4 /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ     −1 ? +2κ /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 + / d / d ρ ˇ 2 2 1   1 1 1 = − − ϑ /d?2 + ζe4 (Φ) − β /d?1 κ − ϑ /d1 /d?1 κ + /d?1 (κ + ϑ) /d?1 κ + ( /d?1 κ)2 2 2 2     −1 ? −1 ? ? −1 −2κϑ /d4 /d3 /d2 + /d2 /d1 ρˇ + (κϑ + 2α) /d4 /d3 /d2 + /d?2 /d−1 ˇ 1 ρ h  i +ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 , e4 ρ     3 3 −1 ? +ϑ /d4 /d?3 /d−1 + / d / d κˇ ρ + ρˇ κ − Err[e ρ ˇ ] 2 4 2 1 2 2 1 1 − ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) − ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 ? ? + /d3 /d2 (−2β − /d1 κ + κζ − ϑζ) /d3 ϑ + (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2 2 In view of Corollary 8.36 with a = 4, we have Z     2 max sup r06 dk /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ k≤J+1 r0 ≥4m0

.



{r=r0 } 2

ˇ R] ˇ N ≥4m0 [J, Γ,

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 2 to estimate the terms of the RHS with one derivative of κ ˇ, • the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the identity     d/ /d?1 κ = d/ /d−1 ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 1 2 + / 1 ρ     −1 ? ? −1 ? + d/ /d−1 / d / d κ − ϑ / d / d / d + / d / d ρ ˇ 1 1 4 3 2 2 1 1 to estimate the terms of the RHS with two angular derivatives of κ ˇ.

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

469

Step 11. Recall that we have     e4 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 2κ eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ 3 1 = − µeθ (κ) − ϑeθ (µ) − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β 2h 2 i
? ? −ϑ /d2 /d2 ( /d1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ       1 1 ? −1 2 + /d?3 ϑ /d?2 /d−1 ρ ˇ − e ϑ / d / d −ˇ µ + ϑϑ − e ϑ κϑ + ζ θ 2 1 θ 1 4 8   3 3 −2ζ /d1 /d?1 κ − 2eθ (κ) /d?2 ζ − 2(κζ + β + ϑζ)ˇ ρ − 2ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2  
3 +2β /d?2 ζ + eθ κζ 2 + 2κ ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ . 2 In view of Corollary 8.36 with a = 4, we have Z

2 6 max sup r0 dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ k≤J+1 r0 ≥4m0 {r=r0 } Z  2

2 ≥4m0 2 ˇ ˇ . N [J, Γ, R] +  d≤J+1 eθ (κ) − 4β , (ext) M(≥4m ) 0

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 2 to estimate the terms of the RHS with one derivative of κ ˇ, • the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the fact that eθ (κ) = (eθ (κ) − 4β) + 4β to estimate the term with one angular derivative of κ, • the identity     −1 ? −1 ? d/ /d?1 κ = d/ /d−1 ϑ / d / d / d + / d / d ρ ˇ 4 3 2 1 2 1     −1 ? ? −1 + d/ /d1 /d1 /d1 κ − ϑ /d4 /d3 /d2 + /d?2 /d−1 ˇ 1 ρ and the estimates of Step 10 to estimate the terms of the RHS with two angular derivatives of κ ˇ. Step 12. Recall that we have e4 (eθ (κ) − 4β) + κ(eθ (κ) − 4β) 1 1 = 2eθ (µ) + 12ρζ − κeθ (κ) + 4ϑβ − ϑeθ (κ) − eθ (ϑϑ) + 2eθ (ζ 2 ) 2 2   1 ? ? −1 = 2 eθ (µ) + ϑ /d2 /d2 ( /d1 /d1 ) β + 2ζ ρˇ + 12ρζ − κeθ (κ) − 2ϑ /d2 /d?2 ( /d?1 /d1 )−1 β 2 1 +2ζ ρˇ + 4ϑβ − ϑ(eθ (κ) − 4β) − 2ϑβ − eθ (ϑϑ) + 2eθ (ζ 2 ). 2

470

CHAPTER 8

In view of Corollary 8.36 with a = 2, we have Z

2 max sup r02 dk eθ (κ) − 4β k≤J+1 r0 ≥4m0

.

{r=r0 }

2



ˇ R] ˇ N ≥4m0 [J, Γ, Z

2 + r4 d≤J+1 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ , (ext) M(≥4m ) 0

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 2 to estimate the terms of the RHS with one derivative of κ ˇ, • the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the estimate for ζ of Step 3. Together with the estimate of Step 11, we infer Z

2 6 max sup r0 dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ k≤J+1 r0 ≥4m0 {r=r0 } Z

2 2 + max sup r0 dk eθ (κ) − 4β k≤J+1 r0 ≥4m0

.



{r=r0 }

ˇ R] ˇ N ≥4m0 [J, Γ,

2

.

Finally, we have obtained Z max

r04 (dk µ ˇ)2 + r02 (dk ϑ)2 + r02 (dk ζ)2

sup

k≤J+1 r0 ≥4m0

{r=r0 }

+r02 (dk (e3 (ζ)

2

!!

.



+ β)) +

ˇ R] ˇ N ≥4m0 [J, Γ,

max

sup

k≤J+2 r0 ≥4m0

r02

2

r02−δB (dk κ ˇ )2

+ (d ϑ) + (d ω ˇ) +



2 !

k

2

k

2

r0−δB (dk ξ)2

,

Z

k

d {r=r0 }



2 κ− r

.

 2 ˇ R] ˇ , N ≥4m0 [J, Γ,

471

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and (

Z max

sup

k≤J+1 rT ≤r0 ≤4m0

+r06 .



k

d

eθ (µ) +

{r=r0 }

    2 r06 dk /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ

ϑ /d2 /d?2 ( /d?1 /d1 )−1 β



2

2 + 2ζ ρˇ + r02 dk eθ (κ) − 4β

)!

2 ˇ R] ˇ . N ≥4m0 [J, Γ,

ˇ R], ˇ and of the various norms, we In view of the definition (8.8.2) of N ≥4m0 [J, Γ, infer (ext) ≥4m0 0 ˇ 0 ˇ G≥4m GJ+1 [Γ] J+1 [Γ] + (Σ∗ ) ˇ + (Σ∗ ) G0J+1 [Γ] ˇ + (ext) RJ+1 [R] ˇ + GJ+1 [Γ]   ˇ + (ext) G≥4m0 0 [Γ] ˇ +0 (ext) G≥4m0 [Γ] (ext)

.

J+1

(ext)

ˇ GJ [Γ]

J+1

and hence, for 0 small enough, (ext) ≥4m0 0 ˇ 0 ˇ G≥4m GJ+1 [Γ] J+1 [Γ] + (Σ∗ ) ˇ + (Σ∗ ) G0 [Γ] ˇ + (ext) RJ+1 [R] ˇ + GJ+1 [Γ] (ext)

.

J+1

(ext)

ˇ GJ [Γ].

This concludes the proof of Proposition 8.32. 8.8.5

Proof of Proposition 8.33

In the proof below, we will repeatedly use the following estimate: Z max (dk f )2 k≤J+1 (ext) M(r≤4m ) 0 Z   . max (dk f )2 + (dk Nf )2 + (dk e4 f )2 + (dk d/f )2 (8.8.3) k≤J

(ext) M(r≤4m ) 0

which follows from the fact that d = ( d/, re4 , e3 ) and e3 = Υe4 − 2N, where we recall that  1 N = Υe4 − e3 . 2 Also, we introduce the following notation which will constantly appear on the RHS of the equalities below: ˇ R] ˇ N ≤4m0 [J, Γ,

:=

(ext) ≥4m0 0 ˇ 0 ˇ ˇ + G≥4m GJ+1 [Γ] + (ext) RJ+1 [R] J+1 [Γ] +   0 (ext) ≤4m0 ˇ 0 ˇ +0 (ext) G≤4m GJ+1 [Γ] . J+1 [Γ] + (ext)

Step 1. Recall that e4 (ˇ κ) + κˇ κ =

Err[e4 κ ˇ ].

(ext)

ˇ GJ [Γ]

(8.8.4)

472

CHAPTER 8

In view of Corollary 8.40, we have Z max sup k≤J+2 rT ≤r0 ≤4m0

(dk κ ˇ )2

.



ˇ R] ˇ N ≤4m0 [J, Γ,

2

{r=r0 }

where we have used the null structure equations for e4 (ϑ), e3 (ϑ) and /d2 ϑ to avoid losing one derivative. Step 2. Next, recall that 3 3 e4 (ˇ µ) + κˇ µ = − µˇ κ + Err[e4 µ ˇ]. 2 2 In view of Corollary 8.40, we have Z max sup k≤J+1 rT ≤r0 ≤4m0

(dk µ ˇ)2

.



2 ˇ R] ˇ N ≤4m0 [J, Γ,

{r=r0 }

where we have used the estimates for κ ˇ of Step 1. Step 3. Next, recall that e4 (ζ) + κζ

−β − ϑζ.

=

In view of Corollary 8.40, we have Z   max sup (dk e4 ζ)2 + (dk ζ)2 k≤J rT ≤r0 ≤4m0

.



2 ˇ R] ˇ . N ≤4m0 [J, Γ,

{r=r0 }

Also, commuting first with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ . max sup (dk Nζ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

Furthermore, in view of the definition of µ and a Poincar´e inequality for /d1 , we have Z  2 ˇ R] ˇ max sup (dk d/ζ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used a trace estimate and the estimate for µ ˇ of Step 2. The above estimates, together with (8.8.3), imply Z  2 ˇ R] ˇ . max sup (dk ζ)2 . N ≤4m0 [J, Γ, k≤J+1 rT ≤r0 ≤4m0

{r=r0 }

Step 4. Recall that e4 (ϑ) + κϑ

=

−2α.

In view of Corollary 8.40, we have Z   max sup (dk e4 ϑ)2 + (dk ϑ)2 k≤J rT ≤r0 ≤4m0

{r=r0 }

.



2 ˇ R] ˇ . N ≤4m0 [J, Γ,

473

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Also, commuting first one time with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ . max sup (dk Nϑ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

Furthermore, in view of Codazzi for ϑ, and a Poincar´e inequality for /d2 , we have Z  2 ˇ R] ˇ max sup (dk d/ϑ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used a trace estimate, and the estimate for κ ˇ and ζ respectively in Step 1 and Step 3. The above estimates, together with (8.8.3), imply Z  2 ˇ R] ˇ . max sup (dk ϑ)2 . N ≤4m0 [J, Γ, k≤J+1 rT ≤r0 ≤4m0

{r=r0 }

Step 5. Recall that we have 1 1 e4 (ˇ κ) + κˇ κ = − κˇ κ − 2 /d1 ζ + 2ˇ ρ + Err[e4 κ ˇ] 2 2 1 1 = − κˇ κ + 2ˇ µ + 4ˇ ρ − ϑϑ + Err[e4 κ ˇ ]. 2 2 In view of Corollary 8.40, we have Z   max sup (dk e4 κ ˇ )2 + (dk κ ˇ )2 k≤J rT ≤r0 ≤4m0

.



2 ˇ R] ˇ N ≤4m0 [J, Γ,

{r=r0 }

where we have used the estimates for κ ˇ and µ ˇ derived respectively in Step 1 and Step 2. Also, commuting first one time with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ max sup (dk Nˇ κ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used the estimates for κ ˇ and µ ˇ derived respectively in Step 1 and Step 2. Furthermore, commuting the equation for e4 (κ) once with eθ , we have 1 1 e4 (eθ (κ)) + κeθ (κ) = − κeθ (κ) + 2eθ (µ) + 4eθ (ρ) − eθ (ϑϑ) + 2eθ (ζ 2 ) − ϑeθ (κ). 2 2 Together with the Bianchi identity for e4 (β), we infer e4 (eθ (κ) − 4β) + κ(eθ (κ) − 4β)

1 = − κeθ (κ) + 2eθ (µ) + 12ρζ 2 1 +4ϑβ − eθ (ϑϑ) + 2eθ (ζ 2 ) − ϑeθ (κ). 2

In view of Corollary 8.40, we have Z   max sup (dk e4 (eθ (κ) − 4β))2 + (dk (eθ (κ) − 4β))2 .

k≤J rT ≤r0 ≤4m0

{r=r0 }



2

ˇ R] ˇ N ≤4m0 [J, Γ,

474

CHAPTER 8

where we have used the estimates for κ ˇ, µ ˇ and ζ derived respectively in Step 1, Step 2 and Step 3. The above estimates, together with (8.8.3), imply Z max sup (dk κ ˇ )2 k≤J+1 rT ≤r0 ≤4m0 {r=r } 0 Z  2 ≤4m0 ˇ R] ˇ . N [J, Γ, + max sup (dk β)2 k≤J rT ≤r0 ≤4m0

.



N

≤4m0

ˇ R] ˇ [J, Γ,

{r=r0 }

2

where we have used a trace estimate on {r = r0 } for rT ≤ r0 ≤ 4m0 . Step 6. Recall that we have e4 (ˇ ω)

= ρˇ + Err[e4 ω ˇ ].

In view of Corollary 8.40, we have Z   max sup (dk e4 ω ˇ )2 + (dk ω ˇ )2 k≤J rT ≤r0 ≤4m0

.



2 ˇ R] ˇ . N ≤4m0 [J, Γ,

{r=r0 }

Also, commuting first one time with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ . max sup (dk Nˇ ω )2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

Step 7. Recall that we have

=

e4 (e3 (ζ) + β) + κ(e3 (ζ) + β)   1 κβ + κκ + 2µ − 6ρ ζ 2   1 1 −ϑβ + ϑβ − ξα − ϑζ + 2ωϑζ − 4ζ /d?1 /d−1 µ ˇ + ρ ˇ − ϑϑ + ϑϑ − 2ζ 3 . 1 4 4

Commuting first one time with N, and in view of Corollary 8.40, we have Z  2 ˇ R] ˇ max sup (dk N(e3 (ζ) + β))2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used the estimate for ζ in Step 3. Step 8. Recall that we have e4 (ξ)

= −e3 (ζ) + β − κζ − ζϑ.

In view of Corollary 8.40, we have Z   max sup (dk e4 ξ)2 + (dk ξ)2 k≤J rT ≤r0 ≤4m0

{r=r0 }

.



2 ˇ R] ˇ N ≤4m0 [J, Γ,

475

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

where we have used the estimates for ζ derived in Step 3. Also, commuting first one time with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ max sup (dk Nξ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used the estimates for e3 (ζ) + β derived in Step 7. Step 9. Recall 2 /d?1 ω

1 1 e3 ζ + κζ − β − κξ + ϑζ − ϑξ. 2 2

=

Using a Poincar´e inequality for /d?1 , we infer Z max sup (dk d/ω ˇ )2 k≤J rT ≤r0 ≤4m0

.



ˇ R] ˇ N ≤4m0 [J, Γ,

2

{r=r0 }

where we have used a trace estimate and the estimate for ζ and ξ respectively in Step 3 and Step 8. The above estimates, together with the estimates for ω ˇ of Step 6 and (8.8.3), imply Z  2 ˇ R] ˇ . max sup (dk ω ˇ )2 . N ≤4m0 [J, Γ, k≤J+1 rT ≤r0 ≤4m0

{r=r0 }

Step 10. Recall that we have ˇ e4 (Ω) In view of Corollary 8.40, we have Z max sup k≤J+1 rT ≤r0 ≤4m0

ˇ = −2ˇ ω+κ ˇ Ω.

ˇ 2 (dk Ω)

.

 2 ˇ R] ˇ N ≤4m0 [J, Γ,

{r=r0 }

where we have used the estimates for ω ˇ derived in Step 9. Step 11. Recall  2 /d1 ξ

=

e3 (ˇ κ) + κ κ ˇ + 2ω κ ˇ + 2κ ω ˇ−

Using a Poincar´e inequality for /d1 , we infer Z max sup (dk d/ξ)2 k≤J rT ≤r0 ≤4m0

.

 1 ˇ − Err[e3 κ κκ − 2ρ Ω ˇ ]. 2



2 ˇ R] ˇ N ≤4m0 [J, Γ,

{r=r0 }

ˇ respectively in Step 5, Step 9 where we have used the estimates for κ ˇ, ω ˇ and Ω and Step 10. The above estimates, together with the estimates for ξ of Step 8 and (8.8.3), imply Z  2 ˇ R] ˇ . max sup (dk ξ)2 . N ≤4m0 [J, Γ, k≤J+1 rT ≤r0 ≤4m0

{r=r0 }

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CHAPTER 8

Step 12. Recall that 1 e4 (ϑ) + κϑ 2

= =

1 2 /d?2 ζ − κϑ + 2ζ 2 2  1 1 1 ? −1 2 /d2 /d1 −ˇ µ − ρˇ + ϑϑ − ϑϑ − κϑ + 2ζ 2 . 4 4 2

In view of Corollary 8.40, we have Z    2 ˇ R] ˇ max sup (dk e4 ϑ)2 + (dk ϑ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used the estimate for µ ˇ and ϑ respectively in Step 2 and Step 4. Also, commuting first one time with N, and proceeding analogously, we infer Z  2 ˇ R] ˇ max sup (dk Nϑ)2 . N ≤4m0 [J, Γ, k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used the estimate for µ ˇ and ϑ respectively in Step 2 and Step 4. Furthermore, in view of Codazzi for ϑ, and a Poincar´e inequality for /d2 , we have Z  2 ˇ R] ˇ N ≤4m0 [J, Γ, max sup (dk d/ϑ)2 . k≤J rT ≤r0 ≤4m0

{r=r0 }

where we have used a trace estimate and the estimate for κ ˇ and ζ respectively in Step 5 and Step 3. The above estimates, together with (8.8.3), imply Z  2 ˇ R] ˇ . max sup (dk ϑ)2 . N ≤4m0 [J, Γ, k≤J+1 rT ≤r0 ≤4m0

{r=r0 }

Step 13. Recall that we have     e4 /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 ρ     −1 ? +2κ /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 + / d / d ρ ˇ 2 2 1   1 1 1 = − − ϑ /d?2 + ζe4 (Φ) − β /d?1 κ − ϑ /d1 /d?1 κ + /d?1 (κ + ϑ) /d?1 κ + ( /d?1 κ)2 2 2 2     −1 ? −1 ? ? −1 −2κϑ /d4 /d3 /d2 + /d2 /d1 ρˇ + (κϑ + 2α) /d4 /d3 /d2 + /d?2 /d−1 ˇ 1 ρ h  i +ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 2 + / 1 , e4 ρ     3 3 −1 ? +ϑ /d4 /d?3 /d−1 + / d / d κˇ ρ + ρˇ κ − Err[e ρ ˇ ] 2 4 2 1 2 2 1 1 − ϑ /d4 /d?3 /d−1 d?1 κ + κζ − ϑζ) − ϑ /d?2 (− /d?1 κ + κζ − ϑζ) 2 (− / 2 2 1 ? −1 1 ? ? + /d3 /d2 (−2β − /d1 κ + κζ − ϑζ) /d3 ϑ + (−2β − /d?1 κ + κζ − ϑζ) /d2 ϑ. 2 2

477

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

In view of Corollary 8.40, we have Z     2 −1 ? max sup dk /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 + / d / d ρ ˇ 2 2 1 k≤J+1 rT ≤r0 ≤4m0

.



{r=r0 }

2 ˇ R] ˇ N ≤4m0 [J, Γ,

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 1 to estimate the terms of the RHS with one derivative of κ ˇ, • the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the identity     −1 ? −1 ? d/ /d?1 κ = d/ /d−1 ϑ / d / d / d + / d / d ρ ˇ 4 3 2 2 1 1     + d/ /d−1 /d1 /d?1 κ − ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 1 2 + / 1 ρ to estimate the terms of the RHS with two angular derivatives of κ ˇ. Step 14. Recall that we have     e4 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 2κ eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ 3 1 = − µeθ (κ) − ϑeθ (µ) − (κϑ + 2α) /d2 /d?2 ( /d?1 /d1 )−1 β 2h 2 i
? ? −ϑ /d2 /d2 ( /d1 /d1 )−1 , e4 β − ϑ /d2 /d?2 ( /d?1 /d1 )−1 κβ + 3ρζ + ϑβ       1 1 ? ? −1 ? −1 2 + /d3 ϑ /d2 /d1 ρˇ − eθ ϑ /d2 /d1 −ˇ µ + ϑϑ − eθ ϑ κϑ + ζ 4 8   3 3 ? ? −2ζ /d1 /d1 κ − 2eθ (κ) /d2 ζ − 2(κζ + β + ϑζ)ˇ ρ − 2ζ κˇ ρ + ρˇ κ − Err[e4 ρˇ] 2 2  
3 +2β /d?2 ζ + eθ κζ 2 + 2κ ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ . 2 In view of Corollary 8.40, we have Z max sup k≤J+1 rT ≤r0 ≤4m0

.





2 dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ

{r=r0 }

Z 2 ˇ R] ˇ N ≤4m0 [J, Γ, + 2 (ext) M(≤4m

0)

d≤J+1 eθ (κ) − 4β



2

,

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 1 to estimate the terms of the RHS with one derivative of κ ˇ,

478

CHAPTER 8

• the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the fact that eθ (κ) = (eθ (κ) − 4β) + 4β to estimate the term with one angular derivative of κ, • the identity     d/ /d?1 κ = d/ /d−1 ϑ /d4 /d?3 /d−1 d?2 /d−1 ˇ 1 2 + / 1 ρ     −1 ? ? −1 ? + d/ /d−1 / d / d κ − ϑ / d / d / d + / d / d ρ ˇ 1 1 4 3 2 1 2 1 and the estimates of Step 13 to estimate the terms of the RHS with two angular derivatives of κ ˇ. Step 15. Recall that we have e4 (eθ (κ) − 4β) + κ(eθ (κ) − 4β) 1 1 = 2eθ (µ) + 12ρζ − κeθ (κ) + 4ϑβ − ϑeθ (κ) − eθ (ϑϑ) + 2eθ (ζ 2 ) 2 2   1 = 2 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ + 12ρζ − κeθ (κ) − 2ϑ /d2 /d?2 ( /d?1 /d1 )−1 β 2 1 +2ζ ρˇ + 4ϑβ − ϑ(eθ (κ) − 4β) − 2ϑβ − eθ (ϑϑ) + 2eθ (ζ 2 ). 2 In view of Corollary 8.40, we have Z max sup k≤J+1 rT ≤r0 ≤4m0

.

{r=r0 }

dk eθ (κ) − 4β



2

 2 ˇ R] ˇ N ≤4m0 [J, Γ, Z

2 + d≤J+1 eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ . (ext) M(≤4m

0)

where we have used • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the fact that d/ϑ = d/ /d−1 d2 ϑ and Codazzi for ϑ to estimate the terms of the RHS 2 / with one angular derivative of ϑ, • the estimates of Step 1 to estimate the terms of the RHS with one derivative of κ ˇ, • the fact that d/ζ = d/ /d−1 d1 ζ and the definition of µ to estimate terms of the 1 / RHS with one angular derivative of ζ, • the estimate for ζ of Step 3.

479

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

Together with the estimate of Step 14, we infer Z

2 max sup dk eθ (µ) + ϑ /d2 /d?2 ( /d?1 /d1 )−1 β + 2ζ ρˇ k≤J+1 rT ≤r0 ≤4m0 {r=r } Z 0

2 + max sup dk eθ (κ) − 4β k≤J+1 rT ≤r0 ≤4m0



.

{r=r0 }

2 ˇ R] ˇ . N ≤4m0 [J, Γ,

ˇ R], ˇ and of In view of Step 1 to Step 15, of the definition (8.8.4) of N ≤4m0 [J, Γ, the various norms, we infer (ext)

.

0 ˇ G≤4m J+1 [Γ] +

(ext)

0 0 ˇ G≤4m J+1 [Γ]

(ext) ≥4m0 0 ˇ 0 ˇ ˇ + G≥4m GJ+1 [Γ] + (ext) RJ+1 [R] J+1 [Γ] +   0 (ext) ≤4m0 ˇ 0 ˇ +0 (ext) G≤4m GJ+1 [Γ] J+1 [Γ] + (ext)

(ext)

ˇ GJ [Γ]

and hence, for 0 small enough,

.

(ext)

0 ˇ G≤4m J+1 [Γ] +

(ext)

(ext)

0 ˇ G≥4m J+1 [Γ] +

(ext)

0

0 ˇ G≤4m J+1 [Γ]

0 0 ˇ G≥4m J+1 [Γ] +

(ext)

ˇ + RJ+1 [R]

(ext)

ˇ GJ [Γ].

This concludes the proof of Proposition 8.33.

8.9

PROOF OF PROPOSITION 8.12

To prove Proposition 8.12, we rely on the following proposition. Proposition 8.42. Let J such that ksmall − 2 ≤ J ≤ klarge − 1. Then, we have (int)

ˇ + GJ+1 [Γ]

(int)

ˇ . G0J+1 [Γ]

ˇ + (ext) G0J+1 [Γ] ˇ + GJ+1 [Γ] Z  12 ˇ 2 + |dJ+1 ((ext) R)| , (ext)

(int)

ˇ RJ+1 [R]

T

ˇ has been introduced in Proposition 8.32, and where where the notation (ext) G0J+1 [Γ] we have introduced the notation Z h
2
2 i (int) 0 ˇ Gk [Γ] := dk eθ (κ) + (d≤k µ ˇ )2 + dk (e4 (ζ) − β) . (int) M

The proof of Proposition 8.42 is postponed to section 8.9.2. It will rely in particular on basic weighted estimates for transport equations along e3 in (int) M derived in section 8.9.1. We now conclude the proof of Proposition 8.12. In view of Proposition 8.42, we

480

CHAPTER 8

have (int)

ˇ . GJ+1 [Γ]

ˇ + (ext) G0J+1 [Γ] ˇ + GJ+1 [Γ] Z  12 ˇ 2 + |dJ+1 ((ext) R)| . (ext)

(int)

ˇ RJ+1 [R]

T

Also, we have in view of Proposition 8.31, Proposition 8.32 and the iteration assumption (8.3.13) (ext)

ˇ + GJ+1 [Γ]

(ext)

ˇ . G0J+1 [Γ]

(ext)

ˇ + B [J]. RJ+1 [R]

We infer (int)

ˇ GJ+1 [Γ]

.

(int)

ˇ + RJ+1 [R]

(ext)

ˇ + B [J] + RJ+1 [R]

Z

ˇ 2 |dJ+1 ((ext) R)|

T

 12 .

Together with Proposition 8.11, we deduce (int)

ˇ GJ+1 [Γ]

. B [J] + 0



(En) NJ+1

+

(match) NJ+1



Z

J+1 (ext)

+ T

|d

(

ˇ 2 R)|

 12

which concludes the proof of Proposition 8.12. The rest of this section is dedicated to the proof of Proposition 8.42. 8.9.1

Weighted estimates for transport equations along e3 in

Lemma 8.43. Let the following transport equation in

(int)

(int)

M

M

a e3 (f ) + κf = h 2 where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies Z Z Z f2 . f2 + h2 . (int) M

T

(int) M

Proof. Multiply by f to obtain 1 a e3 (f 2 ) + f 2 = hf. 2 r Next, integrate over Su,r to obtain ! Z Z 1 1 2 e3 f = (e3 (f 2 ) + κf 2 ) 2 Su,r Su,r 2 Z Z a−1 2 = − κf + hf 2 Su,r Su,r Z Z Z a−1 a−1 = − κ f2 − κ ˇf 2 + hf 2 2 Su,r Su,r Su,r

481

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and hence !   Z Z Z Z 1 1 b a−1 b b 2 b 2 2 b e3 r f + a+1+ κr f =− r κ ˇf + r hf 2 2 2 2 Su,r Su,r Su,r Su,r where we used the fact that 2e3 (r) = rκ. Also, choosing b = −2a, we obtain ! Z Z Z Z 1 1 a − 1 −2a −2a 2 e3 r f + κr−2a f2 = − r κ ˇ f 2 + r−2a hf. 2 2 2 Su,r Su,r Su,r Su,r Next, let 1 ≤ u ≤ u∗ . We now integrate in r and along C u in (int) M. Since r is bounded on (int) M from above and below, we obtain, for 0 > 0 small enough, Z rT Z Z Z rT Z f2 . f2 + h2 . 2m0 −2m0 δ0

Su,r

We may now integrate in u to deduce Z u ∗ Z rT Z Z u∗ Z f2 . 1

2m0 −2m0 δ0

Su,r

2m0 −2m0 δ0

Su,rT

1

f2 +

u∗

Z

Su,rT

Z

Su,r

rT

Z

2m0 −2m0 δ0

1

h2 .

(8.9.1)

Su,r

Remark 8.44. Note that we have the following consequence of the coarea formula √ ς κ+A dT = √ dµurT du, −κ where we used that T = {r = rT }. Also, we have in dM = We infer, in

(int)

(int)

M

4ς 2 dµu,r dudr. r2 κ2

M, r dT =

1−

2m0 (1 + O(0 )) dµu,r0 du, rT

and dM = (1 + O(0 ))dµu,r dudr. Relying on Remark 8.44 we deduce from (8.9.1) Z Z Z f2 . f2 + (int) M

T

h2

(int) M

as desired. This concludes the proof of the lemma. Corollary 8.45. Let the following transport equation in a e3 (f ) + κf = h 2

(int)

M

482

CHAPTER 8

where a ∈ R is a given constant, and f and h are scalar functions. Then, f satisfies for 5 ≤ l ≤ klarge + 1 Z Z Z Z (dk f )2 . (d≤k f )2 + (d≤k−1 f )2 + (d≤k h)2 (int) M

(int) M

T

 +

(int) M

2  ≤k−5 (int) ˇ + sup |d f| Gk−1 [Γ]

(int)

2 Gk [ˇ κ] .

(int) M

Proof. The proof is based on Lemma 8.43. It is similar to the one of Corollary 8.36 and left to the reader. 8.9.2

Proof of Proposition 8.42

We introduce the following notation which will constantly appear on the RHS of the equalities below: ˇ R] ˇ N (int) [J, Γ,

ˇ + (ext) G0J+1 [Γ] ˇ + (int) RJ+1 [R] ˇ GJ+1 [Γ] Z  12 ˇ 2 + |dJ+1 ((ext) R)| T  ˇ + (int) G0J+1 [Γ] ˇ . +0 (int) GJ+1 [Γ] (ext)

:=

(8.9.2)

Step 1. In view of Lemma 7.44 relating the Ricci coefficients and curvature components of (int) M to the ones of (ext) M on the timelike hypersurface T , we have Z Z J+1 (int) 2 ˇ . ˇ 2. d ( Γ) |dJ+1 ((ext) Γ)| T

T

Also, using again Lemma 7.44, we have Z   2 J+1 (int) (int) (int) (int) (int) eθ ( κ), µ ˇ, e4 ( ζ − (int) β) d ZT   2 J+1 (ext) (int) . eθ ( κ) − 4(ext) β, (int) µ ˇ, (int) e3 ((int) ζ) + (int) β d T Z ˇ 2. + |dJ+1 ((ext) R)| T

We deduce, using that T = {r = rT } and the definitions of the various norms on (ext) M, Z Z   2 J+1 (int) 2 J+1 (int) (int) (int) (int) (int) ˇ + d ( Γ) eθ ( κ), µ ˇ, e4 ( ζ − (int) β) d T

.

(ext)

T

ˇ + GJ+1 [Γ]

(ext)

ˇ + G0J+1 [Γ]

Z T

ˇ 2 |dJ+1 ((ext) R)|

 12

483

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

and hence, in view of (8.9.2), Z Z   2 J+1 (int) 2 J+1 (int) (int) (int) (int) (int) ˇ + d ( Γ) eθ ( κ), µ ˇ, e4 ( ζ − (int) β) d T

.



T

2 ˇ R] ˇ . N (int) [J, Γ,

From now on, we only consider the frame of (int) M. The previous estimate can be written as Z  max (dk µ ˇ )2 + (dk ζ)2 + (dk κ ˇ )2 + (dk ϑ)2 + (dk κ ˇ )2 + (dk ϑ)2 k≤J+1

T

k

2

k

2

k

kˇ 2

2

+(d (e4 (ζ) − β)) + (d ξ) + (d ω ˇ ) + (d Ω) .



!

 2 ˇ R] ˇ N (int) [J, Γ,

and Z max

k≤J+1

(dk eθ (κ))2

.



ˇ R] ˇ N (int) [J, Γ,

2

.

T

Step 2. We have obtained all the desired estimates on T for the foliation of (int) M in Step 1. We now derive the desired estimates on (int) M. To this end, we rely on the transport equations in the e3 directions which we estimate thanks to Corollary 8.45. The initial data on T is estimated thanks to Step 1. In particular, we proceed in the following order: • From e3 (ˇ κ) + κ κ ˇ =

Err[e3 κ ˇ]

and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk κ ˇ )2 . N (int) [J, Γ, k≤J+1

(int) M

• From 3 e3 (eθ (κ)) + κeθ (κ) 2

1 1 = − ϑeθ (κ) − eθ (ϑ2 ) 2 2

and the bootstrap assumptions, we infer Z max (dk eθ (κ))2 . k≤J+1



ˇ R] ˇ N (int) [J, Γ,

2

.

(int) M

• From 3 3 e3 (ˇ µ) + κ µ ˇ = − µκ ˇ + Err[e3 µ ˇ ], 2 2 the above control of κ ˇ and eθ (κ) (the control of eθ (κ) is needed to estimate

484

CHAPTER 8

Err[e3 µ ˇ ]), and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk µ ˇ )2 . N (int) [J, Γ, k≤J+1

(int) M

• From e3 (ϑ) + κ ϑ and the control of α, we infer Z max (dk ϑ)2 k≤J+1

=

.

−2α

 2 ˇ R] ˇ . N (int) [J, Γ,

(int) M

• From e3 (ζ) + κζ

= β − ϑζ

the control of β, and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk ζ)2 . N (int) [J, Γ, k≤J+1

(int) M

• From 1 1 e3 (ˇ κ) + κˇ κ = − κˇ κ + 2 /d1 ζ + 2ˇ ρ + Err[e3 κ ˇ] 2 2 1 1 1 = − κˇ κ + 2ˇ µ + 4ˇ ρ − ϑϑ + ϑϑ + Err[e3 κ ˇ ], 2 2 2 the control of ρˇ, the above control of κ ˇ and µ ˇ , and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk κ ˇ )2 . N (int) [J, Γ, k≤J+1

(int) M

• From 1 e3 (ϑ) + κϑ 2

= =

1 2 /d?2 ζ − κϑ + 2ζ 2 2  1 1 1 ? −1 2 /d2 /d1 µ ˇ + ρˇ − ϑϑ + ϑϑ − κϑ + 2ζ 2 , 4 4 2

the control of ρˇ, the above control of ϑ and µ ˇ , and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk ϑ)2 . N (int) [J, Γ, k≤J+1

(int) M

• From e3 (ˇ ω)

=

ρˇ + Err[e3 ω ˇ ],

485

INITIALIZATION AND EXTENSION (THEOREMS M6, M7, M8)

the control of ρˇ, and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk ω ˇ )2 . N (int) [J, Γ, k≤J+1

(int) M

• From e3 (e4 (ζ) − β) + κ(e4 (ζ) − β)   1 = −κβ + κκ + 2µ − 6ρ ζ 2   1 1 +ϑβ − ϑβ + ξα − ϑζ + 2ωϑζ − 4ζ /d?1 /d−1 µ ˇ + ρ ˇ − ϑϑ + ϑϑ − 2ζ 3 , 1 4 4 the control of β, the above control of ζ, and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk (e4 (ζ) − β))2 . N (int) [J, Γ, k≤J+1

(int) M

• From e3 (ξ)

=

(e4 (ζ) − β) + 2β + κζ + ϑζ,

the control of β, the above control of e4 (ζ) − β and ζ, and the bootstrap assumptions, we infer Z  2 ˇ R] ˇ . max (dk ξ)2 . N (int) [J, Γ, k≤J+1

(int) M

ˇ R], ˇ and In view of the above estimates, of the definition (8.9.2) of N ≤4m0 [J, Γ, of the various norms, we infer (int)

ˇ + GJ+1 [Γ]

(int)

ˇ . G0J+1 [Γ]

ˇ + (ext) G0J+1 [Γ] ˇ + (int) RJ+1 [R] ˇ GJ+1 [Γ] Z  12 ˇ 2 + |dJ+1 ((ext) R)| T   ˇ + (int) G0 [Γ] ˇ +0 (int) GJ+1 [Γ] J+1 (ext)

and hence, for 0 small enough, (int)

ˇ + GJ+1 [Γ]

(int)

ˇ . G0J+1 [Γ]

ˇ + (ext) G0 [Γ] ˇ + GJ+1 [Γ] J+1 Z  12 ˇ 2 + |dJ+1 ((ext) R)| . (ext)

(int)

ˇ RJ+1 [R]

T

This concludes the proof of Proposition 8.42.

8.10

PROOF OF PROPOSITION 8.13

Lemma 4.16 corresponds to the particular case J = klarge − 1 of Proposition 8.13. Its proof in section 4.6.2 extends immediately to the case ksmall −2 ≤ J ≤ klarge −1 which thus yields the proof of Proposition 8.13.

Chapter Nine GCM Procedure 9.1

PRELIMINARIES

We consider an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: • On R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. • We denote by (e3 , e4 , eθ ) the null frame adapted to the outgoing geodesic foliation (u, s) on R. • Let ◦

◦ ◦

S

:= S(u, s)

(9.1.1)

◦

◦

and r the area radius of S, where S(u, s) denote the 2-spheres of the outgoing geodesic foliation (u, s) on R. • In adapted coordinates (u, s, θ, ϕ) with b = 0, see Proposition 2.75, the spacetime metric g in R takes the form, with Ω = e3 (s), b = e3 (θ), 2 1 g = −2ςduds + ς Ωdu + γ dθ − ςbdu + e2Φ dϕ2 , 2 2



2

(9.1.2)

where θ is chosen such that b = e4 (θ) = 0. • The spacetime metric induced on S(u, s) is given by g/ = γdθ2 + e2Φ dϕ2 .

(9.1.3)

• The relation between the null frame and coordinate system is given by e4 = ∂s ,

e3 =

2 ∂u + Ω∂s + b∂θ , ς

eθ = γ −1/2 ∂θ .

(9.1.4)

◦

• We denote the induced metric on S by ◦

◦

g/ = γ dθ2 + e2Φ dϕ2 . ◦

◦

◦ ◦

Definition 9.1. Let 0 < δ ≤  two sufficiently small constants. Let (u, s) real numbers so that ◦

1 ≤ u < +∞,

◦

4m0 ≤ s < +∞.

(9.1.5)

◦ ◦

We define R = R(δ, ) to be the region n ◦ R := |u − u| ≤ δR ,

o ◦ |s − s| ≤ δR ,

◦ ◦
− 1 2

δR := δ 

,

(9.1.6)

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GCM PROCEDURE ◦

such that assumptions A1–A3 below with constant  on the background foliation of ◦

R are verified. The smaller constant δ controls the size of the GCMS quantities as it will be made precise below. In this section we define the renormalized Ricci and curvature components,  

2 2Υ m ˇ ˇ:= κ Γ ˇ , ϑ, ζ, η, κ − , κ + ,κ ˇ , ϑ, ξ, ω ˇ , ω − 2 , Ω, Ω+Υ , ς +1 , r r r   ˇ : = α, β, ρˇ, ρ + 2m , β, α . R r3 Since our foliation is outgoing geodesic we also have ξ = ω = 0,

η + ζ = 0.

(9.1.7)

ˇ = Γg ∪ Γb where We decompose Γ   2 2Υ Γg = κ ˇ , ϑ, ζ, κ ˇ, κ − , κ + , r r n

o m ˇ r−1 ςˇ, r−1 Ω + Υ , r−1 ς − 1 . Γb = η, ϑ, ξ, ω ˇ , ω − 2 , r−1 Ω, r

(9.1.8)

Given a p-reduced scalar f ∈ sp (M), with respect to the given geodesic foliation on R, we consider the following norms on spheres S = S(u, r) ⊂ R, kf k∞ (u, r) : = kf k kf k∞,k (u, r) =

k X i=0

kf k2 (u, r) := kf k


,

L∞ S(u,r)


,

L2 S(u,r)

kdi f k∞ (u, r),

kf k2,k (u, r) =

k X i=0

kdi f k2 (u, r),

(9.1.9)

where, we recall, that di stands for any combination of length i of operators of the form e3 , re4 , d/. Recall that ( p r2p 4 / k, if s = 2p, s d/ f = (9.1.10) p 2p+1 r /dk 4 / k, if s = 2p + 1. On a given polarized surface S ⊂ R, not necessarily a leaf S of the given foliation, we define kf khqs (S) :

=

s X i=0

k d/S


i

f kLq (S)

(9.1.11)

where d/S is defined as above with respect to the intrinsic metric on S. In the particular case when q = 2 we omit the upper index, i.e., hs (S) = h2s (S).

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9.1.1

Main assumptions

Given an integer smax , we assume the following:1 A1. For all k ≤ smax , we have on R ◦

kΓg kk,∞ . r−2 ,

(9.1.12)

◦

kΓb kk,∞ . r−1 ,

and ◦

kα, β, ρˇ, µ ˇkk,∞ . r−3 , ◦

ke3 (α, β)kk−1,∞ . r−4 ,

(9.1.13)

◦

kβkk,∞ . r−2 , ◦

kαkk,∞ . r−1 . A2. We have, with m0 denoting the mass of the unperturbed spacetime, m ◦ sup − 1 . . (9.1.14) R m0 A3. The metric coefficients are assumed to satisfy the following assumptions in R, for all k ≤ smax ,

 

γ

eΦ ◦

r 2 − 1, b, −1 + kΩ + Υk∞,k + kς − 1k∞,k . . (9.1.15)

r r sin θ ∞,k Remark 9.2. The above assumptions imply in particular the following: |e4 (r)|, |e3 (r)| . 1, ◦

◦

◦

e4 (s) = 1 + O(),

e3 (u) = 2 + O(),

e4 (u) = 0.

◦ ◦

Hence, since r = r at (u, s), we infer ◦

◦

|r − r| .

◦

|s − s| + |u − u|,

and thus, in view of the definition (9.1.6) of R, ◦

sup |r − r| .

◦ ◦
− 1 2

δ 

.

(9.1.16)

R

We will make use of the following lemma, see Lemmas 4.9 and 4.10. Lemma 9.3. Under the assumption A3 for the metric coefficients we have 2 1 r eθ (Φ) ≤ sinθ , ≤ 2 (r|eθ Φ| + 1) . (9.1.17) sin θ 1 In applications, s max = ksmall + 4 in Theorem M7, and smax = klarge + 5 in Theorem M0 and Theorem M6.

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Moreover, for any reduced 1-scalar h, we have sup S

9.1.2

|h| eΦ



. r−1 supS (|h| + | d/h|),

h eΦ L2 (S)

. r−1 khkh1 (S) .

(9.1.18)

Elliptic Hodge lemma

We shall often make use of the results of Proposition 2.33 and Lemma 2.38 which we rewrite as follows. Lemma 9.4. Under the assumptions A1, A3 the following elliptic estimates hold true for the Hodge operators /d1 , /d2 , /d?1 , /d?2 , for all k ≤ smax : 1. If f ∈ s1 (S) k d/f khk (S) + kf khk (S) . rk /d1 f khk (S) . 2. If f ∈ s2 (S) k d/f khk (S) + kf khk (S) . rk /d2 f khk (S) . 3. If f ∈ s0 (S) k d/f khk (S) . rk /d?1 f khk (S) . 4. If f ∈ s1 (S) kf khk+1 (S)

Z rk /d?2 f khk (S) + r−2 eΦ f .

.

S

5. If f ∈ s1 (S)

R

f eΦ Φ

S

f − R 2Φ e

e S

hk+1 (S)

. rk /d?2 f khk (S) .

We shall often make use of the following non-sharp product estimate on S, see Proposition 2.43. Lemma 9.5. The following estimates hold true on a given polarized surface S ⊂ R, for any contraction between two reduced scalars ψ1 , ψ2 , k ≥ 2, kψ1 · ψ2 khk (S) 9.2 9.2.1

.

r−1 kψ1 khk (S) kψ2 khk (S) .

DEFORMATIONS OF S SURFACES Deformations ◦

◦ ◦

Recall that S = S(u, s) is a fixed sphere of the (u, s) outgoing geodesic foliation of ◦ ◦

a fixed spacetime region R = R(, δ). ◦

◦

Definition 9.6. We say that S is an O() Z-polarized deformation of S if there

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CHAPTER 9 ◦

exists a map Ψ : S → S of the form ◦  ◦ ◦ ◦ Ψ(u, s, θ, ϕ) = u + U (θ), s + S(θ), θ, ϕ

(9.2.1) ◦

where U, S are functions defined on the interval [0, π] of amplitude at most , leading to a smooth surface S. We denote by ψ the reduced map defined on the interval [0, π], ◦

◦

ψ(θ) = (u + U (θ), s + S(θ), θ).

(9.2.2)

We restrict ourselves to deformations which fix the south pole, i.e., U (0) = S(0) = 0. 9.2.2

(9.2.3)

Pullback map ◦

We recall that given a scalar function f on S one defines its pullback on S to be the function f # := Ψ# f = f ◦ Ψ. ◦

On the other hand, given a vectorfield X on S one defines its push-forward Ψ# X to be the vectorfield on S defined by Ψ# X(f ) = X(Ψ# f ) = X(f ◦ Ψ). ◦

Given a covariant tensor U on S, one defines its pullback to S to be the tensor Ψ# U (X1 , . . . , Xk ) = U (Ψ# X1 , . . . , Ψ# Xk ). ◦

Lemma 9.7. Given a Z-invariant deformation Ψ : S → S, we have: #

1. Let g/ S the induced metric on S and g/ S,# = γ S,# dθ2 + e2Φ dϕ2 its pullback to ◦

S. The metric coefficients γ S and γ S,# are related by ◦

◦

γ S,# (θ) = γ S (ψ(θ)) = γ S (u + U (θ), s + S(θ), θ)

(9.2.4)

where γ S is defined implicitly by S #

#

(γ ) = γ + ς


# 2

 # 1 2 Ω+ b γ (U 0 )2 − 2ς # U 0 S 0 − (γςb)# U 0 , 4

(9.2.5)

that is, γ S (ψ(θ))

  1 = γ(ψ(θ)) + ς 2 (ψ(θ)) Ω(ψ(θ)) + (b(ψ(θ)))2 γ(ψ(θ)) (U 0 (θ))2 4 0 0 − 2ς(ψ(θ))U (θ)S (θ) − γ(ψ(θ))ς(ψ(θ))b(ψ(θ))U 0 (θ).

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GCM PROCEDURE

2. The Z-invariant vectorfield ∂θS := Ψ# (∂θ ) is tangent to S and h   i ς ς ς √  . ∂θS |Ψ(p) = ∂θ S − Ω∂θ U e4 + ∂θ U e3 + γ 1 − b∂θ U eθ 2 2 2 Ψ(p)

(9.2.6)

3. If f ∈ sk (S) and P S is a geometric operator acting on f , then (P S [f ])#

= P S,# [f # ]

(9.2.7) ◦

where P S,# is the corresponding geometric operator on S with respect to the metric g/ S,# and f # = ψ # f . 4. The L2 norm of f # = ψ # f with respect to the metric g/ S,# is the same as the L2 norm of f with respect to the metric g/ S , i.e., Z Z # 2 |f | dag/ S,# = |f |2 dag/ S . ◦ S

S

5. If f ∈ hk (S) and f # is its pullback by ψ then kf # k

◦

hk (S, g / S,# )

= kf khk (S) .

Proof. If ∂θ denotes the coordinate derivative ∂θ = Ψ# (∂θ )|Ψ(p)

∂ ∂θ

◦

then, at every point p ∈ S,

∂θ U ∂u |Ψ(p) + ∂θ S∂s |Ψ(p) + ∂θ |Ψ(p) ,

=

Ψ# (∂ϕ ) = ∂ϕ .

In view of (9.1.4) we have ∂s = e4 ,

∂u =

 ς e3 − Ωe4 − bγ 1/2 eθ , 2

∂θ =

√

γeθ .

Hence, at a point Ψ(p) on S we have    ς ς ς √  Ψ# (∂θ ) = ∂θ S − Ω∂θ U e4 + ∂θ U e3 + γ 1 − b∂θ U eθ . 2 2 2 ◦

We denote by g/ # = Ψ# (g/ S ) the pullback to S of the metric g/ S on S, i.e., at any ◦

point p ∈ S, g/ # (∂θ , ∂θ ) g/ # (∂θ , ∂ϕ ) #

g/ (∂ϕ , ∂ϕ )

= g/ S (Ψ# ∂θ , Ψ# ∂θ ) = g(∂θ U ∂u + ∂θ S∂s + ∂θ , ∂θ U ∂u + ∂θ S∂s + ∂θ ) =

(∂θ U )2 guu + 2∂θ U ∂θ Sgus + 2∂θ U guθ + gθθ ,

=

0, #

= e2Φ ,

where
1 guu = ς 2 Ω + γb2 , 4

gus = −ς,

ς guθ = − γb, 2 ◦

Hence the pullback metric Ψ# (g/ S ) on S is given by #

γ S,# dθ2 + e2Φ dϕ2

gss = gsθ = 0,

gθθ = γ.

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CHAPTER 9

where γ S,# = (γ S )# ,

(9.2.8)

with γ S defined by S #

#

# 2

(γ ) = γ + (ς )



1 Ω + b2 γ 4

#

(U 0 )2 − 2ς # U 0 S 0 − (γςb)# U 0 .

(9.2.9)

Note that the vectorfield eS θ :=

1 Ψ# (∂θ ) (γ S )1/2

is tangent, Z-invariant and forms together with eϕ an orthonormal frame on S. Note that we can also write ◦

(γ )1/2 := S 1/2 Ψ# (eθ ) (γ )

eS θ

◦

◦

where γ is the coefficient in front of dθ2 of the metric induced by g on S, ◦

◦

g/ = γ dθ2 + e2Φ dϕ2 . In general, any geometric calculation on S can be reduced to a geometric cal◦

culation on S with respect to the metric g/ S,# . Moreover the L2 norm on S with respect to the metric g/ S is the same as the L2 norm of f # = ψ # f with respect to the norm g/ S,# . This concludes the proof of the lemma. 9.2.3

Comparison of norms between deformations ◦ ◦

◦

Lemma 9.8. Let Ψ : S → S a Z-invariant deformation in R(, δ) with U, S verifying the bounds sup



0≤θ≤π

|U 0 (θ)| + |S 0 (θ)|



◦

.

δ,

(9.2.10)

as well as the bound (9.1.15) for the coordinates system (u, s, θ, ϕ) of R. The following hold true: 1. We have ◦◦ S,# ◦ γ − γ . δ r.

(9.2.11)

2. For every f ∈ sk (S) we have kf # k

◦

L2 (S,g / S,# )

=

kf # k

◦ ◦

L2 (S,g /)

 ◦  1 + O(r−1 δ) .

(9.2.12)

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3. As a corollary of (9.2.12) (choosing f = 1) we deduce2 rS ◦

◦ −1 ◦

= 1 + O(r

δ)

(9.2.13)

r ◦

◦

where rS is the area radius of S and r that of S. Proof. Recall γ

  1 2 (u, s, θ) = γ(ψ(θ)) + ς (ψ(θ)) Ω(ψ(θ)) + (b(ψ(θ))) γ(ψ(θ)) (U 0 (θ))2 4 0 0 − 2ς(ψ(θ))U (θ)S (θ) − γ(ψ(θ))ς(ψ(θ))b(ψ(θ))U 0 (θ).

S,# ◦ ◦

2

In view of our assumptions on U 0 and S 0 as well as our estimates (9.1.15) for γ, Ω and b and ς, we infer ◦ ◦1/2 ◦

|γ S,# − γ| . |γ # − γ| + r

δ.

Also, we have ◦ ◦

◦ ◦

γ # (u, s, θ) − γ(u, s, θ)

◦

◦

◦ ◦

= γ(u + U (θ), s + S(θ), θ) − γ(u, s, θ) Z 1 i d h ◦ ◦ = γ(u + λU (θ), s + λS(θ), θ) dλ 0 dλ Z 1 ◦ ◦ = U (θ) ∂u γ(u + λU (θ), s + λS(θ), θ)dλ 0

Z +S(θ)

1

◦

◦

∂s γ(u + λU (θ), s + λS(θ), θ)dλ. 0

In view of our estimates (9.1.15) for γ, the assumption (9.2.10) on (U 0 , S 0 ) and the fact that  ς e3 − Ωe4 − bγ 1/2 eθ , ∂s = e4 , ∂u = 2 we infer3 |γ # − γ| .

◦◦

rδ.

We have finally obtained ◦◦

◦◦

|γ S,# − γ| . |γ # − γ| + r δ . r δ. ◦

◦ ◦

also from (9.1.16) that r − r = O(δ()−1/2 ). 3 Note that we also use the assumption U (0) = S(0) = 0 to estimate (U, S) from (U 0 , S 0 ). 2 Recall

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CHAPTER 9

To prove the second part of the lemma we write p Z Z γ S,# # 2 # 2 |f | dag/ S,# = |f | q da ◦ ◦ ◦ g / ◦ S S γ p  S,# γ |f # |2 da ◦ + ◦ |f # |2  q − 1 da ◦ ◦ g / g / ◦ S S γ

Z =

Z

which yields, in view of the first part, Z Z   ◦ −1 ◦ # 2 |f | dag/ S,# = |f # |2 da ◦ 1 + O(r δ) . ◦ ◦ S

g /

S

This concludes the proof of the lemma. ◦

Remark 9.9. In view of (9.2.13) and (9.1.16), r , rS and the value of r along S are all comparable. Corollary 9.10. Under the assumptions of Lemma 9.8 the following estimate4 holds true for an arbitrary scalar f ∈ s0 (R), Z   Z ◦◦ ≤1 f− f . δ r sup |d% f | + sup r|e3 f | . ◦ S

R

S

R

Proof. We have Z S

f−

Z

p

Z ◦

f

=

S

◦

f#

S

γ S,# q − ◦ γ

 Z S,# γ #  q f= ◦f − 1 + ◦ (f # − f ). ◦ ◦ S S S γ

Z

p

Z

Hence, Z Z f− f ◦ S

S

◦◦

. δ r sup |f | + S

Z ◦

S

# f − f .

Now, proceeding as in the proof of (9.2.11), ◦

◦

◦ ◦

Z

1

i d h ◦ ◦ f (u + λU (θ), s + λS(θ), θ) dλ 0 dλ Z 1 ◦ ◦ ∂u f (u + λU (θ), s + λS(θ), θ)dλ = U (θ)

f (u + U (θ), s + S(θ)) − f (u, s) .

0

Z +S(θ)

1

◦

0

4 Recall

◦

that R := {|u − u| ≤ δR ,

◦

◦

∂s f (u + λU (θ), s + λS(θ), θ)dλ.

|s − s| ≤ δR }, see (9.1.6).

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Therefore, Z Z f− f ◦ S

S

  ◦◦ ◦◦ . rδ sup |f | + δ r sup |d% f | + sup r|e3 f | R R S   ◦◦ ≤1 . δ r sup |d% f | + sup r|e3 f | R

R

as stated. To compare higher order Sobolev spaces, we will need the following lemma. ◦ ◦

◦

Lemma 9.11. Let S ⊂ R = R(, δ) as in Definition 9.1 verifying the assumptions ◦

A1–A3. Let Ψ : S → S be Z-invariant deformation. Assume the bound ◦

◦

k(U 0 , S 0 )k

◦

L∞ (S)

+ (r )−1 k(U 0 , S 0 )k

◦ ◦

hsmax −1 (S,g /)

. δ.

(9.2.14)

Then, we have for any reduced scalar h defined on R khkhs (S) . r sup |d≤s h|, R

for 0 ≤ s ≤ smax .

Also, if f ∈ hs (S) and f # is its pullback by ψ, we have kf khs (S) = kf # k

◦

hs (S, g / S,# )

◦

= kf # k

◦ ◦

hs (S,g /)

(1 + O(r−1 δ)) for 0 ≤ s ≤ smax − 1.

Proof. See section C.1. Corollary 9.12. Under the same assumptions as Lemma 9.11, we have, for all j, k ≥ 0 with 0 ≤ j + k ≤ smax , ◦

kd≤j Γg khk (S) . r−1 ,

(9.2.15)

◦

kd≤j Γb khk (S) . ,

◦

kd≤j (α, β, ρˇ, µ ˇ) khk (S) . r−2 , ◦

kd≤j βkhk (S) . r−1 , ≤j

kd

(9.2.16)

◦

αkhk (S) . ,

 

≤j γ

eΦ ◦

d

− 1, b, − 1 . ,

2 r r sin θ hk (S)

≤j

≤j

◦

d (Ω + Υ)

+ d (ς − 1) h (S) . r. h (S) k

(9.2.17)

k

Proof. In view of Lemma 9.11 and assumptions A1–A3 we have, for j, k ≥ 0 with 0 ≤ j + k ≤ smax ,

≤j

d Γg h

k (S)

◦ . r sup d≤k d≤j Γg . r sup d≤smax Γg . r−1 . R

R

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CHAPTER 9

The other estimates are proved in the same manner. 9.2.4

Adapted frame transformations

We consider general null transformations introduced in Lemma 2.87,   1 e04 = λ e4 + f eθ + f 2 e3 , 4     1 1 1 1 e0θ = 1 + f f eθ + f e4 + f 1 + f f e3 , 2 2 2 4      1 1 1 1 e03 = λ−1 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 . 2 16 4 4

(9.2.18)

◦

Definition 9.13. Given a deformation Ψ : S → S we say that a new frame (e03 , e04 , e0θ ), obtained from the standard frame (e3 , e4 , eθ ) via the transformation (9.2.18), is S-adapted if we have e0θ = eS θ =

1 (γ S )1/2

Ψ# (∂θ ). ◦

(9.2.19) ◦ ◦

Proposition 9.14. Consider a deformation Ψ : S → S in R = R(, δ) verifying the assumption A3. The following statements hold true. 1. A new frame e03 , e0θ , e04 generated by (f, f , λ = ea ) according to (9.2.18) is adapted ◦

◦

to S = S(u + U, s + S) provided that, at all points θ ∈ [0, π],    p 
ς# # 0 1 S # 1/2 # # γ 1− b U = (γ ) 1 + (f f ) , 2 2  
1 # 0 S # 1/2 # # ς U = (γ ) f 1 + (f f ) , 4  
1/2 # ς# # 0 2 S0 − Ω U = (γ S )# f , 2

(9.2.20)

where (γ S )#

 # 1 = γ # + (ς # )2 Ω + b2 γ (∂θ U )2 − 2ς # ∂θ U ∂θ S − (γςb)# ∂θ U 4

and # denotes the pullback by ψ of the corresponding reduced scalars, i.e., for ◦ ◦ example, f # (θ) = f (u + U (θ), s + S(θ), θ). 2. There exists a small enough constant5 δ1 such that for given f, f on R satisfying   sup |f | + |f | ≤ r−1 δ1 , R

5 In

later applications, we will have ◦

sup (|f | + |f |) . r−1 δ. R

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we can uniquely solve the system (9.2.20) for U, S subject to the initial conditions, U (0) = 0,

S(0) = 0. ◦

◦ ◦

Thus, if (u, s, 0) corresponds to the south pole of S and f, f are given there exists a unique deformation S ⊂ R, given by U, S : [0, π] → R, adapted to frames generated by6 (f, f ) which passes through the same south pole. Moreover, sup |(U 0 , S 0 )| .

[0,π]

◦

r sup |f | + |f |




(9.2.21)

S

and, for 2 ≤ s ≤ smax − 1, ◦

k(U 0 , S 0 )k

◦

L∞ (S)

+ (r )−1 k(U 0 , S 0 )k

◦ ◦

hs (S,g /)

. kf, f khs (S)

(9.2.22)

with kf, f khs (S) = kf khs (S) + kf khs (S) . 3. As a consequence of (9.2.22) the deformation thus obtained verifies the conclusions of Lemmas 9.8–9.11 and Corollary 9.12. In particular, a) We have S,# ◦ −γ . γ

◦

δ1 r .

b) We have S r −1 − 1 . r◦ δ1 . ◦ r Proof. In view of Lemma 9.7, The Z-invariant vectorfield eS θ := be expressed by the formula eS θ

=

1 (γ S )1/2 ◦

1 Ψ (∂ ) (γ S )1/2 # θ

can

h   i ς ς ς √  ∂θ S − Ω∂θ U e4 + ∂θ U e3 + γ 1 − b∂θ U eθ 2 2 2 ◦

where ψ(p) = (u + U (θ), s + S(θ), θ) and U 0 = ∂θ U (θ), S 0 = ∂θ S(θ). On the other hand, according to (9.2.18), at Ψ(p) ∈ S,     1 1 1 1 e0θ = 1 + f f eθ + f 1 + f f e3 + f e4 . 2 2 4 2 We deduce, at every θ ∈ [0, π],  p  ς# # 0 γ# 1 − b U 2 ς #U 0   ς# # 0 2 S0 − Ω U 2 6 Note

= = =

that a is not restricted in this result.

 1 1 + (f f )# , 2  
1 1/2 # (γ S )# f 1 + (f f )# , 4
1/2 # (γ S )# f , (γ S )#


1/2



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CHAPTER 9

as desired. To prove the second part of the lemma we first check for the compatibility of the three equations in (9.2.20). Note that, if we denote   1 1 A = 1 + (f f )# , B = f # 1 + (f f )# , C = f # , 2 4 we have A2 − BC = 1. Hence, squaring the first equation and subtracting the product of the other two we derive (γ S )#

= = =

 2   ς# # 0 ς# # 0 γ# 1 − b U − 2U 0 ς # S 0 − Ω U 2 2  
1 2 γ # 1 − (ςb)# U 0 + (ςb)# U 0 − 2ς # U 0 S 0 + (ς # )2 Ω# (U 0 )2 4   1 #
2 # # # # 2 γ + (ς ) Ω + b γ (U 0 )2 4 p

−2ς # U 0 S 0 − γ # ς # b# U 0

(9.2.23)

which coincides with the formula (9.2.5). It thus suffices to only consider the last two equations in (9.2.20) which we write in the form   1 U 0 = (ς # )−1 ((γ S )# )1/2 f # 1 + (f f )# , 4   (9.2.24) 1 1 1 S 0 = ((γ S )# )1/2 f # + Ω# ((γ S )# )1/2 f # f # + (f f )# , 2 2 4 i.e., 0

U (θ) S 0 (θ)

   1 ◦ ◦ −1 S 1/2 = ς (γ ) f 1 + (f f ) (u + U (θ), s + S(θ), θ), 4    1 S 1/2 1 1 ◦ ◦ S 1/2 = (γ ) f + Ω(γ ) f 1 + f f (u + U (θ), s + S(θ), θ). 2 2 4 ◦ −1

Thus under the assumption supR (|f | + |f |) ≤ r δ1 , with δ1 sufficiently small, making use also of the expression (9.2.23) of γ S , and the estimates (9.1.15) for ◦ (γ, b, Ω), for  sufficiently small, we can uniquely solve for U, S subject to the initial conditions U (0) = 0,

S(0) = 0.

Moreover the solution verifies
◦ sup |(U 0 , S 0 )| . r sup |f | + |f |

[0,π]

S

according to Definition 9.6. Estimate (9.2.22) can be easily derived by taking higher derivatives and using Lemma 9.11 and A1–A3. This concludes the proof of the lemma. We now provide a lemma analogous to Proposition 9.14 in the particular case

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GCM PROCEDURE

when f is only bounded in r, unlike the rest of the chapter where it decays like r−1 . This lemma is not needed for the construction of GCM spheres in this chapter. It is used in the proof of Theorem M0 in the region (ext) L0 ∩ (ext) M of the initial data layer, see Step 8 in section 4.1. Lemma 9.15. There exists a small enough constant δ1 such that for given f, f on R satisfying kf khsmax −1 (S) + (rS )−1 kf khsmax −1 (S)

≤ δ1 ,

the following holds: 1. We have ◦

kU 0 k

◦

L∞ (S)

+ (r )−1 kS 0 k

◦

◦

L∞ (S)

+ (r )−1 kU 0 k

◦ ◦

hsmax −1 (S,g /)

◦

+(r )−2 kS 0 k

◦ ◦

hsmax −1 (S,g /)

.

δ1 .

In particular, we have ◦

sup |u − u| . δ1 , S

◦

◦

sup |s − s| . rδ1 . S

2. We have S,# ◦ −γ . γ

◦

δ1 (r )2 .

3. We have S S r − 1 + sup r − 1 . δ1 . ◦ r S r 4. The following estimate holds true for an arbitrary scalar h ∈ s0 (R), # h − h . δ1 sup |d≤1 h|. R

5. The following estimate holds true for an arbitrary scalar h ∈ s0 (R), Z Z ◦ h− h . δ1 (r)2 sup |d≤1 h|. ◦ S

R

S

6. We have for any reduced scalar h defined on R khkhs (S) . r sup |d≤s h|,

for 0 ≤ s ≤ smax .

R

7. If h ∈ hs (S) and h# is its pullback by ψ, we have kf khs (S) = kf # k

◦

hs (S, g / S,# )

= kf # k

◦ ◦

hs (S,g /)

(1 + O(δ1 )) for 0 ≤ s ≤ smax − 1.

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CHAPTER 9

Proof. Recall from (9.2.20) that we have in particular  
1/2 # 1 ς # U 0 = (γ S )# f 1 + (f f )# , 4   #
1/2 # ς 2 S0 − Ω# U 0 = (γ S )# f . 2 In view of the assumptions on (f, f ), and the control of the background foliation of R, we immediately obtain the first claim of the lemma concerning the control of ◦ ◦ (U, S). Note that the estimate for u − u and s − s follows then from ◦

sup |U | . sup |U 0 | . δ1 ,

sup |u − u| . S

◦

◦

S

S

◦

◦

sup |S| . sup |S 0 | . rδ1 .

sup |s − s| . S

◦

◦

S

S

The first claim then yields the second and third claims by a straightforward adaptation of the proof of Proposition 9.14. Also, the fifth claim follows from the second and the fourth claims, by a simple adaptation of the proof of Corollary 9.10. The sixth and seventh claims follow from the other claims by a simple adaptation of Lemma 9.11. Finally, we focus on the fourth claim. We have for an arbitrary scalar h ∈ s0 (R), ◦ ◦

◦ ◦

h# (u, s, θ) − h(u, s, θ)

◦

◦

◦ ◦

= h(u + U (θ), s + S(θ), θ) − h(u, s, θ) Z 1 i d h ◦ ◦ = h(u + λU (θ), s + λS(θ), θ) dλ 0 dλ Z 1 ◦ ◦ = U (θ) ∂u h(u + λU (θ), s + λS(θ), θ)dλ 0

Z +S(θ)

1

◦

◦

∂s h(u + λU (θ), s + λS(θ), θ)dλ. 0

In view of our estimates (9.1.15) for γ, the assumption (9.2.10) on (U 0 , V 0 ) and the fact that  ς ∂s = e4 , ∂u = e3 − Ωe4 − bγ 1/2 eθ , 2 we infer7 together with the first claim |h# − h| . .

sup |U | sup |dh| + r−1 sup |S| sup |re4 (h)| R

◦

S

◦

S

R

δ1 sup |dh| R

as desired. Lemma 9.15 yields the following corollaries.

7 Note

that we also use the assumption U (0) = S(0) = 0 to estimate (U, S) from (U 0 , S 0 ).

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GCM PROCEDURE

Corollary 9.16. Assume that there exists a small enough constant δ1 such that we have ◦

kU 0 k

◦

L∞ (S)

◦

+ (r )−1 kS 0 k

◦

L∞ (S)

+ (r )−1 kU 0 k

◦ ◦

hsmax −1 (S,g /)

◦

+(r )−2 kS 0 k

◦ ◦

hsmax −1 (S,g /)

≤ δ1 .

Then, we have, for all j, k ≥ 0 with 0 ≤ j + k ≤ smax , ◦

kd≤j Γg khk (S) . r−1 ,

(9.2.25)

◦

kd≤j Γb khk (S) . ,

◦

kd≤j (α, β, ρˇ, µ ˇ) khk (S) . r−2 , ◦

kd≤j βkhk (S) . r−1 ,

(9.2.26)

◦

kd≤j αkhk (S) . ,

 

≤j γ

eΦ ◦

d

− 1, b, − 1 . ,

2 r r sin θ hk (S)

≤j

≤j

◦

d (Ω + Υ)

+ d (ς − 1) h (S) . r. h (S) k

(9.2.27)

k

Proof. The proof is similar to the one of Corollary 9.12 and relies on property 6 of Lemma 9.15 and the control A1–A3 of the background foliation. Corollary 9.17. Let 3 ≤ s ≤ smax . There exists a small enough constant δ1 such that given f, f on R satisfying kf khs (S) + (rS )−1 kf khs (S)

≤ δ1 ,

then sup K S − S

◦ 1 δ1 +  . , (rS )2 (rS )2

S

K −

◦ 1 δ1 + 

. , (rS )2 hs−1 (S) rS

and Z S

e2Φ =

4π S 4 ◦ (r ) (1 + O(δ1 + )). 3

Proof. Using 1 1 K S = −ρS − κS κS + ϑS ϑS , 4 4

1 1 K = −ρ − κκ + ϑϑ, 4 4

the change of frame formulas for ρS , κS , κS , ϑS and ϑS , and the assumptions

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CHAPTER 9

(9.4.23) for8 (f, f ), we infer sup K S − K . S

δ1 , (rS )2

S

δ1

K − K . S. hs−1 (S) r

Together with the control A1–A3 for the background foliation, we deduce ◦ S 1 δ1  sup K − 2 . S 2 + 2 , r (r ) r S

◦

S

δ1 

K − 1 . + .

2 S r hs−1 (S) r r

Also, in view of the assumptions (9.4.23) for (f, f ), we may apply Lemma 9.15. Using property 3 of that lemma on the control of r − rS , we easily infer sup K S − S

◦ 1 δ1 +  . , (rS )2 (rS )2

S

K −

Also, using property 5 of that lemma, we have Z Z 2Φ e2Φ − e . ◦ S

◦ 1 δ1 + 

. . (rS )2 hs−1 (S) rS

δ1 (rS )4

S

which together with the control A3 for the background foliation implies Z 4π S 4 ◦ e2Φ = (r ) (1 + O(δ1 + )). 3 S This concludes the proof of the corollary. Corollary 9.18. Let 2 ≤ s ≤ smax . There exists a small enough constant δ1 such that given f, f on R satisfying kf khs (S) + (rS )−1 kf khs (S)

≤ δ1 ,

then, for any scalar function D = D(u, s) on R depending only on the coordinates (u, s) of the background foliation, we have

  S

. r kf khs (S) + r−1 kf khs (S) sup |d≤s D|.

D − D hs (S)

R

Proof. We have, using a Poincar´e inequality,



S

. rS eS

D − D θ (D) hs−1 (S) , hs (S)

s ≥ 1.

S Thus, we need to compute eS θ (D). Decomposing eθ on the background frame, we 8 Note that the change of frame formulas for ρS , κS κS and ϑS ϑS do not involve λ, and involve at most one tangential derivative to S of (f, f ).

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GCM PROCEDURE

have eS θ (D)

 = =

1 1 + ff 2



1 1 eθ (D) + f e4 (D) + f 2 2   1 1 1 f e4 (D) + f 1 + f f e3 (D) 2 2 4

  1 1 + f f e3 (D) 4

where we have used in the last inequality D = D(u, s) and eθ (u) = eθ (s) = 0. We infer, for 2 ≤ s ≤ smax ,

 



1 S

S 1 . r f e4 (D) + f 1 + f f e3 (D)

D − D

2 4 hs (S) hs−1 (S) . kf khs (S) ke4 (D)khs−1 (S)


+kf khs (S) 1 + r−2 kf khs (S) kf khs (S) ke3 (D)khs−1 (S)   . r kf khs (S) + r−1 kf khs (S) sup |d≤s D|, R

where we have used in the last inequality the control on (f, f ), as well as property 6 of Lemma 9.15 with h = e4 (D) and h = e3 (D). Corollary 9.19. Assume that (f, f ) given on R satisfy for a small enough constant δ1 kf khsmax −1 (S) + (rS )−1 kf khsmax −1 (S)

≤ δ1 .

Then, we have ◦

◦

|mS − m| . δ1 + ()2 . Proof. According to the identity (2.2.12), we have Z ρ

S

=

S

Z ◦

ρ =

S

Z 8πmS 1 − S + ϑS ϑS , r 4 S Z ◦ 8π m 1 − ◦ + ϑϑ. 4 S r

In view of the transformation formulas for ϑS and ϑS , and noticing that the product ϑS ϑS only involves (f, f ) but not λ, we infer from the assumptions A1–A3 for the background foliation of R, and the assumptions on (f, f ) that ◦

S S

|ϑ ϑ | + |ϑϑ| .

()2 . r3

We infer Z ρ

S

=

S

8πmS − S +O r ◦

Z ρ = ◦ S

−

8π m ◦

r

+O

! ◦ ()2 , r ! ◦ ()2 , r

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CHAPTER 9

and hence ◦ S m − m .

Z Z S ◦ ◦ S r ρ − r ◦ ρ + ()2 . S

S

Next, we apply Lemma 9.15 and infer in particular Z Z δ1 ◦ ◦ ρ− |rS − r| . rδ1 , ρ . ◦ . ◦ S S r We deduce ◦ S m − m

Z . r

S


◦ ρ − ρ + δ1 + ()2 . S

Together with the transformation formula for ρS , which only involves (f, f ) but not λ, we infer from the assumptions A1–A3 for the background foliation of R, and the assumptions on (f, f ) that ◦ ◦ S m − m . δ1 + ()2 as desired.

9.3

FRAME TRANSFORMATIONS

For the convenience of the reader we start by recalling the transformation formulas recorded in Proposition 2.90. Proposition 9.20 (Transformation formulas-GCM). Under a general transformation of type (9.2.18) with λ = ea the Ricci coefficients and curvature components transform as follows:   1 1 ξ 0 = λ2 ξ + λ−1 e04 (f ) + ωf + f κ + λ2 Err(ξ, ξ 0 ), 2 4 1 Err(ξ, ξ 0 ) = f ϑ + l.o.t., 4   (9.3.1) 1 1 0 ξ = λ−2 ξ + λe03 (f ) + ω f + f κ + λ−2 Err(ξ, ξ 0 ), 2 4 1 1 Err(ξ, ξ 0 ) = − λf 2 e03 (f ) + f ϑ + l.o.t., 8 4

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GCM PROCEDURE

1 ζ 0 = ζ − e0θ (log(λ)) + (−f κ + f κ) + f ω − f ω + Err(ζ, ζ 0 ), 4 1 0 1 0 Err(ζ, ζ ) = f eθ (f ) + (−f ϑ + f ϑ) + l.o.t., 2 4 1 0 1 0 η = η + λe3 (f ) + κf − f ω + Err(η, η 0 ), 2 4 1 0 Err(η, η ) = f ϑ + l.o.t., 4 1 1 0 η = η + λ−1 e04 (f ) + κf − f ω + Err(η, η 0 ), 2 4 1 2 −1 0 1 0 Err(η, η ) = − f λ e4 (f ) + f ϑ + l.o.t., 8 4

(9.3.2)

κ0 = λ (κ + /d1 0 (f )) + λErr(κ, κ0 ), 1 Err(κ, κ0 ) = f (ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω + l.o.t.,
4 (9.3.3) κ0 = λ−1 κ + /d1 0 (f ) + λ−1 Err(κ, κ0 ), 1 1 Err(κ, κ0 ) = − f 2 e0θ (f ) + f (−ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω + l.o.t., 4 4 ϑ0 = λ (ϑ − /d?2 0 (f )) + λErr(ϑ, ϑ0 ), 1 Err(ϑ, ϑ0 ) = f (ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t.
4 −1 (9.3.4) 0 −1 ?0 ϑ =λ ϑ − /d2 (f ) + λ Err(ϑ, ϑ0 ), 1 1 Err(ϑ, ϑ0 ) = − f 2 e0θ (f ) + f (−ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t., 4 4   1 ω 0 = λ ω − λ−1 e04 (log(λ)) + λErr(ω, ω 0 ), 2 1 1 1 1 1 1 Err(ω, ω 0 ) = f e04 (f ) + ωf f − f η + f ξ + f ζ − κf 2 + 4 2 2 2 2 8 + l.o.t.,   1 ω 0 = λ−1 ω + λe03 (log(λ)) + λ−1 Err(ω, ω 0 ), 2 1 1 1 1 1 Err(ω, ω 0 ) = − f e03 (f ) + ωf f − f η + f ξ − f ζ − κf 2 + 4 2 2 2 8 + l.o.t.

1 1 f f κ − ωf 2 8 4 (9.3.5)

1 1 f f κ − ωf 2 8 4

The lower order terms we denote by l.o.t. are linear with respect to the Ricci coefficients {ξ, ξ, ϑ, κ, η, η, ζ, κ, ϑ} of the background, and quadratic or higher order in f, f , and do not contain derivatives of these latter.

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CHAPTER 9

Also, α0 = λ2 α + λ2 Err(α, α0 ), 3 Err(α, α0 ) = 2f β + f 2 ρ + l.o.t., 2   3 0 β = λ β + ρf + λErr(β, β 0 ), 2 1 Err(β, β 0 ) = f α + l.o.t., 2 ρ0 = ρ + Err(ρ, ρ0 ), 3 Err(ρ, ρ0 ) = ρf f + f β + f β + l.o.t., 2   3 0 β = λ−1 β + ρf + λ−1 Err(β, β 0 ), 2 1 Err(β, β 0 ) = f α + l.o.t., 2 α0 = λ−2 α + λ−2 Err(α, α0 ), 3 Err(α, α0 ) = 2f β + f 2 ρ + l.o.t. 2

(9.3.6)

The lower order terms we denote by l.o.t. are linear with respect to the curvature quantities α, β, ρ, β, α and quadratic or higher order in f, f , and do not contain derivatives of these latter. In the following lemma we rewrite a subset of these transformations in a more useful form: Lemma 9.21. Under a general transformation of type (9.2.18) with λ = ea we have, in particular, 1 1 ζ 0 = ζ − e0θ (a) − f ω + f ω − f χ + f χ + Err(ζ, ζ 0 ), 2 2   (9.3.7) 1 1 1 1 Err(ζ, ζ 0 ) = f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + (−f ϑ + f ϑ) + l.o.t. 2 4 16 4 κ0 = ea (κ + /d01 f ) + ea Err(κ, κ0 ), 1 1 1 Err(κ, κ0 ) = f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ − f 2 κ 2 4 4 2 + f f ω − f ω + l.o.t.

(9.3.8)


κ0 = e−a κ + /d01 f + e−a Err(κ, κ0 ),     1 0 1 2 2 3 1 0 2 Err(κ, κ ) = − f eθ f f + f f + f f + (f f ) e0θ (f ) 2 8 4 8    

1 1 1 1 + 1 + f f f e0θ f f − f 1 + f f e0θ f 2 4 2 4 4 1 2 + f (−ζ + η) + f ξ − f κ + f f ω − f 2 ω + l.o.t. 4

(9.3.9)

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GCM PROCEDURE

Also, ϑ0 = λ (ϑ − /d?2 0 (f )) + λErr(ϑ, ϑ0 ), 1 1 1 Err(ϑ, ϑ0 ) = f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ + f f κ 2 4 4 + f f ω − f 2 ω + l.o.t.
ϑ0 = λ−1 ϑ − /d?2 0 (f ) + λ−1 Err(ϑ, ϑ0 ),     (9.3.10) 1 0 1 2 2 3 1 0 2 Err(ϑ, ϑ ) = − f eθ f f + f f + f f + (f f ) e0θ (f ) 2 8 4 8    

1 1 1 1 + 1 + f f f e0θ f f − f 1 + f f e0θ f 2 + f (−ζ + η) 4 2 4 4 1 + f ξ + f f κ + f f ω − f 2 ω + l.o.t. 4 The lower order terms we denote by l.o.t. are cubic or higher order in the small quantities ξ, ξ, ϑ, η, η, ζ, ϑ as well as f, f , and do not contain derivatives of these quantities. We also have   3 0 β = λ β + ρf + λErr(β, β 0 ), 2 1 Err(β, β 0 ) = f α + l.o.t., (9.3.11) 2 ρ0 = ρ + Err(ρ, ρ0 ), 3 Err(ρ, ρ0 ) = ρf f + f β + f β + l.o.t. 2 The lower order terms above denoted by l.o.t. are cubic or higher order in the small quantities ξ, ξ, ϑ, η, η, ζ, ϑ as well as a, f, f . Lemma 9.22. The following transformation formula holds true:   1 1 µ0 = µ + ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ + Err(µ, µ0 ), 4 4
1 Err(µ, µ0 ) = − /d 01 Err(ζ, ζ 0 ) − Err(ρ, ρ0 ) + ϑ0 ϑ0 − ϑϑ . 4 ˇ R) ˇ The error term Err(µ, µ0 ) is quadratic or higher order with respect to (f, f , a, Γ, 0 and depends only on at most two angular derivatives eθ of f and one angular derivative e0θ of a, f . Proof. Recall that 1 µ = − /d1 ζ − ρ + ϑϑ. 4

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CHAPTER 9

Therefore, µ0

1 − /d 01 ζ 0 − ρ0 + ϑ0 ϑ0 4   1 1 0 0 − /d 1 ζ − eθ (a) − f ω + f ω − f κ + f κ + Err(ζ, ζ 0 ) − ρ − Err(ρ, ρ0 ) 4 4 1 0 0 + ϑϑ 4   1 1 1 − /d 01 ζ − ρ + ϑϑ − /d01 ( /d?1 )0 a − f ω + f ω − f κ + f κ 4 4 4
1 /d 01 Err(ζ, ζ 0 ) − Err(ρ, ρ0 ) + ϑ0 ϑ0 − ϑϑ . 4

= =

= − Note that

1 − /d 01 ζ − ρ + ϑϑ 4

= =

1 − /d 1 ζ − ρ + ϑϑ + f e3 ζ + f e4 ζ + l.o.t. 4 µ + f e3 ζ + f e4 ζ + l.o.t.

Hence, 0

µ

= µ+

/d 01



−( /d?1 )0 a

 1 1 + f ω − f ω + f κ − f κ + Err(µ, µ0 ) 4 4

where Err(µ, µ0 )

= − /d 01 Err(ζ, ζ 0 ) − Err(ρ, ρ0 ) +


1 0 0 ϑ ϑ − ϑϑ + f e3 ζ + f e4 ζ + l.o.t. 4

In view of the transformation formulas for ϑ, ϑ and the structure of the error terms Err(ζ, ζ 0 ), Err(ρ, ρ0 ), Err(ϑ, ϑ0 ), Err(ϑ, ϑ0 ) in Lemma 9.21 we easily deduce that the error term Err(µ, µ0 ) depends only on at most two angular derivatives e0θ of f and one angular derivative e0θ of a, f . We shall also make use of the following lemma. Lemma 9.23. We have the transformation equations, 1 e0θ (κ0 ) = eθ κ + e0θ /d01 f + κe0θ a − κ(f κ + f κ) + κ(f ω − ωf ) + f ρ 4 + Err(e0θ κ0 , eθ κ), 1 e0θ (κ0 ) = eθ κ + e0θ /d01 f − κe0θ a − κ(f κ + f κ) + κ(f ω − ωf ) + f ρ 4 (9.3.12) + Err(e0θ κ0 , eθ κ),   1 1 3 0 0 0 0 ? 0 eθ (µ ) = eθ µ + eθ ( /d1 ) −( /d1 ) a + f ω − f ω + f κ − f κ + ρ(f κ + f κ) 4 4 4 + Err(e0θ µ0 , eθ µ),

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GCM PROCEDURE

where Err(e0θ κ0 , eθ κ)

  1 1 (ea − 1) eθ κ + e0θ /d 01 f + f e4 κ + f e3 κ 2 2 "   1 + ea e0θ Err(κ, κ0 ) + e0θ (a) /d 01 f + Err(κ, κ0 ) + f f eθ κ 2 #   1 2 1 1 + f f e3 κ + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) 8 2 2   1 1 2 1 1 2 + f f eθ κ + f f e3 κ + f 2 /d1 ξ − ϑ + 2(η + η + 2ζ)ξ , 2 8 2 2

Err(e0θ κ0 , eθ κ)

  1 1 (e−a − 1) eθ κ + e0θ /d 01 f + f e3 κ + f e4 κ 2 2 "   1 + e−a e0θ Err(κ, κ0 ) + e0θ (a) /d 01 f + Err(κ, κ0 ) + f f eθ κ 2 #   1 2 1 1 + f f e3 κ + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) 8 2 2   1 1 2 1 1 2 + f f eθ κ + f f e3 κ + f 2 /d1 ξ − ϑ + 2(η + η − 2ζ)ξ , 2 8 2 2

=

=

and Err(e0θ µ0 , eθ µ)

1 1 = e0θ Err(µ, µ0 ) + f f eθ µ + f 2 f e3 µ 2 8  1  1 f /d1 β − ϑ α − ζ β + 2(η β + ξ β) − 2 2  1  1 − f /d1 β − ϑ α + ζ β + 2(η β + ξ β) 2 2     1 1 1 1 f e4 − /d1 ζ + ϑϑ + f e3 − /d1 ζ + ϑϑ . + 2 4 2 4

Proof. Applying the vectorfield e0θ to κ0

= ea (κ + /d 01 f + Err(κ, κ0 ))

we deduce e0θ (κ0 )

    = ea e0θ κ + e0θ /d 01 f + e0θ Err(κ, κ0 ) + ea e0θ (a) κ + /d 01 f + Err(κ, κ0 ) .

Hence, e−a e0θ (κ0 )

  = e0θ κ + e0θ /d 01 f + e0θ Err(κ, κ0 ) + e0θ (a) κ + /d 01 f + Err(κ, κ0 )

and thus e0θ (κ0 )

1 1 = eθ κ + e0θ (a)κ + e0θ /d 01 f + f e4 κ + f e3 κ + Err1 [eθ (κ), e0θ (κ0 )] 2 2

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CHAPTER 9

with error term Err1 [eθ (κ), e0θ (κ0 )]

  1 1 (ea − 1) eθ κ + e0θ /d 01 f + f e4 κ + f e3 κ 2 2 "   1 + ea e0θ Err(κ, κ0 ) + e0θ (a) /d 01 f + Err(κ, κ0 ) + f f eθ κ 2 # 1 2 + f f e3 κ . 8

=

Now, making use of     1 1 1 1 e0θ κ = 1 + f f eθ κ + f e4 κ + f 1 + f f e3 κ 2 2 2 4 1 1 1 1 2 = eθ κ + f e4 κ + f e3 κ + f f eθ κ + f f e3 κ 2 2 2 8 and the null structure equations, 1 e3 (κ) + κ κ − 2ωκ = 2 1 e4 κ + κ2 + 2ωκ = 2

1 2 /d1 η + 2ρ − ϑ ϑ + 2(ξξ + η 2 ), 2 1 2 2 /d1 ξ − ϑ + 2(η + η + 2ζ)ξ, 2

we deduce     1 1 1 1 e0θ κ = eθ κ + f − κ2 − 2ωκ + f − κ κ + 2ωκ + 2ρ 2 2 2 2   1 1 1 1 + f f eθ κ + f 2 f e3 κ + f 2 /d1 ξ − ϑ2 + 2(η + η + 2ζ)ξ 2 8 2 2   1 1 + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) . 2 2 Hence, e0θ (κ0 )

1 = eθ κ + e0θ (a)κ + e0θ /d01 f + κe0θ a − κ(f κ + f κ) + κ(f ω − ωf ) + f ρ 4 +Err(e0θ κ0 , eθ κ)

where Err(e0θ κ0 , eθ κ)

= +

  1 1 Err1 (e0θ κ0 , eθ κ) + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) 2 2   1 1 2 1 1 f f eθ κ + f f e3 κ + f 2 /d1 ξ − ϑ2 + 2(η + η + 2ζ)ξ 2 8 2 2

as desired. The formula for e0θ (κ0 ) is easily derived by symmetry from the one on e0θ (κ0 ). Note however that a becomes −a
in the transformation.
Applying the operator e0θ = 1 + 12 f f eθ + 12 f e4 + 12 f 1 + 14 f f e3 to the trans-

511

GCM PROCEDURE

formation formula for µ,   1 1 = µ + ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ + Err(µ, µ0 ), 4 4

µ0 we derive e0θ (µ0 )

  1 1 = e0θ (µ) + e0θ ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ + e0θ Err(µ, µ0 ) 4 4   1 1 1 1 = eθ (µ) + f e4 µ + f e3 µ + e0θ ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ 2 2 4 4 1 1 + e0θ Err(µ, µ0 ) + f f eθ µ + f 2 f e3 µ. 2 8

Recalling that µ = − /d1 ζ − ρ + 14 ϑϑ we find 1 1 f e4 µ + f e3 µ = 2 2

  1 1 1 − (f e3 + f e4 )ρ + f e4 − /d1 ζ + ϑϑ 2 2 4   1 1 + f e3 − /d1 ζ + ϑϑ . 2 4

Recalling the Bianchi equations for e3 ρ, e4 ρ 3 e4 ρ + κρ = /d1 β − 12 ϑ α + ζ β + 2(η β + ξ β), 2 3 e3 ρ + κρ = /d1 β − 12 ϑ α − ζ β + 2(η β + ξ β), 2 we further deduce 1 1 f e4 µ + f e3 µ = 2 2 − − +

3 ρ(f κ + f κ) 4  1  1 f /d1 β − ϑ α − ζ β + 2(η β + ξ β) 2 2  1  1 f /d1 β − ϑ α + ζ β + 2(η β + ξ β) 2 2     1 1 1 1 f e4 − /d1 ζ + ϑϑ + f e3 − /d1 ζ + ϑϑ . 2 4 2 4

Therefore, e0θ (µ0 )

  3 1 1 = eθ (µ) + ρ(f κ + f κ) + e0θ ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ 4 4 4 + Err(eθ µ, eθ µ)

512

CHAPTER 9

with Err(e0θ µ0 , eθ µ)

1 1 = e0θ Err(µ, µ0 ) + f f eθ µ + f 2 f e3 µ 2 8  1  1 − f /d1 β − ϑ α − ζ β + 2(η β + ξ β) 2 2  1  1 − f /d1 β − ϑ α + ζ β + 2(η β + ξ β) 2 2     1 1 1 1 + f e4 − /d1 ζ + ϑϑ + f e3 − /d1 ζ + ϑϑ 2 4 2 4

as desired. Finally recalling the definition of the Hodge operators /d1 , /d?1 , ( /d1 )0 , ( /d?1 )0 and noticing that ( /d?1 )0 (κ0 ) ( /d?1 )0 (κ0 )

= =

( /d?1 )0 (ˇ κ0 ), ( /d?1 )0 (ˇ κ0 ),

( /d?1 )(κ) = ( /d?1 )(ˇ κ), ? ? ( /d1 )(κ) = ( /d1 )(ˇ κ),

( /d?1 )0 (µ0 )

=

( /d?1 )0 (ˇ µ0 ),

( /d?1 )(µ) = ( /d?1 )(ˇ µ),

( /d?1 )0 (µ0 )

=

( /d?1 )0 (ˇ µ0 ),

( /d?1 )(µ) = ( /d?1 )(ˇ µ),

we recast the results of Lemma 9.23 in the following form. Lemma 9.24. We have the transformation equations, 1 ( /d?1 )0 (ˇ κ0 ) = /d?1 (ˇ κ) + ( /d?1 )0 ( /d1 )0 f + κ( /d?1 )0 a − ρf + κ(f κ + f κ) 4 − κ(f ω − f ω) − Err1 , 1 ( /d?1 )0 (ˇ κ0 ) = /d?1 (ˇ κ) + ( /d?1 )0 ( /d1 )0 f − κ( /d?1 )0 a − ρf + κ(f κ + f κ) 4 (9.3.13) − κ(f ω − f ω) − Err2 ,   1 1 ( /d?1 )0 (ˇ µ0 ) = /d?1 (ˇ µ) + ( /d?1 )0 ( /d1 )0 −( /d?1 )0 a + f ω − f ω + f κ − f κ 4 4
3 − ρ f κ + f κ − Err3 , 4 where Err1 = Err(e0θ κ0 , eθ κ) = e0θ Err(κ, κ0 ) + ae0θ /d 01 f + e0θ (a) /d 01 f  
1 + a eθ κ + f e4 κ + f e3 κ + f /d1 η + f /d1 ξ + l.o.t., 2 Err2 = Err(e0θ κ0 , eθ κ) = e0θ Err(κ, κ0 ) − ae0θ /d 01 f − e0θ (a) /d 01 f  
1 − a eθ κ + f e4 κ + f e3 κ + f /d1 η + f /d1 ξ + l.o.t., 2
1 Err3 = Err(e0θ µ0 , eθ µ) = e0θ Err(µ, µ0 ) − f /d1 β + f /d1 β 2 1 − (f e3 + f e4 ) /d1 ζ + l.o.t., 2

(9.3.14)

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GCM PROCEDURE

ˇ R ˇ and contain where the terms denoted by l.o.t. are cubic or higher order in a, f, f , Γ, no derivatives of (a, f, f ). 9.3.1

Main GCM equations ◦

Given a deformation Ψ : S → S and adapted frame (e03 , e04 , e0θ ) with e0θ = eS θ we derive an elliptic system for the transition parameters (a, f, f ). The system will later be used in the construction of GCM surfaces. In what follows we denote by /d1S , /d2S , /d1S,? , /d2S,? the basic Hodge operators on S. Noting that the transformation formulae in (9.3.13)–(9.3.14) contain only the operators ( /d1 )0 = /d1S , ( /d?1 )0 = /d1S,? applied to a, f, f we introduce the simplified notation, /d S := ( /d1 )0 ,

/d S,? := ( /d?1 )0 ,

AS := /d S,? /d S ,

/d?:= /d?1 .

(9.3.15)

With these notation (9.3.13) takes the following form: 1 /d S,? κ ˇ S = /d?κ ˇ + AS f + κ /d S,? a − ρf + κ(f κ + f κ) − κ(f ω − f ω) − Err1 , 4 1 S,? S ? S S,? /d κ ˇ = /d κ ˇ + A f − κ /d a − ρf + κ(f κ + f κ) − κ(f ω − f ω) − Err2 , 4  
1 1 3 S,? S ? S S,? /d µ ˇ = /d µ ˇ + A − /d a + f ω − f ω + f κ − f κ − ρ f κ + f κ − Err3 , 4 4 4 or   1 1 3 AS − /d S,? a + f ω − f ω + f κ − f κ − ρ(κf + κf ) = /d S,? µ ˇS − /d?µ ˇ 4 4 4 + Err3 , 1 AS f + κ /d S,? a − ρf + κ(f κ + f κ) − κ(f ω − f ω) = /d S,? κ ˇ S − /d?κ ˇ (9.3.16) 4 + Err1 , 1 AS f − κ /d S,? a − ρf + κ(f κ + f κ) − κ(f ω − f ω) = /d S,? κ ˇ S − /d?κ ˇ 4 + Err2 . Since AS is invertible9 we can write, setting z := κf + κf , /d S,? a

9 We

of

/d S


1 1 3 = f ω − f ω + f κ − f κ − (AS )−1 ρz 4 4 4
+(AS )−1 − /d S,? µ ˇS + /d?µ ˇ − Err3 .

R R have S f AS f = S ( /d S f )2 which in view of the identity (2.1.21) for /d1S and the definition S implies that A is invertible.

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CHAPTER 9

We can thus eliminate /d S,? a from the last two equations,  
1 3 AS f + κκ − ρ f − κ(AS )−1 ρz = /d S,? κ ˇ S − /d?κ ˇ 2 4

AS f +




− κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err4 ,


1 3 κκ − ρ f + κ(AS )−1 ρz = /d S,? κ ˇ S − /d?κ ˇ 2 4


+ κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err5 , where Err4

=

Err1 + κ(AS )−1 Err3 ,

Err5 = Err2 − κ(AS )−1 Err3 .

Therefore the system (9.3.16) is equivalent to the system  
1 3 AS f + κκ − ρ f − κ(AS )−1 ρz = /d S,? κ ˇ S − /d?κ ˇ 2 4

AS f +




− κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err4 ,


1 3 κκ − ρ f + κ(AS )−1 ρz = /d S,? κ ˇ S − /d?κ ˇ 2 4


+ κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err5 ,


3 /d S,? a + (AS )−1 ρz − f ω + f ω 4
1 1 − f κ + f κ = (AS )−1 − /d S,? µ ˇS + /d?µ ˇ − (AS )−1 Err3 . 4 4 We summarize the results of the above calculation in the following lemma. Lemma 9.25. The original system (9.3.13) in (a, f, f ) associated to a deformation sphere S is equivalent to the following:

3 AS + V f = κ(AS )−1 ρz + /d S,? κ ˇ S − /d?κ ˇ 4
− κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err4 ,

3 AS + V f = − κ(AS )−1 ρz + /d S,? κ ˇ S − /d?κ ˇ 4
+ κ(AS )−1 − /d S,? µ ˇS + /d?µ ˇ + Err5 ,
3 1 1 /d S,? a = − (AS )−1 ρz + f ω − f ω + f κ − f κ 4 4 4
+ (AS )−1 − /d S,? µ ˇS + /d?µ ˇ − (AS )−1 Err3 ,

(9.3.17)

where z := κf + κf ,

V :=

1 κκ − ρ. 2

(9.3.18)

The error terms are given by Err1 , Err2 , Err3 , defined in Lemma 9.24, and Err4

= Err1 + κ(AS )−1 Err3 ,

Err5 = Err2 − κ(AS )−1 Err3 . (9.3.19)

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GCM PROCEDURE

Remark 9.26. We note the following remarks concerning the system (9.3.17). 1. The right-hand side of the equations is linear in the quantities /d S,? κ ˇS ,

/d S,? κ ˇS ,

/d S,? µ ˇS ,

as well as

/ˇ d?κ,

/ˇ d?κ,

/ˇ d?µ.

The first group is to be constrained by our GCM conditions in the next section while the second group depends on assumptions regarding the background foliation of R. 2. The error terms contain only S-angular derivatives of (a, f, f ) of order at most equal to the order of the corresponding operators on the left-hand sides, see Lemma 9.27 below. Thus the system is in a standard quasilinear elliptic system form. 3. In order to uniquely solve the equations for f and f , we need to establish the coercivity of the operator AS + V . One can easily show that the potential V is positive for small values of r, i.e., r near rH = 2m0 (1 + δH ) but negative for large r. In fact AS + V has a nontrivial kernel for large r as one can easily see from the following calculation. Since 1 1 K = −ρ − κκ + ϑϑ 4 4

AS = /d?1 S /d1 S = /d2 S /d?2 S + 2K, we deduce 1 AS + V = AS + κκ − ρ 2

=

1 /d2 S /d?2 S − 3ρ + ϑϑ. 2

Thus for large enough r the operator AS + V behaves like /d2S /d2S,? which has a nontrivial kernel. 4. To be able to correct for the lack of coercivity of the system, we need to prescribe the ` = 1 modes of (f, f ). 5. The equations do not provide information on the average of a. For this we will need yet another equation derived in section 9.3.2. Lemma 9.27. The error terms Err1 , . . . , Err5 can be written schematically as follows:

r2 Err1 = ( d/S )2 (f, f , a)2 + d/S (f, f , a)(rΓg ) + l.o.t.,

ˇ + l.o.t., rErr2 = r−1 ( d/S )2 (f, f , a)2 + d/S (f, f , a)(Γ) (9.3.20)

r3 Err3 = ( d/S )3 (f, f , a)2 + ( d/S )2 (f, f , a)(rΓg ) + l.o.t., Err4 , Err5 = Err1 + r−1 (AS )−1 Err3 , ˇ R ˇ and where the lower order terms denoted l.o.t. are cubic with respect to a, f, f , Γ, may involve fewer angular (along S) derivatives of a, f, f . Remark 9.28. Note that Err2 behaves worse in powers of r than Err1 . The reason 0 is the presence of the terms f eθ ξ, eθ (f ξ) in the formula for eS θ (Err(κ , κ)). Proof. Note that in the spacetime region R of interest r and rS are comparable.

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CHAPTER 9

Recall, see (9.3.14), Err1 = Err(e0θ κ0 , eθ κ) = e0θ Err(κ, κ0 ) + ae0θ /d 01 f + e0θ (a) /d 01 f  
1 + a eθ κ + f e4 κ + f e3 κ + f /d1 η + f /d1 ξ + l.o.t., 2 Err2 = Err(e0θ κ0 , eθ κ) = e0θ Err(κ, κ0 ) − ae0θ /d 01 f − e0θ (a) /d 01 f  
1 − a eθ κ + f e4 κ + f e3 κ + f /d1 η + f /d1 ξ + l.o.t., 2
1 1 Err3 = Err(e0θ µ0 , eθ µ) = e0θ Err(µ, µ0 ) − f /d1 β + f /d1 β − (f e3 + f e4 ) /d1 ζ 2 2 + l.o.t., and10 1 1 1 f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω 2 4 4 +l.o.t.,     1 1 3 1 Err(κ, κ0 ) = − f e0θ f f + f 2 f 2 + f f + (f f )2 e0θ (f ) 2 8 4 8    

1 1 1 1 + 1 + f f f e0θ f f − f 1 + f f e0θ f 2 4 2 4 4 1 2 + f (−ζ + η) + f ξ − f κ + f f ω − f 2 ω + l.o.t. 4 Err(κ, κ0 )

=

Also, Err(µ, µ0 )

=

Err(ζ, ζ 0 )

=

Err(ρ, ρ0 )

=


1 −e0θ Err(ζ, ζ 0 ) − Err(ρ, ρ0 ) + ϑ0 ϑ0 − ϑϑ , 4   1 1 1 1 f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + (−f ϑ + f ϑ) + l.o.t., 2 4 16 4 3 ρf f + f β + f β + l.o.t. 2

We write schematically11 Err1

(f, f , a)(r−2 d/S )2 (f, f , a) + (r−1 d/S (f, f , a))2 + r−1 d/S (f, f , a)Γg
1 + r−2 d/S (f 2 ) + a f e4 κ + f e3 κ + l.o.t. 2 =




Making use of 1 e3 (κ) + κ κ − 2ωκ = 2 1 e4 κ + κ2 + 2ωκ = 2 10 Recall

1 2 /d1 η + 2ρ − ϑ ϑ + 2(ξξ + η 2 ), 2 1 2 2 /d1 ξ − ϑ + 2(η + η + 2ζ)ξ, 2

also the outgoing geodesic conditions, i.e., ξ = 0, ζ + η = 0, ζ − η = 0, ω = 0. last term r−2 / dS (f 2 ) on the right of the identity below is due to the term e0θ (f 2 ω) in the expression of e0θ Err(κ, κ0 ). 11 The

517

GCM PROCEDURE

and treating the curvature terms that appear as Γg , we easily derive

r2 Err1 = ( d/S )2 (f, f , a)2 + d/S (f, f , a)(rΓg ) . We obtain a worse estimate for Err2 because of the presence e0θ (f ξ), since ξ ∈ Γb . In fact,

ˇ . rErr2 = r−1 ( d/S )2 (f, f , a)2 + d/S (f, f , a)Γ For Err3 we write similarly, treating the curvature terms that appear as Γg ,

eθ (µ, µ0 ) = r−3 ( d/S )3 (f, f , a)2 + r−3 ( d/S )2 (f, f , a)Γg + l.o.t. Using the null structure equations for ζ we infer that Err3

=

eθ (µ, µ0 ) −

=

r−3 ( d/S )3


1 1 f /d1 β + f /d1 β − (f e3 + f e4 ) /d1 ζ + l.o.t. 2 2

(f, f , a)2 + r−3 ( d/S )2 (f, f , a)Γg + l.o.t.

as stated. Making use of the above lemma and the assumptions A1–A3 we can derive the following. ◦

Lemma 9.29. Assume given a deformation Ψ : S → S in R and adapted frame (e03 , e04 , e0θ ) with e0θ = eS θ with transition parameters a, f, f defined on S. Assume that there exists a small enough constant δ1 such that the following holds true: ◦

◦

(r )−1 kU 0 k

◦ ◦

hsmax −1 (S,g /)

+ (r )−2 kS 0 k

◦ ◦

hsmax −1 (S,g /)

.

δ1 .

Then, for 5 ≤ s ≤ smax + 1,



r−2 (f, f , a)



kErr1 , Err2 khs−2 (S) .



r−3 (f, f , a)



kErr3 khs−3 (S) .



r−2 (f, f , a)



kErr4 , Err5 khs−2 (S) .

hs (S)





 + r−1 f, f , a





 + r−1 f, f , a



,

hs−1 (S)

◦

hs (S)

hs (S)



 + r−1 f, f , a

◦

, (9.3.21)

hs−1 (S)

◦

.

hs−1 (S)

Proof. The proof follows easily from Lemma 9.27, Corollary 9.16, coercivity of AS and obvious product estimates on S. Consider for example the term

ˇ . Err2 = r−2 ( d/S )2 (f, f , a)2 + r−1 d/S (f, f , a)(Γ) We write ( d/S )k Err2



ˇ + l.o.t. = r−2 ( d/S )2+k (f, f , a)2 + r−1 ( d/S )1+k (f, f , a)(Γ)

and ( d/S )2+k (f, f , a)2




=

X i+j=k+2

d/i (f, f , a) · d/j (f, f , a).

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CHAPTER 9

k+2 Thus, dividing the sum into terms with i ≥ [ k+2 2 ] and i < [ 2 ] and using Sobolev estimates for the terms involving fewer derivatives we derive, for [ k+2 2 ] + 2 ≤ k + 1,


k( d/S )2+k (f, f , a)2 kL2 (S)

. r−1 k(a, f, f )khk+1 (S) k(a, f, f )khk+2 (S) .

ˇ Similarly, making use of our assumptions for Γ,
ˇ ˇ h (S) ( d/S )1+k (f, f , a)(Γ) . r−1 k(a, f, f )khk+1 (S) kΓk k+1 .

◦ −1

r

k(a, f, f )khk+1 (S) .

Thus, for all 3 ≤ k ≤ smax − 1 S k

k( d/ ) Err2 kL2 (S) . r

−2



◦

k(f, f , a)khk+2 (S)  + r



(f, f , a)





f, f , a



−1

,

hk+1 (S)

i.e., for 5 ≤ s ≤ smax + 1, kErr2 khs−2 (S)

.

r

−2



(f, f , a)

hs (S)



◦

+r

−1

.

hs−1 (S)

All other terms can be treated similarly. 9.3.2

Equation for the average of a

In the proof of existence and uniqueness of GCMS, see Theorem 9.32, we will need, in addition to the equations derived so far, an equation for the average of a. To achieve this we make use of the transformation formula for κ of Lemma 9.21: κ0

=

Err(κ, κ0 )

=

ea (κ + /d01 f ) + ea Err(κ, κ0 ), 1 1 1 f (ζ + η) + f ξ + f e4 f + f e3 f + f f κ + f f ω − ωf 2 + l.o.t. 2 2 4

which we rewrite in the form     2 2 κS = ea +κ ˇ+ κ− + /d S f + Err(κ, κ0 ) . r r We deduce (ea − 1)

2 r

  2 2 − ea κ ˇ + κ − + /d S f − ea Err(κ, κ0 ) r r     2 2 2 2 S = κS − S + − − κ ˇ + κ − + / d f r rS r r   2 − ea Err(κ, κ0 ) − (ea − 1) κ ˇ + κ − + /d S f r = κS −

519

GCM PROCEDURE

or 2 a r

 2 S − κ ˇ + κ − + /d f r   2 2 a 0 a S − e Err(κ, κ ) − (e − 1) κ ˇ + κ − + /d f − (ea − 1 − a) . r r 2 = κ − S+ r S



2 2 − rS r





We deduce a Err6

     2 rS rS 2 κS − S + 1 − − κ ˇ + κ − + /d S f + Err6 r r 2 r     S r 2 2 = − ea Err(κ, κ0 ) − (ea − 1) κ ˇ + κ − + /d S f − (ea − 1 − a) 2 r r  S  r − a −1 . r =

rS 2



Taking the average on S we infer that aS =

rS 2



S

κS −

2 rS



 S  S rS rS 2 S + 1− − κ ˇ+κ− + Err6 r 2 r

(9.3.22)

S

where h denotes the average of h on S. 9.3.3

Transversality conditions

Lemma 9.30. Assume given a deformed sphere S ⊂ R with adapted null frame S S eS 3 , e4 , eθ and transition functions (a, f, f ). We can extend a, f, f , and thus the S S frame eS 3 , e4 , eθ , in a small neighborhood of S such that the following hold true: ξ S = 0,

ω S = 0,

η S + ζ S = 0.

(9.3.23)

Proof. According to Proposition 9.20 we have   1 1 ξ S = e2a ξ + e−a eS (f ) + f κ + f ω + e2a Err(ξ, ξ S ), 4 2 4 1 1 S ζ S = ζ − eS θ (a) − f ω + f ω − f κ + f κ + Err(ζ, ζ ), 4 4 1 1 S η S = η + e−a eS 4 f − f ω + f κ + Err(η, η ), 2 4   1 ω S = ea ω − e−a eS a + ea Err(ω, ω S ). 4 2 S Clearly the conditions ξ S = 0, ω S = 0 allow us to determine eS 4 f and e4 a on S S S S while the condition η + ζ = 0 allows us to determine e4 f on S.

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CHAPTER 9

Remark 9.31. According to Proposition 9.20 we also have   1 1 ξ S = e−2a ξ + ea eS (f ) + f κ + f ω + e−2a Err(ξ, ξ S ), 3 2 4   1 ω S = e−a ω + ea eS a + e−a Err(ω, ω S ), 3 2 so that we may impose, in addition, vanishing conditions on ξ S and ω S along S. S Indeed these are determined by eS 3 f and e3 a.

9.4

EXISTENCE OF GCM SPHERES

We now impose the GCM conditions on the deformed sphere S: 2 /d2S,? /d1S,? κS = /d2S,? /d1S,? µS = 0, κS = S , r Z Z Φ f e = Λ, f eΦ = Λ, S

(9.4.1)

S

where (f, f ) belong to the triplet (f, f , λ = ea ) which denotes the change of frame ◦

coefficients from the frame of S to the one of S. We are ready to state the first main result of this chapter. ◦

◦ ◦

Theorem 9.32 (Existence of GCM spheres). Let S = S(u, s) be a fixed sphere of the (u, s) outgoing geodesic foliation of a fixed spacetime region R. Assume in addition to A1–A3 that there exists scalar functions C = C(u, s), M = M (u, s), such that the following estimates hold true on R, for all k ≤ smax , with smax ≥ 6, k−1 ?
◦ d /d1 κ − CeΦ . δr−3 , k−1 ?
◦ d /d1 µ − M eΦ . δr−4 , ◦ κ − 2 + dk κ ˇ . δr−2 . r

(9.4.2)

For any fixed Λ, Λ ∈ R verifying ◦

|Λ|, |Λ| . δr2

(9.4.3) ◦

there exists a unique GCM sphere S = S(Λ,Λ) , which is a deformation of S, such that the GCM conditions (9.4.1) are verified. Moreover the following estimates hold true. 1. We have S ◦ r − 1 . r−1 δ. ◦ r ◦

In particular r, r and rS are all comparable in R.

(9.4.4)

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GCM PROCEDURE

2. The unique functions (λ, f, f ) on S, which relate the original frame e3 , e4 , eθ to S S the new frame on eS 3 , e4 , eθ according to (9.2.18), verify the estimates ◦



f, f , log λ

k ≤ smax + 1.

. δ,

hk (S)

(9.4.5)

3. The parameters U, S of the deformation, see Definition 9.6, verify the estimate k(U 0 , S 0 )k

◦

L∞ (S)

+

max

0≤s≤smax −1

◦

r−1 k(U 0 , S 0 )k

◦ ◦

hs (S,g /)

. δ.

(9.4.6)

4. The Hawking mass mS verifies the estimate S ◦ m − m

◦

. δ.

(9.4.7)

5. The curvature components (αS , β S , ρS , β S , αS ), as well as µS and the Ricci coefficients12 (κS , ϑS , ζ S , κS , ϑS ) on S, verify, for all k ≤ smax , ◦

kˇ κS , ϑS , ζ S , κ ˇ S khk (S) . r−1 , ◦

kϑS khk (S) . , ◦

kαS , β S , ρˇS , µ ˇS khk (S) . r−2 ,

(9.4.8)

◦

kβ S khk (S) . r−1 , ◦

kαS khk (S) . .

6. The functions, (λ, f, f ) uniquely defined above, can be smoothly extended to a small neighborhood of S in such a way that the corresponding Ricci coefficients verify the following transversality conditions ξ S = 0,

ω S = 0,

η S + ζ S = 0.

(9.4.9)

In that case, the following estimates hold13 for all k ≤ smax − 1 keS 4 (f, f , log λ)khk (S)

◦

. r−1 δ + r−3 (|Λ| + |Λ|) ,

(9.4.10)

and ◦

keS κS , ϑS , ζ S , κ ˇ S )khk (S) . r−2 , 4 (ˇ ◦

keS 4

S −1 keS , 4 (ϑ )khk (S) . r
◦ αS , β S , ρˇS , µ ˇS khk (S) . r−3 , S keS 4 (β )khk (S)

◦ −2

. r

(9.4.11)

,

◦

S −1 keS . 4 (α )khk (S) . r 12 All

S other Ricci coefficients involve the transversal derivatives eS 3 , e4 of the frame. ◦ ◦ be more precise one should replace r by r in the estimates below. Of course r and r are comparable in R, in particular on S. 13 To

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CHAPTER 9

To prove Theorem 9.32, it will be useful, using the fact that the kernel of /d2S,? is spanned by eΦ , to rewrite the GCM conditions (9.4.1) in the following form: /d1S,? κS = C S eΦ , /d1S,? µS = M S eΦ , Z Z f eΦ = Λ, f eΦ = Λ, S

κS =

2 , rS

(9.4.12)

S

where C S and M S are constants. Proposition 9.33. Assume that there exists constants C S , M S , such that the deformed sphere S verifies the GCM conditions (9.4.12). Then, the deformation parameters (a, f, f ) verify the system


3 AS + V f = κ(AS )−1 ρz − /d?κ ˇ − κ(AS )−1 − M S eΦ + /d?µ ˇ 4 + Err4 ,


3 AS + V f = − κ(AS )−1 ρz + C S eΦ − /d?κ ˇ + κ(AS )−1 − M S eΦ + /d?µ ˇ 4 + Err5 , (9.4.13)
3 S −1 1 1 S,? /d a = − (A ) ρz + f ω − f ω + f κ − f κ 4 4 4
+ (AS )−1 − M S eΦ + /d?µ ˇ − (AS )−1 Err3 ,  S  S rS rS 2 S S a = 1− − κ ˇ+κ− + Err6 , r 2 r and Z

Φ

Z

e f = Λ, S

eΦ f = Λ,

(9.4.14)

S

where we recall that z = κf + κf ,

Err4 = Err1 + κ(AS )−1 Err3 ,

Err6

V =

1 κκ − ρ, 2

Err5 = Err2 − κ(AS )−1 Err3 ,

    rS a 2 2 = − e Err(κ, κ0 ) − (ea − 1) κ ˇ + κ − + /d S f − (ea − 1 − a) 2 r r  S  r − a −1 r

with the error terms Err1 , Err2 , Err3 , defined in Lemma 9.24. Conversely, if there exist constants C S , M S , such that the deformation parameters (a, f, f ) verify the system (9.4.13), (9.4.14), then the deformed sphere S verifies the GCM conditions (9.4.12). Proof. The first statement is an immediate consequence of Lemma 9.25 and (9.3.22).

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GCM PROCEDURE

We then focus on the second statement, i.e., we assume that the deformation parameters (a, f, f ) verify the system (9.4.13), (9.4.14) for some constants C S , M S . Then, subtracting the first three equations of (9.4.13) from (9.3.17) and the last equation of (9.4.13) from (9.3.22), we obtain
/d S,? κ ˇ S − κ(AS )−1 − /d S,? µ ˇS + M S eΦ = 0,
/d S,? κ ˇ S − C S eΦ + κ(AS )−1 − /d S,? µ ˇS + M S eΦ = 0,
(AS )−1 − /d S,? µ ˇS + M S eΦ = 0,   S rS 2 S κ − S = 0, 2 r which, together with (9.4.14), immediately implies (9.4.12). Remark 9.34. In view of Propositions 9.14 and 9.33, to find a GCM sphere amounts to solving the following coupled system:  
1 # 0 S # 1/2 # # ς U = (γ ) f 1 + (f f ) , 4
1/2 # ς# # 0 1 S0 − Ω U = (γ S )# f , 2 2  # (9.4.15) 1 2 S # # # 2 (γ ) = γ + (ς ) Ω + b γ (∂θ U )2 − 2ς # ∂θ U ∂θ S 4 − (γςb)# ∂θ U,

U (0) = S(0) = 0,

where the inputs (a, f, f ) verify (9.4.13), (9.4.14). Recall that for a reduced scalar h defined on S we write ◦ ◦

h# (u, s, θ)

◦

◦

= h(u + U (θ), s + S(θ), θ).

We will solve the coupled system of equations (9.4.13), (9.4.14), (9.4.15) by an iteration argument which will be introduced below. Before doing this however it pays to observe that the system (9.4.13) can be interpreted as an elliptic system on a fixed surface S for (a, f, f ). In the next section we state a result which establishes the coercivity of the corresponding linearized system. The full proof of the theorem is detailed in section 9.4.3 to section 9.6.3.

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CHAPTER 9

9.4.1

The linearized GCM system

We start with the following linearized version of the equations (9.4.13): 
6mS S −1 2  S S S (A ) f − Υ f + M − M (AS )−1 eΦ + F1 , (rS )5 rS 
 6mS ΥS S B S f = − S 5 (AS )−1 f − ΥS f + C S − C eΦ (r ) (9.4.16)  S  2Υ S + S M S − M (AS )−1 eΦ + F2 , r /d S,? a = F3 , BSf = −

aS = b0 , where F1 , F2 , and F3 are given reduced scalar on S, b0 is a given constant, and where we have introduced the notation BS

:=

/d2S /d?2S +

6mS . (rS )3

(9.4.17)

Remark 9.35. Recalling that we have AS + V = /d2S /d?2S − 3ρ + 12 ϑϑ, the GCM system (9.4.13) corresponds, in view of the definition of B S , to the linearized GCM system (9.4.16) with the following choices for F1 , F2 , F3 , b0 :      3mS S −1 2 2ΥS F1 = − S 4 (A ) κ− S f + κ+ S f (r ) r r     
3 2 3 2mS S −1 S −1 + κ − S (A ) ρz − S (A ) ρ + S 3 z − /d?κ ˇ 4 r 2r (r )    
2 2 S − κ − S (AS )−1 − M S eΦ + /d?µ ˇ − S (AS )−1 /d?µ ˇ − M eΦ r r +Err4 ,

F2

     3mS ΥS S −1 2 2ΥS (A ) κ − f + κ + f (rS )4 rS rS     
3ΥS 3 2ΥS 2mS − κ + S (AS )−1 ρz + S (AS )−1 ρ+ S 3 z 4 r 2r (r )     S
2Υ S − /d?κ ˇ − C eΦ + κ + S (AS )−1 − M S eΦ + /d?µ ˇ r   S 2Υ S − S (AS )−1 /d?µ ˇ − M eΦ + Err5 , r

= −

F3

b0


3 1 1 = − (AS )−1 ρz + f ω − f ω + f κ − f κ 4 4
4 +(AS )−1 − M S eΦ + /d?µ ˇ − (AS )−1 Err3 ,  S  S rS rS 2 S = 1− − κ ˇ+κ− + Err6 . r 2 r

525

GCM PROCEDURE

Remark 9.36. To motivate the introduction of the system (9.4.16), let us note that the above particular choices for F1 and F2 in Remark 9.35 correspond to the terms in the first two equations of (9.4.13) which14 • either depend on κ ˇ , /d?κ ˇ − CeΦ , and /d?µ ˇ − M eΦ , • or are nonlinear. The following result plays a main role in the proof of Theorem 9.32. Proposition 9.37. Let a fixed spacetime region R verifying assumptions A1–A3 and (9.4.2). Assume S is a given surface in R such that, for a small enough constant δ1 > 0 and for any 2 ≤ s ≤ smax + 1, Z 1 δ1 1 4π S 4 sup K S − S 2 ≤ S 2 , kK S khs−2 (S) . S , e2Φ = (r ) (1 + O(δ1 )). (r ) (r ) r 3 S S Then, for every Λ, Λ, • Existence and uniqueness. There exists unique constants (C S , M S ) and a unique solution (f, f , λ) of the system (9.4.16), (9.4.14) verifying the estimates S
C − C S + rS M S − M S . (rS )−7 |Λ| + |Λ| + (rS )−2 kF1 kL2 (S) + (rS )−2 kF2 kL2 (S) ,

(9.4.18)


k(f, f )khs (S) . (rS )−2 |Λ| + |Λ| + (rS )2 kF1 khs−2 (S) + (rS )2 kF2 khs−2 (S) ,(9.4.19) kˇ aS khs (S)

. rS kF3 khs−1 (S)

(9.4.20)

and |aS | .

|b0 |.

(9.4.21)

• A priori estimates. If (f, f , λ) verifies the system (9.4.16), (9.4.14) for some constant (C S , M S ), then (M S , C S ) satisfies (9.4.18) and (f, f , λ) satisfies (9.4.19), (9.4.20), and (9.4.21). As a corollary, we derive the following rigidity result for GCM spheres. Corollary 9.38. Let a fixed spacetime region R verifying assumptions A1–A3 and (9.4.2). Assume that S is a deformed sphere in R which verifies the GCM conditions κS = 14 Note

2 , rS

/d?2 S /d?1 S κS = /d?2 S /d?1 S µS = 0 S

(9.4.22)

S

that the terms / d?κ ˇ − C eΦ and / d?µ ˇ − M eΦ can be decomposed as follows:

S

S

/ d?κ ˇ − C eΦ = / d?κ ˇ − CeΦ + (C − C )eΦ , where C − C

S

and M − M

S

S

S

/ d?µ ˇ − M eΦ = / d?µ ˇ − M eΦ + (M − M )eΦ ,

are nonlinear in view of Corollary 9.18 applied to D = C and D = M .

526

CHAPTER 9

and such that for a small enough constant δ1 > 0, the transition functions (f, f , λ) from the background frame of R to that of S verify, for some 4 ≤ s ≤ smax , the bound kf khs (S) + (rS )−1 k(f , a)khs (S)

≤

δ1 .

(9.4.23)

Then (f, f , λ) verify the estimates ◦

S

k(f, f , a ˇ )khs+1 (S)

δ+r

.

−2

Z   Z   f eΦ + f eΦ + rδ1 ◦ + δ1 S

S

and S

r|a | .

◦

δ+r

−2

Z   Z f eΦ + f eΦ + sup |r − rS |. S S

S

Remark 9.39. As mentioned before Lemma 9.15, the anomalous behavior for (f , a) in the assumption (9.4.23) does not appear in the construction of GCM spheres in this chapter. It appears however in the proof of Theorem M0 in the region (ext) L0 ∩ (ext) M of the initial data layer, see Step 8 in section 4.1. The proof of Proposition 9.37 will be given in section 9.5.1 while the proof of Corollary 9.38 will be given in section 9.5.2. 9.4.2

Comparison of the Hawking mass

We establish the estimate (9.4.7) concerning the Hawking mass mS . Recall that   Z rS 1 S S S m = 1+ κ κ , 2 16π S  Z ◦  r 1 ◦ ◦◦ m = 1+ κκ . 2 16π S◦ We write ◦

2

mS m − ◦ rS r

! =

1 16π

Z S


κS κS − κκ +

Z S

κκ −

Z

 Z   ◦◦ κκ − κκ − κ κ ◦ ◦

S

S

= I1 + I2 + I3 . ◦ ◦◦ ◦ ◦ In view of Proposition 9.14 we have |rS − r | . δ and γ S,# − γ . δ r . Making use of Corollary 9.10 and the assumptions A1–A3 for κ, κ we deduce Z Z ◦ I2 = κκ − κκ . δr−1 . ◦ S

S

 Similarly, taking into account the definition of R := I3

◦

. δr−1 .

◦

◦

|u − u| ≤ δ,

 ◦ ◦ |s − s| ≤ δ ,

527

GCM PROCEDURE

Finally, making use also of the transformation formula from the original frame S S (e4 , e3 , eθ ) to the frame (eS 4 , e3 , eθ ) of S

κS κS = κ + /dS f + Err(κ, κS ) κ + /dS f + Err(κ, κS ) and the estimates for (f, f , a = log λ) we deduce S S κ κ − κκ

◦

. r−3 δ.

Hence, I1

◦

. δr−1 .

We infer that ◦ mS m S − ◦ . r r

◦

δr−1

from which the desired estimate (9.4.7) easily follows. 9.4.3

Iteration procedure for Theorem 9.32

We solve the coupled system of equations (9.4.13), (9.4.14), (9.4.15) by an iteration argument as follows. Starting with the septets ◦ ◦

◦ ◦

◦ ◦

◦ ◦

Q(0)

:=

(U (0) , S (0) , a(0) , f (0) , f (0) , C (0) , M (0) ) = (0, 0, 0, 0, 0, C(u, s), M (u, s)),

Q(1)

:=

(U (1) , S (1) , a(1) , f (1) , f (1) , C (1) , M (1) ) = (0, 0, 0, 0, 0, C(u, s), M (u, s)), ◦

corresponding to the undeformed sphere S, we define iteratively the quintet Q(n+1) = (U (n+1) , S (n+1) , a(n+1) , f (n+1) , f (n+1) , C (n+1) , M (n+1) ) from Q(n−1)

=

(U (n−1) , S (n−1) , a(n−1) , f (n−1) , f (n−1) , C (n−1) , M (n−1) ),

Q(n)

=

(U (n) , S (n) , a(n) , f (n) , f (n) , C (n) , M (n) ),

as follows. 1. The pair (U (n) , S (n) ) defines the deformation sphere S(n) and the corresponding ◦

pullback map #n given by the map Ψ(n) : S → S(n), ◦ ◦

◦

◦

(u, s, θ, ϕ) −→ (u + U (n) (θ), s + S (n) (θ), θ, ϕ).

(9.4.24)

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CHAPTER 9

By induction we assume that the following estimates hold true: S(n−1)

r4 |C (n) − C | + r5 |M (n) − M



+ (a(n) , f (n) , f (n) )

S(n−1)

|

hsmax −1 (S(n−1))

◦

. δ,

(9.4.25)

and     k∂θ U (n−1) , S (n−1) k ∞ ◦ + r−1 k∂θ U (n−1) , S (n−1) k ◦ ◦ L (S) hsmax −1 (S,g /)     +k∂θ U (n) , S (n) k ∞ ◦ + r−1 k∂θ U (n) , S (n) k ◦ ◦ L

(S)

hsmax −1 (S,g /)

◦

.

δ.

(9.4.26)

2. We then define the quintet (a(n+1) , f (n+1) , f (n+1) , C (n+1) , M (n+1) ) by solving the system on S(n) consisting of the equations (9.4.27), (9.4.32) and (9.4.33) below.
6mS(n) S(n) −1 (n+1) (A ) f − ΥS(n) f (n+1) S(n) 5 (r ) 2 S(n) + S(n) (M (n+1) − M )(AS(n) )−1 eΦ + E (n+1) , r  6mS(n) ΥS(n) S(n) −1  (n+1) B S(n) f (n+1) = − (A ) f − ΥS(n) f (n+1) S(n) 5 (r ) B S(n) f (n+1) = −

+ (C (n+1) − C

S(n)

)eΦ +

(9.4.27)

2ΥS(n) S(n) (M (n+1) − M )(AS(n) )−1 eΦ rS

+ E (n+1) , e (n+1) , /d?S(n) a(n+1) = E with 3mS(n) S(n) −1 (A ) (rS(n) )4      2 2ΥS(n) (n) × κ − S(n) f (n) + κ + f n−1 n−1 r rS(n)   3 2 (n)
+ κ − S(n) (AS(n) )−1 ρzn−1 4 r    3 2mS(n) (n) S(n) −1 − S(n) (A ) ρ + S(n) 3 zn−1 − /d?κ ˇ 2r (r )  
2 − κ − S(n) (AS(n) )−1 − M (n) eΦ + /d?µ ˇ r 2 S(n) Φ
(n+1) − S(n) (AS(n) )−1 /d?µ ˇ−M e + Err4 , r

E (n+1) : = −

(9.4.28)

529

GCM PROCEDURE

3mS(n) ΥS(n) S(n) −1 (A ) (rS(n) )4      2 2ΥS(n) (n) × κ − S(n) f (n) + κ + f n−1 n−1 r rS(n)   3 2ΥS(n) (n)
− κ + S(n) (AS(n) )−1 ρzn−1 4 r (9.4.29)    S(n) 3Υ 2mS(n) (n) S(n) −1 + S(n) (A ) ρ + S(n) 3 zn−1 2r (r )   S(n)
2Υ S(n) Φ
? − /d κ ˇ−C e + κ + S(n) (AS(n) )−1 − M (n) eΦ + /d?µ ˇ r 2ΥS(n) S(n) −1 ? S(n) Φ
(n+1) − S(n) (A ) /d µ ˇ−M e + Err5 , r

E (n+1) : = −


e (n+1) : = − 3 (AS(n) )−1 ρz (n+1) + f (n+1) ω − f (n+1) ω + 1 f (n+1) κ E 4 4
1 (n+1) S(n) −1 (n+1) Φ ? (9.4.30) − f κ + (A ) −M e + /d µ ˇ 4 (n+1)
− (AS(n) )−1 Err3 , where
(n) fn−1 = f (n) ◦ Ψ(n−1) ◦ (Ψ(n) )−1 ,


f (n) = f (n) ◦ Ψ(n−1) ◦ (Ψ(n) )−1 , n−1 (n)

z (n+1) := κf (n+1) + κf (n+1) ,

(n)

zn−1 := κfn−1 + κf (n) , n−1

and the error terms, (n+1)

Err1

(n+1)

, Err2

(n+1)

, Err3

(n+1)

, Err4

(n+1)

, Err5

,

(9.4.31)

are obtained from the error terms Err1 , Err2 , Err3 , Err4 and Err5 by setting (a, f, f ) = (a(n) , f (n) , f (n) ) and their derivatives by the corresponding ones on S(n − 1), and then composing by Ψ(n−1) ◦ (Ψ(n) )−1 so that the error terms in (9.4.31) are well defined on S(n). We also set Z Z eΦ f (n+1) = Λ, eΦ f (n+1) = Λ, (9.4.32) S(n)

S(n)

and a(n+1)

S(n)

=

(n+1)

 S(n)  S(n) S(n) rS(n) rS(n) 2 (n+1) 1− − κ ˇ+κ− + Err6 ,(9.4.33) r 2 r

where Err6 is obtained from the error terms Err6 , as above in (9.4.31), by setting (a, f, f ) = (a(n) , f (n) , f (n) ) and their derivatives by the corresponding ones on the sphere S(n − 1), and then composing by Ψ(n−1) ◦ (Ψ(n) )−1 so that

530

CHAPTER 9

(n+1)

Err6 is defined on S(n). 3. The system of equations (9.4.27), (9.4.32) and (9.4.33) admits a unique solution (f (1+n) , f (1+n) , a(n+1) , C (n+1) , M (n+1) ) according to Proposition 9.40 below. ◦

4. We then use the new pair (f (n+1) , f (n+1) ) to solve the equations on S, ς #n ∂θ U (n+1) = (γ (n) )1/2 (f (n+1) )#n   1  (n+1) (n+1) #n × 1+ f f , 4 1 1 ∂θ S (n+1) − ς #n Ω#n ∂θ U (n+1) = (γ (n) )1/2 (f (n+1) )#n , 2 2 (9.4.34)   #n
2 1 γ (n) = γ #n + ς #n Ω + b2 γ (∂θ U (n) )2 4   #n − 2ς #n ∂θ U (n) ∂θ S (n) − γςb ∂θ U (n) , U (n+1) (0) = S (n+1) (0) = 0, where, we repeat, the pullback #n is defined with respect to the map ◦ ◦

◦

◦

Ψ(n) (u, s, θ) = (u + U (n) (θ), s + S (n) (θ), θ), and γ (n) := γ S(n),#n . The equation (9.4.34) admits a unique solution (U (n+1) , S (n+1) ) according to Proposition 9.41 below. The new pair (U (n+1) , S (n+1) ) defines the new polarized sphere S(n + 1) and we can proceed with the next step of the iteration. 9.4.4 9.4.4.1

Existence and boundedness of the iterates Existence and boundedness of (f (n+1) , f (n+1) , a(n+1) , C (n+1) , M (n+1) )

Proposition 9.40. The system of equations (9.4.27), (9.4.32) and (9.4.33) admits a unique solution (f (1+n) , f (1+n) , a(n+1) , C (n+1) , M (n+1) ) verifying the estimates S(n)

r4 |C (n+1) − C | + r5 |M (n+1) − M

(n+1)

S(n) (n+1) (n+1) (n+1)

+ (a −a ,f ,f )

S(n)

|

◦

. δ

hsmax −1 (S(n))

and   ◦ S(n) (n+1) . δ + k∂θ U (n) , S (n) k r a ◦ L∞ (S) uniformly for all n ∈ N. Proof. The system (9.4.27), (9.4.32) and (9.4.33) corresponds to the linearized GCM

531

GCM PROCEDURE

system (9.4.16), (9.4.14) with the following choice for F1 , F2 , F3 and b0 : e (n+1) , F1 = E (n+1) , F2 = E (n+1) , F3 = E  S(n)  S(n) S(n) rS(n) rS(n) 2 (n+1) b0 = 1 − − κ ˇ+κ− + Err6 . r 2 r Also, the induction assumptions (9.4.26) for (U (n) , S (n) ) together with Corollary 9.17 imply that the sphere S(n) satisfies in particular the assumptions of Proposition 9.37. We infer from that proposition the existence and uniqueness of the quintet solution (f (1+n) , f (1+n) , a(n+1) , C (n+1) , M (n+1) ) to (9.4.27), (9.4.32) and (9.4.33), as well as the following a priori estimate: S(n)

S(n)

(rS(n) )4 |C (n+1) − C | + (rS(n) )5 |M (n+1) − M |

(n+1) (n+1) + (f ,f ) hsmax −1 (S(n))
S(n) −2 . (r ) |Λ| + |Λ| + (rS(n) )2 kE (n+1) khsmax −3 (S(n)) +(rS(n) )2 kE (n+1) khsmax −3 (S(n)) ,



(n+1) S(n) (n+1)

a

− a

hsmax −1 (S(n))

(9.4.35)

e (n+1) kh . rS(n) kE , smax −2 (S(n))

(9.4.36)

and S(n) a(n+1) .

 S(n) S(n)  S(n) S(n) r r 2 1− + κ ˇ+κ− 2 r r (n+1) S(n) . + Err6 (9.4.37)

We need to control the RHS of the inequalities (9.4.35), (9.4.36), (9.4.37). We (n+1) start with the control of the error terms Errj , j = 3, 4, 5, 6. The induction assumptions (9.4.26) for (U (n) , S (n) ) imply that the sphere S(n) satisfies in particular ◦

the assumptions of Lemma 9.29 with δ1 = δ. We deduce from that lemma

◦ ◦◦

(n+1) (n+1) kErr1 , Err2 khsmax −3 (S(n)) . r−2 (f (n) , f (n) , a(n) ) . r−2 δ, hsmax −1 (S(n−1))



◦ −3 (n) ◦◦

(n+1) (n) (n) . r (f , f , a ) . r−3 δ,

Err3

hsmax −4 (S(n)) hsmax −1 (S(n−1))

◦ ◦◦

(n+1) (n+1) kErr4 , Err5 khsmax −3 (S(n)) . r−2 (f (n) , f (n) , a(n) ) . r−2 δ, hsmax −1 (S(n−1))

where we have also used the induction assumptions (9.4.25) for (f (n) , f (n) , a(n) ), as well as Lemma 9.11 which implies for a reduced scalar h on S(n − 1) ◦
kh ◦ Ψ(n−1) ◦ (Ψ(n) )−1 khs (S(n)) = khkhs (S(n−1)) (1 + O(r−1 δ)), 0 ≤ s ≤ smax − 1.

532

CHAPTER 9

(n+1)

Also, recall that Err6 is given by " (   (n) rS(n) a(n) 2 (n+1) Err6 = − e Err(κ, κ0 ) − (ea − 1) κ ˇ + κ − + /dS(n−1) f (n) 2 r ) #    (n) 2
rS(n) − ea − 1 − a(n) − a(n) −1 ◦ Ψ(n−1) ◦ (Ψ(n) )−1 r r which together with the control A1–A3 of the background foliation, the induction assumptions (9.4.25) for (f (n) , f (n) , λ(n) ), the control of r − rS(n) following from the induction assumptions (9.4.26) for (U (n) , S (n) ) and Lemma 9.8, and Sobolev, yields

◦ ◦◦

(n+1) sup |Err6 | . r−1  (f (n) , f (n) , a(n) ) . r−1 δ. h3 (S(n−1))

S(n)

In view of e (n+1) , • the definition (9.4.28), (9.4.29), (9.4.30) of E (n+1) , E (n+1) and E • the control of the background foliation on S(n) provided by Corollary 9.12, • the assumption (9.4.2) for κ ˇ , /d?1 κ − CeΦ and /d?1 µ − M eΦ , S(n)

S(n)

• the control of C − C and M − M using Corollary 9.18, the control of the background foliation, as well as the induction assumptions (9.4.26) for (U (n) , S (n) ), • the control of r − rS(n) following from the induction assumptions (9.4.26) for (U (n) , S (n) ) and Lemma 9.8, • the control of m − mS(n) thanks to section 9.4.2 which uses the control of S(n) provided by the induction assumptions (9.4.26) for (U (n) , S (n) ), as well as the induction assumptions (9.4.25) for (f (n) , f (n) , λ(n) ), (n+1)

• the above estimates for Errj

, j = 3, 4, 5, 6,

we infer kE (n+1) khsmax −3 (S(n)) + kE (n+1) khsmax −3 (S(n)) 

 . max sup dk κ ˇ + r dk−1 /d?1 κ − CeΦ + r2 dk−1 /d?1 µ − M eΦ k≤smax −2 R ◦ −2 ◦

+r

δ,

e (n+1) kh kE smax −2 (S(n))

.

r3

max


sup dk−1 /d?1 µ − M eΦ

k≤smax −3 R

S(n−1)

+r4 |M (n) − M |

(n) −1 (n) +r (f , f )

hsmax −1 (S(n−1))

◦◦

+ r−1 δ,

533

GCM PROCEDURE

and  S(n) S(n)  S(n) S(n) (n+1) S(n) 2 1− r + r + Err κ ˇ + κ − 2 6 r r   2 ◦◦ . r sup κ − + |ˇ κ| + r−1 sup |r − rS(n) | + r−1 δ. r R S(n) Together with (9.4.35), (9.4.36) and (9.4.37), as well as the assumption (9.4.2) for κ ˇ , /d?1 κ − CeΦ and /d?1 µ − M eΦ , the induction assumptions (9.4.25) for M (n) and (f (n) , f (n) , λ(n) ), and the control of r − rS(n) following from the induction assumptions (9.4.26) for (U (n) , S (n) ) and Lemma 9.8, this implies, uniformly in n, S(n)

r4 |C (n+1) − C | + r5 |M (n+1) − M

(n+1)

S(n) (n+1) (n+1) (n+1) + (a − a , f , f )

S(n)

|

◦

. δ

hsmax −1 (S(n))

and S(n) . r a(n+1)

  ◦ δ + k∂θ U (n) , S (n) k

◦

L∞ (S)

.

This concludes the proof of Proposition 9.40. 9.4.4.2

Existence and boundedness of (U (n+1) , S (n+1) )

Proposition 9.41. The equation (9.4.34) admits a unique solution U (1+n) , S (1+n) verifying the estimate   k∂θ U (n+1) , S (n+1) k

◦

L∞ (S)

  + r−1 k∂θ U (n+1) , S (n+1) k

◦ ◦ ◦

hsmax −1 (S,g /)

. δ

uniformly for all n ∈ N. Proof. The existence and uniqueness part of the proposition is an immediate consequence of the standard results for ODEs. To prove the desired estimate, we use the equations for (U (1+n) , S (1+n) ) and infer, for s = smax − 1, k∂θ U (n+1) k ◦ ◦ hs (S,g /)

 

1  (1+n) (1+n) #n

(n) 1/2 #n
−1 (1+n) #n

. (γ ) ς (f ) 1+ f f

4

. ◦ ◦

hs (S,g /)

◦

◦

Together with the non-sharp product estimate on (S, g/ ), see Lemma 9.5, we infer

534

CHAPTER 9

that, for s = smax − 1,

.

k∂θ U (n+1) k ◦ ◦ hs (S,g /)



#n
−1 (n) 1/2 −1 (n+1) #n r (f ) , (f (n+1) )#n (γ ) ◦ ◦

ς ◦ ◦ hsmax −2 (S,g /) hs (S,g /)  

2

(1+n) #n

× 1 + (f ) , (f (1+n) )#n ◦ ◦ . hs (S,g /)

In view of Lemma 9.11, Corollary 9.12, and the bound for (f (n+1) , f (n+1) ) provided by Proposition 9.40, we deduce

◦
−1 (n) 1/2

k∂θ U (n+1) k ◦ ◦ . δr−1 ς #n (γ ) ◦ ◦ . hs (S,g /)

hs (S,g /)

We recall that γ (n)

= γ #n + ς #n


2

Ω + 14 b2 γ


#n

  #n (∂θ U (n) )2 − 2ς #n ∂θ U (n) − γςb ∂θ U (n) .

In view of our assumptions on the Ricci coefficients and the non-sharp product estimates of Lemma 9.5



  #

#n

1 2
n ◦

+ . r

ς Ω+ b γ

γb

◦ ◦

◦ ◦ 4 hs /) −1 (S,g hsmax −1 (S,g /)

max

we deduce

#n
−1 (n) γ

ς

◦ ◦

hsmax −1 (S,g /)

◦

#

γ n

.

◦ ◦

hsmax −1 (S,g /)

+ δr2 .

Together with Lemma 9.11 and Corollary 9.12, we deduce

#n
−1 (n)
1/2 γ . r2

ς

◦ ◦ hsmax −1 (S,g /)

and therefore, k∂θ U (n+1) k

◦ ◦ ◦

hsmax −1 (S,g /)

. δr.

Proceeding in the same manner with equation 1 1 ∂θ S (1+n) − ς #n Ω#n ∂θ U (1+n) = (γ (n) )1/2 (f (1+n) )#n 2 2 we infer that r−1 k∂θ U (n+1) , ∂θ S (n+1) k

◦ ◦ ◦

hsmax −1 (S,g /)

.

δ.

This, together with the Sobolev inequality, concludes the proof of Proposition 9.41.

535

GCM PROCEDURE

9.4.5

Convergence of the iterates

To finish the proof of Theorem 9.32, it remains to prove convergence of the iterates. Step 1. In order to prove the convergence of the iterative scheme, we introduce the following septets P (n) : ◦ ◦

◦ ◦

◦ ◦

◦ ◦

P (0) = (0, 0, 0, 0, 0, M (u, s), C(u, s)), P (1) = (0, 0, 0, 0, 0, M (u, s), C(u, s)),   P (n) = U (n) , S (n) , (a(n) )#n−1 , (f (n) )#n−1 , (f (n) )#n−1 , C (n) , M (n) , n ≥ 2. Since (a(n) , f (n) , f (n) ) are defined on S(n − 1), their respective pullback by Ψ(n−1) ◦

◦

is defined on S so that P (n) consists of a quintet of functions on S, together with two constants, for any n, and we may introduce the following norms to compare the elements of the sequence:   S(n−1) (n) 4 kP (n) kk : = r−1 k∂θ U (n) , S (n) k −C | ◦ + r |C hk−1 (S)

5

(n)

S(n−1)

+ r |M −M |

 

+ (a(n) )#n−1 , (f (n) )#n−1 , (f (n) )#n−1

(9.4.38) ◦

.

hk−1 (S)

Here are the steps needed to implement a convergence argument. 1. The quintets P (n) are bounded with respect to the norm (9.4.38) for the choice k = smax . 2. The quintets P (n) are contractive with respect to the norm (9.4.38) for the choice k = 2. The precise statements are given in the following propositions. Proposition 9.42. We have, uniformly for all n ∈ N, kP (n) ksmax

◦

. δ.

Proof. The proof is an immediate consequence of Propositions 9.37, 9.41 and the estimate

   

(n) (n) (n)

(n−1) # (n) (n) (n) ) f ,f ,a .

(Ψ

f ,f ,a

◦ hsmax −1 (S)

hsmax −1 (S(n−1))

which is a consequence of Lemma 9.11. Proposition 9.43. We have, uniformly for all n ∈ N, the contraction estimate kP (n+1) − P (n) k2   ◦ .  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 + kP (n−2) − P (n−3) k2 . The proof of Proposition 9.43 is postponed to section 9.6.

536

CHAPTER 9

Step 2. In view of Proposition 9.43, we have   ◦ n−2 kP (n+1) − P (n) k2 . ()b 3 c kP (3) − P (2) k2 + kP (2) − P (1) k2 + kP (1) − P (0) k2 , which in view of Proposition 9.42 yields ◦

◦ n kP (n+1) − P (n) k2 . δ ()b 3 c ,

n ≥ 3. ◦

Together with a simple interpolation argument on S and Proposition 9.42, we infer n ◦ ◦( ssmax −k −2 )b 3 c

kP (n+1) − P (n) kk . δ 

max

2 ≤ k ≤ smax ,

,

n ≥ 3.

We infer the existence of a septet P (∞) such that ◦

kP (∞) ksmax

.

δ

(9.4.39)

and lim kP (n) − P (∞) ksmax −1 = 0.

(9.4.40)

n→+∞

Also, we have   (∞) (∞) (∞) (∞) P (∞) = U (∞) , S (∞) , a0 , f0 , f (∞) , C , M , 0 ◦

where the quintet of functions are defined on S and (C (∞) , M (∞) ) are two constants. The functions (U (∞) , S (∞) ) define a sphere S(∞) and we introduce the map ◦  ◦ ◦ ◦ Ψ(∞) (u, s, θ, ϕ) = u + U (∞) (θ), s + S (∞) (θ), θ, ϕ ◦

so that Ψ(∞) is a map from S to S(∞). Then, let (∞)

a(∞) = a0

(∞)

◦ (Ψ(∞) )−1 ,

f (∞) = f0

◦ (Ψ(∞) )−1 ,

f (∞) = f (∞) ◦ (Ψ(∞) )−1 0

so that a(∞) , f (∞) , f (∞) are defined on S(∞) and (∞)

a0

= (a(∞) )#∞ ,

(∞)

f0

= (f (∞) )#∞ ,

f (∞) = (f (∞) )#∞ . 0

From these definitions, the above control of P (∞) and Lemma 9.11, we infer r−1 k(∂θ U (∞) , ∂θ S (∞) )k

◦

hsmax −1 (S)

+ k(a(∞) , f (∞) , f (∞) )khsmax −1 (S(∞))

◦

.

δ.

In particular, applying Corollary 9.38 twice, first with s = smax − 1, and then with s = smax , we deduce k(a(∞) , f (∞) , f (∞) )khsmax +1 (S(∞))

◦

.

δ.

n ≥ 3,

537

GCM PROCEDURE

Together with the above control for (U (∞) , S (∞) ), we finally obtain 0

◦

0

r−1 k(U (∞) , S (∞) )k

◦

hsmax −1 (S)

+ k(a(∞) , f (∞) , f (∞) )khsmax +1 (S(∞)) . δ. (9.4.41)

Step 3. We proceed to control the area radius rS(∞) and the Hawking mass mS(∞) ◦

of the sphere S(∞). First, note from (9.4.41) and the Sobolev embedding on S that we have ◦

k(U (∞) , S (∞) )k

δ.

.

◦ L∞ (S)

Together with Lemma 9.8, we infer that15 S(∞) r r − 1 .

(9.4.42)

◦

δ.

(9.4.43)

Next, we denote by ΓS(∞) the connection coefficients of S(∞). We have in view of the transformation formula from the original frame (e4 , e3 , eθ ) to the frame S(∞) S(∞) S(∞) (e4 , e3 , eθ ) of S(∞)   κS(∞) κS(∞) = κ + /dS(∞) f (∞) + Err(κ, κS(∞) )   × κ + /dS(∞) f (∞) + Err(κ, κS(∞) ) . Together with the estimate (9.4.41) for f (∞) and f (∞) and the assumptions A1–A3 ˇ corresponding to the original frame (e4 , e3 , eθ ), we infer for Γ ◦ S(∞) S(∞) κ − κκ . δr−3 . κ Recall that (see (9.4.2)) κ −

2 ◦ −2 . δr , r

2 1− κ + r

2m r


◦ . δ.

Thus, since κ = κ + κ ˇ,
◦ 4 1 − 2m 2 r κκ = − + κ ˇ + O(δ)r−2 . 2 r r We deduce
2m 2 S(∞) S(∞) 4 1 − r κ + − κ ˇ κ r2 r 15 Here,

◦

. δr−3 .

we also use the fact that, on S (∞) , we have ◦

◦

|r − r| . k(U (∞) , S (∞) )k

◦

L∞ (S)

. δ.

538

CHAPTER 9

Thus, in view of (9.4.43), Z

S(∞) S(∞)

κ

κ

=

S(∞)

−

Z S(∞)


◦ 4 1 − 2m r + O(δ)r−1 . r2

Making use of the definition of the Hawking mass, i.e., S(∞)

m

rS(∞) = 2

1 1+ 16π

!

Z

S(∞) S(∞)

κ

κ

,

S(∞)

we easily deduce16 S(∞) − m m

◦

. δ.

(9.4.44)

Step 4. We make use of Lemma 9.30 to extend (a(∞) , f (∞) , f (∞) ) as well as the S(∞) S(∞) S(∞)
frame e3 , e4 , eθ in a small neighborhood of S(∞) such that we have ξ S(∞) = 0,

ω S(∞) = 0,

η S(∞) + ζ S(∞) = 0,

(9.4.45)

and then provide estimates for the corresponding Ricci coefficients and curvature ˇ S(∞) , R ˇ S(∞) . More precisely we make use of the assumption A1, components Γ the estimates in (9.4.41) for (a(∞) , f (∞) , f (∞) ), and the transformation formulae to derive the desired estimates (9.4.8) for smax derivative of the Ricci coefficients and curvature components of S(∞). Step 5. Thanks to (9.4.40), we can pass to the limit in (9.4.27), (9.4.32), (9.4.33). In view of Remark 9.35, we deduce that equations (9.4.13), (9.4.14) hold true. Thus, we may apply Proposition 9.33 which implies that (9.4.12) holds true. In particular, the desired GCM conditions (9.4.1) hold true which concludes the proof of Theorem 9.32.

9.5

PROOF OF PROPOSITION 9.37 AND OF COROLLARY 9.38

9.5.1

Proof of Proposition 9.37

Step 1. We start with the proof of existence. Note first that the existence of aS and a ˇS is immediate in view of the last two equations of (9.4.16). We thus focus on the existence of (f, f ). In view of the first two equations of (9.4.16), we have
6mS S −1 2 S (A ) f − ΥS f + S (M S − M )(AS )−1 eΦ + F1 , S 5 (r ) r S S
6m Υ S B S f = − S 5 (AS )−1 f − ΥS f + (C S − C )eΦ (r ) 2ΥS S + S (M S − M )(AS )−1 eΦ + F2 . r BSf = −

16 See

also section 9.4.2.

(9.5.1)

539

GCM PROCEDURE

In particular, subtracting ΥS times the first equation to the second equation, we infer that the existence of (f, f ) is equivalent to the existence of
S B S f − ΥS f = F2 − ΥS F1 + (C S − C )eΦ , BSf = −

(9.5.2)
6mS S −1 2 S S S S −1 Φ (A ) f − Υ f + (M − M )(A ) e + F . 1 (rS )5 rS

Step 2. Next, we differentiate (9.5.2) w.r.t. /d?2S which yields the system   n o
3mS ?S S /d2 /d2 + S 3 /d?2S f − ΥS f = /d?2S F2 − ΥS F1 , (r ) (   S
3m 6mS /d?2S /d2S + S 3 /d?2S f = /d?2S − S 5 (AS )−1 f − ΥS f (r ) (r ) ) 2 S S S −1 Φ + S (M − M )(A ) e + F1 , r

(9.5.3)

where we have used the fact that /d?2S eΦ = 0. Since the operator /d?2S /d2S is coercive 3mS Φ and invertible, so is /d?2S /d2S + (r generates the S )3 . Thus, using also the fact that e ?S ?S S kernel of /d2 and that /d2 is surjective, there exists f − Υ f solution to
/d?2S f − ΥS f = Z

S

S




Φ



/d?2S /d2S +

6mS (rS )3

−1

n o /d?2S F2 − ΥS F1 ,

(9.5.4)

S

f − Υ f e = Λ − Υ Λ.

Step 3. Next, we have, using in particular the assumptions on K S ,   AS (eΦ ) = /d?1S /d1S (eΦ ) = /d2S /d?2S + 2K S eΦ = 2K S eΦ   2 Φ 2 S = e + K − eΦ (rS )2 (rS )2 and hence S −1

(A )

Φ

(e )

=

(rS )2 Φ (rS )2 S −1 e − (A ) 2 2



S

K −

2 (rS )2

 e

Φ

 .

(9.5.5)

In particular, we have, in view of the assumptions of the proposition, Z Z 4π S 4 2π S 6 e2Φ = (r ) (1 + O(δ1 )), eΦ (AS )−1 (eΦ ) = (r ) (1 + O(δ1 )), (9.5.6) 3 3 S S

540

CHAPTER 9

so that these quantities do not vanish. We may thus choose C S and M S as follows:  Z −1 Z  S  Φ 6mS S 2Φ C =C + (Λ − Υ Λ) + Υ F1 − F2 e e , (rS )3 S S     Z rS 6mS 6mS S −1 S MS = M + Λ + (A ) (f − ΥS f ) − F1 eΦ S 5 2 (rS )3 S (r ) Z −1 × eΦ (AS )−1 (eΦ ) , S

S



(9.5.7)

S

where f − ΥS f appearing on the RHS of the above choice of M S is the solution of (9.5.4). Step 4. Next, with f − ΥS f chosen as in (9.5.4) and M S chosen as in (9.5.7), and arguing as in Step 2, there exists f solution to (  −1
6mS 6mS ?S ?S S ?S /d2 f = /d2 /d2 + S 3 /d2 − S 5 (AS )−1 f − ΥS f (r ) (r ) ) 2 S (9.5.8) + S (M S − M )(AS )−1 eΦ + F1 , r Z f eΦ = Λ. S

Now, in view of 1. the fact that (f, f ) satisfies (9.5.3) in view of (9.5.4), (9.5.8), 2. the choice (9.5.7) for the constants C S and M S , 3. the fact that eΦ generates the kernel of /d?2S , we infer that (f, f ) satisfies (9.5.2), and hence (9.5.1), which concludes the existence part of the proof. Step 5. Next, we focus on the proof of the a priori estimates. Note first that the last two equations of (9.4.16) immediately yield the a priori estimates for aS and a ˇS . We then focus on the a priori control of (C S , M S ) and (f, f ). We multiply the first two equations of (9.5.1) by eΦ and integrate on S. Using the fact that eΦ generates the kernel of /d?2S , and that /d?2S is the adjoint of /d2S , we deduce that the constants C S and M S are given by (9.5.7). Together with (9.5.6), we infer the following control for the constants C S and M S :
S |C S − C | . (rS )−7 |Λ| + |Λ| + (rS )−2 kF1 kL2 (S) + (rS )−2 kF2 kL2 (S) , (9.5.9)
S |M S − M | . (rS )−8 |Λ| + |Λ| + (rS )−6 kf − ΥS f kL2 (S) + (rS )−3 kF1 kL2 (S) . Step 6. Next, we multiply the first equation of (9.5.2) by (f − ΥS f ), integrate on

541

GCM PROCEDURE

S, and integrate by parts the term B S (f − ΥS f ). We obtain krS /d?2S (f − ΥS f )k2L2 (S)



.

 (rS )2 kF1 kL2 (S) + (rS )2 kF2 kL2 (S) kf − ΥS f kL2 (S) S

+(rS )2 |C S − C |(|Λ| + |Λ|). S

Together with a Poincar´e inequality for /d?2S and the estimate for C S −C in (9.5.9), we deduce
kf − ΥS f kh1 (S) . (rS )−2 |Λ| + |Λ| + (rS )2 kF1 kL2 (S) + (rS )2 kF2 kL2 (S) .(9.5.10) In particular, together with (9.5.9), we infer S

S

|C S − C | + rS |M S − M | .


(rS )−7 |Λ| + |Λ| + (rS )−2 kF1 kL2 (S) +(rS )−2 kF2 kL2 (S)

(9.5.11)

which is the desired a priori estimate for (C S , M S ). Step 7. Next, we multiply the second equation of (9.5.2) by f , integrate on S, and integrate by parts the term B S f . We obtain  krS /d?2S f k2L2 (S) . (rS )−1 kf − ΥS f kL2 (S) + (rS )2 kF1 kL2 (S)  S +(rS )5 |M S − M | kf kL2 (S) which together with a Poincar´e inequality for /d?2S , (9.5.10), and (9.5.11) yields
kf kh1 (S) . (rS )−3 |Λ| + |Λ| + (rS )2 kF1 kL2 (S) + (rS )2 kF2 kL2 (S) . Together with (9.5.10), we obtain kf kh1 (S) + kf kh1 (S)


. (rS )−2 |Λ| + |Λ| + (rS )2 kF1 kL2 (S) +(rS )2 kF2 kL2 (S) .

(9.5.12)

Step 8. Finally, using the identity /d2S /d?2S = /d?1S /d1S − 2K S , we rewrite (9.5.1) as follows:  
6mS 6mS /d?1S /d1S f = 2K S − S 3 f − S 5 (AS )−1 f − ΥS f (r ) (r ) 2 S + S (M S − M )(AS )−1 eΦ + F1 , r  
6mS 6mS ΥS S −1 S ?S S /d1 /d1 f = 2K S − S 3 f − (A ) f − ΥS f + (C S − C )eΦ S 5 (r ) (r ) +

2ΥS S (M S − M )(AS )−1 eΦ + F2 . rS

Together with (9.5.12), (9.5.11) and the assumptions for K S sup |K S | . S

1 , (rS )2

kK S khs−2 (S) .

1 , rS

542

CHAPTER 9

we deduce by iteration k(f, f )khs (S)

. (rS )2 kF1 khs−2 (S) + (rS )2 kF2 khs−2 (S) + (rS )−2 (|Λ| + |Λ|)

which concludes the part on a priori estimates. The part on uniqueness follows from the linearity of the equations and the a priori estimates. This ends the proof of Proposition 9.37. 9.5.2

Proof of Corollary 9.38

Step 1. First, we introduce for convenience the notation Z Z Λ := f eΦ , Λ := f eΦ . S

S

Then, in view of the assumptions of Corollary 9.38, (f, f , λ) satisfies (9.4.1), and hence, there exist constants (C S , M S ) such that (f, f , λ) satisfies (9.4.12). In particular, from Proposition 9.33, (f, f , λ) satisfies (9.4.13), (9.4.14). In view of Remark 9.35, we deduce that (f, f , λ) satisfies the linearized GCM system (9.4.16) with the following choices for F1 , F2 , F3 , b0 ,      2 3mS 2ΥS F1 = − S 4 (AS )−1 κ− S f + κ+ S f (r ) r r     
3 2 3 2mS + κ − S (AS )−1 ρz − S (AS )−1 ρ + S 3 z − /d?κ ˇ 4 r 2r (r )  

2 2 S − κ − S (AS )−1 − M S eΦ + /d?µ ˇ − S (AS )−1 /d?µ ˇ − M eΦ + Err4 , r r

F2

     3mS ΥS S −1 2 2ΥS = − S 4 (A ) κ− S f + κ+ S f (r ) r r      S
3 2ΥS 3Υ 2mS S −1 S −1 − κ + S (A ) ρz + S (A ) ρ+ S 3 z 4 r 2r (r )   S

2Υ S − /d?κ ˇ − C eΦ + κ + S (AS )−1 − M S eΦ + /d?µ ˇ r S
2Υ S − S (AS )−1 /d?µ ˇ − M eΦ + Err5 , r F3

=

b0

=


3 1 1 − (AS )−1 ρz + f ω − f ω + f κ − f κ 4 4 4
+(AS )−1 − M S eΦ + /d?µ ˇ − (AS )−1 Err3 ,  S  S rS rS 2 S 1− − κ ˇ+κ− + Err6 . r 2 r

Step 2. In view of Corollary 9.17, we may apply Proposition 9.37. In particular,

543

GCM PROCEDURE

the following a priori estimates hold:
S S |C S − C | + rS |M S − M | . (rS )−7 |Λ| + |Λ| + (rS )−2 kF1 kL2 (S)

(9.5.13)


k(f, f )khs+1 (S) . (rS )−2 |Λ| + |Λ| + (rS )2 kF1 khs−1 (S)

(9.5.14)

+ (rS )−2 kF2 kL2 (S) ,

+ (rS )2 kF2 khs−1 (S) ,

kˇ aS khs+1 (S)

. rS kF3 khs (S)

(9.5.15)

and |aS | . |b0 |,

(9.5.16)

where F1 , F2 , F3 and b0 are given in Step 1. Step 3. In view of the a priori estimates of Step 2, we need to estimate F1 , F2 , F3 and b0 . We start with the control of the error terms Errj , j = 3, 4, 5, 6. In view of Lemma 9.29, we have, since 4 ≤ s ≤ smax , ◦  kErr1 , Err2 khs−1 (S) . r−2 k(f, f , a)khs+1 (S)  + r−1 kf, f , akhs (S) , ◦  kErr3 khs−2 (S) . r−3 k(f, f , a)khs+1 (S)  + r−1 kf, f , akhs (S) , ◦  kErr4 , Err5 khs−1 (S) . r−2 k(f, f , a)khs+1 (S)  + r−1 kf, f , akhs (S) . In particular, in view of the assumptions (9.4.23) for (f, f , λ), we deduce kErr1 , Err2 khs−1 (S) . kErr3 khs−2 (S) . kErr4 , Err5 khs−1 (S) .

◦  r−2 k(f, f , a)khs+1 (S)  + δ1 , ◦  r−3 k(f, f , a)khs+1 (S)  + δ1 , ◦  r−2 k(f, f , a)khs+1 (S)  + δ1 .

(9.5.17)

Also, recall that Err6 is given by     rS a 2 2 0 a S a Err6 = − e Err(κ, κ ) − (e − 1) κ ˇ + κ − + /d f − (e − 1 − a) 2 r r  S  r − a −1 r which together with the control A1–A3 of the background foliation, the assumptions (9.4.23) for (f, f , λ), the control of r − rS , and Sobolev, yields ◦

sup |Err6 | . r−1 ( + δ1 )k(f, f , a)kh3 (S) . S

Step 4. We now estimate F1 , F2 , F3 and b0 . In view of

544 • • • • • • • •

CHAPTER 9

the definition of F1 , F2 , F3 and b0 in Step 1, the control A1–A3 for the background foliation, the assumption (9.4.2) for κ ˇ , /d?1 κ − CeΦ and /d?1 µ − M eΦ , S S the control of C − C and M − M using Corollary 9.18 and the control of the background foliation, the control of r − rS in property 3 of Lemma 9.15, the control of m − mS thanks to Corollary 9.19 and property 4 of Lemma 9.15, property 6 of Lemma 9.15, the estimates for Errj , j = 3, 4, 5, 6 of Step 3,

we infer kF1 khs−1 (S) + kF2 khs−1 (S)

h i ◦ S . r−2 δ + r3 |M S − M | + r−2 k(f, f , a)khs+1 (S) ◦  ×  + δ1 ,

◦  ◦ S kF3 khs (S) . r−1 δ + r4 |M S − M | + r−1 k(f, f )khs+1 (S) + r−1  + δ1 kakhs+1 (S) , and ◦

◦

r|b0 | . δ + sup |r − rS | + ( + δ1 )k(f, f , a)kh3 (S) . S

Step 5. In view of the estimates of Step 2 for (C S , M S ) and (f, f , λ), and the estimate for F1 , F2 , F3 and b0 in Step 4, we deduce ◦
S S |C S − C | + rS |M S − M | . (rS )−4 δ + (rS )−7 |Λ| + |Λ| ◦ h i S +  + δ1 rS |M S − M | + (rS )−4 k(f, f , a)khs+1 (S) , ◦
k(f, f )khs+1 (S) . δ + (rS )−2 |Λ| + |Λ| ◦ h i S +  + δ1 (rS )5 |M S − M | + k(f, f , a)khs+1 (S) ,

kˇ aS khs+1 (S)

.

◦  ◦ S δ + (rS )5 |M S − M | + k(f, f )khs+1 (S) +  + δ1 kakhs+1 (S)

and ◦

◦

r|aS | . δ + sup |r − rS | + ( + δ1 )k(f, f , a)kh3 (S) . S

◦

The above estimates for (C S , M S ) and (f, f ) yields for δ1 and  small enough S

S

(rS )4 |C S − C | + (rS )5 |M S − M | + k(f, f )khs+1 (S)  ◦
◦

. δ + (rS )−2 |Λ| + |Λ| +  + δ1 a . hs+1 (S)

545

GCM PROCEDURE ◦

Plugging in the above estimate for a ˇS , we infer for δ1 and  small enough S

.

S

(rS )4 |C S − C | + (rS )5 |M S − M | + k(f, f , a ˇS )khs+1 (S)   ◦
◦ δ + (rS )−2 |Λ| + |Λ| +  + δ1 r aS . ◦

Also, plugging in the above estimate for aS , we infer for δ1 and  small enough S

.

S

(rS )4 |C S − C | + (rS )5 |M S − M | + k(f, f , a ˇS )khs+1 (S)   ◦
◦ δ + (rS )−2 |Λ| + |Λ| +  + δ1 sup |r − rS | S

and ◦
r|aS | . δ + (rS )−2 |Λ| + |Λ| + sup |r − rS |. S

Using the third property of Lemma 9.15 in the first equation to bound r − rS , we finally obtain S

.

S

(rS )4 |C S − C | + (rS )5 |M S − M | + k(f, f , a ˇS )khs+1 (S) ◦  ◦
δ + (rS )−2 |Λ| + |Λ| + rδ1  + δ1

and r|aS | .

◦
δ + (rS )−2 |Λ| + |Λ| + sup |r − rS | S

which are the desired estimates. This concludes the proof of Corollary 9.38.

9.6

PROOF OF PROPOSITION 9.43

9.6.1

Pullback of the main equations

According to Proposition 9.42 we may assume valid the uniform bounds for the quintets P (n) . To establish a contraction estimate we need to compare the quantities, h(n) :

=

(Ψ(n−1) )# f (n) ,

e(n) :

=

(Ψ(n−1) )# a(n) ,

h(n) := (Ψ(n−1) )# f (n) ,

w(n) := (Ψ(n−1) )# z (n) ,

and h(n+1) :

=

(Ψ(n) )# f (n+1) , h(n+1) := (Ψ(n) )# f (n+1) , w(n+1) := (Ψ(n) )# z (n+1) ,

e(n+1) :

=

(Ψ(n) )# a(n+1) .

546

CHAPTER 9

According to Lemma 9.7 we have   (Ψ(n) )# /dS(n) f (n+1) = /d(n) h(n+1) ,   (Ψ(n) )# AS(n) f (n+1) = A(n) h(n+1) ,   (Ψ(n) )# B S(n) f (n+1) = B (n) h(n+1) , ◦

where /d(n) , /d?(n) , A(n) , B (n) are the corresponding Hodge operators on S defined with respect to the metric g/ (n) := (Ψ(n) )# (g/ S(n) ) given by #n

g/ (n) = γ (n) dθ2 + e2Φ

dϕ2 .

Consequently the system (9.4.27) takes the form
6mS(n) (n) −1 (n+1) (A ) h − ΥS(n) h(n+1) S(n) 5 (r ) #n 2 S(n) + S(n) (M (n+1) − M )(A(n) )−1 eΦ + (Ψ(n) )# E (n+1) , r  6mS(n) ΥS(n) (n) −1  (n+1) S(n) (n+1) B (n) h(n+1) = − (A ) h − Υ h (rS(n) )5 B (n) h(n+1) = −

+ (C (n+1) − C

S(n)

(9.6.1)

#n

)eΦ

#n 2ΥS(n) S(n) (M (n+1) − M )(A(n) )−1 eΦ + (Ψ(n) )# E (n+1) , S r e (n+1) . = (Ψ(n) )# E

+ /d?(n) e(n+1)

Equation (9.4.32) takes the form Z #n eΦ h(n+1) = Λ, ◦

Z

#n

eΦ

◦

(S,g / (n) )

h(n+1) = Λ.

(9.6.2)

(S,g / (n) )

Equation (9.4.33) takes the form ◦ ◦

e(n+1)

S,g /



(n)

=

1−

r

 S,g/ S(n) (n)

−

r ◦

(n+1)

+Err6

◦

(n)

S,g /

r

S(n)

2



2 κ ˇ+κ− r

#n S,g/

(n)

(n)

.

(9.6.3)

547

GCM PROCEDURE

Finally the system (9.4.34) takes the form   1 ς #n ∂θ U (1+n) = (γ (n) )1/2 h(1+n) 1 + h(1+n) h(1+n) , 4 1 1 ∂θ S (1+n) − ς #n Ω#n ∂θ U (1+n) = (γ (n) )1/2 h(1+n) , 2 2 γ (n) = γ #n (9.6.4)  
1 2 # + ς #n Ω#n + (b n )2 γ #n (∂θ U (n) )2 4 − 2ς #n ∂θ U (n) ∂θ S (n) − γ #n ς #n b#n ∂θ U (n) ,

U (1+n) (0) = S (1+n) (0) = 0.

We recall, see (9.4.38), the definition of the norm for the quintets P (n) in the particular case k = 2   S(n) kP (n) k2 : = r−1 k∂θ U (n) , S (n) k ◦ + r4 |C (n) − C | h1 (S)

  S(n)

+ r5 |M (n) − M | + (a(n) )#n−1 , (f (n) )#n−1 , (f (n) )#n−1 ◦ . h1 (S)

To prove the estimate kP (n+1) − P (n) k2   ◦ .  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 + kP (n−2) − P (n−3) k2 , we set δw(n+1)

=

δe(n+1)

=

δC (n+1)

=

w(n+1) − w(n) , δh(n+1) = h(n+1) − h(n) , δh(n+1) = h(n+1) − h(n) ,

e(n+1) − e(n) , δU (n+1) = U (n+1) − U (n) , δS (n+1) = S (n+1) − S (n) , C (n+1) − C (n) , δM (n+1) = M (n+1) − M (n) ,

and δw(n) (n)

δe

δC (n)

= w(n) − w(n−1) , = e

(n)

−e

(n−1)

,

= C (n) − C (n−1) ,

δh(n) = h(n) − h(n−1) ,

δU

(n)

=U

(n)

−U

(n−1)

,

δh(n) = h(n) − h(n−1) ,

δS (n) = S (n) − S (n−1) ,

δM (n) = M (n) − M (n−1) .

We will derive in section 9.6.3 the following estimates: r4 |δC (n+1) | + r5 |δM (n+1) |

! ◦

S,g / (n)

(n+1) (n+1) (n+1) (n+1) + δh , δh , δe − δe

◦ h1 (S)   ◦ .  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 ,

(9.6.5)

548

CHAPTER 9

◦ (n) (n+1) S,g/ r δe .


r−1 k∂θ δU (n) , δS (n) k

◦

h1 (S)

  ◦ + kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 , (9.6.6) and
r−1 k∂θ δU (n+1) , δS (n+1) k

kδh(n+1) , δh(n+1) k

.

◦

h1 (S)

◦

h1 (S)

◦

+kP (n) − P (n−1) k2 .

(9.6.7)

Proposition 9.43 is then an immediate consequence of (9.6.5), (9.6.6), (9.6.7). Thus, from now on, we focus on the proof of (9.6.5), (9.6.6), (9.6.7). To this end, we will rely on the following lemmas. 9.6.2

Basic lemmas ◦

Lemma 9.44. Let F be a reduced scalar function defined in a neighborhood of S ◦

in R and define its pullback F (n) = (Ψ(n) )# F to S, i.e., ◦

F (n) (θ)

◦

= F (u + U (n) (θ), s + S (n) (θ), θ), ◦

F (n−1) (θ)

◦

= F (u + U (n−1) (θ), s + S (n−1) (θ), θ).

Then,17 for all 1 ≤ p ≤ ∞, with δn U = U (n+1) − U (n) , δn S = S (n+1) − S (n)  
kδn F k p ◦ . kδn U k p ◦ + kδn Sk p ◦ sup ∂s F + ∂u F . (9.6.8) L (S)

L (S)

L (S)

R

Also, kδn F k

◦

h1 (S)

 . kδn U k

 ◦

h1 (S)

+ kδn Sk

◦

h1 (S)


sup d≤1 ∂s F + d≤1 ∂u F (9.6.9) R

where δn U = U (n+1) − U (n) , δn S = S (n+1) − S (n) . Proof. We write δn F

:= F (u0 + U (n) (θ), s0 + S (n) (θ), θ) − F (u0 + U (n−1) (θ), s0 + S (n−1) (θ), θ) Z 1 d  F u0 + tU (n) (θ) + (1 − t)U (n−1) (θ), = 0 dt  s0 + tS (n) (θ) + (1 − t)S (n−1) (θ), θ ,

17 Recall

∂s = e4 , ∂u =

ς 2


e3 − Ωe4 − bγ 1/2 eθ .

549

GCM PROCEDURE

i.e., denoting δn U = U (n) − U (n−1) , δn S = S (n) − S (n−1) , Z  1 |δn F | . δn U ∂u F u0 + tU (n) (θ) + (1 − t)U (n−1) (θ), 0

 s0 + tS (n) (θ) + (1 − t)S (n−1) (θ), θ + δ n S

Z 0

1

 ∂s F u0 + tU (n) (θ) + (1 − t)U (n−1) (θ),  s0 + tS (n) (θ) + (1 − t)S (n−1) (θ), θ ,

i.e., |δn F | . U (n) (θ) − U (n−1) (θ) sup |∂u F | + S (n) (θ) − S (n−1) (θ) sup |∂s F | ◦

◦◦

◦

from which (9.6.8) easily follows. Similarly,  k d/δn F k 2 ◦ . kδn U k ◦ + kδn Sk L (S)

h1 (S)

◦

S+S

S+δ S

 ◦

h1 (S)


sup d≤1 ∂s F + d≤1 ∂u F . R

Hence, kδn F k

◦

h1 (S)

.

 kδn U k

 ◦

h1 (S)

+ kδn Sk

◦

h1 (S)


sup d≤1 ∂s F + d≤1 ∂u F R

as desired. ◦

Lemma 9.45. Let ψ, h ∈ s1 (S), and δB (n) = B (n) −B (n−1) . The following formula holds true. Z   (n) ψδB h . r−3 k∂θ Ψ(n) − Ψ(n−1) k ◦ kψk ◦ khk ◦ . ◦ h1 (S) h1 (S) h2 (S) (S,g/ (n) ) Proof. Recall that the metric g/ (n) is given by g/ (n) = γ (n) dθ2 + e2Φ (n)

#n

dϕ2 ◦

so that the operator B (n) = /d2 /d?2 (n) , applied to s1 tensors h on S, is given by !

1 1 p B (n) h = p ∂θ + 2∂θ (Φ#n ) −∂θ h + ∂θ (Φ#n )h . γ (n) γ (n)

550

CHAPTER 9

This yields δB (n) h !
1 1 p = −p ∂θ + 2∂θ (Φ#n ) γ (n) γ (n−1) 1

+p ∂θ + 2∂θ (Φ#n ) γ (n−1) 1

#n

1


p

γ (n)

p

−p

#n

−∂θ h + ∂θ (Φ

γ (n)


)h

!

1 γ (n−1)

!
−∂θ h + ∂θ (Φ#n )h !

1




!

1

#n

p +p ∂θ + 2∂θ (Φ ) ∂θ (Φ γ (n−1) γ (n−1) 2 + (n−1) ∂θ (Φ#n − Φ#n−1 )ψ∂θ (Φ#n−1 )h. γ

−Φ

#n−1

)h

◦

Using the previous formula to integrate ψδB (n) h on S with the volume of g/ (n) , and after integration by parts, we infer Z ψδB (n) h ◦ (S,g / (n) )

Z =

p

1

◦

(S,g / (n) )

1 ×p γ (n) Z + ◦

#n

p −∂θ + ∂θ (Φ ) γ (n)
−∂θ h + ∂θ (Φ#n )h

(S,g / (n) )

1 p

γ (n)




1− p

p −∂θ + ∂θ (Φ

#n


)

p

γ (n)

γ (n−1)

γ (n)

γ (n−1)

! ψ

! ! ψ

1

1

!

p −p γ (n) γ (n−1)


× ∂θ h + ∂θ (Φ#n )h ! p Z (n)
1 γ p p + ◦ −∂θ + ∂θ (Φ#n ) ψ γ (n) γ (n−1) (S,g / (n) )
1 ×p ∂θ Φ#n − Φ#n−1 h γ (n−1) Z 2 + ◦ ∂ (Φ#n − Φ#n−1 )ψ∂θ (Φ#n−1 )h. (n−1) θ γ (n) (S,g / ) We now make use of the bounds (9.1.15) for (Ω, b, γ) involved in the definition of γ (n−1) and γ (n) , the uniform bound of P (n) provided by Proposition 9.42 and the Sobolev inequality to deduce Z (n) ψδB h . r−5 kγ (n) − γ (n−1) k ◦ kψk ◦ khk ◦ ◦ h1 (S) h1 (S) h2 (S) (S,g/ (n) )


−2 #n #n−1 + r ∂θ Φ − Φ h 2 ◦ kψk ◦ , (9.6.10) L (S)

h1 (S)

where we have also used Lemma 9.3 to estimate





ψ

ψ

∂θ (Φ#n−1 )ψ ◦



. r . r ◦ .

eΦ 2 ◦ . kψkh1 (S) ◦ Φ (n) 2 L (S,g / ) e L2 (S,g/ (n) ) L (S)

551

GCM PROCEDURE

To estimate the term γ (n) − γ (n−1) we recall that  
2 1 γ (n) = γ #n + ς #n Ω#n + (b#n )2 γ #n (∂θ U (n) )2 4

γ (n−1)

=

−2ς #n ∂θ U (n) ∂θ S (n) − γ #n ς #n b#n ∂θ U (n)  
2 1 γ #n−1 + ς #n−1 Ω#n−1 + (b#n−1 )2 γ #n−1 (∂θ U (n−1) )2 4 −2ς #n−1 ∂θ U (n−1) ∂θ S (n−1) − γ #n−1 ς #n−1 b#n−1 ∂θ U (n−1) .

The principal term γ #n − γ #n−1 can be estimated with the help of Lemma 9.44, the uniform bound of P (n) provided by Proposition 9.42, and the bounds provided18 by A3. All other terms can be estimated in a similar fashion. We derive   kγ (n) − γ (n−1) k ◦ . rk∂θ Ψ(n) − Ψ(n−1) k ◦ (9.6.11) h1 (S)

where   k∂θ Ψ(n) − Ψ(n−1) k

◦

h1 (S)

We deduce Z ψδB (n) h ◦ (S,g/ (n) )

h1 (S)

:= k∂θ (U (n) − U (n−1) )k

+ k∂θ (S (n) − S (n−1) )k

◦

h1 (S)

◦

h1 (S)

  . r−4 k∂θ Ψ(n) − Ψ(n−1) k

◦

h1 (S)


+r−2 ∂θ Φ#n − Φ#n−1 h

kψk

◦

h1 (S)

◦

L2 (S)

kψk

khk

◦

h2 (S)

◦

h1 (S)

. (9.6.12)

The proof of Lemma 9.45 is now an immediate consequence of the following lemma. ◦

Lemma 9.46. The following estimate holds true for a reduced scalar h ∈ s1 (S):  




∂θ Φ#n − Φ#n−1 h ◦ . r−1 k∂θ Ψ(n) − Ψ(n−1) k ◦ khk ◦ . (9.6.13) 2 L (S)

18 Note

h1 (S)

◦

h2 (S)

in particular that A3 implies ∂u (γ) = ∂u (r2 ) + O(r) = O(r) and ∂s (γ) = ∂s (r2 ) + O(r) = O(r). ◦

.

552

CHAPTER 9

Proof. We write
∂θ Φ#n − Φ#n−1     # n 1 1 1 √ (n) (n) (n) (n) = ∂θ S − Ω∂θ U e4 Φ + ∂θ U e3 Φ + γ 1 − b∂θ U eθ Φ 2 2 2 (  1 1 (n−1) (n−1) − ∂θ S − Ω∂θ U e4 Φ + ∂θ U (n−1) e3 Φ 2 2 )#n−1   1 √ (n−1) + γ 1 − b∂θ U eθ Φ 2   1 1 = ∂θ S (n) − Ω#n ∂θ U (n) (e4 Φ)#n + ∂θ U (n) (e3 Φ)#n 2 2   1 √ # + γ n 1 − b#n ∂θ U (n) (eθ Φ)#n 2   1 1 − ∂θ S (n−1) − Ω#n−1 ∂θ U (n−1) (e4 Φ)#n−1 − ∂θ U (n−1) (e3 Φ)#n−1 2 2   1 √ # − γ n−1 1 − b#n−1 ∂θ U (n−1) (eθ Φ)#n−1 , 2 i.e., grouping the terms appropriately,

J1

J2 J3


∂θ Φ#n − Φ#n−1 = J1 + J2 + J3 ,   1 = ∂θ S (n) − Ω#n ∂θ U (n) (e4 Φ)#n 2   1 − ∂θ S (n−1) − Ω#n −1 ∂θ U (n−1) (e4 Φ)#n−1 , 2 1 1 = ∂θ U (n) (e3 Φ)#n − ∂θ U (n−1) (e3 Φ)#n−1 , 2 2   1 #n √ #n = γ 1 − b ∂θ U (n) (eθ Φ)#n 2   1 √ #n−1 − γ 1 − b#n−1 ∂θ U (n−1) (eθ Φ)#n−1 , 2

and J3

=

J31 + J32 , "

J31

=

J32

=

  1 1 − b#n ∂θ U (n) 2  # 1 #n−1 √ #n−1 − γ 1− b ∂θ U (n−1) , 2  
1 √ #n γ 1 − b#n ∂θ U (n) (eθ Φ)#n − (eθ Φ)#n−1 . 2

(eθ Φ)

#n−1

√

γ

#n

The contribution to the estimate of Lemma 9.46 given by J1 , J2 , J31 can be easily estimated by making use of the uniform bound of P (n) provided by Proposition

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GCM PROCEDURE

9.42, the bound (9.1.15) for (Ω, b, γ), and Lemma 9.11, as well as Lemma 9.44. We thus derive k(J1 , J2 , J31 )hk

◦

L2 (S)

. r−1 k∂θ Ψ(n) − ∂θ Ψ(n−1) k

It remains to estimate the term kJ32 hk

◦

L2 (S)

◦

h1 (S)

khk

◦

h2 (S)

.

which presents a difficulty at the axis

of symmetry where sin θ = 0. Clearly, we have


kJ32 hk 2 ◦ . r (eθ Φ)#n − (eθ Φ)#n−1 h

◦

L2 (S)

L (S)


We are thus left to estimate the term (eθ Φ)#n − (eθ Φ)#n−1 h

. ◦

L2 (S)

. Proceeding

as in the proof of Lemma 9.44 we write, for F = eθ Φ, Z  1 |δn F | . δn U ∂u F u0 + tU (n) (θ) + (1 − t)U (n−1) (θ), 0

 s0 + tS (n) (θ) + (1 − t)S (n−1) (θ), θ + δ n S

Z 0

1

 ∂s F u0 + tU (n) (θ) + (1 − t)U (n−1) (θ),  s0 + tS (n) (θ) + (1 − t)S (n−1) (θ), θ .

We need to pay special attention on the axis,19 where sin θ = 0, to the integral term involving ∂u (eθ Φ) =

 1 e3 − Ωe4 − bγ 1/2 eθ eθ Φ. 2

This leads us to consider the integral Z 1 ◦ ◦ [beθ (eθ (Φ))] (u + tU (n) (θ) + (1 − t)U (n−1) (θ), s, θ)dt 0

◦

and the L2 norm of its product with h on S. We recall (see Lemma 2.13) that 4 / Φ = −K, and therefore, eθ (eθ Φ) . r−2 + |eθ Φ|2 . The contribution due to K does not present any difficulties on the axis therefore we are led to consider the integral Z 1   ◦ ◦ I(θ) := b(eθ (Φ))2 (u + tU (n) (θ) + (1 − t)U (n−1) (θ), s, θ)dt 0

◦

and the L2 norm of its product with h on S. Making use of (9.1.17) and then the first estimate of (9.1.18) of Lemma 9.3 together with our assumption A3 we derive 19 Indeed

the term eθ (eθ Φ) is quite singular on the axis.

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CHAPTER 9

the bound r2 I(θ)h(θ)

Z 1  1 ◦ ◦ (n) (n−1) u + tU (θ) + (1 − t)U (θ), s, θ) b( dt |h(θ)| sin2 θ 0 h(θ) b . r2 h(θ) sup b . ◦ h(θ) . . sup Φ eΦ eΦ sin θ R sin θ R e .

Making use of the second estimate in (9.1.17) we then derive

◦ −2 h ◦

kI · hk 2 ◦ . r . r−3 khkh1 (S) .

Φ L (S) e L2 (S) This shows that the behavior along the axis in (9.6.13) is not an issue. This ends the proof of Lemma 9.46. ◦

Lemma 9.47. Let δh(n+1) and δh(n+1) reduced scalars on S satisfying
6mS(n) (n) −1 (A ) δh(n+1) − ΥS(n) δh(n+1) (rS(n) )5 #n 2 + S(n) δM (n+1) (A(n) )−1 eΦ − (δB (n) )h(n) + H (n+1) , r  6mS(n) ΥS(n) (n) −1  (n+1) (9.6.14) S(n) (n+1) B (n) δh(n+1) = − (A ) δh − Υ δh (rS(n) )5 B (n) δh(n+1) = −

#n 2ΥS(n) δM (n+1) (A(n) )−1 eΦ S r − (δB (n) )h(n) + H (n+1) , #n

+ δC (n+1) eΦ

+

as well as Z

#n

= D(n+1) ,

#n

= D(n+1) .

◦

δh(n+1) eΦ

◦

δh(n+1) eΦ

S,g / (n)

Z S,g / (n)

Also, assume the bounds k(h(n) , h(n) )k

◦ ◦

h2 (S)

.

δ.

Then we have r4 |δC (n+1) | + r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k ◦ h1 (S)   ◦ −1 (n) (n−1) 2 (n+1) . δr k∂θ Ψ − Ψ k ◦ + r kH k 2 ◦ + r2 kH (n+1) k 2 ◦ (9.6.15) h1 (S) L (S) L (S)   (n+1) −2 (n+1) +r D + D . Proof. We proceed exactly as for the a priori estimates in Step 5 to Step 7 of the proof of Proposition 9.37, see section 9.5.1, with the exception of the terms involving δB (n) for which we do not use Cauchy-Schwarz. We obtain the following analog of

555

GCM PROCEDURE

(9.5.12), (9.5.9): 

4

(n+1)

5

(n+1)

(n+1)

(n+1)

2

r |δC | + r |δM | + k(δh , δh )k ◦ h1 (S)    (n+1) (n+1) 2 (n+1) 2 −2 (n+1) k 2 ◦ +r . r kH k 2 ◦ + r kH D + D L (S)

L (S)

×k(δh(n+1) , δh(n+1) )k 2 ◦ L (S) Z (n+1) 2 (S(n) (n+1) (n) (n) S(n) (n) +r ◦ (δh −Υ δh )δB (h − Υ h ) (S,g/ (n) ) Z +r2 ◦ δh(n+1) δB (n) h(n) (S,g/ (n) ) !2 Z Z Φ (n) (n) Φ (n) (n) + ◦ e (δB h ) + ◦ e (δB h ) . (S,g/ (n) ) (S,g/ (n) ) Next, we estimate the terms involving δB (n) . Using Lemma 9.45 with the choices • • • •

ψ ψ ψ ψ

= δh(n+1) − Υ(S(n) δh(n+1) and h = h(n) − ΥS(n) h(n) , = δh(n+1) and h = h(n) , = eΦ and h = h(n) , = eΦ and h = h(n) , ◦

which yields, together with the assumption on the h2 (S) norm of h(n) and h(n) , Z (n+1) (S(n) (n+1) (n) (n) S(n) (n) (δh −Υ δh )δB (h − Υ h ) ◦ (S,g/ (n) ) Z + ◦ δh(n+1) δB (n) h(n) (S,g/ (n) )   ◦ . r−3 δk∂θ Ψ(n) − Ψ(n−1) k ◦ k(δh(n+1) , δh(n+1) )k ◦ h1 (S)

h1 (S)

and Z Z !2 Φ (n) (n) Φ (n) (n) e (δB h ) + ◦ e (δB h ) ◦ (S,g/ (n) ) (S,g/ (n) )   2   ◦ . r−1 δk∂θ Ψ(n) − Ψ(n−1) k ◦ . h1 (S)

556

CHAPTER 9

Plugging in the above estimate, we infer 

4

(n+1)

5

(n+1)

(n+1)

(n+1)

2

r |δC | + r |δM | + k(δh , δh )k ◦ h1 (S)    (n+1) (n+1) 2 (n+1) 2 −2 (n+1) . r kH k 2 ◦ + r kH k 2 ◦ +r D + D L (S)

L (S)

(n+1)

×k(δh(n+1) , δh )k 2 ◦ L (S)   ◦ +r−1 δk∂θ Ψ(n) − Ψ(n−1) k ◦ k(δh(n+1) , δh(n+1) )k ◦ h1 (S) h1 (S)  2   ◦ + r−1 δk∂θ Ψ(n) − Ψ(n−1) k ◦ h1 (S)

and hence r4 |δC (n+1) | + r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k ◦ h1 (S)   ◦ . δr−1 k∂θ Ψ(n) − Ψ(n−1) k ◦ + r2 kH (n+1) k 2 ◦ + r2 kH (n+1) k 2 ◦ h (S) L (S) L (S)  1  (n+1) −2 (n+1) +r D + D as desired. 9.6.3

Proof of the estimates (9.6.5), (9.6.6), (9.6.7)

We are now in position to prove (9.6.5), (9.6.6), (9.6.7). Step 1. We start by estimating δh(n+1) , δh(n+1) . To this end, we need to apply Lemma 9.47 to the equations for δh(n+1) , δh(n+1) , derived from the first two equations in (9.6.1) and (9.6.2), and estimate the corresponding terms H (n+1) , H (n+1) , D(n+1) and D(n+1) on the right-hand side. This is tedious but straightforward and one derives   r2 kH (n+1) k 2 ◦ + r2 kH (n+1) k 2 ◦ + r−2 D(n+1) + D(n+1) L (S) L (S)   ◦ .  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 . Remark 9.48. Note that the presence of the inverse operators (A(n) )−1 in the right-hand side of the equations for δh(n+1) , δh(n+1) do not create any difficulties when taking differences. Indeed, we can write   (A(n) )−1 − (A(n−1) )−1 = (A(n) )−1 A(n−1) − A(n) (A(n−1) )−1 and estimate the difference δA(n) = A(n) − A(n−1) similar to the estimate for δB (n) in the proof of Lemma 9.45.

557

GCM PROCEDURE

We infer from Lemma 9.47 and the above estimates

.

r4 |δC (n+1) | + r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k ◦ h1 (S)   ◦  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 .

(9.6.16)

Step 2. Next, we estimate /d?δe(n+1) . Recall (9.6.1) /d?(n) e(n+1)

e (n+1) , = (Ψ(n) )# E

where e (n+1) E


3 1 = − (AS(n) )−1 ρz (n+1) + f (n+1) ω − f (n+1) ω + f (n+1) κ 4 4
1 (n+1) (n+1)
− f κ + (AS(n) )−1 − M (n+1) eΦ + /d?µ ˇ − (AS(n) )−1 Err3 . 4

This yields p ?(n)

/d

(n+1)

δe

=

γ (n)

− 1− p γ (n−1)

! e (n+1) . /d?(n) e(n) + H

e (n+1) is tedious but straightforward and one derives, using in The control of H particular Remark 9.48,

e (n+1) r H

2 ◦ . r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k 2 ◦ L (S) L (S)   ◦ + kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 . Also, in view of the boundedness of e(n) and γ (n) , we have

! p

◦ γ (n)

?(n) (n) r 1− p /d e . r−3 δkγ (n) − γ (n−1) k 2 ◦ (n−1) L (S)

2 ◦ γ L (S)   ◦ . r−2 δk∂θ Ψ(n) − Ψ(n−1) k

◦

h1 (S)

where we have used (9.6.11) in the last inequality. We deduce



r /d?(n) δe(n+1) ◦ . r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k 2 ◦ L (S) L2 (S)   ◦ + kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 and hence, using a Poincar´e inequality,

◦

S,g / (n)

(n+1) − δe(n+1) . r5 |δM (n+1) | + k(δh(n+1) , δh(n+1) )k 2 ◦

δe

L (S)

◦ h1 (S)   ◦ +  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 .

558

CHAPTER 9

Together with (9.6.16), we deduce r4 |δC (n+1) | + r5 |δM (n+1) |

! ◦

S,g / (n)

(n+1) (n+1) (n+1) (n+1) + δh , δh , δe − δe

◦ h1 (S)   ◦ .  kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 , which is the desired estimate (9.6.5). Step 3. Next, we estimate the average of δe(n+1) . Recall from (9.6.3) ◦ ◦

e(n+1)

S,g /

 (n) S,g/ rS(n) = 1− r

(n)

◦

(n)

−

r

S(n)



2

2 κ ˇ+κ− r

(n) S,g/

(n)

◦

+

(n+1) Err6

S,g / (n)

and ◦ ◦

e(n)

S,g / (n)

=

 (n−1) S,g/ rS(n−1) 1− r

(n−1)

◦

 (n) S,g/ rS(n−1) 2 − κ ˇ+κ− 2 r

(n−1)

◦

(n) +Err6

S,g / (n−1)

.

Taking the difference, recalling that we have in the (θ, ϕ) coordinates system dvolg/ (n) =

p

#n

γ (n) eΦ

dθdϕ, 4π(rS(n) )2 =

Z

2π

π

Z

0

#n

p

γ (n) eΦ

Z

π

dθdϕ

0

and dvolg/

(n−1)

Z p Φ#n−1 S(n−1) 2 (n−1) = γ e dθdϕ, 4π(r ) = 0

2π

p

#n−1

γ (n−1) eΦ

dθdϕ

0

and using the uniform bound of P (n) provided by Proposition 9.42 and the bounds ˇ we infer A1 for Γ, ◦ #n−1 #n (n+1) S r δe k 2 ◦ . r−2 kγ (n) − γ (n−1) k 2 ◦ + r−1 keΦ − eΦ L (S) L (S) (n+1)

+kδErr6

k

◦

L2 (S)

.

Arguing as above, we deduce ◦ (n)
(n+1) S,g/ r δe . r−1 k∂θ δU (n) , δS (n) k ◦ h1 (S)   ◦ + kP (n) − P (n−1) k2 + kP (n−1) − P (n−2) k2 which is the desired estimate (9.6.6).

559

GCM PROCEDURE

Step 4. Finally, we focus on (9.6.7). Recall (9.6.4)   1 ς #n ∂θ U (1+n) = (γ (n) )1/2 h(1+n) 1 + h(1+n) h(1+n) , 4 1 1 ∂θ S (1+n) − ς #n Ω#n ∂θ U (1+n) = (γ (n) )1/2 h(1+n) , 2 2  
2 1 (n) γ = γ #n + ς #n Ω#n + (b#n )2 γ #n (∂θ U (n) )2 4 − 2ς #n ∂θ U (n) ∂θ S (n) − γ #n ς #n b#n ∂θ U (n) ,

U (1+n) (0) = S (1+n) (0) = 0.

Taking the difference and arguing as above, we derive r−1 k∂θ δU (1+n) k

◦

h1 (S)

◦

. kδh(n+1) , δh(n+1) k

+ r−3 kγ (n) − γ (n−1) k

◦

h1 (S)

◦

h1 (S)

◦

r−1 k∂θ δS (1+n) k

+kP (n) − P (n−1) k2 ,

◦

h1 (S)

. k∂θ δU (1+n) k

◦

h1 (S)

+ kδh(n+1) , δh(n+1) k

◦

h1 (S)

◦

◦

+r−3 kγ (n) − γ (n−1) k

◦

h1 (S)

+ kP (n) − P (n−1) k2 .

Estimating γ (n) − γ (n−1) as above, we infer
r−1 k∂θ δU (n+1) , δS (n+1) k

◦

h1 (S)

. kδh(n+1) , δh(n+1) k

◦

◦

h1 (S)

+ kP (n) − P (n−1) k2

which is the desired estimate (9.6.7). This concludes the proof of Proposition 9.43.

9.7

A COROLLARY TO THEOREM 9.32

In what follows we prove an important corollary of Theorem 9.32 which makes use of the arbitrariness of Λ, Λ to ensure the vanishing of the ` = 1 modes of β and κ ˇ . The result requires stronger assumptions than those made in A1. Namely we assume that Γb has the same behavior as Γg , i.e., A1-Strong. For k ≤ smax ,



◦



. r−2 ,

Γg

Γb k,∞

k,∞





Γb

◦

. r−1 ,

k,∞

◦

1

. () 3 r−2 .

(9.7.1)

Theorem 9.49 (Existence of GCM spheres). In addition to the assumptions of Theorem 9.32, we assume that A1-Strong holds, and that, for any background sphere S in R, Z Z Z ◦ Φ Φ Φ r βe + r eθ (κ)e + r eθ (κ)e . δ. (9.7.2) S

S

S

◦

Then there exists a unique GCM sphere S, which is a deformation of S, such that

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CHAPTER 9

the following GCM conditions hold true: 2 /d2S,? /d1S,? κS = /d2S,? /d1S,? µS = 0, κS = S , r Z Z S Φ S S Φ β e = 0, eθ (κ )e = 0. S

(9.7.3)

S

Moreover, all other estimates of Theorem 9.32 hold true. Proof. The proof of the theorem follows easily in view of Theorem 9.32 and the following lemma. ◦ R Lemma R9.50. Let S be a deformation of S as in Theorem 9.32 with Λ = S f eΦ and Λ = S f eΦ . The following identities hold true: Z  Z r3 Φ S Φ Λ= βe − β e + F1 (Λ, Λ), ◦ 3m S S (9.7.4) Z  Z Z r3 Φ S S Φ Φ Λ= eθ (κ)e − (eθ κ )e − Υ ◦ eθ (κ)e + ΥΛ + F2 (Λ, Λ), ◦ 6m S S S where F1 , F2 are continuous20 in Λ, Λ, verifying, provided A1-Strong holds, the estimates ◦ F1 | + F2 . (◦) 13 δr2 , ∂Λ,Λ F1 | + ∂Λ,Λ F2 . (◦) 13 r2 .

(9.7.5)

Proof. To prove (9.7.4), we start with the change of frame formula,   3 S a β = e β + ρf + ea Err(β, β S ), 2 1 Err(β, β S ) = f α + l.o.t. 2 We write21 βS

= =

  3 3 β + ρf + (ea − 1) β + ρf + ea Err(β, β S ) 2 2       3 2m 3 2m 3 β+ − 3 f+ ρ + 3 f + (ea − 1) β + ρf + ea Err(β, β S ) 2 r 2 r 2

and deduce βS +

20 In

3mS f (rS )3

= β + Err0 (β, β S )

fact smooth.

21 Here

◦

(r, m) represents the area radius and Hawking mass of S, while (rS , mS ) represent the S

◦

◦

area radius and Hawking mass of S. Since | rr − 1| . δ and |mS − m| . δ, we can interchange freely rS with r and mS with m.

561

GCM PROCEDURE

with error term Err0 (β, β S ),       3mS 3m 3 2m 3 0 S a Err (β, β ) = − 3f + ρ + 3 f + (e − 1) β + ρf (rS )3 r 2 r 2 +ea Err(β, β S ). Making use of the assumptions A1–A3 , the estimates of Theorem 9.32 for (f, f , a) ◦

◦

as well as the bounds for r − rS , m − mS , we deduce ◦◦ 0 Err (β, β S ) . r−1 δ . Thus, 3mS (rS )3

Z

f eΦ

Z =

S

S

Z =

βeΦ −

βeΦ − ◦

Z

β S eΦ +

Z

S

Z

S

Err0 (β, β S )eΦ

S

β S eΦ +

Z

S

Err0 (β, β S )eΦ +

Z

S

S

βeΦ −

Z

βeΦ ◦



S

or 3m r3

Z

Φ

fe

Z =

Φ

S

 +

βe −

Z

S Φ

0

S

Φ

β e + Err (β, β )e + S Z 3m 3m − S 3 f eΦ . r3 (r ) S

◦

S

Z S S

Z S

Φ

βe −

Z

Φ

◦



βe

S

Clearly, Z Err0 (β, β S )eΦ

◦◦

. r−1 δ .

S

Also, proceeding exactly as in Corollary 9.10 we deduce Z   Z ◦
2 ◦ ≤1 Φ βeΦ − d (βeΦ ) + sup r◦ e3 (βeΦ | . βe . δ sup r ◦ % S

R

S

R

Thus, in view of the assumptions A1–A3, Z Z ◦◦ Φ βeΦ − βe . δ r−1 . ◦ S

(9.7.6)

S

We deduce Λ

=

r3 3m

Z

βeΦ − ◦

S

Z

β S eΦ

 + F1 (Λ, Λ)

S

where the error term F1 (Λ, Λ) is a continuous function of Λ, Λ verifying the estimate F1 (Λ, Λ)

.

◦◦ 2

δr .

562

CHAPTER 9

We also recall, see Lemma 9.23, 1 = eθ κ − /d1S,? /d1S f − κeS θ a − κ(f κ + f κ) + κ(f ω − ωf ) + f ρ 4 S +Err(eS θ κ , eθ κ)

S eS θ (κ )

where   1 1 (e−a − 1) eθ κ − /d S,? /d1S f + f e3 κ + f e4 κ 2 2 "   1 S S + e−a eS d1S f + Err(κ, κS ) + f f eθ κ θ Err(κ, κ ) + eθ (a) / 2 #   1 2 1 1 + f f e3 κ + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) 8 2 2   1 1 2 1 1 2 + f f eθ κ + f f e3 κ + f 2 /d1 ξ − ϑ + 2(η + η − 2ζ)ξ . 2 8 2 2

S Err(eS θ κ , eθ κ)

=

Making use of the identity /d1S,? /d1S = /d2S /d2S,? + 2K S we deduce   1 1 2 S eS (κ ) + κκ − ρ + 2K f = eθ κ − /d2S /d2S,? f − κeS θ θ a − κ f − κ ωf 4 4 S −2(K S − K)f + Err(eS θ κ , eθ κ). 2 2 r +(κ− r ),

Using κ = we have

1 κκ − ρ + 2K 4

2Υ 2m 2m κ = − 2Υ r +(κ+ r ), ρ = − r 3 +(ρ+ r 3 ), K =

=

=

1 r2

+(K − r12 ),

      1 4m 1 2Υ Υ 2 2m + 3 + κ+ − κ− + ρ+ 3 r2 r 2r r 2r r r   1 +2 K − 2 r 1 4m ◦ + 3 + O(r−3 ). r2 r

Also, using A1-Strong,   2Υ 2Υ 2Υ ◦ κ = − + κ+ =− + O(r−2 ), r r r   2mΥ ◦ 1 κ ω = − 3 + O r−3 () 3 , r ◦

and, in view of A1-Strong, and since a, f, f = O(r−1 δ), S Err(eS θ κ , eθ κ)

=

◦ ◦

1

O(r−4 δ() 3 ).

We deduce S eS θ (κ )

 +

1 4m + 3 r2 r

 f

=

eθ κ −

/d2 /d?2 f

Υ 1− 2Υ S + eθ a − r r2

4m r


f + Err1

563

GCM PROCEDURE

with error term Err1

◦

1

◦

. () 3 δr−4 .

Projecting on eΦ and proceeding as before,  Z Z Z Z 1 4m 2Υ Φ Φ S S Φ + 3 f e = ◦ (eθ κ)e − (eθ κ )e − (eS a)eΦ r2 r r S θ S S S
Z Υ 1 − 4m r − f eΦ + I1 (Λ, Λ) r2 S

(9.7.7)

where the error term I1 is continuous in Λ, Λ and verifies the estimate ◦ 1◦ I1 . r−1 () 3 δ. We now calculate S eS θ (κ )

R S

Φ (eS θ a)e . Recall from Lemma 9.23

1 S = eθ κ − /d?1S /dS 1 f + κeθ a − κ(f κ + f κ) + κ(f ω − ωf ) + f ρ 4 S S +Err(eθ κ , eθ κ),

where S Err(eS θ κ , eθ κ)

  1 1 S (ea − 1) eθ κ + eS / d f + f e κ + f e κ 4 3 θ 1 2 2 "   1 S S S S + ea eS Err(κ, κ ) + e (a) / d f + Err(κ, κ ) + f f eθ κ θ θ 1 2 #   1 2 1 1 + f f e3 κ + f 2 /d1 η − ϑ ϑ + 2(ξξ + η 2 ) 8 2 2   1 1 1 1 + f f eθ κ + f 2 f e3 κ + f 2 /d1 ξ − ϑ2 + 2(η + η + 2ζ)ξ . 2 8 2 2 =

Using again the identity /d1S,? /d1S = /d2S /d2S,? + 2K S and proceeding as above, we S infer, using also the GCM condition for κS which yields eS θ (κ ) = 0,   2 S 1 2 1 S ?S 0 = eθ κ − /d2 /d2 f + S eθ a − κ f + κκ + κω + 3ρ f r 4 4   2 1 S S S + κ − S eS θ a − 2(K − K)f − ϑϑf + Err(eθ κ , eθ κ). r 2

564

CHAPTER 9

Integrating over S, we deduce Z 2 eS (a)eΦ rS S θ  Z Z Z  1 1 Φ 2 Φ = − eθ (κ)e + κ fe − κκ + κω + 3ρ f eΦ 4 S 4 S S   Z  2 1 S S S S + κ − S eθ a − 2(K − K)f − ϑϑf + Err(eθ κ , eθ κ) eΦ r 2 S   Z 1 1 2m = − ◦ eθ (κ)eΦ + 2 Λ + 2 1 + Λ + I2 (Λ, Λ) r r r S where, using in particular A1-Strong, ◦ 1◦ I2 (Λ, Λ) . r−1 () 3 δ. Indeed, using once more Corollary 9.10, we note that Z   Z ◦ ◦ ◦ 1◦ Φ −1 4 eθ (κ)eΦ − e (κ)e . r δ  + r sup |e (e (κ))| . r−1 () 3 δ, θ 3 θ ◦ R S

S

where we used A1-Strong, the transport equation for e3 (κ) and a commutator formula for [e3 , eθ ] to estimate e3 (eθ (κ)). All other error terms are easily estimated. Back to (9.7.7) we deduce
  Z Z Z Υ 1 − 4m 1 4m 2Υ Φ S S Φ S Φ r + Λ = (e κ)e − (e κ )e + (e a)e − Λ θ θ ◦ r2 r3 r S θ r2 S S +I1 (Λ, Λ)
Z Z Υ 1 − 4m Φ S S Φ r = (eθ κ)e − (eθ κ )e − Λ + I1 (Λ, Λ) ◦ r2 S S  Z    1 1 2m + Υ − ◦ eθ (κ)eΦ + 2 Λ + 2 1 + Λ + I2 (Λ, Λ) r r r S Z Z Z Υ 6mΥ S Φ Φ = (eθ κ)eΦ − (eS Λ θ κ )e − Υ ◦ eθ (κ)e + 2 Λ + ◦ r r3 S S S + I1 (Λ, Λ) + ΥI2 (Λ, Λ). Hence, 6m Λ r3

Z =

(eθ κ)eΦ − ◦

S

+

Z S

S Φ (eS θ κ )e − Υ

Z ◦

eθ (κ)eΦ +

S

6mΥ Λ r3

I1 (Λ, Λ) + ΥI2 (Λ, Λ).

Thus, Λ

=

r3 6m

Z

Φ

◦

S

(eθ κ)e −

Z S

S Φ (eS θ κ )e

−Υ

Z ◦

S

eθ (κ)e

Φ

 + ΥΛ + F2 (Λ, Λ)

565

GCM PROCEDURE

with error term F2 (Λ, Λ) continuous in Λ, Λ and verifying the estimate F2 (Λ, Λ)

◦

1

◦

. () 3 δr2 .

To check the second part in (9.7.5) one needs to revisit the proof of Theorem 9.32 and check the dependence of U, S, f, f , λ on the parameters Λ, Λ. It is tedious but standard to check the following estimates for the derivatives with respect to Λ, Λ.




. 1, k ≤ smax , (9.7.8)

∂Λ,Λ f, f , log λ hk (S)


k∂Λ,Λ U 0 , S 0 k

◦

L∞ (S)

+

max

0≤s≤smax −1


r−1 k∂Λ,Λ U 0 , S 0 k

◦ ◦

hs (S,g /)

.

1. (9.7.9)

Using these estimates and taking into account the structure of the error terms F1 , F2 we derive the second inequality in (9.7.5). This ends the proof of the lemma. Under the assumptions of the theorem, the system Z r3 Λ= βeΦ + F1 (Λ, Λ), 3m S◦ Z  Z r3 Φ Φ e (κ)e − Υ e (κ)e + ΥΛ + F2 (Λ, Λ), Λ= θ θ ◦ ◦ 6m S S has a unique solution Λ0 , Λ0 verifying the estimate ◦

|Λ0 | + |Λ0 | . δr2 . Taking Λ = Λ0 , Λ = Λ0 in (9.7.4) we deduce Z Z S Φ β S eΦ = 0, eS θ (κ )e = 0, S

S

as stated. Corollary 9.51. Let a fixed spacetime region R verifying assumptions A1–A3 and (9.4.2), as well as, for any background sphere S in R, Z Z ◦ βeΦ + eθ (κ)eΦ . δ. (9.7.10) S

S

Assume that S is a sphere in R which verifies the GCM conditions κS =

2 , rS

/d?2 S /d?1 S κS = /d?2 S /d?1 S µS = 0

(9.7.11)

and such that, for a small enough constant δ1 > 0, the transition functions (f, f , λ) from the background frame of R to that of S verifies, for some 4 ≤ s ≤ smax , the bound kf khs (S) + (rS )−1 k(f , log λ)khs (S)

≤

δ1 .

(9.7.12)

566

CHAPTER 9

Assume in addition that we have Z Z S Φ β S eΦ + eS θ (κ )e S

◦

. δ.

(9.7.13)

S

Then the transition functions (f, f , λ) from the background frame of R to that of S verify the estimates ◦  ◦ . r δ + δ1 (  + δ1 )

ˇ S )kh (S) k(f, f , λ s+1 and

◦  S ◦ r|λ − 1| . r δ + δ1 ( + δ1 ) + sup r − rS . S

Proof. Applying Corollary 9.38, we have ˇ S )kh k(f, f , λ smax +1 (S)

◦

◦

. δ + r−2 (|Λ| + |Λ|) + rδ1 ( + δ1 ), ◦ S r|λ − 1| . δ + r−2 (|Λ| + |Λ|) + sup r − rS . S

Thus, to conclude, it suffices to prove the estimate ◦  ◦ |Λ| + |Λ| . r3 δ + δ1 ( + δ1 ) . Now, revisiting the proof of Lemma 9.50 without assuming that A1-Strong holds, and using in particular (9.7.12), we obtain the following analog of (9.7.4): Z  Z   r3 ◦ Φ S Φ Λ= βe − β e + O r 3 δ1 (  + δ1 ) , ◦ 3m S S Z  Z Z   r3 ◦ Φ S S Φ Φ Λ= (eθ κ)e − (eθ κ )e − Υ ◦ eθ (κ)e + ΥΛ + O r3 δ1 ( + δ1 ) . ◦ 6m S S S The desired estimate for (Λ, Λ) follows then immediately from (9.4.2) for κ, (9.7.10) and (9.7.13).

9.8

CONSTRUCTION OF GCM HYPERSURFACES

We are ready to state our main result concerning the construction of GCM hypersurfaces. Theorem 9.52. Let a fixed spacetime region R verifying assumptions A1–A3 and (9.4.2). In addition we assume that Z Z ◦ ◦ Φ 2 Φ ηe . r δ, ξe . r2 δ, (9.8.1) S(u,s) S(u,s)

567

GCM PROCEDURE

and, everywhere on R,   2 2m + Ω − 1 − ς r SP

◦

.δ

(9.8.2)

where SP denotes the south pole, i.e., θ = 0 relative to the adapted geodesic coordinates u, s, θ. ◦ ◦ Let S0 = S0 [u, s, Λ0 , Λ0 ] be a fixed GCMS provided by Theorem 9.32. Then, there exists a unique, local,22 smooth, Z-invariant spacelike hypersurface Σ0 passing through S0 , a scalar function uS defined on Σ0 , whose level surfaces are topological spheres denoted by S, and a smooth collection of constants ΛS , ΛS verifying ΛS0 = Λ0 ,

ΛS0 = Λ0 ,

such that the following conditions are verified: 1. The surfaces S of constant uS verify all the properties stated in Theorem 9.32 for the prescribed constants ΛS , ΛS . In particular they come endowed with null S S frames (eS 4 , eθ , e3 ) such that a) b) c) d)

For each S the GCM conditions (9.4.1) with Λ = ΛS , Λ = ΛS , are verified. The transition functions (f, f , a = log λ) verify the estimates (9.4.5). The transversality conditions (9.4.9) are verified. The corresponding Ricci and curvature coefficients verify the estimates (9.4.8) and (9.4.11).

2. Denoting rS to be the area radius of the spheres S we have, for some constant c∗ , uS + rS = c∗ ,

along

Σ0 .

(9.8.3)

3. Let ν S be the unique vectorfield tangent to the hypersurface Σ0 , normal to S, and S normalized by g(ν S , eS 4 ) = −2. There exists a unique scalar function a on Σ0 S such that ν is given by S S ν S = eS 3 + a e4 .

The following normalization condition holds true at the south pole SP of every sphere S, i.e., at θ = 0, aS

SP

= −1 −

2mS . rS

(9.8.4)

4. We extend uS and rS in a small neighborhood of Σ0 such that the following transversality conditions are verified23 on Σ0 , S eS 4 (u ) = 0,

22 That

S eS 4 (r ) =

rS S κ = 1. 2

(9.8.5)

is, in a neighborhood of S0 . the average of κS is taken on S. In view of the GCM conditions (9.8.16) we deduce S eS 4 (r ) = 1. 23 Here

568

CHAPTER 9

5. In view of (9.8.5) the Ricci coefficients η S , ξ S are well defined for every S ⊂ Σ0 and verify Z Z η S eΦ = ξ S eΦ = 0. (9.8.6) S

S

6. The following estimates hold true for all k ≤ smax , ◦

kη S khk (S)

. ,

kξ khk (S)

S

S

a + 1 + 2m

rS hk (S)

. ,

S

(9.8.7)

◦

(9.8.8)

◦

. .

(9.8.9)

The eS ˇ S , ϑS , ζ S , κ ˇ S , ϑS , αS , β S , ρˇS , µS , β S are well defined on Σ0 3 derivatives of κ and we have, for all k ≤ smax − 1, ◦

keS κS , ϑS , ζ S , κ ˇ S )khk (S) . r−1 , 3 (ˇ ◦

keS 3

S keS 3 (ϑ )khk (S) . ,
◦ αS , β S , ρˇS , µS khk (S) . r−2 , S keS 3 (β )khk (S)

◦ −1

. r

(9.8.10)

,

◦

S keS 3 (α )khk (S) . .

7. The transition functions from the background foliation to that of Σ0 verify ◦

kd≤smax +1 (f, f , log λ)kL2 (S) . δ.

(9.8.11)

Corollary 9.53. Let a fixed spacetime region R verifying assumptions A1–A3 and the small GCM conditions (9.4.2). Assume given a GCM hypersurface Σ0 ⊂ R foliated by surfaces S such that Z Z 2 S ?S ?S S ?S ?S S S Φ η e = 0, ξ S eΦ = 0. κ = S , /d2 /d1 κ = /d2 /d1 µ = 0, r S S 1. If we assume in addition that for a specific sphere S0 on Σ0 , the transition functions f, f from the background foliation to S0 verify k(f, f , log(λ))khsmax +1 (S0 )

◦

. δ,

(9.8.12)

then, the following holds true: ◦

kd≤smax +1 (f, f , log(λ))kL2 (S0 ) . δ.

(9.8.13)

2. If we assume in addition that for a specific sphere S0 on Σ0 , the transition functions f, f from the background foliation to S0 verify kf khsmax +1 (S0 ) + (rS0 )−1 k(f , log λ)khsmax +1 (S0 )

◦

. δ,

(9.8.14)

569

GCM PROCEDURE

then, the following holds true: kd≤smax +1 f kL2 (S0 ) + r−1 kd≤smax +1 (f , log λ)kL2 (S0 ) +kd≤smax eS 3 (f , log λ)kL2 (S0 )

◦

. δ.

(9.8.15)

We give below the proof of Theorem 9.52 and of Corollary 9.53. 9.8.1

Definition of Σ0 ◦

◦

As stated in the theorem, we assume given a region R = {|u− u| ≤ δR , |s− s| ≤ δR } (see definition(9.1.6)) endowed with a background foliation such that the condition A1–A3 hold true. We also assume given a deformation sphere S0 ◦

◦ ◦

:= S[u, s, Λ0 , Λ0 ]

◦ ◦

of a given sphere S = S(u, s) of the background foliation which verify the conclusions of Theorem 9.32. We then proceed to construct, in a small neighborhood of S0 , a spacelike hypersurface Σ0 initiating at S0 verifying all the desired properties mentioned above. In what follows we outline the main steps in the construction. Step 1. According to Theorem 9.32, for every value of the parameters (u, s) in R (i.e., such that the background spheres S(u, s) ⊂ R) and every real numbers (Λ, Λ), there exists a unique GCM sphere S[u, s, Λ, Λ], as a Z-polarized deformation of S(u, s). In particular the following are verified: • S coincides with S(u, s) at their south poles (i.e., for θ = 0 in the adapted coordinates). • On S, the following GCMS conditions hold: κS =

2 , rS

/d2S,? /d1S,? κS = 0, Z

f eΦ = ΛS ,

S

Z

/d2S,? /d1S,? µS = 0,

f eΦ = ΛS ,

(9.8.16)

(9.8.17)

S

where (f, f , λ) are the transition parameters of the frame transformation from the S S background frame (e3 , eθ , e4 ) to the adapted frame (eS 3 , eθ , e4 ). The constants S S Λ , Λ depend smoothly on the surfaces S and ΛS0 = Λ0 ,

ΛS0 = Λ0 .

• There is a map Ξ : S(u, s) → S given by   Ξ : (u, s, θ) = u + U (θ, u, s, Λ, Λ), s + S(θ, u, s, Λ, Λ), θ

(9.8.18)

with U, S vanishing at θ = 0. • The transversality conditions (9.4.9) hold, i.e., ξ S = ω S = ζ S + η S = 0. Note that these specify the eS 4 derivatives of (f, f , λ) on S.

570

CHAPTER 9

• The Ricci coefficients24 κS , κS , ϑS , ϑS , ζ S are well defined on each sphere S of Σ0 , and hence on Σ0 . The same holds true for all curvature coefficients αS , β S , ρS , β S , αS . Taking into account our transversality condition we remark that the only ill defined Ricci coefficients are η S , ξ S , ω S . • Let ν S be the unique vectorfield tangent to the hypersurface Σ0 , normal to S, S and normalized by g(ν S , eS 4 ) = −2. There exists a unique scalar function a on S Σ0 such that ν is given by S S ν S = eS 3 + a e4 .

We deduce that the quantities S g(Dν S eS 4 , eθ )

=

S g(Dν S eS 3 , eθ ) S g(Dν S eS 3 , e4 )

= =

2η S + 2aS ξ S = 2η S , 2ξ S + 2aS η S = 2(ξ S − aS ζ S ), 4ω S − 4aS ω S = 4ω S ,

are well defined on Σ0 . Thus the scalar aS allows us to specify the remaining Ricci coefficients, η S , ξ S , ω S along Σ0 , which we do below. 9.8.2

Extrinsic properties of Σ0

We analyze the extrinsic properties of the hypersurfaces Σ0 defined in Step 1. Step 2. We define the scalar function uS on Σ0 as uS := c0 − rS ,

(9.8.19)

◦ where rS is the area radius of S and the constant c0 is such that uS S0 = u, i.e., ◦ c0 = u + r S . S0

Step 3. We extend uS and rS in a small neighborhood of Σ0 such that the following transversality conditions are verified. S eS 4 (u ) = 0,

S eS 4 (r ) =

rS S κ , 2

(9.8.20)

where the average of κS is taken on S. In view of the GCM conditions (9.8.16) we S deduce eS 4 (r ) = 1. S S Step 4. Note that eS 3 (u , r ) remain undetermined. On the other hand, since 24 Consequently

the Hawking mass mS is also well defined.

571

GCM PROCEDURE

S S S eS θ (u ) = eθ (r ) = 0, we deduce in view of (9.8.20)   1 S S S S S S S S S S S S S S eS (e (u )) = [e , e ]u = (κ + ϑ )e + (ζ − η )e + ξ e θ 3 θ 3 θ 3 4 u 2 S (ζ S − η S )eS 3 (u ),   1 S S S S S S S S S S S = [eS , e ]r = (κ + ϑ )e + (ζ − η )e + ξ e θ 3 θ 3 4 r 2

= S S eS θ (e3 (r ))

=

S S (ζ S − η S )eS 3 (r ) + ξ .

Thus introducing the scalars 2

ςS

:=

:=

2 S S (e (r ) + ΥS ), rS 3

S eS 3 (u )

,

(9.8.21)

and AS

(9.8.22)

we deduce S eS θ (log ς ) S eS θ (A )

(η S − ζ S ), 2ΥS 2 = − S (ζ S − η S ) − S ξ S + (ζ S − η S )AS . r r

=

(9.8.23) (9.8.24)

S S S We infer that eS θ (log ς ) and eθ (A ) are determined in terms of η, ξ.

Step 5. In view of the definition of ν S and ς S we make use of (9.8.20) to deduce S S S S ν S (uS ) = eS 3 (u ) + a e4 (u ) =

2 . ςS

On the other hand, since uS := c0 − rS along Σ0 , S S S S S ν S (uS ) = −ν S (rS ) = −eS 3 (r ) − a e4 (r ) = Υ −

rS S A − aS 2

and therefore, aS

= −

2 rS 2 + Υ S − AS = − S − Ω S S ς 2 ς

(9.8.25)

where S S ΩS := eS 3 (r ) = −Υ −

rS S A . 2

(9.8.26)

Step 6. The following lemma will be used, in particular,25 to determine the AS .

25 It

will also be used below to derive equations for Λ, Λ.

572

CHAPTER 9

Lemma 9.54. For every scalar function h we have the formula Z  Z
νS h = (ς S )−1 ς S ν S (h) + (κS + aS κS )h . S

(9.8.27)

S

In particular ν S (rS )

rS S −1 S S (ς ) ς (κ + aS κS ) 2

=

where the average is with respect to S. Proof. We consider the coordinates uS , θS along Σ0 with ν S (θS ) = 0. In these coordinates we have νS =

2 ∂ S. ςS u

The lemma follows easily by expressing the volume element of the surfaces S ⊂ Σ0 with respect to the coordinates uS , θS (see also the proof of Proposition 2.64). 2 rS

Step 7. Note that the GCM condition κS = Hawking mass implies that κS = −

2ΥS , rS

together with the definition of the

ΥS = 1 −

2mS , rS

where the average is taken with respect to S. Thus in view of Lemma 9.54 we deduce S S eS 3 (r ) + a

= ν S (rS ) =

rS S −1 S S (ς ) ς (κ + aS κS ) 2
ς S κS + ςˇS κ ˇ S + (ς S )−1 ς S aS

=

rS S −1 (ς ) 2

=

−ΥS (ς S )−1 ς S +

S S Since according to (9.8.22) eS 3 (r ) = −Υ +

A

S

=

2 rS



S

S

Υ −a −Υ

S

(ς S )−1 ς S

rS S −1 S S (ς ) ςˇ κ ˇ + (ς S )−1 ς S aS . 2 rS S 2 A ,

we deduce

 rS S −1 S S S −1 S S + (ς ) ςˇ κ ˇ + (ς ) ς a . 2

In particular, multiplying by ς S and taking the average, we infer ς S AS

= ςˇS κ ˇS ,

and hence A

S

=

1 ςS



ςˇS κ ˇS

−

ˇS ςˇS A

 .

Step 8. We summarize the results in Steps 1–7 in the following.

(9.8.28)

573

GCM PROCEDURE

Proposition 9.55. Let Σ0 be a smooth spacelike hypersurface foliated by framed26 2 S S S spheres (S, eS 4 , eθ , e3 ) whose Ricci coefficients verify the GCM condition κ = r S and transversality condition (9.4.9). Define uS as in (9.8.19) such that uS + rS is constant on Σ0 with rS the area radius of the spheres S. Extend uS and rS in a neighborhood of Σ0 such that the transversality conditions (9.8.20) are verified. Then, defining the scalars ς S , AS as in (9.8.21), (9.8.22) we establish the following relations between η S , ξ S and ς S , AS and aS , where the latter scalar is defined in Step 1, S S S eS θ (log ς ) = (η − ζ ), S eS θ (A ) = −

AS =

1

2ΥS S 2 (ζ − η S ) − S ξ S + (ζ S − η S )AS , rS r 

ςS

aS = −

ˇ ςˇS κ ˇ S − ςˇS A

S

,

(9.8.29)

2 rS S S + Υ − A . ςS 2

S S S Remark 9.56. Note that we lack equations for η , ξ and the average of a . The S latter can be fixed by fixing the value of a SP and observing that

aS = aS SP − a ˇS SP .

(9.8.30)

In what follows we state a result which ties η S , ξ S to the other GCM conditions in (9.8.16)–(9.8.17). Step 9. To state the proposition below we split the Ricci coefficients into the following groups.   2 2mS S S S S S S− S+ ΓS = κ ˇ , ϑ , ζ , κ ˇ , r ρ ˇ , κ , ρ , g rS (rS )3   mS S S S S S S ΓS = η , ξ , ω ˇ , ω − , β , α . b (rS )2 Proposition 9.57. The following statements hold true:27 1. Under the same assumptions as in Proposition 9.55, the Ricci coefficients η S , 26 That 27 r S

S S is, differentiable spheres S endowed with adapted null frames (eS 4 , eθ , e3 ). here denotes rS the area radius of S.

574

CHAPTER 9

ξ S , ω S verify the following identities.   2 /d2S,? /d1S,? /d1S /d2S /d2S,? η S = κS C1 + 2 /d2S,? /d1S,? /d1S β S − rS−3 d/3 C2 −4 S + rS−5 ( d/S )≤4 ΓS /S )≤4 (ΓS g + rS ( d b · Γb ) + l.o.t.,   2 /d2S,? /d1S,? /d1S /d2S /d2S,? ξ = C3 − κS C1 + 2 /d2S,? /d1S,? /d1S β S + rS−5 ( d/S )≤4 ΓS g S + rS−4 ( d/S )≤4 (ΓS b · Γb ) + l.o.t.,   1 S /d1S,? ω S = κ + ω S η S − (κS )−1 /d2S /d2S,? η S 4 1 S S 1 S −1 + κ ξ − (κ ) C2 + rS−1 ( d/S )≤1 ΓS g 4 2 S + d/S (ΓS b · Γb ),

(9.8.31)

where C 1 = eS d2S,? /d1S,? µS ), 3(/ S S C 2 = eS 3 (eθ κ ),   C3 = e3 ( /d2S,? /d2S + 2K S ) /d2S,? /d1S,? κS .

(9.8.32)

The quadratic terms denoted by l.o.t. are lower order both in terms of decay as well as in terms of number of derivatives. Also, they contain angular derivatives S d/S , but neither eS 3 nor e4 derivatives. Moreover, we note that the error terms −5 −4 S ≤4 S S ≤4 S rS ( d/ ) Γg and rS ( d/ ) (ΓS b · Γb ) do in fact not contain more than three S derivatives of ω ˇ . 2. If in addition (9.4.8) of Theorem 9.32 hold true then, for k ≤ smax − 7, ◦

k /d2S,? η S kh4+k (S) . rS3 kC1 khk (S) + rkC2 kh3+k (S) + rS−1

S S + rS−1 kΓS g kh4+k (S) + kΓb · Γb kh4+k (S) + l.o.t., ◦

k /d2S,? ξ S kh4+k (S) . rS4 kC3 khk (S) + rS3 kC1 kh3+k (S) + rS−1

S S + rS−1 kΓS g kh4+k (S) + kΓb · Γb kh4+k (S) + l.o.t.,

(9.8.33)

k /d1S,? ω S kh2+k (S) . rS−1 kη S kh4+k (S) + rS−1 kξ S kh2+k (S) + rkC2 kh2+k (S) S S + rS−1 kΓS g kh3+k (S) + kΓb · Γb kh3+k (S) + l.o.t.

3. If in addition the GCM conditions (9.8.16) hold true along Σ0 and the estimates (9.4.11) are also verified then, for k ≤ smax − 7,  



2mS ◦ −5 S −1 S

C1 a +1+ S +r a ˇ h (S) , . r hk−2 (S) k−2 r   S



2m ◦ −3 S −1 S

C2 (9.8.34) . r a +1+ S +r a ˇ h (S) , hk−1 (S) k−1 r   S



S ◦ −5 S

C3 a + 1 + 2m + r−1 a . r ˇ h (S) , S hk−4 (S) k−4 r where aS was defined in Step 1 and can be expressed in terms of ς S and AS by

GCM PROCEDURE

575

formula (9.8.25). Proof. The proof28 of the first two identities in (9.8.31) was derived in Proposition 7.21 in connection to the proof29 of Theorem M4, starting with the following:30   1 S 1 S S S −1 S S 2 /d1S,? ω S = κ + 2ω S η S + eS Γg + Γ S 3 (ζ ) − β + κξ + r b · Γb , 2 2   −2 S S 2 /d2S /d2S,? η S = κS −e3 (ζ S ) + β S − eS /S )≤1 ΓS 3 (eθ (κ )) + rS ( d g (9.8.35) −1 S S + rS d/(Γb · Γb ),   −2 S S 2 /d2S /d2S,? ξ S = κS e3 (ζ S ) − β S − eS /S )≤1 ΓS 3 (eθ (κ )) + rS ( d g S + rS−1 d/S (ΓS b · Γb ).

The last identity in (9.8.31) follows by combining the first two identities in (9.8.35). To prove the estimates for η S in the second part of the proposition we make use of the identity /d1S,? /d1S = /d2S /d2S,? + 2K S to deduce /d2S,? ( /d2S /d2S,? + 2K S ) /d2S /d2S,? η S 1 S = κ C1 + κS /d2S,? ( /d2S /d2S,? + 2K S )β S − rS−3 ( d/S )3 C2 + rS−5 ( d/S )≤4 ΓS g 2 −4 S ≤4 S S +rS ( d/ ) (Γb · Γb ) + l.o.t., i.e.,   /d2S,? ( /d2S /d2S,? + 2K S ) /d2S /d2S,? η S − κS β S =

1 S 1 −4 S κ C1 − rS−3 ( d/S )3 C2 + rS−5 ( d/S )≤4 ΓS /S )≤4 (ΓS g + rS ( d b · Γb ) + l.o.t. 2 2

Similarly for ξ S   /d2S,? ( /d2S /d2S,? + 2K S ) /d2S /d2S,? ξ S + κS β S =

1 1 −4 S C3 − κS C1 + rS−5 ( d/S )≤4 ΓS /S )≤4 (ΓS g + rS ( d b · Γb ) + l.o.t. 2 2

The desired estimates for η S and ξ S follow then by making use of the coercivity of the operator /d2S,? ( /d2S /d2S,? + 2K S ) and the estimate for β = β S in (9.4.11). The estimate for /d1S,? ω S is straightforward from the last identity in (9.8.35). To prove the last part of the proposition we make use of the GCM conditions 28 The equations used in the derivation of these identities only require the transversality conditions (9.4.9). 29 Strictly speaking Proposition 7.21 requires the e Ricci and Bianchi identities of a geodesic 3 foliation. It is easy to justify the application of these equations in our context by using the transversality conditions to generate a geodesic foliation in a neighborhood of Σ0 . 30 These identities were recorded in Proposition 7.12. Note also that / S d(ΓS b ·Γb ) does not contain derivatives of ω ˇ.

576

CHAPTER 9

(9.8.16) on Σ0 to deduce that ν S ( /d2S,? /d1S,? µS ) = 0,

S ν S (eS θ κ ) = 0,

  ν S ( /d2S,? /d2S + 2K S ) /d2S,? /d1S,? κS = 0.

Hence, the quantities C1 , C2 , C3 in (9.8.32) can be expressed in the form
−aS eS d2S,? /d1S,? µS , 4 /

C1

=

C2

= −aS eS (eS κS ), 4 θ  S,? S S,? S,? S S = a S eS ( / d / d + 2K ) / d / d κ . 4 2 2 2 1

C3

Making use of our commutation formulas of Lemma 2.68 and the estimates (9.4.11) and (9.4.8) we easily deduce
keS d2S,? /d1S,? µS khk−2 (S) 4 / S S keS 4 (eθ κ )khk−1 (S)

. .

◦ −5

r

◦ −3

r

, .

Similarly,

 

S

S,? S,? S,?

e4 ( /d2 /d2S + 2K S ) /d2 /d1 κS

hk−4 (S)

◦

. r−5 .

Writing aS = aS + a ˇS and making use of product estimates we deduce  



2mS ◦ −5 S −1 S

C1 . r a +1+ S +r a ˇ h (S) , hk−2 (S) k−2 r   S



S ◦ −3 S

C2 a + 1 + 2m + r−1 a

. r ˇ , hk−1 (S) hk−1 (S) rS   S



S ◦ −5 S

C3 a + 1 + 2m + r−1 a

. r ˇ , hk−4 (S) hk−4 (S) rS as stated. Step 10. Propositions 9.55 and 9.57 provide us with potential31 estimates for /d2S,? η S , /d2S,? ξ S , /d1S,? ω S , /d1S,? ς S . To close we also need to control the ` = 1 modes of η S , ξ S the average of ω S and the average32 of aS . Note that the average of ω S can in fact be derived from the equation 1 S S S S S eS 3 (κ ) + κ κ − 2ω κ 2 31 We

= 2 /d1S η S + 2ρS − 12 ϑS ϑS + 2(η S )2

cannot close the estimates without also being able to estimate the ` = 1 modes of and the average aS . 32 The quantity a ˇS can be determined using Proposition 9.55.

ηS , ξS , ωS

577

GCM PROCEDURE

in terms of AS and η S . Indeed, making use of the GCM condition κS = r2S ,   1 1 S S 1 S S S S S S S S S 2 ω = e (κ ) + κ κ − 2 /d1 η − 2ρ + ϑ ϑ − 2(η ) 2κS 3 2 2   S S 1 Υ r 1 = − e3 (rS ) − S + −2 /d1S η S − 2ρS + ϑS ϑS − 2(η S )2 2 2r 4 2   S 1 r 1 = − AS + −2 /d1S η S − 2ρS + ϑS ϑS − 2(η S )2 . 4 4 2 Thus, recalling the definition of µS , ωS

1 rS S = − AS + µ − (η S )2 4 2

or ωS

mS − S 2 (r )

! 1 S rS mS S S S = − A + µ − S 3 −η ·η . 4 2 (r )

(9.8.36)

Step 11. In view of the above we can determine η S , ξ S , ω S , ς S , AS provided that we control the ` = 1 modes of η S , ξ S and the average of ς S . For this reason we introduce,33 along Σ0 , Z Z 2mS BS = η S eΦ , BS = ξ S eΦ , DS = aS + 1 + S . (9.8.37) r SP S S We are now ready to prove the following: Proposition 9.58. Let Σ0 be a smooth spacelike hypersurface foliated by framed S S spheres (S, eS 4 , eθ .e3 ) which verify the GCM conditions (9.8.16), transversality condition (9.4.9) and the estimates (9.4.8)–(9.4.11) of Theorem 9.32. Let uS as in (9.8.19) such that uS + rS is constant on Σ0 . Extend uS and rS in a neighborhood of Σ0 such that the transversality conditions (9.8.20) are verified. As shown above these allow us to define η S , ξ S , ω S , ς S , AS , aS and the constants B S , B S , DS as in (9.8.37). Finally we assume that
◦1/2 r−2 |B S | + |B S | + |DS | ≤  .

(9.8.38)

Under these assumptions the following estimates hold true for all k ≤ smax − 7: 33 Note that to prove our main theorem we have to construct our hypersurface Σ such that in 0 fact B = B = D = 0.

578

CHAPTER 9

1. The Ricci coefficients η S , ξ S , ω S verify

S ◦

η .  + rS−2 |B S |, h5+k (S)

S ◦

ξ .  + rS−2 |B S |, h5+k (S)  

S ◦

ω ˇ h (S) .  + rS−2 |B S | + |B S | , 3+k   mS ◦ S ω − S 2 .  + rS−2 |B S | + |B S | . (r )

(9.8.39)

2. The scalar aS verifies 2mS S + + 1 + a k+1 (S) rS

S rS−1 a ˇ h

◦

 + rS−2 |B S | + |DS |.

.

(9.8.40)

3. We also have  

S ◦ S −2 S

A .  + r |B | + |B | + |DS |, S hk+1 (S)   ◦ rS−1 kˇ ς S khk+1 (S) + ς S − 1 .  + rS−2 |B S | + |B S | + |DS |.

(9.8.41)

4. We also have, for all k ≤ smax − 4, ◦

keS κS , ϑS , ζ S , κ ˇ S )khk (S) . rS−1 , 3 (ˇ ◦

keS 3

S keS 3 (ϑ )khk (S) . ,
◦ αS , β S , ρˇS , µS khk (S) . rS−2 , ◦

S −1 keS 3 (β )khk (S) . rS , ◦

S keS 3 (α )khk (S) . .

Proof. To simplify the exposition below we make the auxiliary bootstrap assumptions,

S

η h

5+k (S)

◦1/2

+ ξ S h

5+k (S)

.

.

(9.8.42)

We start with the following lemma. Lemma 9.59. The following estimates hold true: 2mS ◦ + aS + 1 + S . DS + kη S khk (S) + kξ S khk (S) + . (9.8.43) r Proof. Since aS = aS + a ˇS we deduce aS SP = aS + a ˇS SP . Hence,

S rS−1 a ˇ h

k+1 (S)

aS

= DS − 1 −

2mS −a ˇS SP . rS

We also have (see Proposition 9.55) aS = −

2 rS S S + Υ − A . ςS 2

(9.8.44)

579

GCM PROCEDURE

Hence, S

a

2 rS S 2 S = − S + Υ − A =− S S 2 ς + ςˇ ς

ςˇS 1− S +O ς



ςˇS ςS

2 !

+ ΥS −

rS S A . 2

Taking the average on S we deduce aS

= −

2 ςS

+ ΥS −

rS S A +O 2



ςˇS ςS

2 .

(9.8.45)

Also, using (9.8.44), a ˇS = 2ˇ ςS −

rS ˇ S A + l.o.t. 2

(9.8.46)

where l.o.t. denotes higher order terms in ςˇS and ς S − 1. Indeed   2 rS 2 rS S a ˇS = aS − aS = − S + ΥS − AS − − + ΥS − A ς 2 2 ςS S S S 2 2 r ˇS 2ˇ ς r ˇS rS ˇ S = − S+ − A = − A = 2ˇ ςS − A + l.o.t. ς 2 2 2 ςS ς Sς S Thus to estimate a ˇS and aS we first need to estimate AS , ςˇS and ς S . Using the equations (see Proposition 9.55) S eS θ (A )

AS

= − =

1

2ΥS S 2 (ζ − η S ) − S ξ S + (ζ S − η S )AS , S r r

ςS

ˇ ςˇS κ ˇ S − ςˇS A

S

,

and the auxiliary assumption we derive

S

A h

k+1 (S)

  ◦ . kη S khk (S) + kξ S khk (S) + r−1 1 + kˇ ς S khk (S) + ς S − 1 .(9.8.47)

From the equation S eS θ (log ς )

= (η S − ζ S ),

we also derive ◦ ◦ rS−1 kˇ ς S khk+1 (S) . kη S khk (S) +  +  ς S − 1 .

(9.8.48)

To estimate ς S − 1 we derive from (9.8.45) and (9.8.46)   2 rS S 2mS rS S S S S S = −a + Υ − A = − D − 1 − S − a ˇ SP + ΥS − A + l.o.t. 2 r 2 ςS S r S = −DS + 2 + a ˇS SP − A + l.o.t. 2 rS  S ˇ S  = −DS + 2 + 2ˇ ς S SP − A + A SP + l.o.t. 2

580

CHAPTER 9

and therefore, rS  S ˇ S  = −DS + 2ˇ ς S SP − A + A SP + l.o.t., 2

2(1 − ς S ) ςS

i.e., ςS − 1

=

1 S rS  S ˇ S  D − ςˇS SP + A + A SP + l.o.t. 2 4

where l.o.t. denote higher order terms in ςˇS and ς S − 1. Thus, ς S − 1 . |DS | + kˇ ς S kL∞ (S) + rS kAS kL∞ (S) . Hence, back to (9.8.48) we derive ◦ rS−1 kˇ ς S khk+1 (S) + ς S − 1 . |DS | + rS kAS kL∞ (S) + rkη S khk (S) + . Combining with (9.8.47) we deduce

S ◦

A . kη S khk (S) + kξ S khk (S) + , hk+1 (S) ◦ rS−1 kˇ ς S khk+1 (S) + ς S − 1 . rkη S khk (S) + . In view of (9.8.46) we also deduce

S rS−1 a ˇ h (S) . k+1

.

rS−1 kˇ ς S khk+1 (S) + AS h

(9.8.49)

k+1 (S)

S

S

◦

kη khk (S) + kξ khk (S) + .

From (9.8.44) we further deduce 2mS S a + 1 + S . r

S ◦ D + kˇ aS kL∞ (S) . DS + kη S khk (S) + kξ S khk (S) + .

Hence, 2mS ◦ S + + 1 + a . DS + kη S khk (S) + kξ S khk (S) +  S k+1 (S) r

S rS−1 a ˇ h as stated.

◦1/2

In view of the lemma above and the assumption |DS | .  the estimates (9.8.34) become  

◦1/2 ◦ −4 S S

C1 , . rS kη khk (S) + kξ khk (S) +  hk (S)  

◦ −2 ◦1/2 S S

C2 . r kη k + kξ k +  , (9.8.50) hk+3 (S) hk+3 (S) S hk+3 (S)  

◦ −4 ◦1/2 S S

C3 . r kη k + kξ k +  . hk (S) hk (S) S h (S) k

581

GCM PROCEDURE

To prove the desired estimate for η S , ξ S , ω S we make use of (9.8.33) and the following lemma. Lemma 9.60. The error term S S = rS−1 kΓS g kh4+k (S) + kΓb · Γb kh4+k (S) ,

Ek

k ≤ smax − 7,

appearing in (9.8.33) verifies the estimate Ek

◦1/2

◦

. rS−1  + rS−1 

 

(η S , ξ S )kh (S) + kˇ ω S khk+3 (S) . 4+k

Proof. Since ΓS g contains only terms estimated by (9.4.8), kΓS g kh4+k (S)

.

◦

rS−1 , S

S S S ΓS ˇ S , ω S − (rmS )2 . b contains ϑ , which is estimated by (9.4.8), as well as η , ξ , ω

S S 1/2 ◦ Thus, in view of the auxiliary estimates η , ξ h (S) .  and the fact that 5+k

the quadratic error terms contain one less derivative of ω ˇ S , we deduce kΓS b

·

ΓS b kh4+k (S)

.

◦ rS−1 

!

S S mS S S S

η , ξ kh (S) + kˇ ω khk+3 (S) + r ω − S 2 . 4+k (r )

1/2

In view of equation (9.8.36), ω S − S ω S − m . (rS )2 . .

mS (r S )2

= − 14 AS +

rS 2



µS −

mS (r S )3

 − ηS · ηS ,

S S A + rS µ − m + |η S |2 (rS )3   ◦ ◦ ◦1/2 rS−1 |DS | + r−1  kη S kh2 (S) + kξ S kh2 (S) + r−2   ◦1/2 ◦ rS−1  kη S kh2 (S) + kξ S kh2 (S) +  .

Hence, kΓS b

·

ΓS b kh4+k (S)

.

◦ rS−1 

1/2

S S ◦

η , ξ kh (S) + kˇ ω S khk+3 (S) +  4+k

!

and Ek

=

S S rS−1 kΓS g kh4+k (S) + kΓb · Γb kh4+k (S)

.

◦ rS−1 

+

◦ rS−1 

1/2

!

S S

η , ξ kh (S) + kˇ ω S khk+3 (S) 4+k

as stated. In view of the lemma and estimates (9.8.50) for C1 , C2 , C3 , the estimates (9.8.33)

582

CHAPTER 9

of Proposition 9.57 become k /d2S,? η S kh4+k (S) k /d2S,? ξ S kh4+k (S) k /d1S,? ω S kh2+k (S)

+

◦ rS−1 

1/2

+

◦ rS−1 

1/2

.

◦ rS−1 

.

◦ rS−1 

.

rS−1 kη S kh4+k (S)

+

◦ rS−1 

+

◦ rS−1 

1/2

!

S S S

η , ξ kh (S) + kˇ ω khk+3 (S) , 4+k !

S S S

η , ξ kh (S) + kˇ ω khk+3 (S) , 4+k +

(9.8.51)

rS−1 kξ S kh2+k (S)

!

S S S

η , ξ kh (S) + kˇ ω kh2+k (S) . 3+k

From the last equation we derive kˇ ω S kh3+k (S)

◦

. kη S kh4+k (S) + kξ S kh2+k (S) + .

Thus the first two equations in (9.8.51) become ◦

◦1/2

◦

◦1/2

rS k /d2S,? η S kh4+k (S) .  +  rS k /d2S,? ξ S kh4+k (S) .  +  from which we deduce

S

η h5+k (S)

S

ξ h5+k (S)

S

ω ˇ h

3+k (S)

. . .


kη S kh4+k (S) + kξ S kh4+k (S) ,
kη S kh4+k (S) + kξ S kh4+k (S) ,

(9.8.52)

◦

 + rS−2 |B S |,

◦

 + rS−2 |B S |,   ◦  + rS−2 |B S | + |B S | ,

as stated. We can then go back to the preliminary estimates obtained above for ς S , AS and aS to derive the remaining statements (1–4) of Proposition 9.58. To prove the last part of the proposition we make use of the corresponding Ricci and Bianchi equations in the eS 3 direction. Corollary 9.61. Under the same assumptions as in the proposition above we have R the more precise estimates, with d(S) = S e2Φ ,

S

◦

η − 1 B S eΦ . ,

d(S) h (S)

5+k

S

◦

ξ − 1 B S eΦ . .

d(S) h5+k (S) Note also that d(S)

=

(rS )4



 8π ◦ + O() . 3

583

GCM PROCEDURE

Proof. In view of (9.8.52), (9.8.39) and auxiliary assumption (9.8.38), we deduce

R S Φ!

η e

S

. rk /d2S,? η S kh4+k (S)

η − RS 2Φ eΦ

e S h5+k (S)

◦1/2

◦

kη S kh4+k (S) + kξ S kh4+k (S)    ◦ ◦1/2 ◦ . +  + r−2 |B S | + |B S | .

.

+




◦

.

We deduce

S

η − B S R 1 eΦ

e2Φ h S

. .



S

ξ − B S R 1 eΦ

2Φ e S h

.

◦

5+k (S)

Similarly, ◦

5+k (S)

as desired. 9.8.3

Construction of Σ0

To construct the spacelike hypersurface of Theorem 9.52 we proceed as follows. Step 12. Let Ψ(s), Λ(s), Λ(s) real valued functions that will be carefully chosen later. We look for the hypersurface Σ0 in the form [ [ Σ0 = S[P (s)] = S[Ψ(s), s, Λ(s), Λ(s)] (9.8.53) ◦

◦

s≥s

s≥s

where P (s) is a curve in the parameter space P given by P (s) = (Ψ(s), s, Λ(s), Λ(s)).

(9.8.54)

◦ ◦

In order for Σ0 to start at S0 = S[u, s, Λ0 , Λ0 ] we impose the conditions ◦

◦

Ψ(s) = u,

◦

Λ(s) = Λ0 ,

◦

Λ(s) = Λ0 .

(9.8.55)

Step 13. We expect Σ0 to be a perturbation of the spacelike hypersurface u+s = c0 for some constant c0 . We thus introduce the notation ψ(s) := Ψ(s) + s − c0 , so that Ψ(s) = −s + c0 + ψ(s) ◦

and expect ψ(s) = O(δ). Step 14. In view of (9.8.18) we can express the collection of spheres Σ0 in the

584

CHAPTER 9

form n Σ0 = Ξ(s, θ),

◦

s ≥ s, θ ∈ [0, π]

o

(9.8.56)

where the map Ξ(s, θ) = Ξ(Ψ(s), s, θ) is defined as   Ξ(s, θ) := Ψ(s) + U (θ, P (s)), s + S(θ, P (s)), θ .

(9.8.57)

At the south pole, i.e., θ = 0, where U (0, P ) = S(0, P ) = 0,   Ξ(s, 0) = Ψ(s), s, 0 .

(9.8.58)

Clearly, ∂s Ξ(s, θ)

=

∂θ Ξ(s, θ)

=



 Ψ0 (s) + ∂P U (θ, P (s))P 0 (s), 1 + ∂P S(θ, P (s))P 0 (s), 0 ,   ∂θ U (θ, P (s)), ∂θ S(θ, P (s)), 1 ,

where ∂P U (·)P 0 (s)

=

Ψ0 (s)∂u U (·) + ∂s U (·) + Λ0 (s)∂Λ U (·) + Λ0 (s)∂Λ U (·),

∂P S(·)P 0 (s)

=

Ψ0 (s)∂u S(·) + ∂s S(·) + Λ0 (s)∂Λ S(·) + Λ0 (s)∂Λ S(·).

Given f a function on Σ0 we have
d f Ξ(s, θ) ds


d f Ξ(s, θ) dθ

=



 Ψ0 (s) + ∂P U (θ, P (s))P 0 (s) ∂u f   + 1 + ∂P S(θ, P (s))P 0 (s) ∂s f

=

X∗ f,

=

∂θ U (θ, P (s))∂s f + ∂θ S(θ, P (s))∂s + ∂θ f

=

Y∗ f,

where X∗ , Y∗ are the following tangent vectorfields along Σ0 :     X∗ (s, θ) : = Ψ0 (s) + ∂P U (θ, P (s))P 0 (s) ∂u + 1 + ∂P S(θ, P (s))P 0 (s) ∂s , (9.8.59) Y∗ (s, θ) : = ∂θ U (θ, P (s))∂s + ∂θ S(θ, P (s))∂s + ∂θ , or     ˘ θ) ∂u + 1 + B(s, ˘ θ)P 0 (s) ∂s , X∗ (s, θ) : = Ψ0 (s) + A(s, ˘ θ)∂u + D(s, ˘ θ)∂s + ∂θ , Y∗ (s, θ) : = C(s,

(9.8.60)

585

GCM PROCEDURE

where ˘ θ) : A(s,

= ∂P U (θ, P (s))P 0 (s) =

∂u U (θ, P (s))Ψ0 (s) + ∂s U (θ, P (s)) + ∂Λ U (θ, P (s))Λ0 (s) +∂Λ U (θ, P (s))Λ0 (s),

˘ θ) : B(s,

= ∂P S(θ, P (s))P 0 (s) =

∂u S(θ, P (s))Ψ0 (s) + ∂s S(θ, P (s)) + ∂Λ U (θ, P (s))Λ0 (s) +∂Λ S(θ, P (s))Λ0 (s),

˘ θ) : C(s, ˘ D(s, θ) :

= ∂θ U (θ, P (s)), = ∂θ S(θ, P (s)).

Step 15. Define the vectorfield, along the south pole of each S ⊂ Σ0 , X∗

h= SP


d h Ξ(s, 0) . ds

(9.8.61)

S S Lemma 9.62. At the south pole we have the relations (recall ν S = eS 3 + a e4 )

1 X∗ SP = ςΨ0 ν S , 2λ SP

aS SP

=

(9.8.62)


2λ2 1 1 − Ψ0 (s)ςΩ |SP , 0 Ψ (s)ς 2

(9.8.63)

or, more precisely, 1 2λ2 a (Ψ(s), s, 0) = 0 Ψ (s) ς S



 1 0 1 − Ψ (s)ςΩ (Ψ(s), s, 0). 2

Here f, f , λ are the transition functions and ς, Ω correspond to the background foliation. Proof. Note that ˘ 0) = B(s, ˘ 0) = C(s, ˘ 0) = D(s, ˘ 0) = 0. A(s, Thus, at the south pole SP, X∗ (s, 0) = Ψ0 (s)∂u + ∂s . Recall that ∂s = e4 ,

∂u =

 1  ς e3 − Ωe4 − bγ 1/2 eθ , 2

∂θ =

√

γeθ ,

or, since b vanishes at the south pole, X∗ (s, 0)

=

1 1 1 Ψ0 ς (e3 − Ωe4 ) + e4 = (1 − Ψ0 (s)ςΩ)e4 + Ψ0 (s)ςe3 . 2 2 2

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CHAPTER 9

On the other hand, since the transition functions f, f vanish at the south pole, −1 eS e3 . 3 =λ

eS 4 = λe4 , Hence, X∗ (s, 0)


1 1 −1 0 λ 1 − Ψ0 (s)ςΩ eS Ψ (s)ςeS 4 + λ 3 2 2   2
S 1 0 1 −1 0 2λ S λ Ψ (s)ς e3 + 0 1 − Ψ (s)ςΩ e4 . 2 Ψ (s)ς 2

= =

In view of the definition of ν S we deduce X∗ (s, 0)

=

1 −1 0 λ Ψ (s)ς ν S SP 2

and34 S

a (s, 0)

=

2λ2 Ψ0 (s)ς



1 1 − Ψ0 (s)ςΩ 2



as stated. Step 16. The transition functions (f, f , λ) are uniquely determined on S by the results of Theorem 9.32 in terms of Λ, Λ. The same holds true for all curvature components and the Ricci coefficients κS , ϑS , ζ S , κS , ϑS . One can easily see from the transformation formulas that the values of the eS 3 derivatives of (f, f , λ) are determined by the transversal Ricci coefficients η S , ξ S , ω S . Indeed, schematically, from the transformation formulas for η, ξ, ω in Proposition 9.20, 1 S eS 3 f = 2(η − η) − κf + f ω + F · Γb + l.o.t., 2 1 S S e3 f = 2(ξ − ξ) − f κ + 4ω) + F · Γb + l.o.t., 2 S eS (log λ) = 2(ω − ω) + Γb · F + l.o.t., 3

(9.8.64)

where F = (f, f , log λ) and l.o.t. denotes terms which are linear in Γg , Γb and linear and higher order in F . Recall also that the eS 4 derivatives of F are fixed by our transversality condition (9.4.9) More precisely we have eS 4 (f )

1 = − κf + l.o.t., 2

eS 4 (f )

=

2eS θ (log λ)

eS 4 (log λ)

=

l.o.t.

  1 − f κ + 2 ω + κ f + l.o.t., 4

It follows that η S , ξ S , ω S can be determined by ν S (f, f , λ) and the scalar aS . More 34 Note

that aS (s, 0) = aS (Ξ(s, 0)).

587

GCM PROCEDURE

precisely, 1 ν S (f ) = 2(η S − η) − (κf + aS κf ) + f ω + F · Γb + l.o.t., 2
1 S S ν (f ) = 2(ξ − ξ) − κ + 4ω)(f − aS f ) + aS 2eS θ (log λ) − f κ 2 + F · Γb + l.o.t.

(9.8.65)

Step 17. We derive equations for Λ(s) = Λ(Ψ(s), s, 0), Λ(s) = Λ(Ψ(s), s, 0) as follows. Lemma 9.63. We have the following identities Z 6 1 0 c(s) 0 Λ (s) = ν S (f )eΦ − S Λ(s) + E(s), Ψ (s) r ZS 1 6 0 c(s) 0 Λ (s) = ν S (f )eΦ − S Λ(s) + E(s), Ψ (s) r S(s)

(9.8.66)

where c(s)

=

 2λ   2λ  (Ψ(s), s, 0) (s) = ς SP ς

and error terms    3 2mS S E(s) = 3ˇ κ −ϑ −a ϑ + S a + 1+ S f eΦ r r S(s)  Z   6 + (ς S )−1 ς S − ς S ν S (f ) − S Λ(s) eΦ + l.o.t., r SP S(s) SP     Z 1 3 2mS S S S S S E(s) = 3ˇ κ −ϑ −a ϑ + S a + 1+ S f eΦ 2 S(s) r r  Z   6 S −1 S S S + (ς ) ς −ς ν (f ) − S Λ(s) eΦ + l.o.t. r SP S(s) SP 1 2



Z

S

S

S S

Proof. According to Lemma 9.54 we have Z  Z
S ν h = (ς S )−1 ς S ν S (h) + (κS + aS κS )h . S

S

Thus, applying the vectorfield ν S SP =



2λ ςΨ0 X∗ SP

to the formulas (9.8.17),

 1  2λ  d Λ(s) = ν S (Λ) = ν S (Λ) SP = ν S 0 Ψ (s) ς SP ds SP Z   = (ς S )−1 ς S ν S (f eΦ ) + (κS + aS κS )f eΦ . SP

Z S

 f eΦ

SP

S(s)

Introducing J(f ) = e−Φ ν S (f eΦ ) + (κS + aS κS )f

(9.8.67)

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CHAPTER 9

we deduce 1 c(s) 0 Ψ (s)

=



Z

S −1

(ς ) Z

SP

ς S J(f )eΦ

S(s) Φ

=



Z

S −1

J(f )e + (ς ) S(s)

SP

S(s)



ς S − ς S



J(f ).

SP

On the other hand, since e3 Φ = 12 (κ − ϑ), e4 Φ = 12 (κ − ϑ),
S S S S S = ν S (f ) + eS f 3 Φ + a e4 Φ + κ + a κ   1 = ν S (f ) + 3κS − ϑS + aS (3κS − ϑS ) f 2 
1 S 3 S = ν S (f ) + κ + aS κS − ϑ + aS ϑS . 2 2

J(f )

Since κS = J(f )

2 rS

S

and κS = κS + κ ˇ S = − 2Υ +κ ˇ S , we deduce rS


3 1 S S S S S S −Υ + a f + 3ˇ κ − ϑ − a ϑ f rS 2   3 2mS = ν S (f ) + S −ΥS − (1 + S ) f r r     1 3 2mS S S S S S + 3ˇ κ −ϑ −a ϑ + S a + 1+ S f 2 r r     6 1 3 2mS S S S S S S = ν (f ) − S Λ(s) + 3ˇ κ −ϑ −a ϑ + S a + 1+ S f. r 2 r r = ν S (f ) +

We deduce Z

1 c(s) 0 Ψ (s)

= S

ν S (f )eΦ −

6 Λ(s) + E(s) rS

where E(s)

= +

   3 2mS 3ˇ κS − ϑS − aS ϑS + S aS + 1 + S f eΦ r r S(s) Z   (ς S )−1 ς S − ς S J(f ) 1 2

Z



SP

= +

SP

S(s)

   3 2mS S 3ˇ κ −ϑ −a ϑ + S a + 1+ S f eΦ r r S(s)  Z   6 (ς S )−1 ς S − ς S ν S (f ) − S Λ(s) eΦ + l.o.t. r SP S(s) SP 1 2

Z



S

S

S S

The proof for Λ is exactly the same. Step 18. We make use of the estimates for F = (f, f , log λ) and eS 4 (F ) derived in S S S S S Theorem 9.32 as well as the estimates for a , ς , η , ξ , ω derived in Proposition 9.58 to evaluate the right-hand sides of (9.8.66). Recall that in Proposition 9.58 we

589

GCM PROCEDURE

have made the auxiliary assumption (9.8.38), i.e.,
◦1/2 rS−2 |B S | + |B S | + |DS | ≤  . Proposition 9.64. The following equations hold true for the functions35 Λ(s)

=

Λ(Ψ(s), s, 0),

Λ(s) = Λ(Ψ(s), s, 0),

B(s)

=

Λ(Ψ(s), s, 0),

B(s) = Λ(Ψ(s), s, 0),

r(s) = r(Ψ(s), s, 0),

1 1 7 Λ0 (s) = B(s) − r(s)−1 Λ(s) − r(s)−1 Λ(s) + O(r−1 )Λ(s) 0 −1 + ψ (s) 2 2 + N (B, B, D, Λ, Λ, ψ)(s), (9.8.68) 1 7 1 0 −1 −1 Λ (s) = B(s) − r(s) Λ(s) + r(s) Λ(s) −1 + ψ 0 (s) 2 2
−1 + O(r ) Λ(s) + Λ(s) + N (B, B, D, Λ, Λ, ψ)(s). The expressions N, N verify the following properties. • They depend on B, B, D, Λ, Λ, ψ, F = (f, f , λ − 1), the background Ricci coeffiˇ = {α, β, ρˇ, β, α}. cients Γb , Γg and curvature R • N, N vanish at (B, B, D, Λ, Λ, ψ) = (0, 0, 0, 0, 0, 0). In fact, ◦

|N, N | . r2 δ. ◦

• The linear part in B, B, D has O() coefficients, i.e., coefficients which depend ˇ F and Λ, Λ, ψ. on the quantities Γb , Γg , R, ◦ • The linear part in Λ, Λ, ψ has O() coefficients. Proof. To prove the desired result we make use of (9.8.65) to check the following: Z ◦ ν S (f )eΦ = 2B(s) − r−1 Λ(s) − r−1 Λ(s) + O(r−1 )Λ(s) + O(r2 δ), S(s) Z (9.8.69) ◦
ν S (f )eΦ = 2B(s) − r−1 Λ(s) + r−1 Λ(s) + O(r−1 ) Λ(s) + Λ(s) + O(r2 δ). S(s)

Combining this with (9.8.66), 1 Λ0 (s) 0 Ψ (s) 1 c(s) 0 Λ0 (s) Ψ (s) c(s)

Z

6 Λ(s) + E(s), rS ZS 6 = ν S (f )eΦ − S Λ(s) + E(s), r S(s) =

ν S (f )eΦ −

and the following estimates for the error terms E, E, |E(s)| + |E(s)| . 35 Note

also that rS(s) = rS(s) = r|SP (S(s)) = r(s).

◦

r2 δ,

(9.8.70)

590

CHAPTER 9

we deduce 1 1 Λ0 (s) = Ψ0 (s) c(s) 1 Ψ0 (s)

Λ0 (s) =

  ◦ −1 −1 −1 2 2B(s) − r Λ(s) − 7r Λ(s) + O(r )Λ(s) + O(r δ) ,


1 2B(s) − 7r−1 Λ(s) + r−1 Λ(s) + O(r−1 ) Λ(s) + Λ(s) c(s) ! ◦

+ O(r2 δ) . ◦

According to our assumptions ς = 1 + O(). Also according to Theorem 9.32 ◦ λ = 1 + O(r−1 ). Thus, ◦  2λ  2(1 + O(r−1 )) ◦ c(s) = = 2 + O(). (s) = ◦ ς SP 1 + O()

Hence, 1 Λ0 (s) Ψ0 (s)

◦  1 2B(s) − r−1 Λ(s) − 7r−1 Λ(s) + O(r−1 )Λ(s) + O(r2 δ) 2 ◦ 1 7 = B(s) − r−1 Λ(s) − r−1 Λ(s) + O(r−1 )Λ(s) + O(r2 δ). 2 2

=

Setting Ψ(s) = −s + ψ(s) + c0 and recalling the structure of the error terms we ◦

have denoted by O(r2 δ)

1 Λ0 (s) −1 + ψ 0 (s)

1 7 = B(s) − r−1 Λ(s) − r−1 Λ(s) + O(r−1 )Λ(s) 2 2 + N (B, B, D, Λ, Λ, ψ)(s)

where N verifies the properties mentioned in the proposition. In the same manner we derive 1 Λ0 (s) −1 + ψ 0 (s)


7 1 = B(s) − r−1 Λ(s) + r−1 Λ(s) + O(r−1 ) Λ(s) + Λ(s) 2 2 + N (B, B, D, Λ, Λ, ψ)(s)

as stated in the proposition. It remains to check (9.8.69) and (9.8.70). According to (9.8.65) and our assumptions on the Ricci coefficients κ, κ, ω, we have along the sphere S   1 2 m S S S 2Υ ν (f ) = 2(η − η) − f −a f + f 2 + F · Γb + l.o.t. 2 r r r    m 2m = 2(η S − η) − r−1 f + r−1 + aS 1 − f + F · Γ. r r According to (9.8.43) and auxiliary assumption (9.8.38)     S S 1/2 a + 1 + 2m . ◦ + r−2 |B S | + |B S | + |DS | . ◦ . rS

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GCM PROCEDURE

Thus, ν S (f )

    ◦ ◦1/2 m m2 2(η S − η) − r−1 f − r−1 1 − − 2 f + r−2 O δ  . r r

=

◦

Since r and rS are comparable along S, i.e., |r − rS | ≤ δ, we deduce, recalling the definition of B,   Z Z m m2 S Φ Φ −1 −1 ν (f )e = 2B(s) − 2 ηe − r Λ(s) − r 1− − 2 Λ(s) r r S(s) S(s)   ◦ ◦1/2 +rO δ  . Making use of the assumption (9.8.1) for η as well as Corollary 9.10 we easily deduce Z ◦ ηeΦ . r2 δ. S(s) Hence, Z

ν S (f )eΦ

=

◦
2B(s) − r−1 Λ(s) − r−1 1 + O(r−1 ) Λ(s) + O(r2 δ)

=

2B(s) − r−1 Λ(s) − r−1 Λ(s) + O(r−1 )Λ(s) + O(r2 δ).

S(s)

◦

Similarly, starting with ν S (f )

= 2(ξ S − ξ) −

we deduce Z

S

ν (f )e S(s)

Φ

1 2


κ + 4ω)(f − aS f ) + aS 2eS θ (log λ) − f κ + F · Γb + l.o.t.

=

2B(s) − 2

Z

Φ

ξe + r

−1

S(s)



8m 1+ r

 Λ(s)

  2m 8m2 + r−1 1 − − 2 Λ(s) r r  Z   ◦ ◦1/2 2m Φ − 2 1+ eS (log λ)e + rO δ  . θ r S(s) Making use of the assumption (9.8.1) for ξ, as well as Corollary 9.10, Z ◦ Φ ξe . r2 δ. S(s) Also, in view of the estimates of Theorem 9.32, Z ◦ Φ 2 eS θ (log λ)e . r δ. S(s)

(9.8.71)

592

CHAPTER 9

We deduce Z ◦
ν S (f )eΦ = 2B(s) − r−1 (1 + O(r−1 ))Λ(s) + r−1 1 + O(r−1 ) Λ(s) + O(r2 δ) S(s)

as stated. The estimates for E, E in (9.8.70) can also be easily checked. This ends the proof of Proposition 9.64. Step 19. We derive an equation for ψ. The main result is stated in the proposition below. Proposition 9.65. The function ψ(s) = Ψ(s) + s − c0 defined in Step 13 verifies the following equation: ψ 0 (s)

= − 12 D(s) + O(D(s)2 ) + M (s)

(9.8.72)

where M (s) is a function which depends only on Γ, R of the background foliation, ψ and (f, f , λ − 1), such that M (s)

◦

. δr(s)−1 .

Proof. In view of (9.8.63) and the definition of c(s) = 0

Ψ (s)

2λ2 · ς aS SP (s) 1

=





2λ ς

1 1 − Ψ0 ςΩ 2





(s) we have SP

(s) SP

or 

0

Ψ (s)

=

2λ2 1 ς aS + λ2 Ω

=

 2λ2 1 . ς aS + λ2 Ω SP

Now, we have 2 1 + O(λ − 1) ς aS + Ω aS + 2ς + Ω = −1 + + O(λ − 1). aS + Ω

Hence, "

0

ψ (s)

=

=

# aS + 2ς + Ω Ψ (s) + 1 = + O(λ − 1) aS + Ω SP " # 2 S ◦ a + ς +Ω + O(r−1 δ). S a +Ω SP 0

We have, see (9.8.37), aS SP (s)

= D(s) − 1 −

2mS . rS

(9.8.73)

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GCM PROCEDURE

Hence,
aS + Ω SP (s)

= = =

2mS + Ω SP (s) S r   2mS 2m ◦ D(s) − 1 − S − 1 − + O() r r D(s) − 1 −

◦

D(s) − 2 + O().

In view of the assumption (9.8.2)   2 2m + Ω − 1 − ς r SP

◦

. r−1 δ

we deduce   2 S a + + Ω SP (s) ς

◦ 2m = aS SP + 1 + + O(r−1 δ) r

= D(s) +

◦ 2m 2mS − S + O(r−1 δ) r r ◦

= D(s) + O(r−1 δ). Hence, " 0

ψ (s)

=

# ◦ ◦ aS + 2ς + Ω 1 + O(r−1 δ) = − D(s) + O(D(s)2 ) + O(r−1 δ) aS + Ω 2 SP

as stated. Step 20. We combine Propositions 9.64 and 9.65 to derive the closed system of equations in Λ, Λ, ψ, 1 1 7 Λ0 (s) = B(s) − r(s)−1 Λ(s) − r(s)−1 Λ(s) + O(r−1 )Λ(s) 0 −1 + ψ (s) 2 2 + N (B, B, D, Λ, Λ, ψ)(s), 1 7 1 (9.8.74) Λ0 (s) = B(s) − r(s)−1 Λ(s) + r(s)−1 Λ(s) −1 + ψ 0 (s) 2 2
+ O(r−1 ) Λ(s) + Λ(s) + N (B, B, D, Λ, Λ, ψ)(s), 1 ψ 0 (s) = − D(s) + O(D(s)2 ) + M (s), 2 with initial conditions ◦

ψ(s) = 0,

◦

Λ(s) = Λ0 ,

◦

Λ(s) = Λ0 .

(9.8.75)

Recall also that r(s) is a smooth function of ψ(s). The system (9.8.74) is verified for all choices of (Λ, Λ, Ψ). We now make a suitable particular choice for (Λ, Λ, Ψ) as follows. Consider in particular the system obtained from(9.8.74) by setting B, B, D to

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CHAPTER 9

zero ψ 0 (s) = M (s), 1 1 7 Λ0 (s) = − r(s)−1 Λ(s) − r(s)−1 Λ(s) −1 + ψ 0 (s) 2 2 −1 e (Λ, Λ, ψ)(s), + O(r )Λ(s) + N

(9.8.76)

1 7 1 Λ0 (s) = − r(s)−1 Λ(s) + r(s)−1 Λ(s) −1 + ψ 0 (s) 2 2
−1 e (Λ, Λ, ψ)(s), + O(r ) Λ(s) + Λ(s) + N where e (Λ, Λ, ψ) N e (Λ, Λ, ψ) N

= N (0, 0, 0, Λ, Λ, ψ), = N (0, 0, 0, Λ, Λ, ψ). ◦

We initialize the system at s = s as in (9.8.75), i.e., ◦

◦

Λ(s) = ψ(s) = 0,

◦

Λ0 ,

Λ(s) = Λ0 . ◦

◦

The system admits a unique solution ψ(s) defined in a small neighborhood I of s. The function Ψ(s) = −s + ψ(s) + c0 defines the desired hypersurface Σ0 . Step 21. It remains to show that the function B, B, D vanish on the hypersurface Σ0 defined above. Since the system (9.8.74) is verified for all functions Λ, Λ, ψ we deduce, along Σ0 , D

=

0,

B

=

B

=

N (B, B, D, Λ, Λ, ψ)(s) − N (0, 0, 0, Λ, Λ, ψ)(s), N (B, B, D, Λ, Λ, ψ)(s) − N (0, 0, 0, Λ, Λ, ψ)(s).

In view of the properties of N, N we deduce N (B, B, D, Λ, Λ, ψ)(s) − N (0, 0, 0, Λ, Λ, ψ)(s) N (B, B, D, Λ, Λ, ψ)(s) − N (0, 0, 0, Λ, Λ, ψ)(s)

.

◦

   sup |B(s)| + |B(s)| , ◦

I

.

◦

   sup |B(s)| + |B(s)| . ◦

I

Hence, ◦
sup |B(s)| + sup |B(s)| . δ sup |B(s)| + sup |B(s)| . ◦

◦

◦

◦

I

I

I

I

Hence B, B, D vanish identically on Σ0 . Step 22. We have dr ◦ − 1 . . ds

(9.8.77)

595

GCM PROCEDURE

Indeed, according to Step 15 and Lemma 9.54 we have d r(s) ds

1 1 ςΨ0 ν S (rS ) = (−1 + ψ 0 (s)) ςν S (rS ) 2λ 2λ SP SP SP   S 1 r (−1 + ψ 0 (s)) ς (ς S )−1 ς S (κS + aS κS ) . 2λ 2 SP

= X∗ =

rS =

◦ In view of Proposition 9.65, with D = 0, ψ 0 . r−1 δ. We deduce

d r(s) ds

 = −

 ◦ 1 rS S −1 S S ς (ς ) ς (κ + aS κS ) + O(δ). 2λ 2 SP

Step 23. Therefore the functions B, B, D vanish identically on the hypersurface Σ0 defined by the function Ψ(s) = −s + ψ(s) + c0 which accomplishes the main task of Theorem 9.52. More precisely we have produced a local hypersurface Σ0 , as defined in Step 12, foliated by the function uS , defined in Step 2 and extended in Step 3, such that the items 2–5 of the theorem are verified. The estimates in items 6–7 are an immediate consequence of Proposition 9.58. It only remains to prove the smoothness of the function Ξ(s, θ) in (9.8.56), Step 14 and the estimates for F = (f, f , log λ) in the last part of the theorem. To check the differentiability properties recall that   ∂s Ξ(s, θ) = Ψ0 (s) + ∂P U (θ, P (s))P 0 (s), 1 + ∂P S(θ, P (s))P 0 (s), 0 ,   ∂θ Ξ(s, θ) = ∂θ U (θ, P (s)), ∂θ S(θ, P (s)), 1 , where ∂P U (·)P 0 (s)

=

Ψ0 (s)∂u U (·) + ∂s U (·) + Λ0 (s)∂Λ U (·) + Λ0 (s)∂Λ U (·),

∂P S(·)P 0 (s)

=

Ψ0 (s)∂u S(·) + ∂s S(·) + Λ0 (s)∂Λ S(·) + Λ0 (s)∂Λ S(·).

Thus to prove the smoothness of Ξ we need to appeal to the smoothness of U, S with respect to the parameters Λ, Λ and u, s. Though tedious, this can be easily done by appealing to the coupled system of equations (9.4.13), (9.4.14), (9.4.15), as in the proof of Theorem 9.32, and studying its dependence on these parameters. Step 24. It only remains to derive the estimates (9.8.11) for the transition functions F = (f, f , log λ). To start with we have, in view of the construction of Σ0 and the estimates for F = (f, f , log λ) of Theorem 9.32, for every S ⊂ Σ0 ◦

kF khsmax +1 (S) . δ.

(9.8.78)

To derive the remaining tangential derivatives of F along Σ0 we commute the S S GCM system (9.4.13) of Proposition 9.33 with respect to ν = ν S = eS 3 + a e4 and then proceed, as in the proof of the a priori estimates of Theorem 9.37 to derive

596

CHAPTER 9

recursively the estimates, for K = smax + 1, Z   Z ◦ kν l (F )khK−l (S) . δ + r−2 ν l (f )eΦ + ν l (f )eΦ S

◦

+ δr

−1

S

≤l−1 S

ν a h

K−l+1 (S)

+ kν

≤l−1

(9.8.79)

F khK−l+1 (S) .

We already have estimates for the ` = 1 modes of F = (f, f ). To estimate the ` = 1 modes of ν l (f, f ), l ≥ 1, we make use of the equations (9.8.65) and the vanishing of the ` = 1 modes of η S , ξ S along Σ0 to derive, recursively, for all 1 ≤ l ≤ K, Z   Z   ◦ ◦

r−2 ν l (f )eΦ + ν l (f )eΦ . δ + r−1 δ ν ≤l−1 a, ΩS , ς S S

hK−l+1 (S)

S

◦

+ δ ν ≤l−1





ξS , ηS , ω ˇS

(9.8.80)

hK−l+1 (S)

+ r−1 kν ≤l−1 (F )khK−l+1 (S) . We can then proceed as in the proof of Proposition 9.58 to derive, recursively, the estimates

   



r−1 ν ≤l−1 aS , ΩS , ς S + ν ≤l−1 ξ S , η S , ω ˇS hK−l+1 (S)

. 1 + r−1 kF khK (S) +

l X j=1

hK−l+1 (S)

kν j (F )khK−j (S) .

(9.8.81)

Combining (9.8.79), (9.8.80), (9.8.81), we obtain kν l (F )khK−l (S)

◦

.

δ+

l−1 X j=0

kν j (F )khK−j (S) ,

which, together with (9.8.78), yields the desired estimate for all tangential derivatives K X j=0

kν j (F )khK−j (S)

◦

.

δ.

(9.8.82)

To complete the desired estimate for all derivatives we make use of the equations S for eS 4 (F ), due to the transversality conditions (9.4.9). The e3 derivatives can then S S S be derived from ν S = eS + a e and the estimates for a . This concludes the proof 3 4 of Theorem 9.52. Step 25. We now prove Corollary 9.53. Consider first the simpler case where k(f, f , log(λ))khsmax +1 (S0 )

◦

.

δ,

so that the estimate (9.8.78) holds true for S0 . We then proceed exactly as in Step 24 to derive the estimates (9.8.79), (9.8.80), (9.8.81) for our distinguished sphere S0 . Note that S0 can be viewed as a deformation of the unique background sphere sharing the same south pole.

597

GCM PROCEDURE

It remains to prove Corollary 9.53 in the more difficult case where kf khsmax +1 (S0 ) + (rS0 )−1 k(f , log λ)khsmax +1 (S0 )

◦

δ.

.

◦

In view of Lemma 9.15, with δ1 = δ, we infer S S r 0 r 0 ◦ − 1 . δ ◦ − 1 + sup r S0 r so that r and rS0 are comparable, and hence kf khsmax +1 (S0 ) + r−1 k(f , log(λ))khsmax +1 (S0 )

◦

.

δ.

(9.8.83)

Next, we introduce as in Step 24 the notation K = smax + 1. We claim the following analog of (9.8.82): K X j=1

◦

kν j (F )khK−j (S0 )

δ.

.

(9.8.84)

To complete the desired estimate for all derivatives we then make use, as in Step 0 24, of the equations for eS 4 (F ), due to the transversality conditions (9.4.9), and S0 S0 S 0 recover the e3 derivatives from ν S0 = eS 3 + a e4 , which concludes the proof of Corollary 9.53. It thus remains to prove (9.8.84). Note that S0 can be viewed as a deformation of the unique background sphere sharing the same south pole. We proceed exactly as in Step 24 to derive the estimates (9.8.80), (9.8.81) for our distinguished sphere S0 , which yields, for all 1 ≤ l ≤ K, Z   Z −2 l Φ l Φ r ν (f )e + ν (f )e S

◦

.

δ + r−1 kF khK (S) +

S l X j=1

kν j (F )khK−j (S)

(9.8.85)

and

r−1 ν ≤l−1 aS h

K−l+1 (S)

. 1 + r−1 kF khK (S) +

l X j=1

kν j (F )khK−j (S) .

We now claim the following sharpened version of (9.8.79): Z   Z ◦ kν l (F )khK−l (S) . δ + r−2 ν l (f )eΦ + ν l (f )eΦ S S ◦

−1 ≤l−1 S + δr ν a h + kf khK (S) (S) K−l+1

+ r−1 k(f , log(λ))khK (S) +

l−1 X j=1

kν j (F )khK−j (S) .

(9.8.86)

(9.8.87)

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CHAPTER 9

Then, (9.8.85), (9.8.86) and (9.8.87) imply kν l (F )khK−l (S)

◦

. δ + kf khK (S) + r−1 k(f , log(λ))khK (S) +

l−1 X j=1

kν j (F )khK−j (S) .

Together with (9.8.83), we deduce (9.8.84) by iteration. Finally, it remains to prove (9.8.87). As for the proof of (9.8.79), we commute S S the GCM system (9.4.13) of Proposition 9.33 with respect to ν S = eS 3 + a e4 and then proceed, as in the proof of the a priori estimates of Theorem 9.37 to derive (9.8.87) recursively. To obtain a stronger conclusion than (9.8.79), we need to analyze the differentiation w.r.t. ν S more carefully. First, note that the commutator [ d/S , ν S ]F satisfies, in view of Lemma 2.68, ◦

◦

k[ d/S , ν S ]F khl (S) . r−1 kF khl+1 (S) + kν S F khl (S) where the important observation is that the first term on the right-hand side gains a power of r−1 which is consistent with (9.8.87). It remains to analyze the differentiation of the error terms Err1 , . . . , Err6 of the GCM system w.r.t. ν S . To this end, in what follows, we single out all the terms that lose one power of r−1 in view of the anomalous behavior of (f , log(λ)) compared to the one in Step 24, and denote by · · · all terms that behave as before. We have by direct check Err(κ, κ0 ) = · · · , Err(ζ, ζ 0 ) = · · · ,

1 Err(κ, κ0 ) = − κf 2 + · · · , 4 Err(ρ, ρ0 ) = · · · , Err(µ, µ0 ) = · · · .

Then, in view of the identities in Lemma 9.23 1 (ea − 1) f e4 κ + · · · , 2  1 0 0 −a Err(eθ κ , eθ κ) = (e − 1) e0θ /d01 f + f e4 κ 2      1 1 + e−a e0θ − κf 2 + e0θ (a) /d 01 f + − κf 2 + ··· , 4 4 Err(e0θ µ0 , eθ µ) = · · · . Err(e0θ κ0 , eθ κ)

=

In view of (9.3.14), we deduce 1 a (e − 1)f + · · · , r2

 1  Err2 = 2 (e−a − 1) ( d/S )2 f + f + e−a d/S (f 2 ) + d/S (a) d/S f + f 2 + ··· , r Err3 = · · · . Err1 = −

We infer, in view of the expression of Err4 and Err5 in section 9.3.1, 1 a (e − 1)f + · · · , r2

 1  Err5 = 2 (e−a − 1) ( d/S )2 f + f + e−a d/S (f 2 ) + d/S (a) d/S f + f 2 + ··· . r Err4 = −

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GCM PROCEDURE

Finally, in view of the expression of Err6 in section 9.3.2, we have  S  rS a r Err6 = (e − 1 − a) − a − 1 + ··· . r r Thus, to conclude the proof of (9.8.87), it suffices to show that all terms singled out in the above expression of Err1 , . . . , Err6 gain a power of r−1 when differentiated w.r.t. ν S . Now, they are all quadratic expressions involving a, f , r and rS . Since ν S (a) and ν S (f ) gain r−1 compared to a and f in view of (9.8.84), the conclusion then follows from the straightforward estimate |ν S (rS )| |ν S (r)| + rS r

. r−1 .

Chapter Ten Regge-Wheeler Type Equations The goal of this chapter is to prove Theorem 5.17 and Theorem 5.18 concerning the weighted estimates for the solution ψ to 2 ψ = V ψ + N,

V = −κκ.

Recall that these theorems were used in Chapter 5 to prove Theorem M1. The structure of the chapter is as follows. • In section 10.1, we prove basic Morawetz estimates for ψ. • In section 10.2, we prove rp -weighted estimates in the spirit of Dafermos and Rodnianski [23] for ψ. In particular, we obtain as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ with derivatives). • In section 10.3, we use a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0 (i.e., without commutating the equation of ψˇ with derivatives). • In section 10.4, commuting the equation of ψ with derivatives, we complete the proof of Theorem 5.17 by controlling higher order derivatives of ψ, i.e., for s ≤ ksmall +30. Also, commuting the equation of ψˇ with derivatives, we complete ˇ i.e., for the proof of Theorem 5.18 by controlling higher order derivatives of ψ, s ≤ ksmall + 29. 10.1

BASIC MORAWETZ ESTIMATES

Recall (trap)



(trap)

• the definitions in section 5.1.1 of M, M, τ , Σ(τ ) and (trap) Σ, • the main quantities involved in the energy and Morawetz estimates, e.g., E[ψ](τ ), s Mor[ψ](τ1 , τ2 ), Morr[ψ](τ1 , τ2 ), F [ψ](τ1 , τ2 ), Jδ [ψ, N ](τ1 , τ2 ) and B˙ p;R [ψ](τ1 , τ2 ), introduced in section 5.1.4. The following theorem claims basic Morawetz estimates for the solution ψ of the wave equation (5.3.5). Theorem 10.1 (Morawetz). Let ψ a reduced 2-scalar solution to 2 ψ = V ψ + N,

V = −κκ.

Let δ > 0 be a fixed small constant verifying 0 <   δ. The following estimates hold true in M(τ1 , τ2 ), 0 ≤ τ1 < τ2 ≤ τ∗ , E[ψ](τ2 ) + Mor[ψ](τ1 , τ2 ) + F [ψ](τ1 , τ2 ) . E[ψ](τ1 ) + Jδ [ψ, N ](τ1 , τ2 ) (10.1.1) + O()B˙ δ ; 4m [ψ](τ1 , τ2 ). 0

601

REGGE-WHEELER TYPE EQUATIONS

Also, E[ψ](τ2 ) + Morr[ψ](τ1 , τ2 ) + F [ψ](τ1 , τ2 ) . E[ψ](τ1 ) + Jδ [ψ, N ](τ1 , τ2 ) (10.1.2) + B˙ δ ; 4m [ψ](τ1 , τ2 ). 0

Remark 10.2. Note that the bulk term B˙ δ ; 4m0 [ψ](τ1 , τ2 ) cannot yet be absorbed on the left-hand side of the inequality. To do that we will rely on the rp -weighted estimates of Theorem 10.37. Remark 10.3. In addition to  and δ, the proof of Theorem 10.1 will involve b δ1 , δH , H , Λ−1 and Λ−1 . These smallness several smallness constants: C −1 , δ, H constants will be chosen such that b δH , H , Λ−1 , Λ−1  δ1  C −1 . 0 <   δ, H

(10.1.3)

b H , Λ−1 and Λ−1 will in fact be chosen towards the end of the proof In addition, δ, H as explicit powers of δH , see (10.1.63), (10.1.65) and Proposition 10.30. The goal of this section is to prove Theorem 10.1. This will be achieved in section 10.1.15. 10.1.1

Structure of the proof of Theorem 10.1

To prove Theorem 10.1, we proceed as follows: • In section 10.1.2, we introduce a simplified set of assumptions of the Ricci coefficients which is sufficient in order to prove Theorem 10.1. • In section 10.1.3, we discuss notations concerning functions depending on m and r. • In section 10.1.4, we compute the deformation tensor of the vectorfields R, T , and X = f (r, m)R. • In section 10.1.5, we introduce the basic integral identities for wave equations that will be used repeatedly in the proof of Theorem 10.1. • In section 10.1.6, we derive the main Morawetz identity. • In section 10.1.7, we derive a first estimate. This estimate is insufficient due to – a lack of positivity of the bulk in the region 3m ≤ r ≤ 4m,

– a log divergence of a suitable choice of vectorfield at r = 2m, – a degeneracy at r = 2m. • In section 10.1.8, we add a correction and rely on a Poincar´e inequality to obtain a positive estimate also on the region 3m ≤ r ≤ 4m. • In section 10.1.9, we perform a cut-off to remove above mentioned log divergence at r = 2m. • In section 10.1.10, we introduce the redshift vectorfield to remove the above mentioned degeneracy at r = 2m. • In section 10.1.11, we combine the previous estimates with the redshift vectorfield to obtain a bulk term suitable on the whole spacetime M. • In section 10.1.12, we prove the positivity of the boundary terms arising from adding a large multiple of the energy estimate to the Morawetz estimate. • In section 10.1.13, combining the good properties of the bulk and of the bound-

602

CHAPTER 10

ary terms established so far, we obtain a first Morawetz estimate providing in particular the control of the quantity Mor[ψ]. • In section 10.1.14, we analyze an error term appearing in the right-hand side of the above mentioned Morawetz estimate. • Finally, in section 10.1.15, we add a correction to upgrade the control of Mor[ψ] to the control of the quantity Morr[ψ], hence concluding the proof of Theorem 10.1. 10.1.2

A simplified set of assumptions

To prove Theorem 10.1, it suffices to make a simplified set of assumptions. Define ( 0 7m0 1+τ for r ∈ [ 5m 2 , 2 ], utrap = (10.1.4) 0 7m0 1 for r ∈ / [ 5m 2 , 2 ]. For k = 0, 1, we assume the following. ˇ ≤k verify, on M = Mor1. The renormalized Ricci coefficients Γ dec ˇ ≤k | . r−1 u−1−δ |Γ , trap   m ≤k dec ω + 2 , ξ . r−2 u−1−δ . d trap r

Mor2. The Gauss curvature K of S and ρ verify  2m  ≤k dec ρ + 3 . r−2 u−1−δ , d trap r   1 ≤k dec K − 2 . r−2 u−1−δ . d trap r

(int)

M∪

(ext)

M,

(10.1.5)

(10.1.6)

Mor3. We also assume

≤k

|d

|m − m0 | . m0 ,

dec (e3 m, r2 e4 m)| .  u−1−δ . trap

(10.1.7)

Remark 10.4. Note that in the case when the bootstrap constant  = 0, i.e., in Schwarzschild, the assumptions made above are consistent with the behavior relative to the regular frame (near horizon) e3 = Υ−1 ∂t − ∂r , e4 = ∂t + Υ∂r . 10.1.3

Functions depending on m and r

In order to prove Theorem 10.1, we will adapt the derivation of the Morawetz estimate for the wave equation in Schwarzschild. In particular, we will need to consider various scalar functions, used to define suitable analogs of the vectorfields in Schwarzschild, which depend on m and r. Now, m is a scalar function unlike the Schwarzschild case where it is constant. To take this into account, we will rely on the following lemma.

603

REGGE-WHEELER TYPE EQUATIONS

Lemma 10.5. Let f = f (r, m) a C 1 function of r and m. Then, we have
dec e4 f (r, m) = ∂r f (r, m)e4 (r) + O(r−2 u−1−δ |∂m f |), trap
−1−δdec e3 f (r, m) = ∂r f (r, m)e3 (r) + O(utrap |∂m f |),

2 e4 e3 f (r, m) = ∂r f (r, m)e4 (r)e3 (r) + ∂r f (r, m)e4 (e3 (r)) e3 e4 f (r, m)





2 dec +O(r−2 u−1−δ (r|∂r ∂m f | + |∂m f |)), trap

= ∂r2 f (r, m)e4 (r)e3 (r) + ∂r f (r, m)e3 (e4 (r))


eθ f (r, m) = 0.

2 dec +O(r−2 u−1−δ (r|∂r ∂m f | + |∂m f |)), trap

Proof. Straightforward verification using (10.1.7). Remark 10.6. Note that in the sequel, ∂r f will not denote a spacetime coordinate vectorfield applied to f , but instead the partial derivative with respect to the variable r of the function f (r, m). 10.1.4

Deformation tensors of the vectorfields R, T, X

Recall the definition (5.1.10) of the regular vectorfields1 T

=

1 (e4 + Υe3 ) , 2

R=

1 (e4 − Υe3 ) . 2

Note that −g(T, T ) = g(R, R) = Υ,

g(T, R) = 0.

Note also that R(r) = 1 −

2m dec + O(u−1−δ ), trap r

dec T (r) = O(u−1−δ ). trap

Lemma 10.7. The following hold true. 1. The components of the deformation tensor of R = 4m (R) π 34 + 2 r 2 (R) π(eA , eB ) − ΥδAB r (R) π 33 (R) π 3θ (R) π 4θ

1 2

(e4 − Υe3 ) are given by

dec . r−1 u−1−δ , trap dec . r−1 u−1−δ , trap dec . r−1 u−1−δ , trap dec . r−1 u−1−δ , trap dec . r−1 u−1−δ . trap

1 In Schwarzschild, in standard coordinates, we have T = ∂ , R = Υ∂ which are regular near t r the horizon.

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CHAPTER 10

Moreover, (R) π 44

dec . r−2 u−1−δ . trap

2. If V := −κ κ, we have   8 3m dec 1 − + O()r−3 u−1−δ , trap r3 r   8Υ 3m dec e4 (V ) = − 3 1 − + O()r−3 u−1−δ , trap r r e3 (V ) =

(10.1.8)

and 

3m 1− r



R(V )

=

8Υ − 3 r

T (V )

=

dec O()r−3 u−1−δ . trap

dec + O()r−3 u−1−δ , trap

3. All components of the deformation tensor of T = 12 (e4 + Υe3 ) can be bounded by dec O(r−1 u−1−δ ). Moreover, trap (T ) −1−δ π 44 . r−2 utrap dec . Proof. We have (R)

π 44

=

(R)

π 34

= =

(R) (R)

π 33

=

π AB

= = =

g(D4 (e4 − Υe3 ), e4 ) = 2e4 (Υ) + 4Υω, 1 1 g(D3 (e4 − Υe3 ), e4 ) + g(D4 (e4 − Υe3 ), e3 ) 2 2 e3 (Υ) − 2Υω + 2ω,

g(D3 (e4 − Υe3 ), e3 ) = −4ω, 1 1 g(DA (e4 − Υe3 ), eB ) + g(DB (e4 − Υe3 ), eA ), 2 2 (1+3) χAB − Υ (1+3)χAB 1 (κ − Υκ)δAB + (1+3)χ bAB − Υ (1+3)χ b AB . 2

Note that 

e3 (Υ)

= =

e4 (Υ)

= =



2m 2e3 m m dec e3 (r) − = (κ + A) + O(r−1 u−1−δ ) trap r2 r r m 2m dec dec κ + O(r−1 u−1−δ ) = − 2 + O(r−1 u−1−δ ), trap trap r r  2m 2m 2e4 m m −1−δdec e4 1 − = 2 e4 (r) − = (κ + A) + O(r−2 utrap ) r r r r m 2m dec dec κ + O(r−2 u−1−δ ) = 2 Υ + O(r−2 u−1−δ ). trap trap r r e3

2m 1− r

=

605

REGGE-WHEELER TYPE EQUATIONS

Thus (R)

π 44

(R)

π 33

(R)

π 34

(R)

π AB

dec = O(r−2 u−1−δ ), trap dec = O(r−1 u−1−δ ), trap m dec = −4 2 + O(r−1 u−1−δ ), trap r 2Υ dec = δAB + O(r−1 u−1−δ ). trap r

Also, in view of −1−δ ξ, ξ, η, η, ζ . r−1 utrap dec , we deduce (R) π 3θ ,

(R)

dec π 4θ . r−1 u−1−δ trap

as desired. To prove the second part of the lemma we write e3 (V )

= −e3 (κ)κ − κe3 (κ)   1 −2 −1−δdec = − − κκ + 2ωκ + 2ρ + O(r utrap ) κ 2   1 dec −κ − κ2 − 2ω κ + O(r−2 u−1−δ ) trap 2 =

dec (κκ − 2ρ)κ + O(r−3 u−1−δ ). trap

On the other hand,  κκ − 2ρ = −

2Υ dec + O()r−1 u−1−δ trap r



2 dec + O()r−1 u−1−δ trap r

 +

dec +O(r−3 u−1−δ ) trap   4 3m dec = − 2 1− + O(r−2 u−1−δ ). trap r r

Hence, e3 (V )

= =

dec (κκ − 2ρ)κ + O(r−3 u−1−δ ) trap   8 3m dec 1− + O(r−3 u−1−δ ) trap r3 r

and similarly for e4 (V ). Thus, R(V )

=

T (V )

=

as desired.

  1 8Υ 3m dec (e4 − Υe3 )V = − 3 1 − + O(r−3 u−1−δ ), trap 2 r r 1 dec (e4 + Υe3 )V = O()r−3 u−1−δ , trap 2

4m r3

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CHAPTER 10

To prove the last part of the lemma we write (T )

π 44

(T )

π 34

(T )

π 33

(T )

π AB

= g (D4 (e4 + Υe3 ), e4 ) = −2e4 (Υ) − 4Υω, 1 1 = g(D3 (e4 + Υe3 ), e4 ) + g(D4 (e4 + Υe3 ), e3 ) 2 2 = −e3 (Υ) + 2Υω + 2ω,

= g (D3 (e4 + Υe3 ), e3 ) = −4ω, 1 1 = g (DA (e4 + Υe3 ), eB ) + g (DB (e4 + Υe3 ), eA ) , 2 2 = (1+3)χAB + Υ (1+3)χAB 1 = (κ + Υκ)δAB + (1+3)χ bAB + Υ (1+3)χ b AB , 2

and the proof continues as above in view of our assumptions. Consider now X = f (r, m)R and following lemma.

(X)

Lemma 10.8. Let X = f (r, m)R and

(X)

(X)

π=

(X)

π its deformation tensor. We have the π its deformation tensor. We have

π˙ +  (X) π ¨

where2 • The only nonvanishing components of

(X)

π˙ are

(X)

π˙ 33

=

2∂r f,

(X)

π˙ 44

=

(X)

π˙ 34

=

π˙ AB

=

2∂r f Υ2 , 4m − 2 f − 2∂r f Υ, r 2Υ f δAB . r

(X)

• All components of (X) π ¨ verify (X) dec π ¨ . r−1 u−1−δ (|f | + r|∂m f | + r2 |∂r f |). trap Moreover, (X) π ¨44

dec . r−2 u−1−δ (|f | + r|∂m f | + r2 |∂r f |). trap

Proof. Clearly, (X)

πµν = f (R) π µν + eµ f Rν + eν f Rµ .

2 Recall from Remark 10.6 that ∂ f does not denote a spacetime coordinate vectorfield applied r to f , but instead the partial derivative with respect to the variable r of the function f (r, m).

607

REGGE-WHEELER TYPE EQUATIONS

Therefore, since g(R, e3 ) = −1, g(R, e4 ) = Υ and −1−δ e4 (r) − Υ, e3 (r) + 1 .  utrap dec , and using Lemma 10.5, we deduce (X)

π33

= =

(X)

π44

= =

(X)

π34

=

f (R) π 33 − 2e3 (f ) = f (R) π 33 − 2∂r f e3 (r) − 2∂m f e3 (m)   dec 2∂r f + O r−1 u−1−δ (|f | + r|∂m f | + r2 |∂r f |) trap f (R) π 44 + 2Υe4 (f ) = f (R) π 44 + 2Υ∂r f e4 (r) + 2Υ∂m f e4 (m)   2 dec 2∂r f Υ2 + O r−2 u−1−δ (|f | + r|∂ f | + r |∂ f |) m r trap f (R) π 34 + e3 (f )Υ − e4 (f )

= f (R) π 34 + (∂r f e3 (r) + ∂m f e3 (m))Υ − (∂r f e4 (r) + ∂m f e4 (m))   4m dec = − 2 f − 2∂r f Υ + O r−1 u−1−δ (|f | + r|∂m f | + r2 |∂r f |) . trap r This concludes the proof of the lemma. 10.1.5

Basic integral identities

We recall, see section 2.4.1, that wave equations for ψ ∈ s2 (M) of the form 2 ψ = V ψ + N [ψ],

V = −κκ,

(10.1.9)

can be lifted to the spacetime version3 ˙ = V Ψ + N [Ψ] Ψ

(10.1.10)

where Ψ ∈ S2 (M) and N [Ψ] ∈ S2 (M) are defined according to Proposition 2.106. In fact, Ψθθ Nθθ [Ψ]

= −Ψϕϕ = ψ,

= Nϕϕ [Ψ] = N (ψ),

Ψθϕ = 0. N [Ψ]θϕ = 0.

All estimates for (10.1.10) derived in this section can be easily transferred to estimates for (10.1.9) and vice versa. Consider wave equations of the form ˙ gΨ = V Ψ + N 

(10.1.11)

with Ψ ∈ S2 (M) and N a given symmetric traceless tensor, i.e., N ∈ S2 (M). Proposition 10.9. Assume Ψ ∈ S2 (M) verifies (10.1.10). Then, 3 See section 2.4.1.1 and Appendix D for the precise definition of the covariant derivative D ˙ ˙ on S2 (M). and wave operator 

608

CHAPTER 10

1. The energy-momentum tensor Q = Q[Ψ] given by   ˙ µΨ · D ˙ ν Ψ − 1 gµν D ˙ λΨ · D ˙ λΨ + V Ψ · Ψ D 2 1 ˙ ˙ = Dµ Ψ · Dν Ψ − gµν L(Ψ) 2

Qµν :

=

Dν Qµν

=

verifies ˙ µ Ψ · N [Ψ] + D ˙ ν ΨA RABνµ ΨB − 1 Dµ V Ψ · Ψ. D 2

2. The null components of Q are given by Q33

=

Q34

=

Q44

=

|e3 Ψ|2 , |e4 Ψ|2 ,

|∇ / Ψ|2 + V |Ψ|2 ,

and gµν Qµν

=

−L(Ψ) − V |Ψ|2 .

Also, |L(Ψ)| .

|e3 Ψ| |e4 Ψ| + |∇ / Ψ|2 + V |Ψ|2

and |QAB |

≤

|e3 Ψ||e4 Ψ| + |∇ / Ψ|2 + |V ||Ψ|2 ,

|QA4 |

≤

|e4 Ψ||∇ / Ψ|.

|QA3 |

≤

|e3 Ψ||∇ / Ψ|,

3. Introducing b34 := Q34 − V |Ψ|2 = |∇ Q / Ψ|2 , we have b34 + Qθθ + Qϕϕ = −L(Ψ). −Q 4. Let X = ae3 + be4 . Then, since RAB34 = 0 in an axially symmetric polarized spacetime, Dµ (Qµν X ν )

=

1 Q· 2

(X)

1 π + X(Ψ) · N [Ψ] − X(V )Ψ · Ψ. 2

5. Let X = ae3 + be4 as above, w a scalar function and M a 1-form. Define Pµ

=

1 ˙ µ Ψ − 1 |Ψ|2 ∂µ w + 1 |Ψ|2 Mµ . Pµ [X, w, M ] = Qµν X ν + wΨD 2 4 4

609

REGGE-WHEELER TYPE EQUATIONS

Then, 1 1 1 1 Q · (X) π − X(V )Ψ · Ψ + wL[Ψ] − |Ψ|2 g w 2 2  2  4 (10.1.12) 1 ˙µ 1 2 + D (|Ψ| Mµ ) + X(Ψ) + wΨ · N [Ψ]. 4 2

Dµ Pµ [X, w, M ] =

Proof. See sections D.1.4 and D.2 in the appendix. Notation. For convenience we introduce the notation   1 E[X, w, M ](Ψ) := Dµ Pµ [X, w, M ] − X(Ψ) + wΨ · N [Ψ]. 2

(10.1.13)

Thus equation (10.1.12) becomes E[X, w, M ](Ψ)

=

1 1 1 Q · (X) π − X(V )Ψ · Ψ + wL[Ψ] 2 2 2 1 2 1 ˙µ 2 − |Ψ| g w + D (|Ψ| Mµ ). 4 4

(10.1.14)

When M = 0 we simply write E[X, w](Ψ). 10.1.6

Main Morawetz identity

Lemma 10.10. Let f (r, m) a function of r and m, and let X a vectorfield defined by X = f (r, m)R. Then, we have4     2m 2Υ 2Υ Q · (X) π˙ = f − 2 + |∇ / Ψ|2 + 2∂r f |RΨ|2 − f + Υ∂r f LΨ r r r 2m − 2 f V |Ψ|2 r where

(X)

π˙ has been defined in Lemma 10.8.

Proof. In view of Lemma 10.8, we have Q·

(X)

π˙

1 1 1 Q34 π˙ 34 + Q44 π˙ 33 + Q33 π˙ 44 + QAB π˙ AB 2 4 4 2m 1 1 2Υ AB = − 2 f Q34 − ∂r f ΥQ34 + Q44 ∂r f + Q33 Υ2 ∂r f + f δ QAB r 2 2 r
2m 2Υ AB 1 = − 2 f Q34 + f δ QAB + ∂r f Q44 − 2ΥQ34 + Υ2 Q33 . r r 2 =

Note that
Q44 − 2ΥQ34 + Υ2 Q33 = 4QRR 4 Recall from Remark 10.6 that ∂ f does not denote a spacetime coordinate vectorfield applied r to f , but instead the partial derivative with respect to the variable r of the function f (r, m).

610

CHAPTER 10

and, since gµν Qµν = −L(Ψ) − V |Ψ|2 , δ AB QAB

=

b34 − L. Q34 − L − V |Ψ|2 = Q

Hence, Q·

(X)

π˙

 2m 2Υ  b f Q34 + f Q34 − L + 2∂r f QRR 2 r r  2Υ   2m  b b34 − L + 2∂r f QRR = − 2 f Q34 + V |Ψ|2 + f Q r r  2m 2Υ b 2Υ 2m = f − 2 + Q34 + 2∂r f QRR − f L − 2 f V |Ψ|2 . r r r r = −

Finally, QRR

1 1 = |RΨ|2 − g(R, R)L = |RΨ|2 − ΥL. 2 2

Hence, Q · (X) π˙     m Υ b 1 2Υ 2m 2 = 2f − 2 + Q34 + 2∂r f |RΨ| − ΥL − f L − 2 f V |Ψ|2 r r 2 r r     m Υ 2Υ 2m = 2f − 2 + |∇ / Ψ|2 + 2∂r f |RΨ|2 − f + ∂r f Υ L − 2 f V |Ψ|2 . r r r r This concludes the proof of the lemma. We shall also make use of the following lemma. Lemma 10.11. If f = f (r, m), then g (f (r, m))

h dec = r−2 ∂r (r2 Υ∂r f ) + O(r−2 u−1−δ ) r2 |∂r2 f (r, m)| + r|∂r f (r, m)| trap i 2 +r|∂r ∂m f (r, m)| + |∂m f (r, m)| .

Proof. Recall from Lemma 2.102 that, for a general scalar f ,   1 1 g f = − (e3 e4 + e4 e3 )f + 4 / f + (1+3)ω − (1+3)trχ e4 f 2 2   1 + (1+3)ω − (1+3)trχ e3 f. 2 Recall that (1+3)

trχ =

2χ − ϑ,

(1+3)

trχ = 2χ − ϑ,

and 4 / f = eθ eθ f + (eθ Φ)2 eθ f.

(1+3)

ω = ω,

(1+3)

ω=ω

611

REGGE-WHEELER TYPE EQUATIONS

Using Lemma 10.5, we deduce, for a function f = f (r, m),     1 1 1 g f = − (e3 e4 + e4 e3 )f + ω − κ e4 f + ω − κ e3 f 2 2 2 1 = −∂r2 f (r, m)e3 (r)e4 (r) − ∂r f (r, m) (e3 e4 (r) + e4 e3 (r)) 2   1 1 dec − κ∂r f (r, m)e4 r + ω − κ ∂r f (r, m)e3 (r) + O(r−2 u−1−δ ) trap 2 2 h i 2 × r2 |∂r2 f (r, m)| + r|∂r f (r, m)| + r|∂r ∂m f (r, m)| + |∂m f (r, m)|   m Υ dec = −∂r2 f (r, m) −Υ + O(u−1−δ ) + ∂r f (r, m) 2 + ∂r f (r, m) trap r r r−m −2 −1−δdec + 2 ∂r f (r, m) + O(r utrap ) h r i =

2 × r2 |∂r2 f (r, m)| + r|∂r f (r, m)| + r|∂r ∂m f (r, m)| + |∂m f (r, m)|   2 2m dec Υ∂r2 f (r, m) + ∂r f (r, m) − 2 + O(r−2 u−1−δ ) trap r r h i 2 × r2 |∂r2 f (r, m)| + r|∂r f (r, m)| + r|∂r ∂m f (r, m)| + |∂m f (r, m)|

dec = r−2 ∂r (r2 Υ∂r f ) + O(r−2 u−1−δ ) trap h i 2 × r2 |∂r2 f (r, m)| + r|∂r f (r, m)| + r|∂r ∂m f (r, m)| + |∂m f (r, m)|

as desired. According to equation (10.1.14) we have E[X, w](Ψ)

=

1 Q· 2

(X)

1 1 1 π − X(V )|Ψ|2 + wL(Ψ) − |Ψ|2 g w. 2 2 4

In the next proposition we choose X to be of the form X = f (r, m)R and make a choice of w as a function of f . Proposition 10.12. Assume X = f (r, m)R and w(r, m) = r−2 Υ∂r (r2 f ). Then, E[X, w](Ψ)

˙ = E[X, w] + E [X, w]

b34 := Q34 − V |Ψ|2 = |∇ where, with Q / Ψ|2 ,   ˙ R, w](Ψ) = 1 1 − 3m f Q b34 + ∂r f |R(Ψ)|2 − 1 r−2 ∂r (r2 Υ∂r w)|Ψ|2 E[f r r 4 r − 4m + 4Υ f |Ψ|2 , r4 (10.1.15)  1 (X) −3 −1−δdec 2 3 2 E [f R, w](Ψ) =  Q · π ¨ + O r utrap |f | + r |∂r w| + r |∂r w| 2 
2 + r2 |∂r ∂m w| + r|∂m w| |Ψ|2 .

612

CHAPTER 10

Proof. According to Lemma 10.8 and equation (10.1.14) we have E[X, w](Ψ)

=

1 1 1 1 Q · ( (X) π˙ +  (X) π ¨ ) − X(V )|Ψ|2 + wL(Ψ) − |Ψ|2 g w. 2 2 2 4

Hence, in view of Lemmas 10.8 and 10.10, 1 E[X, w](Ψ) −  Q · 2

(X)

π ¨

1 Q· 2

1 1 1 π˙ − X(V )|Ψ|2 + wL(Ψ) − |Ψ|2 g w 2 2 4  m Υ f − 2+ |∇ / Ψ|2 + ∂r f |RΨ|2 r r   Υ 1 m f + Υ∂r f L(Ψ) − 2 f V |Ψ|2 r 2 r 1 1 1 X(V )|Ψ|2 + wL(Ψ) − |Ψ|2 g w. 2 2 4

= = − −

Thus, assuming w = r−2 Υ∂r (r2 f ) = 1 E[X, w](Ψ) −  Q · 2

(X)

π ¨

(X)

2Υ r f

+ ∂r f Υ,   3m −1 = r f 1− |∇ / Ψ|2 + ∂r f |RΨ|2 r   m 1 1 − f V + X(V ) + g w |Ψ|2 . r2 2 4

Note that, in view of Lemma 10.7, X(V )

=

f R(V ) = −8Υf

r − 3m dec + O(r−3 u−1−δ |f |) trap r4

and m 1 f V + X(V ) r2 2

 4m r − 3m dec Υ − 4Υ + O(r−3 u−1−δ |f |) trap r4 r4 r − 4m dec = −4f Υ + O(r−3 u−1−δ |f |). trap r4 

= f

Note also that, in view of Lemma 10.11, g (w)

= r−2 ∂r (r2 Υ∂r w) h i 2 dec +O(r−2 u−1−δ ) r2 |∂r2 w| + r|∂r w| + r|∂r ∂m w| + |∂m w| . trap

Thus, m 1 1 f V + X(V ) + g w 2 r 2 4

=

r − 4m 1 f + r−2 ∂r (r2 Υ∂r w) 4 r 4h dec +O(r−3 u−1−δ ) |f | + r3 |∂r2 w| + r2 |∂r w| trap i 2 +r2 |∂r ∂m w| + r|∂m w| −4Υ

613

REGGE-WHEELER TYPE EQUATIONS

and hence 1 E[X, w](Ψ) −  Q · 2

(X)

π ¨

= r

−1

 f

3m 1− r



|∇ / Ψ|2 + ∂r f |RΨ|2

1 2 −2 r − 4m |Ψ| r ∂r (r2 Υ∂r w) + 4Υ f |Ψ|2 4 r4

−

dec + O r−3 u−1−δ |f | + r2 |∂r w| + r3 |∂r2 w| trap
 2 +r2 |∂r ∂m w| + r|∂m w| |Ψ|2

as desired. 10.1.7

A first estimate

We concentrate our attention on the principal term   ˙ R, w](Ψ) = 1 1 − 3m f Q b34 + ∂r f |R(Ψ)|2 − 1 r−2 ∂r (r2 Υ∂r w)|Ψ|2 E[f r r 4 r − 4m +4Υ f |Ψ|2 r4 and choose f = f (r, m) such that the right-hand side is positive definite. Consider the quadratic forms, b34 + B|RΨ|2 + r−2 W |Ψ|2 , E˙0 (Ψ) : = AQ r − 4m ˙ E(Ψ) : = E˙0 (Ψ) + 4Υ f |Ψ|2 , r4 with the coefficients   3m −1 A := r f 1 − , r

B := ∂r f,

1 W := − ∂r (r2 Υ∂r w). 4

(10.1.16)

(10.1.17)

The goal is to show that there exist choices of f, w verifying the condition of Propo˙ sition 10.12, i.e., w = r−2 ∂r (r2 f ), which makes E(Ψ) positive definite, for all smooth S-valued tensorfields Ψ defined in the region r ≥ 2m0 (1 − δH ), which decay reasonably fast at infinity. We look first for choices of f, w such that the coefficient A, B, W are non-negative. Note in particular that f must be increasing as a function of r and f = 0 on r = 3m. Following J. Stogin [62] we choose w first to ensure that W is non-negative and then choose f , compatible with the equation ∂r (r2 f ) =

r2 w, Υ

f = 0 on r = 3m.

(10.1.18)

To ensure that A = r−2 f (r−3m) is positive we need a non-negative w which verifies (modulo error terms5 ) W = − 14 ∂r (r2 Υ∂r w) ≥ 0. It is more difficult to choose w such that B = ∂r f is also non-negative. Stogin defines w based on the following lemma.

5 That

is, terms which vanish in Schwarzschild.

614

CHAPTER 10

Lemma 10.13. The scalar function w defined by ( 1 if r ≤ 4m, 4m , w(r, m) = 2Υ if r ≥ 4m, r , is C 1 , non-negative and such that W = − 14 ∂r (r2 Υ∂r w) verifies ( 0, if r < 4m,
W (r, m) = m 8m 3 − , if r > 4m. r2 r

(10.1.19)

Proof. For r ≥ 4m, we have w(r, m) =

2Υ , r

∂r w(r, m) = −

2 8m + 3, r2 r

∂r2 w(r, m) =

4 24m − 4 . r3 r

In particular, we have w=

1 , 4m

∂r w = 0 at r = 4m

so that w is indeed C 1 . Furthermore, we also have   2 2m −2 2 2 r ∂r (r Υ∂r w) = Υ∂r w(r) + ∂r w(r) − 2 r r      4 24m 2 8m 2 2m = Υ − 4 + − 2+ 3 − 2 r3 r r r r r   4m 8m = − 4 3− r r so that, for r ≥ 4m, 1 W = − ∂r (r2 Υ∂r w) 4

=

  8m 3− r

m r2

as desired. Once w is defined we can evaluate f as follows. Lemma 10.14. Let w(r, m) defined as in Lemma 10.13. Then, the function f (r, m) given by (
2 2 2m2 log r−2m + (r − 3m) r +6mr+30m , for r ≤ 4m, 2 m 12m r f (r, m) := (10.1.20) C∗ m2 + r2 − (4m)2 , for r ≥ 4m, with the constant C∗ given by6 C∗ 6C

∗

:=

2 log(2) +

35 , 6

is chosen so that f is continuous across r = 4m.

C∗ ∼ 7.22,

615

REGGE-WHEELER TYPE EQUATIONS

is C 2 and satisfies (10.1.18), i.e., we have ∂r (r2 f ) = r2 Υ−1 w,

f = 0 on r = 3m.

Proof. By direct check,7 we have for r ≤ 4m ∂r (r2 f )(r, m)

= = =

2m2 r2 + 6mr + 30m2 2r + 6m + + (r − 3m) (r − 2m) 12m 12m 3 r 4m(r − 2m) r2 1 Υ 4m

and for r ≥ 4m ∂r (r2 f )(r, m)

=

2r,

as well as f = 0 on r = 3m so that, in view of the definition of w(r) in Lemma 10.13, we infer ∂r (r2 f ) = r2 Υ−1 w,

f = 0 on r = 3m

as desired. Note also that w being C 1 , f is thus indeed C 2 . Next, we derive a lower bound on ∂r f for r ≤ 4m. Lemma 10.15. We have for all r and m r3 ∂r f ≥ 16m2 . Also, there exists a constant C > 0 such that for all r and m 

3m 1− r

 f ≥C

−1



3m 1− r

2 .

Proof. We have ∂r (r3 ∂r f )

= ∂r (r∂r (r2 f )) − 2∂r (r2 f ).

Using the identity ∂r (r2 f ) = r2 Υ−1 w, we infer ∂r (r3 ∂r f )

= ∂r (r3 Υ−1 w) − 2r2 Υ−1 w.

7 Recall from Remark 10.6 that ∂ f does not denote a spacetime coordinate vectorfield applied r to f , but instead the partial derivative with respect to the variable r of the function f (r, m).

616

CHAPTER 10

For r ≤ 4m, we have w = (4m)−1 and hence ∂r (r3 ∂r f )


1 ∂r (r3 Υ−1 ) − 2r2 Υ−1 4m 1 3 r ∂r (Υ−1 ) 4m r − 2. 2Υ

= = =

In particular, r3 ∂r f is decreasing in r on r ≤ 4m and hence r3 ∂r f ≥ (4m)3 ∂r f (r = 4m, m) on r ≤ 4m. On the other hand, we have, in view of the definition of f in (10.1.20), ∂r (r2 f )(r = 4m, m)

(4m)2 ∂r f (r = 4m, m) + 8mf (r = 4m, m) m (4m)2 ∂r f (r = 4m, m) + C∗ 2

= =

and hence (4m)2 ∂r f (r = 4m, m)

 =

8−

C∗ 2

 m

so that r3 ∂r f ≥ 2(16 − C∗ )m2 on r ≤ 4m. Since C∗ ∼ 7.22 < 8, we deduce r3 ∂r f ≥ 16m2 on r ≤ 4m. Also, for r ≥ 4m, we have f =1−

(16 − C∗ )m2 r2

so that ∂r f =

2(16 − C∗ )m2 . r3

Since C∗ ∼ 7.22 < 8, we deduce r3 ∂r f ≥ 16m2 on r ≥ 4m which together with the case r ≤ 4m above yields for all r and m the desired estimate for ∂r f r3 ∂r f ≥ 16m2 . In particular, ∂r f > 0 and hence is strictly increasing. On the other hand, f = 0 on r = 3 and converges to 1 as r → +∞. We deduce the existence of a constant

617

REGGE-WHEELER TYPE EQUATIONS

C > 0 such that  1−

3m r



 2 3m f ≥ C −1 1 − r

as desired. We summarize the results in the following. Proposition 10.16. There exist functions f ∈ C 2 , w ∈ C 1 verifying the relation w = r−2 Υ∂r (r2 f ) and such that (
2 2 2m2 log r−2m + (r − 3m) r +6mr+30m , for r ≤ 4m, 2 m 12m r f= (10.1.21) 2 2 2 C∗ m + r − (4m) , for r ≥ 4m, where C∗ is a constant satisfying 7 < C∗ < 8. In particular, (
2m2 r−2m + O( r−3m for r ≤ 4m, 2 log r m m ), f= 2 m 1 + O( r2 ), for r ≥ 4m,

(10.1.22)

and, for some C > 0 and all r ≥ 2m, 

3m 1− r

Also, w is given by ( w=



1 4m , 2 r 1

f ≥C

−

2m r

−1






3m 1− r

2 ,

,

∂r f ≥

16m2 . r3

(10.1.23)

for

r ≤ 4m,

(10.1.24)

for

Moreover W = − 14 ∂r (r2 Υ∂r w) verifies ( 0,
W = m 8m , r2 3 − r

r ≥ 4m.

if r < 4m, if r > 4m,

(10.1.25)

and   3m 2 −2 2 −1 ˙ b34 , E0 [f R, w](Ψ) = ∂r f |R(Ψ)| + r W |Ψ| + r 1− fQ r ˙ R, w](Ψ) = E0 [f R, w](Ψ) + 4Υ r − 4m f |Ψ|2 . E[f r4

(10.1.26)

Recall also that b34 Q

=

|∇ / Ψ|2 .

Remark 10.17. The estimates obtained so far have two major deficiencies. 2 1. The quadratic form E˙0 [f R, w](Ψ) + 4Υ r−4m r 4 f |Ψ| fails to be positive definite in 2 the region 3m ≤ r ≤ 4m because of the potential term Υ r−4m r 4 f |Ψ| . (int) 2. The function f blows up logarithmically at r = 2m in M.

618

CHAPTER 10

In the next section we deal with the first issue. We handle the second problem in the following two sections. 10.1.8

Improved lower bound in 4f Υ r−4m r4

(ext)

M

Note that the term is negative for 3m ≤ r ≤ 4m and positive everywhere else. An improvement can be obtained by using the following Poincar´e inequality. Lemma 10.18. We have, for Ψ ∈ S2 (M), Z  Z |∇ / Ψ|2 ≥ 2r−2 1 − O() Ψ2 daS . S

(10.1.27)

S

Proof. See Proposition 2.35. According to Proposition 10.16 we deduce Z Z Z ˙ R, w](Ψ) ≥ E[f E˙1 − O(r−3 ) Ψ2 daS , S S S   (10.1.28) 3m r − 4m 2 −2 2 −3 2 E˙1 := ∂r f |R(Ψ)| + r W |Ψ| + 2r 1− f |Ψ|2 + 4Υ f |Ψ| , r r4 with W defined in (10.1.25). It is easy to see however that E˙1 still fails to be positive for 3m < r < 4m. To achieve positivity we also need to modify the original energy density E[f R, w](Ψ) by considering instead the modified energy density E[f R, w, M ](Ψ) (see (10.1.12) and notation (10.1.13)) with M = 2hR for a function h = h(r, m) supported for r ≥ 3m and constant for r ≥ 4m. E[f R, w, M ](Ψ)

1 ˙µ = E[f R, w](Ψ) + D (|Ψ|2 Mµ ) 4 1 1 = E[f R, w](Ψ) + (Dµ Mµ )|Ψ|2 + ΨM (Ψ) 4 2 1 µ 2 = E[f R, w](Ψ) + D (hRµ )|Ψ| + hΨR(Ψ). 2

To take into account the additional terms in the modified E[f R, w, M ](Ψ) we first derive the following. Lemma 10.19. Let h(r, m) a C 1 function of r and m. We have 
 dec Dµ (hRµ ) = r−2 ∂r (Υr2 h) + O  r−1 u−1−δ r|∂r h| + |h| + r|∂m h| .(10.1.29) trap Proof. In view of Lemma 10.7, which computes the components of

(R)

π, as well as

619

REGGE-WHEELER TYPE EQUATIONS

Lemma 10.5 to compute R(h), we calculate Dµ (hRµ )

= =

= =

R(h) + h(Dµ Rµ ) =

1 1 (e4 (h) − Υe3 (h)) + h tr ( (R) π) 2 2

1 dec (e4 (r) − Υe3 (r))∂r h + O( u−1−δ |∂m h|) trap 2   1 + h − (R) π 34 + (R) π θθ + (R) π ϕϕ 2   
 1 4m Υ dec Υ∂r h + + 4 h + O  u−1−δ |∂r h| + r−1 |h| + |∂m h| trap 2 2 r r 
 dec r−2 ∂r (Υr2 h) + O  u−1−δ |∂r h| + r−1 |h| + |∂m h| trap

as desired. In view of the lemma we write ˙ R, w, 2hR](Ψ) + E [f R, w, 2hR](Ψ), E[f R, w, 2hR](Ψ) = E[f ˙ R, w, 2hR](Ψ) : = E[f ˙ R, w](Ψ) + 1 r−2 ∂r (Υr2 h)|Ψ|2 + hΨR(Ψ), E[f 2 (10.1.30) E [f R, w, 2hR](Ψ) : = E [f R, w](Ψ) 
 dec + O r−1  u−1−δ r|∂r h| + |h| + r|∂m h| |Ψ|2 . trap The main result of this section is stated below. Proposition 10.20. There exists a function h = h(r, m) with bounded derivative h0 , supported in r ≥ 3m such that h = O(r−2 ), h0 = O(r−3 ) for r ≥ 4m such that ˙ R, w, 2hR](Ψ) + E [f R, w, 2hR](Ψ), E[f R, w, 2hR](Ψ) = E[f   (10.1.31) 1 dec E [f R, w, 2hR](Ψ) =  Q · (X) π ¨ + O r−3 u−1−δ (|f | + 1) |Ψ|2 , trap 2 and, for sufficiently large universal constant C > 0, in the region r ≥ Z S

˙ R, w, 2hR](Ψ) ≥ E[f

C

−1

Z S

5m 2 ,

 2 m2 3m 2 −1 |R(Ψ)| + r 1− |∇ / Ψ|2 r3 r ! m 2 + 4 |Ψ| . (10.1.32) r

Proof. We first derive the weaker inequality  Z Z  2 m m 2 5m −1 2 ˙ E[f R, w, 2hR](Ψ) ≥ C |R(Ψ)| + 4 |Ψ| on r ≥ 3 r r 2 S S by making full use of the Poincar´e inequality above, i.e.,     Z Z 3m 3m −1 2 −3 r 1− f (r, m)|∇ / Ψ| ≥ (2 − O())r 1− f (r, m)|Ψ|2 . r r S S

620

CHAPTER 10

The result will easily follow by writing instead, with a sufficiently small µ > 0,   Z 3m r−1 1 − f (r, m)|∇ / Ψ|2 r S     Z Z 3m 3m = µ r−1 1 − f (r, m)|∇ / Ψ|2 + (1 − µ) r−1 1 − f (r, m)|∇ / Ψ|2 r r S S     Z Z 3m 3m −1 2 ≥ µ r 1− f (r, m)|∇ / Ψ| + (1 − µ) 2r−3 1 − f (r, m)|Ψ|2 r r S S and then proceeding exactly as below. We start with ˙ R, w, 2hR](Ψ) E[f

˙ R, w](Ψ) + 1 r−2 ∂r (Υr2 h)|Ψ|2 + hΨR(Ψ). = E[f 2

Recalling the definition of E˙1 in (10.1.28),   3m r − 4m E˙1 := ∂r f |R(Ψ)|2 + r−2 W |Ψ|2 + 2r−3 1 − f |Ψ|2 + 4Υ f |Ψ|2 , r r4 and setting E˙2

1 := E˙1 + r−2 (Υr2 h)0 |Ψ|2 + hΨR(Ψ) (10.1.33) 2   3m r − 4m = ∂r f |R(Ψ)|2 + 2r−3 1 − f |Ψ|2 + 4Υ f |Ψ|2 + r−2 W |Ψ|2 r r4 1 −2 + r (Υr2 h)0 |Ψ|2 + hΨR(Ψ), 2

we deduce, from (10.1.28), Z Z Z −3 ˙ ˙ E[f R, w, 2hR](Ψ) ≥ E2 − O(r ) |Ψ|2 . S

S

S

We now substitute h =

4Υr−4 e h.

Hence, 1 −2 r ∂r (Υr2 h)|Ψ|2 + hΨR(Ψ) 2

= =

1 −2 r ∂r (4Υ2 r−2 e h)|Ψ|2 + 4Υr−4 e hΨR(Ψ) 2 1 −2 r ∂r (4Υ2 r−2 )e h|Ψ|2 + 2r−4 Υ2 ∂r e h|Ψ|2 2 +4Υr−4 e hΨR(Ψ)

or, since 12 r−2 ∂r (4Υ2 r−2 ) = −4r−2 Υ r−4m r4 , 1 −2 r ∂r (Υr2 h)|Ψ|2 + hΨR(Ψ) 2

=

r − 4m e 2 h|Ψ| + 2r−4 Υ2 ∂r e h|Ψ|2 r4 +4Υr−4 e hΨR(Ψ). −4r−2 Υ

621

REGGE-WHEELER TYPE EQUATIONS

Thus we have E˙2

= +

 3m r − 4m ∂r f |R(Ψ)| + 2r 1− f |Ψ|2 + 4Υ (f − r−2 e h)|Ψ|2 r r4 2r−4 Υ2 ∂r e h|Ψ|2 + 4Υr−4 e hΨR(Ψ) + r−2 W |Ψ|2 . −3

2



We also express 4Υr−4 e hΨR(Ψ)

=

2e h 2e h 2e h (R(Ψ) + Υr−1 Ψ)2 − 3 |R(Ψ)|2 − 5 Υ2 |Ψ|2 3 r r r

and, therefore, E˙2

2e h (∂r f − 2r−3 e h)|R(Ψ)|2 + 3 (R(Ψ) + Υr−1 Ψ)2 + r−2 W |Ψ|2 r     3m r − 4m −3 −2 e −4 −5 2 e e + 2r 1− f + 4Υ (f − r h) + 2r Υ∂ h − 2r Υ h |Ψ|2 . r r r4 =

We choose e h(r, m) as the following continuous and piecewise C 1 function,   0, r ≤ 5m  2 , 
 5m 5m   δ − r ≤ r ≤ 11m ,  eh 2 2 4 , 11m e h= δeh (r − 3m), 4 ≤ r ≤ 3m,   2  r f, 3m ≤ r ≤ 4m,     (4m)2 f (4m, m), r ≥ 4m, where the constant δeh > 0 will be chosen small enough. We consider the following cases: e Case 1 ( 5m 2 ≤ r ≤ 3m). In view of the definition of h and since W = 0, we deduce "   3m r − 4m 2 −3 ˙ E2 = ∂r f |R(Ψ)| + 2r 1− f + 4Υ (f − r−2 e h) r r4 #   −4 2 11m +2δeh r Υ 1 11m − 1 5m |Ψ|2 + δeh O(1)ΨR(Ψ)1 5m . 4 ≤r≤3m 2 ≤r≤ 4 2 ≤r≤3m In view of (10.1.23), we may assume, choosing for δeh > 0 small enough, that ˜ ≤ − 1 |f | on r ≤ 3m. f −h 2

(10.1.34)

We infer, using also that f < 0 on r ≤ 3m, "   3m r − 4m 2 E˙2 ≥ ∂r f |R(Ψ)| + 2r−3 1 − f + 2Υ f r r4 #   −4 2 11m +2δeh r Υ 1 11m − 1 5m |Ψ|2 + δeh O(1)ΨR(Ψ)1 5m . 4 ≤r≤3m 2 ≤r≤ 4 2 ≤r≤3m

622

CHAPTER 10

Since we have ∂r f & 1,

2r

−3

  3m r − 4m 1− + 4Υ . −1, r r4

3m f . − 1 − r

on r ≤ 3m,

where we have used in particular Lemma 10.15 and Proposition 10.16, we infer   3m 11m 5m 11m E˙2 & |R(Ψ)|2 + 1 − + δ 1 − O(1)δ 1 |Ψ|2 e e h h 2 ≤r≤ 4 4 ≤r≤3m r −δeh O(1)ΨR(Ψ)1 5m 2 ≤r≤3m   1 3m 11m ≥ |R(Ψ)|2 + 1 − + δ 1 − O(1)δ e e h h 1 4 ≤r≤3m 2 r ! 11m −O(1)δeh 1 5m |Ψ|2 . 2 ≤r≤ 4

Thus, for δeh > 0 small enough, there exists some large C > 0 such that   5m E2 ≥ C −1 |R(Ψ)|2 + |Ψ|2 on ≤ r ≤ 3m. 2

(10.1.35)

Case 2 (3m ≤ r ≤ 4m). Since e h = r2 f and W = 0, using in particular e h ≥ 0 on 3m ≤ r ≤ 4m, we deduce E˙2

≥ (∂r f − 2r−3 (r2 f ))|R(Ψ)|2     3m −3 −4 2 −5 2 2 + 2r 1− f + 2r Υ∂r (r f ) − 2r Υ (r f ) |Ψ|2 r =

(∂r f − 2r−1 f )|R(Ψ)|2     3m −3 −4 2 2 −3 2 + 2r 1− f + 2r Υ (2rf + r ∂r f ) − 2r Υ f |Ψ|2 r

=

(∂r f − 2r−1 f )|R(Ψ)|2     3m + 2r−3 1 − f + 2r−2 Υ2 ∂r f + 2r−3 Υ2 f |Ψ|2 . r

Note that the second term is strictly positive. It remains to analyze the first term. Lemma 10.21. In the interval [3m, 4m] we have ∂r f − 2r−1 f > 0. Proof. Recall from Proposition 10.16 that w = r−2 Υ∂r (r2 f ) = [3m, 4m]. Using also f = 0 on r = 3m, we deduce ∂r (r2 f )

=

r2 1 . Υ 4m

1 4m

in the interval

623

REGGE-WHEELER TYPE EQUATIONS

We compute   (r − 3m)r2 ∂r r2 f − 4mΥ

= = ≤0

 2 (r − 3m) r ∂r 4m Υ (r − 3m)(r − 4m) − 2mΥ2 on 3m ≤ r ≤ 4m, −

so that the differentiated quantity decays in r on [3m, 4m]. Since it vanishes on r = 3m, we infer f

≤

(r − 3m) on 3m ≤ r ≤ 4m. 4mΥ

Thus, we deduce, using again ∂r (r2 f ) = 2 ∂r f − f r

r2 1 Υ 4m ,

  = r−2 ∂r (r2 f ) − 4rf 1 4 − f 4mΥ r 1 (r − 3m) ≥ − 4mΥ  rmΥ   1 3m ≥ 1−4 1− 4mΥ r > 0 on 3m ≤ r < 4m. =

On the other hand, we have by direct check at r = 4m, using (10.1.21),     2 1 1 1 C∗ ∂r f − f = − fr=4m = 1− >0 r r=4m 2m m 2m 8 since C∗ < 8. Hence, we infer ∂r f − 2r−1 f > 0 on 3m ≤ r ≤ 4m as desired. We thus conclude, for some C > 0, in the interval [3m, 4m]   E˙2 ≥ C −1 |R(Ψ)|2 + |Ψ|2 .

(10.1.36)

Case 3 (r ≥ 4m). Since e h is constant and positive on r ≥ 4m, we deduce E˙2

≥ (∂r f − 2r−3 e h)|R(Ψ)|2     3m r − 4m −2 e −5 2 e −2 f + 4Υ (f − r h) − 2r Υ h + r W |Ψ|2 . + 2r−3 1 − r r4

We examine the first term. In view of the formula for f for r ≥ 4m, see (10.1.21), ∂r f =

2 (16 − C∗ )m2 , r3

e h = (4m)2 f (4m, m) = C∗ m2

624

CHAPTER 10

and hence ∂r f − 2r−3 e h =

2(16 − 2C∗ )m2 r3

and hence, since C∗ < 8, we have ∂r f − 2r−3 e h &

m2 r3

for r ≥ 4m.

It remains to analyze the sign of   3m r − 4m −3 2r 1− f + 4Υ (f − r−2 e h) − 2r−5 Υ2 e h r r4     3m r − 4m = 2r−3 1 − + 4Υ (f − r−2 e h) r r4     3m + 2r−3 1 − − 2r−3 Υ2 r−2 e h. r The first term, which can be written in the form      3m r − 4m −2  2 −3 2 2r 1− + 4Υ r r f (r, m) − (4m) f (4m, m) , r r4 is manifestly positive for r ≥ 4m. To evaluate the sign of the second term we calculate   3m −3 2r 1− − 2r−3 Υ2 = 2mr−5 (r − 4m). r Thus, for r ≥ 4m,   3m r − 4m 2r−3 1 − f + 4Υ (f − r−2 e h) − 2r−5 Υ2 e h ≥ 0. r r4
Also, since W = rm2 3 − 8m r , we have r−2 W

&

1 . r4

Thus, in view of the above, we have, for some C > 0 and for r ≥ 4m,   E˙2 ≥ C −1 r13 |R(Ψ)|2 + r14 |Ψ|2 . Gathering (10.1.35), (10.1.36) and (10.1.37), we infer for some C > 0,   E˙2 ≥ C −1 r13 |R(Ψ)|2 + r14 |Ψ|2 on r ≥ 5m 2 . Recalling Z S

˙ R, w, 2hR](Ψ) ≥ E[f

Z S

E˙2 − O(r−3 )

Z S

|Ψ|2 ,

(10.1.37)

625

REGGE-WHEELER TYPE EQUATIONS

we infer  Z Z  Z 1 1 −1 2 2 −3 ˙ E[f R, w, 2hR](Ψ) ≥ C |R(Ψ)| + 4 |Ψ| − O(r ) |Ψ|2 3 r S S r S and hence, for  > 0 small enough,  Z Z  1 1 5m 2 2 ˙ R, w, 2hR](Ψ) ≥ 1 C −1 E[f |R(Ψ)| + |Ψ| on r ≥ 3 4 2 r r 2 S S as desired. It remains to analyze the error term, E [f R, w, 2hR](Ψ)

= E [f R, w](Ψ)   dec +O r−1  u−1−δ (r|∂ h| + |h| + r|∂ h|) |Ψ|2 r m trap 1 dec  Q (X) π ¨ + O(r−3 u−1−δ (|f | + r2 |∂r w| + r3 |∂r2 w| trap 2 2 +r2 |∂r ∂m w| + r|∂m w|))|Ψ|2   dec +O r−3  u−1−δ (r3 |∂r h| + r2 |h| + r3 |∂m h|) |Ψ|2 . trap

=

Recall that ( w=

1 4m , 2 r 1

−

2m r




for r ≤ 4m, for r ≥ 4m,

,

and h = 4Υr−4 e h, with   0,  
 5m    δeh 2 − r , e h= δeh (r − 3m),    r2 f,     (4m)2 f (4m, m),

r ≤ 5m 2 , ≤ r ≤ 11m 4 , ≤ r ≤ 3m, 3m ≤ r ≤ 4m, r ≥ 4m. 5m 2 11m 4

We deduce E [f R, w, 2hR](Ψ)

  dec = Q (X) π ¨ + O r−3 u−1−δ (|f | + 1) trap

which concludes the proof of Proposition 10.20. 10.1.9

Cut-off correction in

(int)

M


So far we have found a triplet (X = f R, w = r−2 Υ∂r r2 f , M = 2hR) with f defined in Proposition 10.16 and h in Proposition 10.20 allowing for the lower R ˙ R, w, M ](Ψ). The main problem which remains to be bound (10.1.32) on S E[f addressed is: 1. f blows up logarithmically near r = 2m. R ˙ R, w, 2hR](Ψ) does not control e3 (Ψ) near r = 2m. 2. The lower bound for S E[f

626

CHAPTER 10

In this section, we deal with the first problem, while the second problem will be treated in section 10.1.10. To correct for the first problem, i.e., the fact that f blows up logarithmically near r = 2m, we have to modify our choice of f and w there. Introducing u := r2 f, we have f = r−2 u,

w = r−2 Υ∂r u.

(10.1.38)

Warning. The auxiliary function u introduced here, and used only in this section, has of course nothing to do with our previously defined optical function on (ext) M. Definition 10.22. For a given δb > 0 we define the following functions of (r, m) ! m2 δb uδb := − F − 2u , fδb := r−2 uδb, m δb
1 wδb := r−2 Υ∂r uδb, Wδb := − ∂r r2 Υ∂r wδb , 4 where F : R → R is a fixed, increasing, smooth function such that ( x for x ≤ 1, F (x) = 2 for x ≥ 3. We now derive useful properties satisfied by fδb, wδb and Wδb. Lemma 10.23. Let fδb, wδb and Wδb introduced in Definition 10.22. Then, we have fδb ∈ C 2 (r > 0), wδb ∈ C 1 (r > 0), and we have for δb > 0 sufficiently small fδb = f

wδb = w,

Wδb = W

for r ≥

5m . 2

Also, we have for all r > 0 r

−1

 fδb

3m 1− r

 ≥C

−1 −1

r

 2 3m 1− r

(10.1.39)

and ∂r (fδb) ≥

16m2 . r3

(10.1.40)

Proof. Note first that wδb = r−2 Υ∂r uδb = r−2 ΥF 0

! δb − 2 u ∂r u = wF 0 m

! δb − 2u . m

627

REGGE-WHEELER TYPE EQUATIONS

In view of the definition of uδb, fδb, wδb and Wδb, we have uδb = u, uδb = −

fδb = f, 2m2 , δb

wδb = w,

fδb = −

2m2 , b2 δr

wδb = 0,

Also, according to (10.1.22) ( 2m2 log r−2m + O(m(r − 3m)), m u= 2 2 r + O(m ), and hence, for δb > 0 sufficiently small  n o  1 m2 r ≥ 2m + e− 3δb ∪ u ≥ − , δb

for u ≥ −

Wδb = W

Wδb = 0

for

u≤−

for r ≤ 4m, for r ≥ 4m,

 n o  2 3m2 r ≤ 2m + e− δb ⊂ u ≤ − . δb

This yields fδb = f

wδb = w,

Wδb = W

for r ≥

5m . 2

Also, we have (

2

− 2m b 2 , δr

fδb =

f,

2

r ≤ 2m + e− δb ,

for

1

r ≥ 2m + e− 3δb ,

for

and fδb &

1 δb

on

2

1

2m + e− δb ≤ r ≤ 2m + e− 3δb ,

and thus, there exists C > 0 such that, for all r > 0,    2 3m 3m r−1 fδb 1 − ≥ C −1 r−1 1 − r r which is (10.1.39). 2 For u ≤ − 3m b , δ

∂r (fδb)

m2 , δb

= ∂r (r−2 uδb) = −2r−3 uδb + r−2 ∂r (uδb) =

4m2 −3 r . δb

3m2 . δb

628

CHAPTER 10 2

2

m For − 3m b ≤u≤− b δ

δ

∂r (fδb)

= ∂r (r−2 uδb) = −2r−3 uδb + r−2 ∂r (uδb) ! δb −3 −2 0 = −2r uδb + r F − 2 u ∂r u m ! δb −3 −2 0 = −2r uδb + r F − 2 u r2 Υ−1 w, m

and since w ≥ 0 and F 0 ≥ 0, we deduce ∂r (fδb) ≥

−2r−3 uδb ≥ 2δb−1 m2 r−3 .

2

For u ≥ − mb , using Lemma 10.15, we have δ

∂r (fδb)

∂r f ≥

=

16m2 . r3

Hence, for all r ≥ 2m, δb > 0 sufficiently small, ∂r (fδb) ≥

16m2 r3

which is (10.1.40). It remains to evaluate Wδb. This is done in the following lemma. Lemma 10.24. Let W δb(r, m)

:=

1r≤ 5m |Wδb|. 2

(10.1.41)

Then, W δb is supported, for δ > 0 small enough, in the region 2

2m + e− δb ≤ r ≤

9m . 4

Moreover its primitive, Z fb(r, m) := W δ

r

2m

W δb(r0 , m)dr0 ,

(10.1.42)

verifies the pointwise estimate b fb(r, m) . δ. W δ

(10.1.43)

1 Proof. Recall that we have chosen w = 4m to be constant in the region r ≤ 4m. Hence, in that region, ! !! 1 0 δb 1 δb 0 wδb = F − 2u , ∂r wδb = ∂r F − 2 u . 4m m 4m m

629

REGGE-WHEELER TYPE EQUATIONS

Hence, 1 − r−2 ∂r 4

Wδb =

−

=

Now, setting δ0 =

1 2 r Υ∂r 4m

1 −2 r ∂r 16m

r2 Υ∂r

F

0

F0

!!! δb − 2u m !!! δb − 2u . m

b δ m2

for convenience below,

r2 Υ∂r (F 0 (−δ0 u)) = −δ0 F 00 (−δ0 u)r−2 ∂r r2 Υ∂r u

r−2 ∂r

+δ02 F 000 (−δ0 u)Υ(∂r u)2 .

Note that, since r−2 Υ∂r u = w and w = (4m)−1 is constant in r in the region of interest,  4

r r r−2 ∂r r2 Υ∂r u = r−2 ∂r r4 r−2 Υ∂r u = r−2 ∂r = . 4m m Hence, r−2 ∂r r2 Υ∂r (F 0 (−δ0 u))





−δ0 F 00 (−δ0 u)r−2 ∂r r2 Υ∂r u

=

+δ02 F 000 (−δ0 u)Υ(∂r u)2 r −δ0 F 00 (−δ0 u) + δ02 F 000 (−δ0 u)Υ(∂r u)2 . m

= Hence, for r ≤ 4m, with δ0 =

δ02 |Υ||F 000 (−δ0 u)|(∂r u)2 + δ0 |F 00 (−δ0 u)|

|Wδb| . or, since |∂r u| .

1 r−2m ,

b δ m2 ,

in the region of interest,

|Wδb| .

δ02 |F 000 (−δ0 u)| + δ0 |F 00 (−δ0 u)|. |r − 2m|

Since F 00 (−δ0 u) and F 000 (−δ0 u) are supported in the region 1 ≤ −δ0 u ≤ 3, i.e., in − δ30 ≤ u ≤ − δ10 , for δb > 0 sufficiently small 2

1

e− δb ≤ r − 2m ≤ e− 3δb ≤

m . 4

Hence,  W δb = 1r≤ 52 m |Wδb| . δb

 δ + 1 κδb (r − 2m) r − 2m 2

1

with κδb(x) the characteristic function of the interval [e− δb , e− 3δb ]. Note that the primitive of W δb, i.e., Z fb(r, m) = W δ

r

2m

W δb(r0 , m)dr0 ,

630

CHAPTER 10

is a positive, increasing function. Moreover, Z fb(r) . W δ

4m

2m

W δb(r)dr . δ + δ 2

Z

e

− 1 b 3δ

−2 e δb

1 dx . δ x

as desired. We now recall that, see (10.1.16), ˙ R, w](Ψ) = E˙0 [f R, w](Ψ) + 4Υ r − 4m f |Ψ|2 , E[f r4 

3m E˙0 [f R, w](Ψ) = ∂r f |R(Ψ)|2 + r−2 W |Ψ|2 + r−1 1 − r

 b34 . fQ

Using the functions fδb, wδb and Wδb introduced in Definition 10.22, we have   1 3m b ˙ E0 [fδbR, wδb](Ψ) = fb 1 − Q34 + ∂r (fδb)|RΨ|2 + Wδb|Ψ|2 . r δ r Note that in view of the estimates (10.1.39), (10.1.40), and Lemma 10.24, we immediately deduce the existence of a constant C > 0 independent of δb such that   ˙ bR, wb](Ψ) ≥ C −1 |RΨ|2 + |∇ E[f / Ψ|2 + Υ|Ψ|2 δ δ 5m −W δb|Ψ|2 on r ≤ (10.1.44) 2 2

where W δb is a non-negative potential supported in the region 2m + e− δb ≤ r ≤ 9m 4 , R b Combining this with fb(r) = r W b(r0 m)dr0 verifies W fb . δ. whose primitive W δ δ δ 2m estimates of the previous section we derive the following. Proposition 10.25. There exists a constant C > 0, and for any small enough δb > 0, there exist functions fδb ∈ C 2 (r > 0), wδb ∈ C 1 (r > 0) and h ∈ C 2 (r > 0) verifying, for all r > 0, |fδb(r)| . δb−1 ,

wδb . r−1 ,

h . r−4 ,

such that ˙ bR, wb, 2hR] + E [fbR, wb, 2hR](Ψ) = E[f δ δ δ δ

E[fδbR, wδb, 2hR](Ψ) satisfies Z S

Z "

m2 |R(Ψ)|2 r3 S #  2   3m m2 m 2 −1 2 2 +r 1− |∇ / Ψ| + 2 |T Ψ| + 4 |Ψ| r r r Z − W δb|Ψ|2 ,

˙ bR, wb, 2hR] ≥ C −1 E[f δ δ

S

E [fδbR, wδb, 2hR]

1 =  Q· 2

(fδb R)

dec π ¨ + O(r−3 u−1−δ (1 + |fδb|))|Ψ|2 , trap

631

REGGE-WHEELER TYPE EQUATIONS 2

where W δb is non-negative, supported in the region 2m + e− δb ≤ r ≤ R b fb(r) = r W b verifies W fb . δ. that its primitive W δ δ δ 2m

9m 4 ,

and such

Proof. We choose h to be the function of (r, m) introduced in Proposition 10.20, fδb to be the function of (r, m) introduced in Definition 10.22, and W δb, introduced in Lemma 10.24. Also, by an abuse of notation, we denote by wδ,0 b the function denoted by wδb in Definition 10.22. Then, combining Proposition 10.20 in the region 5m r ≥ 5m 2 with the estimate (10.1.44) in the region r ≤ 2 , we immediately obtain Z Z " 2 m −1 ˙ E[fδbR, wδ,0 |R(Ψ)|2 b , 2hR] ≥ C r3 S S #  2 3m m +r−1 1 − |∇ / Ψ|2 + 4 |Ψ|2 r r Z − W δb|Ψ|2 . (10.1.45) S

(10.1.45) corresponds to the desired estimate without the presence of the term |T Ψ|2 on the right-hand side. To get the improved estimate of Proposition 10.25, we set wδb := wδ,0 b − δ1 w1 ,

(10.1.46)

for a small parameter δ1 > 0 to be chosen later, where wδ,0 b is our previous choice introduced in Definition 10.22, and where w1 (r, m) := r−1

 2 m2 3m Υ 1 − . r2 r

(10.1.47)

We evaluate (modulo the same type of error terms as before which we include in E ) ˙ bR, wb, 2hR](Ψ) E[f δ δ

˙ b, wb , 2hR] − 1 δ1 w1 L(Ψ) + δ1 |Ψ|2 r−2 ∂r (r2 Υ∂r w1 ). = E[X δ δ,0 2 4

Now, since L(Ψ)

= −e3 Ψ · e4 Ψ + |∇ / Ψ|2 + V |Ψ|2
−1 2 = Υ −|T Ψ| + |RΨ|2 + |∇ / Ψ|2 + V |Ψ|2 ,

632

CHAPTER 10

we have

=

=

1 δ1 − δ1 w1 L(Ψ) + |Ψ|2 r−2 ∂r (r2 Υ∂r w1 ) 2 4  2 2 1 m 3m − δ1 r−1 2 Υ 1 − L(Ψ) 2 r r  2 !! 2 δ1 3m 2 −2 2 −1 m + |Ψ| r ∂r r Υ∂r r Υ 1− 4 r2 r  2 2 1 −1 3m m δ1 r 1− |T Ψ|2 2 r r2 !  2 m2 3m m 2 2 −1 2 +O(δ1 ) |R(Ψ)| + r 1− |∇ / Ψ| + 4 |Ψ| r3 r r

and hence ˙ bR, wb, 2hR](Ψ) E[f δ δ

 2 2 m ˙ b, wb , 2hR] + 1 δ1 r−1 1 − 3m = E[X |T Ψ|2 δ δ,0 2 r r2 !  2 m2 3m m 2 2 −1 2 +O(δ1 ) |R(Ψ)| + r 1− |∇ / Ψ| + 4 |Ψ| .(10.1.48) r3 r r The desired estimate now follows from (10.1.45) and (10.1.48) provided δ1 > 0 is chosen small enough compared to the constant C > 0 of (10.1.45) so that the last term O(δ1 ) in the above identity can be absorbed. 10.1.10

The redshift vectorfield

Note that the vectorfields T and R both become proportional to e4 for Υ = 0 which means that the estimate of Proposition 10.25 degenerates along Υ = 0, i.e., it does not control e3 (Ψ) there. In this section we make use of the Dafermos-Rodnianski redshift vectorfield to compensate for this degeneracy. The crucial ingredient here is the favorable sign of ω in a small neighborhood of r = 2m. Lemma 10.26. Let π (3) , π (4) denote the deformation tensors of e3 , e4 . In the region r ≤ 3m all components are O() with the exception of (3)

π44

(3) πϕϕ (4)

π34

(4) πϕϕ

8m + O(), r2 2 = κ − ϑ = − + O(), r 4m = 4ω = − 2 + O(), r 2Υ = κ−ϑ= + O(). r = −8ω =

2 (3) πθθ = κ + ϑ = − + O(), r

(4)

πθθ = κ + ϑ =

Proof. Immediate verification in view of our assumptions.

2Υ + O(), r

633

REGGE-WHEELER TYPE EQUATIONS

Lemma 10.27. Given the vectorfield Y = a(r, m)e3 + b(r, m)e4 ,

(10.1.49)

and assuming 

sup r≤3m

 |a| + |∂r a| + |∂m a| + |b| + |∂r b| + |∂m b|

. 1,

we have, for r ≤ 3m,     2m 2m Qαβ (Y ) παβ = a − Υ∂ a Q + ∂ bQ + ∂ a − b − Υ∂ b Q34 r 33 r 44 r r r2 r2 2 Υ + (bΥ − a)e3 Ψ · e4 Ψ + 8 3 (a − Υb)|Ψ|2 r r
+ O() |Q(Ψ)| + r−2 |Ψ|2 . Moreover, with the notation (10.1.14), E[Y, 0](Ψ) =

1 αβ (Y ) r − 3m Q παβ + 4 (−a + bΥ)|Ψ|2 + O()r−2 |Ψ|2 . (10.1.50) 2 r4

Proof. In view of |e4 (r) − Υ, e3 (r) + 1| .

,

Lemma 10.5, and the assumptions on the derivatives of a and b w.r.t. (r, m), we have e4 (a) = Υ∂r a + O(), e4 (b) = Υ∂r b + O(),

e3 (a) = −∂r a + O(), e3 (b) = −∂r b + O(),

eθ (a) = eθ (b) = 0.

We infer Qαβ (Y ) παβ

=

(3)

(4)

aQαβ παβ − (Q33 e4 a + Q43 e3 a) + bQαβ παβ −(Q34 e4 b + Q44 e3 b) + O()|Q(Ψ)|

=

(3)

(4)

aQαβ παβ + bQαβ παβ − Q33 Υ∂r a − Q34 (−∂r a + Υ∂r b) +Q44 ∂r b + O()|Q(Ψ)|.

Note that Qθθ + Qϕϕ = e3 Ψ · e4 Ψ − V |Ψ|2 = e3 Ψ · e4 Ψ − 4

Υ |Ψ|2 + O()r−2 |Ψ|2 .(10.1.51) r2

634

CHAPTER 10

Hence, (3)

(3) = Q44 π44 + Qθθ πθθ + Qϕϕ πϕϕ + O()|Q(Ψ)| 1 8m 2 = Q33 2 − (Qθθ + Qϕϕ ) + O()|Q(Ψ)| 4 r r
2m 2 Υ = Q33 − e3 Ψ · e4 Ψ + 8 3 |Ψ|2 + O() |Q(Ψ)| + r−2 |Ψ|2 , r2 r r

(4)

=

Qαβ παβ

Qαβ παβ

(3)

(3)

(4)

(4)

(4) 2Q34 π34 + Qθθ πθθ + Qϕϕ πϕϕ + O()|Q(Ψ)|

1 m 2Υ Q34 (−4 2 ) + (Qθθ + Qϕϕ ) + O()|Q(Ψ)| 2 r r
2m 2Υ Υ2 = − 2 Q34 + e3 Ψ · e4 Ψ − 8 3 |Ψ|2 + O() |Q(Ψ)| + r−2 |Ψ|2 . r r r =

Therefore, Qαβ (Y ) παβ



 2m 2 Υ 2 Q − e Ψe Ψ + 8 |Ψ| 33 3 4 r2 r r3   2Υ Υ2 2m +b − 2 Q34 + e3 Ψ · e4 Ψ − 8 3 |Ψ|2 − Q33 Υ∂r a r r r

=

a

−

Q34 (−∂r a + Υ∂r b) + Q44 ∂r b + O() |Q(Ψ)| + r−2 |Ψ|2   2m 2 a − Υ∂ a Q33 + Q44 ∂r b + (bΥ − a)e3 Ψe4 Ψ r 2 r r   2m Υ ∂r a − 2 b − Υ∂r b Q34 + 8 3 (a − Υb)|Ψ|2 r r
−2 2 O() |Q(Ψ)| + r |Ψ| .

= + +




To prove the second part of the lemma we recall (see (10.1.14)), E[Y, 0](Ψ)

=

1 1 αβ (Y ) Q παβ − Y (V )|Ψ|2 2 2

and, relying on Lemma 10.5, we have on r ≤ 3m r − 3m (−a + bΥ)∂r V + O() = (−a + bΥ) −8 r4 

Y (V )

=

 + O()

which concludes the proof of the lemma. Corollary 10.28. If we choose a(2m, m) = 1,

b(2m, m) = 0,

∂r a(2m, m) ≥

1 , 4m

∂r b(2m, m) ≥

5 , 4m

then, at r = 2m, we have Qαβ (Y ) παβ ≥


1 (|e3 Ψ|2 + |e4 Ψ|2 + Q34 ) + O() |Q(Ψ)| + r−2 |Ψ|2 (10.1.52) 4m

635

REGGE-WHEELER TYPE EQUATIONS

and 1 8m



1 E[Y, 0](Ψ) ≥ |e3 Ψ| + |e4 Ψ| + Q34 + 2 |Ψ|2 m
+O() |Q(Ψ)| + r−2 |Ψ|2 . 2

2



Moreover the estimates remain true if we add to Y a multiple of T =

(10.1.53) 1 2

(e4 + Υe3 ).

Proof. Recall from Lemma 10.27 that we have, for r ≤ 3m,     2m 2m αβ (Y ) Q παβ = a − Υ∂r a Q33 + ∂r bQ44 + ∂r a − 2 b − Υ∂r b Q34 r2 r 2 Υ + (bΥ − a)e3 Ψ · e4 Ψ + 8 3 (a − Υb)|Ψ|2 r r
+ O() |Q(Ψ)| + r−2 |Ψ|2 . Hence, at r = 2m, using Υ = 0, a = 1, b = 0, ∂r a ≥ (4m)−1 and ∂r b ≥ 5(4m)−1 , we deduce Qαβ (Y ) παβ

=

≥

1 1 Q33 + ∂r bQ44 + ∂r aQ34 − e3 Ψ · e4 Ψ 2m m
+O() |Q(Ψ)| + r−2 |Ψ|2 1 5 1 1 |e3 (Ψ)|2 + |e4 (Ψ)|2 + Q34 − e3 Ψ · e4 Ψ 2m 4m 4m m
+O() |Q(Ψ)| + r−2 |Ψ|2

from which the desired lower bound in (10.1.52) follows. Also, at r = 2m, using (10.1.50), Υ = 0, a = 1, and b = 0, we have E[Y, 0](Ψ)

= = ≥

1 αβ (Y ) r − 3m Q παβ + 4 (−a + bΥ)|Ψ|2 + O()r−2 |Ψ|2 2 r4 1 αβ (Y ) 1 Q παβ + |Ψ|2 2  4m3 
1 1 2 2 2 |e3 Ψ| + |e4 Ψ| + Q34 + 2 |Ψ| + O() |Q(Ψ)| + r−2 |Ψ|2 8m m

which yields (10.1.53). We are now ready to prove the following result. Proposition 10.29. Given a small parameter δH > 0 there exists a smooth vec1 10 torfield YH supported in the region |Υ| ≤ 2δH such that the following estimate holds:   1 2 2 b34 + m−2 |Ψ|2 1 E[YH , 0](Ψ) ≥ 1 |e Ψ| + |e Ψ| + Q 3 4 16m |Υ|≤δH10   1 1 − 10 2 2 −2 2 b 1 1 − δH 1 10 |e Ψ| + |e Ψ| + Q + m |Ψ| 3 4 34 10 δH ≤Υ≤2δH m
1 + O()1 |Q(Ψ)| + m−2 |Ψ|2 . 10 |Υ|≤2δH

636

CHAPTER 10 1

10 Moreover, for |Υ| ≤ δH , we have 1 10 YH = e3 + e4 + O(δH )(e3 + e4 ).

Proof. We introduce the vectorfield Y(0) := ae3 + be4 + 2T,

a(r, m) := 1 +

5 (r − 2m), 4m

b(r, m) :=

5 (r − 2m), 4m

with T = 12 (e4 + Υe3 ). Also, we pick positive bump function κ = κ(r), supported in the region in [−2, 2] and equal to 1 for [−1, 1] and define, for sufficiently small δH > 0, ! Υ YH := κH Y(0) , κH := κ . (10.1.54) 1 10 δH We have E[YH , 0](Ψ)

= = =

Q·

(YH )

κH Q ·

π − YH (V )|Ψ|2

(Y0 )

π + Q(Y(0) , dκH ) + κH Y(0) (V )|Ψ|2

κH E[Y(0) , 0](Ψ) −

1

+O(δH 10 )1

1 10 δH



1 10 ≤Υ≤2δH

 b34 + m−2 |Ψ|2 . |e3 Ψ|2 + |e4 Ψ|2 + Q

Note from the definition of Y(0) and the choice of a and b that Corollary 10.28 applies to Y(0) . In particular, we deduce from (10.1.53) for δH > 0 small enough, E[YH , 0](Ψ) ≥ − +

  1 2 2 b34 + m−2 |Ψ|2 1 1 |e Ψ| + |e Ψ| + Q 3 4 16m |Υ|≤δH10   1 1 − 10 2 2 −2 2 b 1 1 δH 1 10 |e Ψ| + |e Ψ| + Q + m |Ψ| 3 4 34 10 δH ≤Υ≤2δH m
1 O()1 |Q(Ψ)| + m−2 |Ψ|2 10 |Υ|≤2δH

as desired. 10.1.11

Combined estimate

We consider the combined Morawetz triplet (X, w, M ) := (Xδb, wδb, 2hR) + H (YH , 0, 0),

(10.1.55)

with H > 0 sufficiently small to be determined later. Here (Xδb = fδbR, wδb, 2hR) is the triplet given by Proposition 10.25 and YH the vectorfield of Proposition 10.29.

637

REGGE-WHEELER TYPE EQUATIONS

˙ bR, wb, 2hR](Ψ) verifies Recall, see Proposition 10.25, that E˙δb := E[f δ δ "  2   Z Z 2 m2 3m 2 −1 b34 + m |T Ψ|2 E˙δb ≥ C −1 |R(Ψ)| + 1 − r Q r3 r r2 S S # Z m 2 +Υ 4 |Ψ| − W δb|Ψ|2 . r S According to Proposition 10.29, we write, for EH = E(YH , 0, 0)(Ψ), EH E˙H

EH,

= E˙H + EH, ,   1 2 2 b34 + m−2 |Ψ|2 1 ≥ 1 |e Ψ| + |e Ψ| + Q 3 4 8m |Υ|≤δH10   1 1 − 10 2 2 b34 + m−2 |Ψ|2 , 1 1 − δH 1 10 |e Ψ| + |e Ψ| + Q 3 4 10 δH ≤Υ≤2δH m
−2 1 . = O() |Q(Ψ)| + m |Ψ|2 1 10 |Υ|≤2δH

1

10 Note that, for |Υ| ≥ δH , we have

|RΨ|2 + |T Ψ|2 =

1 1 15 (|e4 Ψ|2 + Υ2 |e3 Ψ|2 ) ≥ δH (|e4 Ψ|2 + |e3 Ψ|2 ). 2 2

We now proceed to find a lower bound for the expression E˙δb + H E˙H . For brevity the S integration is omitted below. 1

1

10 10 Region δH ≤ |Υ| ≤ 2δH .

E˙δb + H E˙H

h 1 i 5 ≥ m−1 C −1 δH (|e4 Ψ|2 + |e3 Ψ|2 ) + m−2 |Ψ|2 + |∇ / Ψ|2 − W δb|Ψ|2 − H

1
1 − 10 δ |e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + m−2 |Ψ|2 . m H 3

10 Therefore, choosing H ≤ (2C)−1 δH , we deduce

E˙δb + H E˙H

1
5 ≥ m−1 δH (2C)−1 |e4 Ψ|2 + |e3 Ψ|2 + |∇ / Ψ|2 + m−2 |Ψ|2 − W δb|Ψ|2 . 1

10 Region |Υ| ≤ δH .

H E˙H + E˙δb ≥

H

 1  b34 + m−2 |Ψ|2 − W b|Ψ|2 . |e3 Ψ|2 + |e4 Ψ|2 + Q δ 16m

1

10 Region Υ ≥ 2δH . In this region E˙δb + H E˙H = E˙δb. Hence (ignoring the S integration),

E˙δb + H E˙H

≥ C

−1

!  2   m2 3m m2 m 2 2 −1 2 b |R(Ψ)| + 1 − r Q34 + 2 |T Ψ| + 4 |Ψ| r3 r r r

− W δb|Ψ|2 . To combine these three cases together we modify the vectorfields R, T near

638

CHAPTER 10

r = 2m according to (5.1.11), i.e., ˘ := θ 1 (e4 − e3 ) + (1 − θ)Υ−1 R = R 2 1 ˘ T := θ (e4 + e3 ) + (1 − θ)Υ−1 T = 2

i 1 h˘ θe4 − e3 , 2 i 1 h˘ θe4 + e3 , 2 1

1

10 10 where θ a smooth bump function equal 1 on |Υ| ≤ δH vanishing for |Υ| ≥ 2δH , and where ( 1 10 1, for |Υ| ≤ δH , −1 ˘ θ = θ + Υ (1 − θ) = 1 −1 10 Υ , for |Υ| ≥ 2δH .

Note that ˘ 2 + |T˘Ψ|2 ) = |e3 Ψ|2 + θ˘2 |e4 Ψ|2 . 2(|RΨ| 1

10 ˘ 2 + |T˘Ψ|2 ) and Thus in the region |Υ| ≤ δH we have |e3 Ψ|2 + |e4 Ψ|2 = 2(|RΨ| therefore,

E˙δb + H E˙H

≥ =

 1  b34 + m−2 |Ψ|2 − W b|Ψ|2 |e3 Ψ|2 + |e4 Ψ|2 + Q δ 16m  1  ˘ 2 b34 + m−2 |Ψ|2 − W b|Ψ|2 . H |RΨ| + |T˘Ψ|2 + Q δ 16m H

1

1

−1

10 10 ˘ 2 + |T˘Ψ|2 . |e3 Ψ|2 + δ 5 |e4 Ψ|2 . In the region δH ≤ |Υ| ≤ 2δH , we have |RΨ| H 3

10 Hence, for H ≤ (2C)−1 δH , we deduce

E˙δb + H E˙H

1
5 ≥ m−1 δH (2C)−1 |e4 Ψ|2 + |e3 Ψ|2 + |∇ / Ψ|2 + m−2 |Ψ|2 − W δb|Ψ|2  1    1 ˘ 2 + |T˘Ψ|2 + |∇ ≥ m−1 δ 5 (2C)−1 δ 5 |RΨ| / Ψ|2 + m−2 |Ψ|2

H

H

2

−W δb|Ψ| . 1

10 ˘ = Υ−1 R, T˘ = Υ−1 T . Hence, Finally, for Υ ≥ 2δH we have R "  2   2 3m m2 −1 m 2 −1 2 ˙ ˙ b Eδb + H EH ≥ C |R(Ψ)| + 1 − r Q34 + 2 |T Ψ| r3 r r # m + Υ 4 |Ψ|2 − W δb|Ψ|2 r "  2   1 m2 2 m2 3m −1 2 −1 2 5 10 ˘ b ˘ ≥ C δH 3 |R(Ψ)| + 1 − r Q34 + δH 2 |T Ψ| r r r # m + Υ 4 |Ψ|2 − W δb|Ψ|2 . r

We deduce the following. Proposition 10.30. Let C > 0 the constant of Proposition 10.25. Consider the

639

REGGE-WHEELER TYPE EQUATIONS

combined Morawetz triplet (X, w, M ) := (Xδb, wδb, 2hR) + H (YH , 0, 0), 2

(10.1.56)

3

5 10 with C −1 δH ≤ H ≤ (2C)−1 δH where, for given fixed δb > 0, (Xδb, wδb, 2hR) is the triplet of Proposition 10.25 and YH the vectorfield of Proposition 10.29, supported 1 10 b Let E˙b, E˙H be the in |Υ| ≤ 2δH with δH > 0 sufficiently small, independent of δ. δ principal parts of E[fδbR, wδb, 2hR](Ψ) and respectively EH [YH , 0, 0](Ψ) and Eδ, b , EH, the corresponding error terms, i.e.,

E[fδbR, wδb, 2hR](Ψ) = E˙δb + Eδ, b ,

EH [YH , 0, 0](Ψ) = E˙H + EH, .

Then, provided δH > 0 is sufficiently small, we have: 2

1

−1 −1 5 10 1. In the region −2δH ≤ Υ, r ≤ 5m δH > 0 2 , we have with a constant ΛH := C Z Z   2 ˘ (E˙δb + H E˙H ) ≥ m−1 Λ−1 |R(Ψ)| + |T˘Ψ|2 + |∇ / Ψ|2 + m−2 |Ψ|2 H S S Z − W δb|Ψ|2 . S

˙b + H E˙H = E˙b and W b = 0, we have the same 2. In the region r ≥ 5m 2 , where Eδ δ δ estimate as in Proposition 10.25, i.e., Z Z " 2 m −1 (E˙δb + H E˙H ) ≥ C |R(Ψ)|2 r3 S S #  2   2 3m m m +r−1 1 − |∇ / Ψ|2 + 2 |T Ψ|2 + 4 |Ψ|2 . r r r 3. The -error terms verify the upper bound estimate,   b−1 u−1−δdec r−2 |e3 Ψ|2 + r−1 (|e4 Ψ|2 + |∇ Eδ, / Ψ|2 ) b + H EH, . Cδ trap +

dec −1 Cδb−1 u−1−δ r |e3 Ψ| (|e4 Ψ| + |∇ / Ψ|) trap

dec −3 + Cδb−1 u−1−δ r |Ψ|2 . trap

Proof. It only remains to check the last part. In view of Proposition 10.20 we have Eδ, b

1 = E [fδbR, wδb, 2hR](Ψ) =  Q · 2

(Xδb )


dec π ¨ + O r−3 u−1−δ (|fδb| + 1) |Ψ|2 trap

and |fδb| . δb−1 . Hence,  Eδ, b

= E [fδbR, wδb, 2hR](Ψ) = 

1 Q· 2

Xδb

 −1 −3 −1−δdec 2 b π ¨ + O(δ r utrap )|Ψ| .

We write, with π ¨ =(Xδb) π ¨ for simplicity, Q· π ¨

=

1 1 (Q33 π ¨44 + 2Q34 π ¨34 + Q44 π˙ 33 ) − (QA3 π ¨A4 + QA4 π ¨A3 ) + QAB π ¨AB . 4 2

640

CHAPTER 10

Thus, recalling parts 1 and 2 of Proposition 10.9, and Lemma 10.8, Q· π ¨

.

dec dec r−2 u−1−δ |e3 Ψ|2 + r−1 u−1−δ |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 trap trap




dec + r−1 u−1−δ |e3 Ψ| (|e4 Ψ| + |∇ / Ψ|) . trap

1

10 Finally, since r ∼ 2m and utrap = 1 on |Υ| ≤ 2δH , the error terms generated by the redshift vectorfield YH ,
1 EH, = O()1 |Q(Ψ)| + m−2 |Ψ|2 , 10

|Υ|≤2δH

can easily be absorbed on the right-hand side to derive the desired estimate. 10.1.11.1

Elimination of W δb

We now proceed to eliminate the potential W δb by a procedure analogous to that used in section 10.1.8. More precisely we set, in view of (10.1.14), Eδb = E[fδbR, wδb, 2hR](Ψ),

˘ Eδb0 = E[fδbR, wδb, 2(hR + h2 R)](Ψ),

and ˘ + 1 Dµ (h2 R ˘ µ )|Ψ|2 , Eδb0 = Eδb + h2 ΨRΨ 2 where h2 is a smooth, compactly supported function supported8 in the region r ≤ 9m 4 . Thus, we have in view of Proposition 10.30, ignoring the integration on S, E˙δb0 + H E˙H

= ≥

˘ + 1 Dµ (h2 R ˘ µ )|Ψ|2 E˙δb + H E˙H + h2 ΨRΨ (10.1.57) 2   1 ˘ −1 −1 2 2 2 −2 2 ˘ I(Ψ) + m ΛH |R(Ψ)| + |T Ψ| + |∇ / Ψ| + m |Ψ| 2

where I(Ψ)

:=

1 −1 −1 ˘ ˘ + 1 Dµ (h2 R ˘ µ )|Ψ|2 − W b|Ψ|2 Λ m |R(Ψ)|2 + Ψh2 RΨ δ 2 H 2

so that we have I(Ψ) ≥

i 1h µ ˘ µ ) − 2W b − mΛH h2 |Ψ|2 . D (h2 R 2 δ 2

(10.1.58)

We focus on the coefficient in front of |Ψ|2 on the RHS of (10.1.58). Ignoring the error terms in  (which can easily be incorporated in the upper bound for Eδ, b + H EH, of the previous proposition), we have ˘= DivR

 1  1 1 µ ˘ ˘ (4) − trπ (3) + 1 e4 (θ) ˘ = O(δ − 10 ) D (θ(e4 )µ ) − Dµ (e3 )µ = θtrπ H 2 4 2

8 Recall

that W δb is supported in the region 2m < r ≤

5m . 2

641

REGGE-WHEELER TYPE EQUATIONS

and, using in particular Lemma 10.5, ˘µ) Dµ (h2 R

˘ 4 r − e3 r) + h2 DivR ˘ 2 + h2 DivR ˘ = 1 ∂r h2 (θe ˘ Rh 2 1 1 ˘ + 1) + h2 O(δ − 10 ) ∂r h2 (θΥ H 2 1 − 1 ∂r h2 + h2 O(δH 10 ). 2

= = ≥

Together with (10.1.58), we infer i 1h − 1 ∂r h2 − 4W δb − 4mΛH h22 + h2 O(δH 10 ) |Ψ|2 . 4

I(Ψ) ≥

(10.1.59)

We now consider the choice of the function h2 = h2 (r, m). Recall (see Lemma 2 10.24) that W δb is supported in the region 2m+e− δb ≤ r ≤ 9m 4 and that its primitive Rr −2 b fb(r) := f W W verifies W . m δ. We choose b b δ δ δ 2m ( h2

=:

fb, 4W δ

for r ≤

0,

for r ≥

b we may extend h2 in fb . m−2 δ, and since W δ we have for all r > 0 b |h2 | . m−2 δ,

9m 4

9m 4 5m 2

≤r≤

5m 2

(10.1.60) such that h2 is C 1 and

b |∂r h2 | . m−3 δ.

(10.1.61)

In view of (10.1.59), this choice of h2 yields I(Ψ) ≥

 1 1  b H )− 10 − O m−1 δb + ΛH m−1 δb2 + δ(δ |Ψ|2 . 4

1

1

2

−1 −1 5 10 2 b Hence, for δb  δH Λ−1 δH ) and h2 defined as H , i.e., δ  δH (recall that ΛH = C above, we infer

I(Ψ) ≥

1 −2 − m−1 Λ−1 |Ψ|2 H m 2

which together with (10.1.57) finally yields Z Z 0 ˙ ˙ (Eδb + H EH ) ≥ I(Ψ) S S  Z  1 ˘ 2 2 −2 2 ˘Ψ|2 + |∇ +m−1 Λ−1 | R(Ψ)| + | T / Ψ| + m |Ψ| H 2 S  Z  1 ˘ 1 −2 2 2 2 ˘Ψ|2 + |∇ ≥ m−1 Λ−1 | R(Ψ)| + | T / Ψ| + m |Ψ| . H 2 2 S 10.1.11.2

Summary of results so far

We summarize the result in the following.

642

CHAPTER 10

Proposition 10.31. Consider the combined Morawetz triplet ˇ (X, w, M ) := (fδbR, wδb, 2hR) + H (YH , 0, 0) + (0, 0, 2h2 R)

(10.1.62)

with (fδbR, wδb, 2hR) the triplet of Proposition 10.25, YH the redshift vectorfield of Proposition 10.29 (corresponding to the small parameter δH ) and h2 the C 1 ˙ function above satisfying (10.1.60), (10.1.61). Let E[X, w, M ] the principal part of E[X, w, M ] (independent of ) and E [X, w, M ] the error term in  such that E = E˙ + E . We choose the small strictly positive parameters H , δH , δ such that9 7

3

20 H = δ H ,

5 δb = δH .

(10.1.63)

Then, there holds10 Z

1 2

S

E[X, w, M ](Ψ) ≥ δH +

E [X, w, M ](Ψ)

10.1.12

Z " S

 2   m2 ˘ 2 3m m2 ˘ 2 −1 b |RΨ| + r 1− Q34 + 2 |T Ψ| r3 r r #

m 2 |Ψ| , r4

(10.1.64)

−1 −1−δdec ≤ δH utrap r−2 |e3 Ψ|2 + r−1 (|e4 Ψ|2 −1 −1−δdec −1 + δH utrap r |e3 Ψ| (|e4 Ψ| + |∇ / Ψ|) −1 −1−δdec −3 2 + δH utrap r |Ψ| .



2

+ |∇ / Ψ| )



Lower bounds for Q

In this section we prove lower bounds for Q(X + 2ΛT, e3 ) and Q(X + 2ΛT, e4 ) in the region rH ≤ r, for rH to be determined and Λ sufficiently large. Proposition 10.32. Under the assumptions of Proposition 10.31, and with the choice Λ :=

1 − 13 δ 20 , 4 H

(10.1.65)

the following inequalities hold true for r ≥ 2m0 (1 − δH ). 1

10 1. For the region such that r ≥ 2m0 (1 − δH ) and Υ ≤ δH , we have

Q(X + ΛT, e3 ) ≥ Q(X + ΛT, e4 ) ≥ 9 Note 3 10

1 ΛQ34 , 2 1 ΛQ44 . 2 2

5 that (10.1.63) verifies all the restrictions we have encountered so far, i.e., δH  H  1 2

δH and 0 < δb  δH . 10 Note

1 H Q33 + 4 1 H Q34 + 4

1

2

−1 −1 δ 5 ) and δ −1  δ 2 b−1 in view of (10.1.63). that δH  Λ−1 H (recall that ΛH = C H H

643

REGGE-WHEELER TYPE EQUATIONS 1

10 2. For the region δH ≤ Υ ≤ 13 , we have

−1

Q(X + ΛT, e3 ) ≥ δH 2 (Q33 + Q34 ) , −1

Q(X + ΛT, e4 ) ≥ δH 2 (Q44 + Q34 ) . 3. For the region r ≥ 3m, we have Q(X + ΛT, e3 ) ≥ Q(X + ΛT, e4 ) ≥

1 Λ (Q33 + Q34 ) , 4 1 Λ (Q44 + Q34 ) . 4

4. The null components of Q are given by (recall Proposition 10.9), Q33

= |e3 Ψ|2 ,

Q44 = |e4 Ψ|2 ,

Q34 = |∇ / Ψ|2 +

4Υ (1 + O())|Ψ|2 . r2

Proof. Since X = fδbR + H YH and T = 12 (e4 + Υe3 ), R = 12 (e4 − Υe3 ), we write Q(X + 2ΛT, e3 )

= Q(X, e3 ) + ΛQ(e4 + Υe3 , e3 ) = Q(X, e3 ) + Λ (Q34 + ΥQ33 ) 1 = H Q(YH , e3 ) + Λ (Q34 + ΥQ33 ) + fδb (Q34 − ΥQ33 ) . 2

In the region 2m0 (1 − δH ) ≤ r ≤ 2m we have YH = e3 + e4 + O(δH )(e3 + e4 ), Υ ≥ 0 and fδb < 0. Hence, in that region,     1 1 1 Q(X + 2ΛT, e3 ) ≥ H (Q33 + Q34 ) + Λ − |fδb| Q34 − |Υ| Λ + |fδb| Q33 2 2 2      1 1 1 1 ≥ H − |Υ| Λ + |fδb| Q33 + H + Λ − |fδb| Q34 . 2 2 2 2 Thus, we need to choose Λ such that 1 1 H 1 |fδb| ≤ Λ ≤ − |fδb|. 2 4 δH 2 b −1 ). Thus it suffices Now, recall (10.1.63) as well as the fact that |fδb| is of size O((δ) to choose Λ such that −3

O(δH 5 ) ≤ Λ ≤

1 − 13 −3 δH 20 − O(δH 5 ), 2

i.e., it suffices to choose, for δH > 0 small enough, Λ=

1 − 13 δ 20 , 4 H

to deduce the inequality, Q(X + 2ΛT, e3 ) ≥

1 1 H Q33 + ΛQ34 . 4 2

644

CHAPTER 10 1

10 Next, in the region 0 ≤ Υ ≤ δH , the sign of Υ is more favorable and we have     1 1 1 Q(X + 2ΛT, e3 ) ≥ H (Q33 + Q34 ) + Λ − |fδb| Q34 + |Υ| Λ + |fδb| Q33 2 2 2      1 1 1 1 ≥ H + |Υ| Λ + |fδb| Q33 + H + Λ − |fδb| Q34 . 2 2 2 2

In particular, we simply need Λ  δb−1 , which is in particular satisfied by (10.1.65), to deduce the same inequality, Q(X + 2ΛT, e3 ) ≥

1 1 H Q33 + ΛQ34 . 4 2

1

10 In the region δH ≤ Υ ≤ 13 , where fδb ≤ 0, and using the fact that |fδb| is of size b −1 ) O((δ)

Q(X + 2ΛT, e3 )

1 = H Q(YH , e3 ) + Λ (Q34 + ΥQ33 ) + fδb (Q34 − ΥQ33 ) 2   1 −1 10 ≥ Λ Q34 + δH Q33 − O(δb )Q34 .

Hence, for the choice (10.1.65), and in view of (10.1.63), we infer −1

Q(X + 2ΛT, e3 ) ≥ δH 2 (Q33 + Q34 ) . Finally, for r ≥ 3m where we have 0 ≤ fδb . 1, Q(X + 2ΛT, e3 )

= ≥

1 3

≤ Υ ≤ 1 and YH = 0,

1 Λ (Q34 + ΥQ33 ) + fδb (Q34 − ΥQ33 )   2 1 Λ Q34 + Q33 − O(1)Q33 3

and hence, (10.1.65) implies Q(X + 2ΛT, e3 ) ≥

1 Λ (Q34 + Q33 ) 4

as desired. The proof for Q(X + ΛT, e4 ) is similar. 10.1.13

First Morawetz estimate

We are now ready to state our first Morawetz estimate which is simply obtained by integrating the pointwise inequality in Proposition 10.30 on our spacetime domain M = (int) M ∪ (ext) M described at the beginning of the section, with X replaced by X + ΛT for Λ > 0 sufficiently large. In view of the choice of τ , note that we have NΣ = ae3 + be4 ,

0 ≤ a, b ≤ 1,

a + b ≥ 1,

(10.1.66)

645

REGGE-WHEELER TYPE EQUATIONS

with b = 0, a = 1 on

(int)

M,

a = 0, b = 1 on Mr≥4m0 ,

a, b ≥

1 on 4

(trap)

M.

We recall the following quantities for Ψ in regions M(τ1 , τ2 ) ⊂ M in the past of Σ(τ2 ) and future of Σ(τ1 ). 1. Morawetz bulk quantity "

m2 ˘ 2 m 2 |Rψ| + 4 |Ψ| r3 r M(τ1 ,τ2 )  2  # 3m 1 m2 ˘ 2 2 + 1− |∇ / ψ| + 2 |T ψ| . r r r Z

Mor[Ψ](τ1 , τ2 )

=

2. Basic energy quantity   Z 1 1 E[Ψ](τ ) = (NΣ , e3 )2 |e4 Ψ|2 + (NΣ , e4 )2 |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 . 2 2 Σ(τ ) 3. Flux through A and Σ∗ Z F [Ψ](τ1 , τ2 ) = A(τ1 ,τ2 )

−1 δH |e4 Ψ|2 + δH |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2

Z + Σ∗ (τ1 ,τ2 )





|e4 Ψ|2 + |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 ,

with A(τ1 , τ2 ) = A ∩ M(τ1 , τ2 ) and Σ∗ (τ1 , τ2 ) = Σ∗ ∩ M(τ1 , τ2 ). The following theorem is our first Morawetz estimate. ˙ = V Ψ + N , with the Theorem 10.33. Consider the equation (10.1.10), i.e., Ψ potential V = −κκ and a domain M(τ1 , τ2 ) ⊂ M. Then, we have E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) . E[Ψ](τ1 ) + J[N, Ψ](τ1 , τ2 ) + Err (τ1 , τ2 )[Ψ], Z h ˘ + |T˘Ψ| J[N, Ψ](τ1 , τ2 ) : = |RΨ| M(τ1 ,τ2 )

i + r |Ψ| |N |, Z Err [Ψ](τ1 , τ2 ) = E [Ψ], −1

M(τ1 ,τ2 )

where h −1−δdec E [Ψ] . utrap r−2 |e3 Ψ|2 + r−1 (|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 i +|e3 Ψ| (|e4 Ψ| + |∇ / Ψ|)) .

(10.1.67)

646

CHAPTER 10

Proof. Recall that, see (10.1.13), E[X, w, M ](Ψ)

  1 := Dµ Pµ [X, w, M ] − X(Ψ) + wΨ · N [Ψ] 2

where Pµ

1 ˙ µ Ψ − 1 |Ψ|2 ∂µ w + 1 |Ψ|2 Mµ Pµ [X, w, M ] = Qµν X ν + wΨD 2 4 4

=

with triplet ˘ (X, w, M ) := (fδbR, wδb, 2hR) + H (YH , 0, 0) + (0, 0, 2h2 R) ˇ = X + ΛT in the calculation above given in Proposition 10.30. Replacing X by X we deduce ˙ µ Ψ − 1 |Ψ|2 ∂µ w + 1 |Ψ|2 Mµ . ˇ w, M ] = Qµν X ˇ ν + 1 wΨD = Pµ [X, 2 4 4

Pˇµ

By the divergence theorem we have Z Z Z Z Z ˇ ˇ ˇ P · NA + P · NΣ + E+ P · NΣ∗ = Pˇ · NΣ A Σ2 M(τ1 ,τ2 ) Σ∗ Σ1 Z (10.1.68) 1 ˇ − (X(Ψ) + wΨ)N [Ψ] 2 M(τ1 ,τ2 ) ˇ w, M ](Ψ). Now, where E = E[X, ˇ w, M ](Ψ) E[X,

1 = E[X, w, M ](Ψ) + ΛQ · 2

(T )

1 π − T (V )|Ψ|2 . 2

dec According to Lemma 10.7, T (V ) = O()r−3 u−1−δ , and all components of trap −1 −1−δdec (T ) dec π are O(r utrap ) except for π 44 which is O(r−2 u−1−δ ). We easily trap deduce

(T )

Λ|Q ·

(T )

π| + |T (V )||Ψ|2

.

ΛE .

Thus in view of Proposition 10.31, we have11 "  2   Z Z 1 m2 ˘ 2 3m m2 ˘ 2 −1 2 2 E ≥ δH |RΨ| + r 1− |∇ / Ψ| + 2 |T Ψ| r3 r r M(τ1 ,τ2 ) M(τ1 ,τ2 ) # Z
m −1 + 4 |Ψ|2 − O δH E , r M(τ1 ,τ2 ) i.e., Z M(τ1 ,τ2 )

11 Recall

E

1
−1 2 ≥ δH Mor[Ψ](τ1 , τ2 ) − O δH Err (τ1 , τ2 ).

from (10.1.65) that we have Λ =

13 1 − 20 δ 4 H

−1  δH .

(10.1.69)

647

REGGE-WHEELER TYPE EQUATIONS

We now analyze the boundary terms in (10.1.68). 10.1.13.1

Boundary term along A

Along the spacelike hypersurface A, i.e., r = 2m0 (1 − δH ), the unit normal NA is given by   1 e4 (r) q NA = e4 + e3 e3 (r) 2 ee43 (r) (r)   1 p = e4 + (δH + O())e3 , 2 δH + O() and we have h, h2 = 0 as well as w = −δ1 w1 where δ1 > 0 is a small constant and w1 is given by (10.1.47) w1 (r, m) =

m2 r−1 2 Υ r



3m 1− r

2 .

In particular, we have on A in view of the formula for w1 and for NA p |w1 | . δH , |NA (w1 )| . δH . Hence, P · NA

δ1 δ1 w1 ΨNA (Ψ) + |Ψ|2 NA (w1 ) 2 4 p Q(X + ΛT, e4 ) + 2 δH + O()Q(X + ΛT, e3 )

= Q(X + ΛT, NA ) − =

2 p δH + O() p p 3 2 −O( δH )Ψe4 (Ψ) − O(δH )Ψe3 (Ψ) − O( δH )|Ψ|2 .

Thus, in view of Proposition 10.32, we infer   2 1 1 P · NA ≥ p H Q34 + ΛQ44 2 δH + O() 4   p 1 1 +2 δH + O() H Q33 + ΛQ34 4 2 p p 3 2 −O( δH )Ψe4 (Ψ) − O(δH )Ψe3 (Ψ) − O( δH )|Ψ|2 .

648

CHAPTER 10

Using in particular (10.1.63) and (10.1.65), we deduce     1 13 7 2 1 20 − P · NA ≥ p δH |∇ / Ψ|2 + O(δH )|Ψ|2 + δH 20 |e4 Ψ|2 8 δH + O() 4    p 7 1 20 1 − 13 2 2 2 20 +2 δH + O() δ |e3 Ψ| + δH |∇ / Ψ| + O(δH )|Ψ| 4 H 8 p p 3 2 −O( δH )Ψe4 (Ψ) − O(δH )Ψe3 (Ψ) − O( δH )|Ψ|2 3 1 − 20 1 − 23 1 17 20 ≥ δH |∇ / Ψ|2 + δH 20 |e4 Ψ|2 + δH |e3 Ψ|2 2 8 4 p p 3 2 −O( δH )Ψe4 (Ψ) − O(δH )Ψe3 (Ψ) − O( δH )|Ψ|2 . Recalling the Poincar´e inequality (10.1.27), Z Z
|∇ / Ψ|2 ≥ 2r−2 1 − O() Ψ2 daS , S

S

we deduce, in this region, Z Z   1 −1 P · NA ≥ δH |e4 Ψ|2 + δH |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 8 A(τ1 ,τ2 ) A(τ1 ,τ2 ) as desired in view of the definition of the flux along A. 10.1.13.2

Boundary terms along Σ(τ1 ), Σ(τ2 )

Along a hypersurface Σ(τ ) with timelike unit future normal NΣ(τ ) = ae3 + be4 , we have 1 1 1 2 ˘ P · NΣ = Q(X + ΛT, NΣ ) + wΨNΣ (Ψ) − NΣ (w)|Ψ|2 + NΣ · (hR + h2 R)|Ψ| 2 4 2 and Z E[Ψ](τ )

=



Σ(τ )

 2b2 |e4 Ψ|2 + 2a2 |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 . 1

10 b and NΣ = e3 1. In the region r ≥ 2m0 (1 − δH ), Υ ≤ δH we have h = 0, h2 = O(δ) (i.e., a = 1, b = 0). Also, we have w = −δ1 w1 , where δ1 > 0 is a small constant and w1 is given by (10.1.47)

w1 (r, m) =

m2 r−1 2 Υ r



3m 1− r

2 .

In particular, we have in the region of interest, in view of the formula for w1 and for NΣ 1

10 |w1 | . δH ,

|NΣ (w1 )| = |e3 (w1 )| . 1.

649

REGGE-WHEELER TYPE EQUATIONS

We infer P · NΣ

δ1 δ1 1 ˘ 2 w1 Ψe3 (Ψ) + e3 (w1 )|Ψ|2 + h2 e3 · R|Ψ| 2 4 2

=

Q(X + ΛT, e3 ) −

=

10 Q(X + ΛT, e3 ) − O(δH )w1 Ψe3 (Ψ) − O(1)|Ψ|2

1

˘ = where we used the fact that R according to Proposition 10.32, P · NΣ

1 2 (e4

− e3 ) in the region of interest. Thus,

1 1 1 10 H Q33 + ΛQ34 − O(δH )|Ψ||e3 (Ψ)| − O(1)|Ψ|2 . 4 2

≥

Using in particular (10.1.63) and (10.1.65), we deduce P · NΣ ≥

7 1 1 20 1 − 13 10 δH |e3 Ψ|2 + δH 20 (|∇ / Ψ|2 + O()|Ψ|2 ) − O(δH )|Ψ||e3 Ψ| − O(1)|Ψ|2 . 4 8

Together with the Poincar´e inequality (10.1.27), we deduce Z P · NΣ Σ

1 r≥2m0 (1−δH ), Υ≤δ 10 H

(τ )

Z

≥

7 1 20 δ 8 H

≥

7 1 20 1 [Ψ](τ ). δ E 8 H r≥2m0 (1−δH ), Υ≤δH10



Σ

1 r≥2m0 (1−δH ), Υ≤δ 10 H

(τ )

|e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2



1

10 2. In the region Υ ≥ δH , we have w = O(r−1 ), NΣ (w) = O(r−2 ), h = O(r−4 ) and −4 h2 = O(r ). We infer

P · NΣ

= aQ(X + ΛT, e3 ) + bQ(X + ΛT, e4 ) − O(r−1 )|Ψ|(a|e3 Ψ| + b|e4 Ψ|) −O(r−2 )|Ψ|2 .

Thus, according to Proposition 10.32, P · NΣ

−1

≥ δH 2 (aQ33 + bQ44 + (a + b)Q34 ) − O(1)(a2 |e3 Ψ|2 + b2 |e4 Ψ|2 ) =

≥

−O(r−2 )|Ψ|2    4Υ −1 δH 2 a|e3 Ψ|2 + b|e4 Ψ|3 + (a + b) |∇ / Ψ|2 + 2 |Ψ|2 r   −O(1) a2 |e3 Ψ|2 + b2 |e4 Ψ|2 + r−2 |Ψ|2 !! 1 10 4δH − 12 2 3 2 2 δH a|e3 Ψ| + b|e4 Ψ| + (a + b) |∇ / Ψ| + 2 |Ψ| r   −O(1) a2 |e3 Ψ|2 + b2 |e4 Ψ|2 + r−2 |Ψ|2 .

Hence, for δH > 0 sufficiently small, and since a2 ≤ a, b2 ≤ b and a + b ≥ 1, we

650

CHAPTER 10

infer in this region P · NΣ

−1

Z



≥

δH 5

=

δH 5 E[Ψ]

Σ

1 Υ≥δ 10 H

(τ )

−1

1

10 Υ≥δH

2b2 |e4 Ψ|2 + 2a2 |e3 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2



(τ ). 1

1

10 10 In view of the above estimates in r ≥ 2m0 (1 − δH ), Υ ≤ δH and in Υ ≥ δH , we deduce, everywhere, Z 7 1 20 P · NΣ ≥ δ H E[Ψ](τ ). (10.1.70) 8 Σ(τ )

10.1.13.3

Boundary terms along Σ∗

On Σ∗ , we have  m = T +O + (e3 + e4 ), r

NΣ∗

w = O(r−1 ), NΣ∗ (w) = O(r−2 ), h = O(r−4 ) and h2 = 0. Proceeding as before, we have along Σ∗ ,    1 m P · NΣ = +O + Q(X + ΛT, e3 ) 2 r    1 m + +O + Q(X + ΛT, e4 ) 2 r ≥

−O(r−1 )|Ψ|(|e3 Ψ| + |e4 Ψ|) − O(r−2 )|Ψ|2   1 1 Q(X + ΛT, e3 ) + Q(X + ΛT, e4 ) − O |e3 Ψ|2 + |e4 Ψ|2 + r−2 |Ψ|2 . 4 4

Thus, according to Proposition 10.32, we have P · NΣ

≥ =

≥

≥ and hence Z Σ∗ (τ1 ,τ2 )

  1 Λ (Q33 + Q44 + 2Q34 ) − O |e3 Ψ|2 + |e4 Ψ|2 + r−2 |Ψ|2 16    1 4Υ 2 2 2 2 Λ |e3 Ψ| + |e4 Ψ| + 2 |∇ / Ψ| + 2 |Ψ| 16 r   −O |e3 Ψ|2 + |e4 Ψ|2 + r−2 |Ψ|2   1 − 13 δH 20 |e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 64   −O |e3 Ψ|2 + |e4 Ψ|2 + r−2 |Ψ|2   −1 δH 2 |e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2

−1

P · NΣ∗ ≥ δH 2

Z



Σ∗ (τ1 ,τ2 )

 |e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 .(10.1.71)

651

REGGE-WHEELER TYPE EQUATIONS

10.1.13.4

The inhomogeneous term

R M(τ1 ,τ2 )

ˇ (X(Ψ) + 12 wΨ)N [Ψ]

ˇ = X +ΛT = fbR+YH +ΛT . We easily check, recalling the properties Recall that X δ of fδb, w and Λ and the definition of J[N, Ψ], Z   1 ˇ X(Ψ) + wΨ N [Ψ] 2 M(τ1 ,τ2 ) Z   3 − ˘ + |T˘Ψ| + r−2 |Ψ|2 |N (Ψ)| ≤ δH 4 |RΨ| M(τ1 ,τ2 )

=

−3 δH 4 J[N, Ψ](τ1 , τ2 ).

(10.1.72)

Going back to (10.1.68) we deduce Z E[Ψ](τ2 ) + E + F [Ψ](τ1 , τ2 ) ≤

−

7

δH 20 (E[Ψ](τ1 ) + J[N, Ψ](τ1 , τ2 )) .

M(τ1 ,τ2 )

In view of (10.1.69) we obtain
−1 E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) ≤ δH E[Ψ](τ1 ) + J[N, Ψ](τ1 , τ2 )  3 − +O δH 2 Err (τ1 , τ2 ). This concludes the proof of Theorem 10.33. Analysis of the error term E R Recall that Err (τ1 , τ2 ) = M(τ1 ,τ2 ) E where 10.1.14

E

.

h dec u−1−δ r−2 |e3 Ψ|2 + r−1 |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 trap
i +|e3 Ψ| (|e4 Ψ| + |∇ / Ψ|) . 5m 2

• In the trapping region Mtrap , i.e., is strictly spacelike, we have Z Z −1−δdec E . τtrap Σtrap (τ )

Σtrap (τ )

.

7m 2 ,

≤r≤

where utrap = 1 + τ and Σ(τ )

|e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + |Ψ|2




−1−δdec τtrap E[Ψ](τ ).

Thus, Z Mtrap (τ1 ,τ2 )

E

Z

τ2

. 

−1−δdec τtrap E[Ψ](τ )

τ1

Z

τ2

. 

(1 + τ )

−1−δ

sup τ ∈[τ1 ,τ2 ]

sup τ ∈[τ1 ,τ2 ]

τ1

. 

 E[Ψ](τ )

E[Ψ](τ )

and therefore, for small  > 0, the integral

R Mtrap (τ1 ,τ2 )

E can be absorbed on

652

CHAPTER 10

the left-hand side of (10.1.67). • In the non-trapping region M  we write, with a fixed δ > 0, trap

E

.

Hence, Z M

 (τ1 ,τ2 )

E


r−1−δ |e3 Ψ|2 + r−1+δ |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 . Z .  Mr≥4m0 (τ1 ,τ2 )

trap

Z +  Mr≥4m0 (τ1 ,τ2 )

r−1−δ |e3 Ψ|2 r−1+δ |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2

Z + 



(trap)

Mr≤4m0 (τ1 ,τ2 )





|e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + |Ψ|2 .

Note that for  > 0 sufficiently small, the last integral, on absorbed by the left-hand side of (10.1.67).



(trap)

Mr≤4m0 , can be

As a consequence we deduce the following. Corollary 10.34. The statement of Theorem 10.33 remains true if we replace Err in the statement of the theorem with Z Err = E , Mr≥4m0 (τ1 ,τ2 )

E


. r−1−δ |e3 Ψ|2 + r−1+δ |e4 Ψ|2 + |∇ / Ψ|2 + |r−2 |Ψ|2 ,

for a fixed δ > 0. Remark 10.35. Note that the error terms Err cannot yet be absorbed to the lefthand side of (10.1.67). In fact we need additional estimates. The Morawetz bulk quantity (5.1.13), " Z m2 ˘ 2 m 2 Mor[Ψ](τ1 , τ2 ) = |RΨ| + 4 |Ψ| r3 r M(τ1 ,τ2 )  2  # 3m 1 m2 ˘ 2 2 + 1− |∇ / Ψ| + 2 |T Ψ| , r r r ˘ 2 ˘ 2 is quite weak for r large with regard to the terms using the
R |RΨ| and |T−1Ψ| , while, Poincar´e inequality, Mor[Ψ] controls the term Mr≥4m (τ1 ,τ2 ) r |∇ / Ψ|2 + r−2 |Ψ|2 . 0 R In the next section we show how we can estimate the term M≥R (τ1 ,τ2 ) r−1−δ |e3 Ψ|2 0 R by M≥R (τ1 ,τ2 ) r−1−δ |e4 Ψ|2 and then, we provide estimates for the remaining terms. 0

Note also that the weight r−1−δ is optimal in estimating e3 Ψ in the wave zone region.

653

REGGE-WHEELER TYPE EQUATIONS

10.1.15

Proof of Theorem 10.1

We are now ready to prove Theorem 10.1. Note that it suffices to improve the previous Morawetz estimate of Theorem 10.33 by replacing the quantity Mor[Ψ](τ1 , τ2 ) with Z Morr[Ψ](τ1 , τ2 ) := Mor[Ψ](τ1 , τ2 ) + r−1−δ |e3 (Ψ)|2 . Mf ar (τ1 ,τ2 )

In view of the Morawetz estimate (10.1.67) and Corollary 10.34 we have E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) . E[Ψ](τ2 ) + J[N, Ψ](τ1 , τ2 ) + Err (τ1 , τ2 ), Z ˘ + |T˘Ψ| + r−1 |Ψ|)|N |, J[N, Ψ](τ1 , τ2 ) : = (|RΨ| M(τ1 ,τ2 )

with error term Z Err .  M≥4m0 (τ1 ,τ2 )


r−1−δ |e3 Ψ|2 + r−1+δ |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 .

We divide J[N ] = J[N, Ψ] as follows: J[N ] = J[N ]trap + J[N ] 

trap

where Z J[N ]trap :

= Mtrap

˘ + |T˘Ψ| + r−1 |Ψ|)|N |, (|RΨ|

Z J[N ]  : trap

=



(trap)

M

˘ + |T˘Ψ| + r−1 |Ψ|)|N |. (|RΨ|

For the trapping region, where the hypersurfaces Σ(τ ) are strictly spacelike, we write Z τ2 Z ˘ + |T˘Ψ| + r−1 |Ψ|)|N | J[N ]trap (τ1 , τ2 ) = dτ (|RΨ| τ1

≤

Z

Σtrap (τ )

τ2

E[Ψ](τ )

!1/2

Z

1/2

τ1

2

Σtrap (τ )

≤

τ ∈[τ1 ,τ2 ]

.

λ

sup

1/2

Z

τ2

|N |

!1/2

Z

2

E[Ψ](τ )

sup τ ∈[τ1 ,τ2 ]

τ1

E[Ψ](τ ) + λ−1

Σtrap (τ )

Z

|N |

2

τ2

τ1

kN kL2 (Σtrap (τ ))

.

654

CHAPTER 10

Hence, for λ > 0 sufficiently small, we deduce E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) Z .

trap

2

τ2

E[Ψ](τ2 ) + Err (τ1 , τ2 ) + J  [N, Ψ](τ1 , τ2 ) + τ1

kN kL2 (Σtrap (τ ))

.

On the other hand we have Z J[N ]  (τ1 , τ2 )

=

trap

M

 (τ1 ,τ2 )

˘ + |T˘Ψ| + r−1 |Ψ|)|N | (|RΨ|

trap

≤ λ

Z M



˘ 2 + |T˘Ψ|2 + r−2 |Ψ|2 ) r−1−δ (|RΨ|

trap

−1

Z

+λ

M



r1+δ |N |2 .

trap

The first integral on the right can be divided further into integrals for r ≤ 4m0 and r ≥ 4m0 . The first integral can then be easily absorbed by the term M or[Ψ](τ1 , τ2 ), if λ > 0 is sufficiently small. We are thus led to the estimate E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) . E[Ψ](τ2 ) + Err (τ1 , τ2 ) + Iδ [N ](τ1 , τ2 ) Z + r−1−δ (|e3 Ψ|2 + |e4 Ψ|2 + r−2 |Ψ|2 ) Mr≥4m0

where Iδ [N ](τ1 , τ2 ) :

Z = M

 (τ1 ,τ2 )

r1+δ |N |2 +

Z

2

τ2

τ1

dτ kN kL2 (Σtrap (τ ))

.

trap

Recalling the definition of Err in Corollary 10.34, we deduce E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) . E[Ψ](τ2 ) + Err (τ1 , τ2 ) + Iδ [N ](τ1 , τ2 ).

(10.1.73)

To eliminate the term in e3 Ψ from the error term we appeal to the following proposition. Proposition 10.36. Assume Ψ = V Ψ+N and consider the vectorfield X = f−δ T 0 with f−δ := r−δ for r ≥ 4m0 and compactly supported in r ≥ 7m 2 . With the notation of Proposition 10.9, let Pµ [f−δ T, 0, 0] E[f−δ T, 0, 0]

Then:

= f−δ Qαµ T µ ,

= Dµ Pµ [f−δ T, 0, 0] − f−δ T (Ψ)N.

655

REGGE-WHEELER TYPE EQUATIONS

1. We have, for r ≥ 4m0 , E[f−δ T, 0, 0]

=

Υ2 −1−δ 1 δr |e3 Ψ|2 − δr−1−δ |e4 Ψ|2 4 4

+O r−1−δ |DΨ|2 + r−2 |Ψ|2 .

(10.1.74)

2. We have P[f−δ T, 0, 0] · e4 = f−δ Q(T, e4 ) ≥ 0,

P[f−δ T, 0, 0] · e3 = f−δ Q(T, e3 ) ≥ 0.

We postpone the proof of Proposition 10.36 and continue the proof of Theorem 10.1. By integration, the proposition provides a bound for12 Z r−1−δ |e3 Ψ|2 M≥4m0 (τ1 ,τ2 )

in terms of E[Ψ](τ1 ),

R M

7m0 ≥ 2

(τ1 ,τ2 )

r−1−δ |e4 Ψ|2 and

R M

≥

7m0 2

(τ1 ,τ2 )

r−δ T (Ψ)N , as

well as the error terms. The second bulk integral involving the inhomogeneous term N can be estimated exactly like before. Thus combining the new estimate with that in Corollary 10.34 we derive the desired estimates, both (10.1.1) and (10.1.2), hence concluding the proof of Theorem 10.1. 10.1.15.1

Proof of Proposition 10.36

We consider vectorfields of the form X = f (r)T with T = 12 (Υe3 + e4 ). Recall, see Lemma 10.7, that all components of the deformation tensor (T ) π can be bounded by O(r−1 ). Since f = O(r−δ ), we deduce (X)

παβ = f (T ) π αβ + Dα f Tβ + Dβ f Tα = Dα f Tβ + Dβ f Tα + O(r−1−δ ).

Also, e3 (f ) = f 0 e3 (r) = −f 0 + O(r−1−δ ),

e4 (f ) = f 0 e4 (r) = Υf 0 + O(r−1−δ ).


Thus, modulo error terms of the form O()r−1−δ |e3 Ψ|2 +|e4 Ψ|2 +|∇ / Ψ|2 +r−2 |Ψ|2 , we have
Q · (X) π = 2Qαβ Tα Dβ f = 2 Q3β Tβ e3 f + Q4β Tβ e4 f = −Q(e4 , T )e3 f − Q(e3 , T )e4 f 1 1 = Q(e4 , e4 + Υe3 )f 0 − f 0 ΥQ(e3 , e4 + Υe3 ) 2 2
1 0 2 2 2 = f |e4 Ψ| − Υ |e3 Ψ| . 2

We now apply Proposition 10.9, as well as (10.1.13), (10.1.14), with X = f−δ (r)T , w = 0, M = 0 so that Pµ [f−δ T, 0, 0] = f−δ Qαµ T µ , 12 Note

that Υ2 ≥

1 4

in r ≥ 4m0 .

E[f−δ T, 0, 0] := Dµ Pµ [f−δ T, 0, 0] − f−δ T (Ψ)N

656

CHAPTER 10

and E[f−δ T, 0, 0]

1 1 Q · (X) π − f−δ T (V )|Ψ|2 2 2


1 0 f−δ (r) |e4 Ψ|2 − Υ2 |e3 Ψ|2 + O r−1−δ |DΨ|2 + r−2 |Ψ|2 4

= =

with |DΨ|2 = |e3 Ψ|2 + |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 . Since f−δ (r) = r−δ for r ≥ 4m0 , we deduce, for r ≥ 4m0 , E[f−δ T, 0, 0]

=

Υ2 −1−δ 1 δr |e3 Ψ|2 − δr−1−δ |e4 Ψ|2 4 4

+O r−1−δ |DΨ|2 + r−2 |Ψ|2 .

On the other hand, P[f−δ T, 0, 0] · e4

P[f−δ T, 0, 0] · e3

=

f−δ Q(T, e4 ) ≥ 0,

= f−δ Q(T, e3 ) ≥ 0,

as desired. This concludes the proof of Proposition 10.36.

10.2

DAFERMOS-RODNIANSKI rp -WEIGHTED ESTIMATES

For convenience, we work in this section with the renormalized frame (e03 , e04 ) defined in (10.2.6) instead of the original frame (e3 , e4 ). To simplify the exposition, we still denote it as (e3 , e4 ). Recall that the two are frames are equivalent up to lower terms in m/r. In this section we rely on the Morawetz estimates proved in the previous section to establish rp -weighted estimates in the spirit of Dafermos-Rodnianski [23]. The following theorem claims rp -weighted estimates for the solution ψ of the wave equation (5.3.5). Theorem 10.37 (rp -weighted estimates). Consider a fixed δ > 0 and let R    δ. The following estimates hold true for all δ ≤ p ≤ 2 − δ,

m0 δ ,

E˙ p ; R [ψ](τ2 ) + B˙ p ; R [ψ](τ1 , τ2 ) + F˙p [ψ](τ1 , τ2 ) . Ep [ψ](τ1 ) + Jp [ψ, N ](τ1 , τ2 ).

(10.2.1)

Remark 10.38. Note that Theorem 10.1 on Morawetz estimates and Theorem 10.37 on rp -weighted estimates immediately yield for all δ ≤ p ≤ 2 − δ, sup

Ep [ψ](τ ) + Bp [ψ](τ1 , τ2 ) + Fp [ψ](τ1 , τ2 ) . Ep [ψ](τ1 ) + Jp [ψ, N ](τ1 , τ2 ),

τ ∈[τ1 ,τ2 ]

which corresponds to Theorem 5.17 in the case s = 0. Theorem 10.37 will be proved in section 10.2.3. We will need in this section stronger assumptions in the region r ≥ 4m0 , away from trapping, than those in (10.1.5)–(10.1.7) of the previous section. For convenience we express our conditions

657

REGGE-WHEELER TYPE EQUATIONS

with respect to the weights13 wp,q (u, r) = r−p (1 + τ )−q−δdec +2δ0 . RP0. The assumptions Mor1–Mor3 made in the previous section hold true. RP1. The Ricci coefficients verify, for r ≥ 4m0 , ξ, ϑ, ϑ, η, η, ζ, ω . w1,1 , 2 1 κ + , χ + , e3 Φ − χ . w1,1 , r r 2Υ Υ κ − , χ − , e4 Φ − χ .  min{w1,1 , w2,1/2 }, r r m ω + 2 , |ξ| .  min{w2,1 , w3,1/2 }. r RP2. The derivatives of r verify, for r ≥ 4m0 , e3 (r) + 1 . w0,1 , e4 (r) − Υ . w1,1 , 2m e3 e4 (r) + 2 , e4 e3 (r) . w1,1 . r

(10.2.2)

(10.2.3)

RP3. For r ≥ 4m0 , 2m ρ + 3 . w3,1 , r 1 K − 2 . r−2 , r eθ (Φ) . r−1 .

(10.2.4)

RP4. We also assume, for r ≥ 4m0 , |m − m0 | . ,

|e3 m, r2 e4 m| . w0,1 ,

(10.2.5)

|e3 e4 (m), e4 e3 (m)| . w1,1 . Since the estimates we are establishing are restricted to the far region r > R it is convenient, in this section, to work with the renormalized frame e03 = Υe3 ,

e04 = Υ−1 e4 ,

e0θ = eθ .

(10.2.6)

Relative to the new frame (e03 , e04 , e0θ ) we have ξ 0 = Υ−2 ξ,

ξ 0 = Υ2 ξ,

ζ 0 = ζ,

η 0 = η,

χ0 = Υ−1 χ,

χ0 = Υχ

13 The assumptions are consistent with the global frame used in Theorem M1, see Lemma 5.1. In particular, δ0 > 0 is such that δdec − 2δ0 > 0 which is the only needed property of δdec − 2δ0 to derive the rp -weighted estimates.

658

CHAPTER 10

and ω

0

= =

ω0

= =

   1 −1 −1 m −1 e4 (m) Υ ω + e4 (log Υ) = Υ ω+Υ e4 (r) − Υ 2 r2 r    m m e4 (m) Υ−1 ω + 2 + Υ−1 2 (e4 (r) − Υ) − Υ−1 , r r r     1 −1 m −1 e3 (m) Υ ω − e3 (log Υ) = Υ ω − Υ e3 (r) + Υ 2 r2 r   m m e3 (m) + Υ ω − Υ−1 2 (e3 (r) + 1) + Υ−1 . r2 r r −1



Thus in the new frame we have, for r ≥ 4m0 : RP1’. The Ricci coefficients with respect to the null frame (e03 , e04 , e0θ ) verify, for r ≥ 4m0 : 0 0 ξ , ϑ , ϑ, η 0 , η 0 , ζ 0 , ω 0 − m . w1,1 , 2 r 0 2Υ 0 Υ 0 κ + , χ + , e3 Φ − χ0 . w1,1 , r r (10.2.7) 0 2 0 1 0 0 κ − , χ − , e4 Φ − χ .  min{w1,1 , w2,1/2 }, r r 0 0 ω , |ξ | .  min{w2,1 , w3,1/2 }. RP2’. The derivatives of r verify 0 e3 (r) + Υ . w0,1 , 0 e4 (r) − 1 . w1,1 , 2m 0 0 e3 e4 (r), e04 e03 (r) + 2 . w1,1 . r RP3’. The Gauss curvature K of S and ρ verify 2m ρ + 3 . r−3 , r 1 K − 2 . r−2 . r

(10.2.8)

(10.2.9)

RP4’. We also assume |m − m0 | . ,

|e03 m,

|e03 e04 (m),

r2 e04 m| . w0,1 ,

e04 e03 (m)|

(10.2.10)

. w1,1 .

Remark 10.39. In the far region r ≥ 4m0 all norms we are using in our estimates are equivalent when expressed relative to the null frame (e3 , e4 , eθ ) or (e03 , e04 , e0θ ). Convention. For the remainder of this section we shall do all calculations with respect to the renormalized frame (e03 , e04 , e0θ ). For convenience we shall drop the primes, throughout this section, since there is no danger of confusion. Note however that the main results, which include the interior region r ≤ R, are always expressed

659

REGGE-WHEELER TYPE EQUATIONS

with respect to the original frame. 10.2.1

Vectorfield X = f (r)e4

Lemma 10.40. Consider the vectorfield X = f (r)e4 . 1. We have the decomposition (X)

π=

with symmetric tensor

(X)

(X)

Λg +

(X)

π e,

Λ=

2 f r

π e which verifies

4f + O()w1,1 (|f | + r|f 0 |) r = 4f 0 Υ − 4Υ0 + O()w1,1 (|f | + r|f 0 |)

(X)

π e43 = −2f 0 +

(X)

π e33

(X)

π e4θ = O()w2,1/2 |f |

(X)

(X)

(10.2.11)

π eAB = O()w2,1/2 |f |

(X)

π e3θ = O()w1,1 |f |.

2. We have  (X) Λ

=

m 
2 00 f + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | . (10.2.12) r r

Proof. See Lemma D.14 in appendix. 10.2.2

Energy densities for X = f (r)e4

We start with the following proposition. ˙ g Ψ = V Ψ + N and let Proposition 10.41. Assume Ψ verifies the equation  2f (X) X = f e4 and w = Λ = r and let E := E[X, w] = E[X = f e4 , w = 2f r ]. 1. We have E

=

 
1 0 1 2f 1 f |e4 Ψ|2 + −f 0 + |∇ / Ψ|2 + V |Ψ|2 − f 00 |Ψ|2 2 2 r 2r  m  +Err , , f (Ψ) r

where  m  Err , , f (Ψ) mr m 
= O 2 (|f | + r|f 0 |)|e4 Ψ|2 + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2 r r
+ O()w1,1 (|f | + r|f 0 |) |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 .

660

CHAPTER 10

2. The current Pµ

=

1 1 Pµ [X, w] = Qµν X ν + wΨ · Dµ Ψ − |Ψ|2 ∂µ w 2 4

verifies P · e4 P · e3

2 1 1 = f e4 Ψ + Ψ − r−2 e4 (rf |Ψ|2 ) + O(r−3 )f |Ψ|2 , r 2 1 −2 = f Q34 + r e3 rf ψ 2 ) + r−1 f 0 ψ 2 + O(mr−3 + r−2 )|rf 0 | |Ψ|2 . 2

3. Let θ = θ(r) supported for r ≥ R/2 with θ = 1 for r ≥ R such that fp = θ(r)rp . Let (p) P := P[fp e4 , wp ]. Then, for all r ≥ R, (p)

p 1 P · e4 + r−2 e4 (θrp+1 |Ψ|2 ) ≥ rp−2 (p − 1)2 |Ψ|2 . 2 8

Before proceeding with the proof of Proposition 10.41, we first establish the following lemma. Lemma 10.42. We have    m 
2f Q · (X) π e = f 0 + O 2 (|f | + r|f 0 |) |e4 Ψ|2 + −f 0 + |∇ / Ψ|2 + V |Ψ|2 r r   0 2 2 −2 + O()w1,1 (|f | + r|f |) |e4 Ψ| + |∇ / Ψ| + r |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 . Proof. Recall from Proposition 10.9 that we have Q33 = |e3 Ψ|2 ,

Q44 = |e4 Ψ|2 ,

Q34 = |∇ / Ψ|2 + V |Ψ|2 ,

and |QAB | ≤ |e3 Ψ||e4 Ψ| + |∇ / Ψ|2 + |V ||Ψ|2 , Hence, in view of Lemma D.14 for Q·

(X)

π e

=

= + + = +

(X)

|QA3 | ≤ |e3 Ψ||∇ / Ψ|,

|QA4 | ≤ |e4 Ψ||∇ / Ψ|.

π e, we have

1 1 1 1 Q44 (X) π e33 + Q34 (X) π e34 − Q4A (X) π e3A − Q3A (X) π e4A 4 2 2 2 +QAB (X) π eAB   f 0 Υ − Υ0 f + O()w1,1 (|f | + r|f 0 |) Q44   2f 0 0 −f + + O()w1,1 (|f | + r|f |) Q34 r O()w1,1 |f | Q4A + O()w2,1/2 |f | (Q3A + QAB )    m  2f 0 0 0 f + O 2 (|f | + r|f |) Q44 + −f + Q34 r r O()(|f | + r|f 0 |)w1,1 (Q44 + Q4A ) + O()w2,1/2 |f | (QAB + Q34 ) +O()w3,1/2 |f |Q3A

661

REGGE-WHEELER TYPE EQUATIONS

from which we deduce    m 
2f (X) 0 0 2 0 Q· π e = f + O 2 (|f | + r|f |) |e4 Ψ| + −f + |∇ / Ψ|2 + V |Ψ|2 r r   + O()w1,1 (|f | + r|f 0 |) |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 as desired. We are now ready to prove Proposition 10.41. Proof of Proposition 10.41. If Q = Q[Ψ] is the energy-momentum tensor of Ψ (re˙ = V Ψ + N ) and call Ψ (X)

π=

(X)

Λg +

(X)

π e

we deduce Q·

(X)

π

=

(X)

ΛtrQ + Q ·

Hence, for X = f e4 and w = 1 Q· 2

(X)

(X)

(X)

Λ=

1 π + wL[Ψ] 2

π e=

(X)

Λ (−L(Ψ) − V |Ψ|2 ) + Q ·

(X)

2f r ,

= −

1 (X) 1 ΛV |Ψ|2 + Q · 2 2

(X)

π e.

In view of (10.1.14), we infer E

:= E[X, w = (X) Λ, M = 0] 1 1 1 = Q · (X) π e − |Ψ|2 g (X) Λ − (X(V ) + 2 4 2

(X)

ΛV )|Ψ|2 .

Recall that V = −κκ. Hence, X(V ) +

(X)

ΛV

  2f 2 = f e4 (V ) + V = −f e4 (κκ) + κκ r r   2 = f κ2 κ − 2κρ − κκ + O()w3,1 r m  = O 4 + w3,1 f. r

Hence, in view of the computation (10.2.12) of  (X) Λ, 1 1 − |Ψ|2 g (X) Λ − (X(V ) + (X) ΛV )|Ψ|2 4 2 m 
1 00 2 = − f |Ψ| + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2 . 2r r We deduce E=

1 Q· 2

(X)

π e−

m 
1 00 2 f |Ψ| + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2 . 2r r

π e.

662

CHAPTER 10

Using Lemma 10.42, we deduce    m 
1 1 2f 1 E = f 0 |e4 Ψ|2 + −f 0 + |∇ / Ψ|2 + V |Ψ|2 − f 00 |Ψ|2 + Err , , f (Ψ) 2 2 r 2r r where  m  Err , , f (Ψ) r

m

(|f | + r|f 0 |)|e4 Ψ|2 
+ O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2 r   + O()w1,1 (|f | + r|f 0 |) |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 = O

2

 rm

which is the first part of Proposition 10.41. To prove the second part of Proposition 10.41, we compute P · e4

1 1 = f Q44 + f Ψ · e4 Ψ − e4 (r−1 f )|Ψ|2 r 2   1 1 = f |e4 Ψ|2 + Ψ · e4 Ψ − e4 (r−1 f )|Ψ|2 r 2 2 1 1 1 = f e4 Ψ + Ψ − f Ψ · e4 Ψ − r−2 f |Ψ|2 − e4 (r−1 f )|Ψ|2 r r 2 2 1 1 1 = f e4 Ψ + Ψ − r−2 e4 (rf |Ψ|2 ) + r−2 e4 (rf )|Ψ|2 − r−2 f |Ψ|2 r 2 2 − =

1 e4 (r−1 f )|Ψ|2 2 2 1 1 f e4 Ψ + Ψ − r−2 e4 (rf |Ψ|2 ) + r−2 (e4 (r) − 1)f |Ψ|2 . r 2

Since e4 (r) =

r (κ + A), 2

we have14 e4 (r) − 1 = O(r−1 ). Thus, as desired, P · e4

=

2 1 1 f e4 Ψ + Ψ − r−2 e4 (rf |Ψ|2 ) + O(r−3 )f |Ψ|2 . r 2

14 Note that so far we have only used the weaker version e (r) − 1 = O(). This is the first time 4 we need the stronger version of the estimate in this chapter.

663

REGGE-WHEELER TYPE EQUATIONS

Also, P · e3

1 = f Q34 + r−1 f Ψ · e3 Ψ − e3 (r−1 f )|Ψ|2 2 1 −1 1 2 = f Q34 + r f e3 (|Ψ| ) − e3 (r−1 f )|Ψ|2 2 2  1 1  −2 2 = f Q34 + r e3 rf |Ψ| ) − r−2 e3 (rf )|Ψ|2 − e3 (r−1 f )|Ψ|2 2 2 1 −2 2 −1 0 2 −1 0 = f Q34 + r e3 rf |Ψ| ) + r f Υ|Ψ| − r f (e3 (r) + Υ)|Ψ|2 2
1 = f Q34 + r−2 e3 rf |Ψ|2 ) + r−1 f 0 |Ψ|2 + O mr−3 + r−2 (r|f 0 |)|Ψ|2 2

as desired. It remains to prove the last part of Proposition 10.41. We have, for r ≥ R, (p)

P · e4

1 = rp |e4 Ψ|2 + rp−1 Ψe4 Ψ − e4 (rp−1 )|Ψ|2 2

and (p)

p P · e4 + r−2 e4 (θrp+1 |Ψ|2 ) 2

1 = rp |e4 Ψ|2 + rp−1 Ψ · e4 Ψ − e4 (rp−1 )|Ψ|2 2 p(p + 1) p−2 p−1 + pr Ψ · e4 Ψ + r e4 (r)|Ψ|2 2 = rp |e4 Ψ|2 + (p + 1) rp−1 Ψ · e4 Ψ p2 + 1 + e4 (r)rp−2 |Ψ|2 2 " # 2 p + 1 (p − 1)2 p 2 = r e4 Ψ + Ψ + |Ψ| 2r 4r2 p2 + 1 (e4 (r) − 1)rp−2 |Ψ|2 2 (p − 1)2 p2 + 1 p−3 2 ≥ rp−2 |Ψ|2 − O() r |Ψ| . 4 2 +

This concludes the proof of Proposition 10.41. In applications we would like to apply Proposition 10.41 to f = rp , 0 < p < 2. We note however that the presence of the term − 12 r−1 f 00 |Ψ|2 on the right-hand side of the E identity requires an additional correction if p > 1. This additional correction is taken into account by the following proposition. Proposition 10.43. Assume Ψ verifies the equation g Ψ = V Ψ + N and let −1 0 X = f (r)e4 , w = (X) Λ = 2f f e4 . Then: r and M = 2r 1. We have, with eˇ4 = r−1 e4 (r·), 1 1 E[X, w, M ] = f 0 |ˇ e4 (Ψ)|2 + 2 2



  m  2f 0 − f Q34 + Err , ; f [Ψ] (10.2.13) r r

664

CHAPTER 10

with error term  m  Err , , f (Ψ) mr m 
= O 2 (|f | + r|f 0 |)|e4 Ψ|2 + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2 r r
+ O()w1,1 (|f | + r|f 0 |) |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 . 2. The current Pµ

1 1 1 = Pµ [X, w, M ] = Qµν X ν + wΨDµ Ψ − |Ψ|2 ∂µ w + Mµ |Ψ|2 2 4 4

verifies 1 = f (ˇ e4 Ψ)2 − r−2 e4 (rf |Ψ|2 ) + O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ), 2 1 −2 = f Q34 + r e3 rf |Ψ|2 ) + O(mr−3 + r−2 )(|f | + r|f 0 |)|Ψ|2 . 2

P · e4 P · e3

3. Let θ = θ(r) supported for r ≥ R/2 with θ = 1 for r ≥ R such that fp = θ(r)rp . Let (p) P := P[fp e4 , wp , Mp ]. Then, for all r ≥ R, (p)

p 1 P · e4 + r−2 e4 (θrp+1 |Ψ|2 ) ≥ rp−2 (p − 1)2 |Ψ|2 . 2 8

Proof. We start with the first part of Proposition 10.43. To this end, we use (X)

π43 = −2e4 f + 4f ω,

(X)

πAB = 2f (1+3)χAB ,

so that tr (X) π

= =

− (X) π43 + gAB (X) πAB

2e4 f − 4f ω + 2f κ,

and we compute D µ Mµ µ

= D (2r = = = = =

−1 0

µ



2f 0 Xµ rf



2f 0 = DivX + X rf



2f 0 rf



f e4 ) µ = D  0 f 0 (X) 2f tr π+X rf rf  0 0 f f (2e4 f − 4f ω + 2f κ) + 2f e4 rf rf   00  m  f0 4f f rf − f 0 (f + rf 0 ) 2e4 (r)f 0 + + O 3 + w2,1 + 2f e4 (r) rf r r r2 f 2 m  4f 0 2(f 0 )2 f 00 2f 0 2(f 0 )2 + + 2 − − + O + w (|f | + r|f 0 | + r2 |f 00 |) 3,1 r2 rf r r2 rf r4 m  2f 0 2f 00 + + O 4 + w3,1 (|f | + r|f 0 | + r2 |f 00 |). 2 r r r

665

REGGE-WHEELER TYPE EQUATIONS

We also have 1 Ψ · Dµ ΨM µ 2

= r−1 f 0 Ψ · D4 Ψ.

Since we have E[X, w, M ]

=

1 1 E[X, w] + (Dµ Mµ )|Ψ|2 + Ψ · Dµ ΨM µ , 4 2

we infer

=

E[X, w, M ]  0  m  f f 00 0 2 00 E[X, w] + + + O 4 + w3,1 (|f | + r|f | + r |f |) |Ψ|2 2r2 2r r +r−1 f 0 Ψ · D4 Ψ.

Together with Proposition 10.41, this yields E[X, w, M ]

 1 0 f |e4 Ψ|2 + 2r−1 Ψ · D4 Ψ + r−2 |Ψ|2 2  
1 2f + −f 0 + |∇ / Ψ|2 + V |Ψ|2 2 r  m  m  +Err , , f (Ψ) + O 4 + w3,1 (|f | + r|f 0 | + r2 |f 00 |)|Ψ|2 r  r 
1 0 1 2f −1 2 0 = f |e4 Ψ + r Ψ| + −f + |∇ / Ψ|2 + V |Ψ|2 2 2 r  m  m  +Err , , f (Ψ) + O 4 + w3,1 (|f | + r|f 0 | + r2 |f 00 |)|Ψ|2 r r  
1 0 1 2f −1 = f |ˇ e4 Ψ + r (1 − e4 (r))Ψ|2 + −f 0 + |∇ / Ψ|2 + V |Ψ|2 2 2 r  m  m  +Err , , f (Ψ) + O 4 + w3,1 (|f | + r|f 0 | + r2 |f 00 |)|Ψ|2 r r

=

and hence E[X, w, M ] =

1 0 1 f |ˇ e4 Ψ|2 + 2 2

where  m  Err , , f (Ψ) r

= O +

O



−f 0 +

2f r



 m 
|∇ / Ψ|2 + V |Ψ|2 + Err , , f (Ψ) r

m 2

 rm

(|f | + r|f 0 |)|e4 Ψ|2 
+ w3,1 |f | + r|f 0 | + r2 |f 00 | |Ψ|2

r4
+ O()w1,1 (|f | + r|f 0 |) |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   + O()w2,1/2 |f | |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 . This is the desired estimate (10.2.13).

666

CHAPTER 10

Next, we consider the second part of Proposition 10.43. Pµ [X, w, M ]

=

1 1 Pµ [X, w] + |Ψ|2 Mµ = Pµ [X, w] + r−1 f 0 |Ψ|2 e4 . 4 2

Hence, in view of the results in part 2 of Proposition 10.41, 2 1 P4 [X, w, M ] = P4 [X, w] = f ˇ e4 Ψ + (1 − e4 (r))Ψ − r−2 e4 (rf |Ψ|2 ) 2 −3 2 +O(r )f |Ψ| 1 = f |ˇ e4 Ψ|2 − r−2 e4 (rf |Ψ|2 ) + O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ), 2 P3 [X, w, M ] = P3 [X, w] − r−1 f 0 |Ψ|2 1 = f Q34 + r−2 e3 rf |Ψ|2 ) + r−1 f 0 |Ψ|2 − r−1 f 0 |Ψ|2 2 + O(r−2 )(|f | + r|f 0 |)|Ψ|2 1 = f Q34 + r−2 e3 rf |Ψ|2 ) + O(r−2 )(|f | + r|f 0 |)|Ψ|2 2 as desired. The last part follows from the third part of Proposition 10.41. Lemma 10.44. On Σ∗ , we have P · NΣ∗

=

  1 1 1 f Q34 + f (ˇ e4 Ψ)2 + div Σ∗ r−1 f |Ψ|2 νΣ∗ 2 2 2 −1 +O(mr + )(|f | + r|f 0 |)(|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 ).

Proof. Recall that there exists a constant c∗ such that u + r = c∗ on Σ∗ . In particular, the unit normal NΣ∗ is collinear to −2gαβ ∂α (u + r)∂β

=

e4 (u + r)e3 + e3 (u + r)e4

=

e4 (r)e3 + (e3 (u) + e3 (r))e4

and since   g e4 (r)e3 + (e3 (u) + e3 (r))e4 , e4 (r)e3 + (e3 (u) + e3 (r))e4 = −4e4 (r)(e3 (u) + e3 (r)), we infer p NΣ∗

=

e4 (r)

p e3 + 2 e3 (u) + e3 (r)

p

e3 (u) + e3 (r) p e4 . 2 e4 (r)

In particular, we have P · NΣ∗

=

p p e4 (r) e3 (u) + e3 (r) p p P · e3 + P · e4 . 2 e3 (u) + e3 (r) 2 e4 (r)

667

REGGE-WHEELER TYPE EQUATIONS

Now, recall from Proposition 10.43 that we have P · e4 P · e3

1 = f (ˇ e4 Ψ)2 − r−2 e4 (rf |Ψ|2 ) + O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ), 2 1 −2 = f Q34 + r e3 rf |Ψ|2 ) + O(mr−3 + r−2 )(|f | + r|f 0 |)|Ψ|2 . 2

We deduce P · NΣ∗

=

p e4 (r)

1 p f Q34 + r−2 e3 rf |Ψ|2 ) 2 2 e3 (u) + e3 (r) ! +O(mr−3 + r−2 )(|f | + r|f 0 |)|Ψ|2 p +

e3 (u) + e3 (r) 1 p f (ˇ e4 Ψ)2 − r−2 e4 (rf |Ψ|2 ) 2 2 e4 (r) !

+O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ) =

p e4 (r)

p

e3 (u) + e3 (r) p p f Q34 + f (ˇ e4 Ψ)2 2 e3 (u) + e3 (r) 2 e4 (r) p p e4 (r) e3 (u) + e3 (r) 1 −2 1 −2 2 p + p r e3 rf |Ψ| ) − r e4 (rf |Ψ|2 ) 2 2 2 e3 (u) + e3 (r) 2 e4 (r)

+O(mr−3 + r−2 )(|f | + r|f 0 |)|Ψ|2 + O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ) √ 1 2−Υ √ = (1 + O())f Q34 + (1 + O())f (ˇ e4 Ψ)2 2 2 2−Υ 1 + r−2 νΣ∗ rf |Ψ|2 ) + O(mr−3 + r−2 )(|f | + r|f 0 |)|Ψ|2 2 +O(r−1 )f (|e4 Ψ|2 + r−2 |Ψ|2 ) 1 1 1 = f Q34 + f (ˇ e4 Ψ)2 + r−2 νΣ∗ rf |Ψ|2 ) 2 2 2 +O(mr−1 + )(|f | + r|f 0 |)(|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 ) where we used e4 (r) = 1 + O(),

e3 (r) = −Υ + O(),

e3 (u) = 2 + O(),

and where νΣ∗ denotes the vectorfield p p e4 (r) e3 (u) + e3 (r) p p νΣ∗ = e3 − e4 . 2 e3 (u) + e3 (r) 2 e4 (r) Next, note from the formula that νΣ∗ is unitary and orthogonal to NΣ∗ so that νΣ∗ is a unit vectorfield, tangent to Σ∗ and normal to eθ . Furthermore, since (νΣ∗ , eθ , eϕ ) is an orthonormal frame of Σ∗ , we have div

Σ∗ (νΣ∗ )

= g(DνΣ∗ νΣ∗ , νΣ∗ ) + g(Deθ νΣ∗ , eθ ) + g(Deϕ νΣ∗ , eϕ ).

668

CHAPTER 10

Since νΣ∗ is a unit vector, the first term vanishes, and hence div

Σ∗ (νΣ∗ )

= g(Deθ νΣ∗ , eθ ) + g(Deϕ νΣ∗ , eϕ ) p p e4 (r) e3 (u) + e3 (r) p p = g(Dθ e3 , eθ ) − g(Dθ e4 , eθ ) 2 e3 (u) + e3 (r) 2 e4 (r) +νΣ∗ (Φ) p p e4 (r) e3 (u) + e3 (r) p p = κ− κ. 2 e3 (u) + e3 (r) 2 e4 (r)

In particular, we have   div Σ∗ r−1 f |Ψ|2 νΣ∗

= r−2 νΣ∗ rf |Ψ|2 ) + νΣ∗ (r−2 )rf |Ψ|2 +div

−1

f |Ψ|2  2 rf |Ψ| ) + div

Σ∗ (νΣ∗ )r

 2νΣ∗ (r) = r νΣ∗ r−1 f |Ψ|2 Σ∗ (νΣ∗ ) − r p   e4 (r) 2e3 (r) −2 2 p = r νΣ∗ rf |Ψ| ) + κ− r 2 e3 (u) + e3 (r) p  ! e3 (u) + e3 (r) 2e4 (r) p − κ− r−1 f |Ψ|2 r 2 e4 (r) −2

and hence r−2 νΣ∗ rf |Ψ|2 )

=

div

 Σ∗

 r−1 f |Ψ|2 νΣ∗ + O(r−2 )f |Ψ|2 .

We finally obtain P · NΣ∗

=

=

1 1 1 f Q34 + f (ˇ e4 Ψ)2 + r−2 νΣ∗ rf |Ψ|2 ) 2 2 2 +O(mr−1 + )(|f | + r|f 0 |)(|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 )   1 1 1 f Q34 + f (ˇ e4 Ψ)2 + div Σ∗ r−1 f |Ψ|2 νΣ∗ 2 2 2 +O(mr−1 + )(|f | + r|f 0 |)(|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 )

which concludes the proof of the lemma. 10.2.3

Proof of Theorem 10.37

Consider the function fp = fp,R defined by ( rp , if r ≥ R, fp = 0, if r ≤ R2 ,

(10.2.14)

where R is a fixed, sufficiently large constant which will be chosen in the proof. We also consider Xp = fp e4 ,

wp =

2fp , r

Mp =

2fp0 e4 . r

669

REGGE-WHEELER TYPE EQUATIONS

The proof relies on Proposition 10.43. Step 0. (Reduction to the region r ≥ R) In view of the definition of E[Xp , wp , Mp ], see (10.1.13), and in view of the choice of Xp and wp , we have Dµ Pµ [Xp , wp , Mp ]

= E[Xp , wp , Mp ] + fp (r)ˇ e4 Ψ · N.

We integrate this identity on the domain M(τ1 , τ2 ) to derive Z Z Z P · NΣ + P · NΣ∗ + E Σ(τ2 ) Σ∗ (τ1 ,τ2 ) M(τ1 ,τ2 ) Z Z = P · NΣ − fp eˇ4 Ψ · N. M(τ1 ,τ2 )

Σ(τ1 )

Denoting the boundary terms, Z K≥R (τ1 , τ2 ) : = Σ≥R (τ2 )

Z K≤R (τ1 , τ2 ) :

= Σ≤R (τ1 )

P · e4 +

Z

P · NΣ −

Σ∗ (τ1 ,τ2 )

Z Σ≤R (τ2 )

P · NΣ∗ −

Z Σ≥R (τ1 )

P · e4 ,

M(τ1 ,τ2 )

fp eˇ4 Ψ · N.

P · NΣ ,

we write Z K≥R (τ1 , τ2 ) + M≥R (τ1 ,τ2 )

E = K≤R (τ1 , τ2 ) −

Z M≤R (τ1 ,τ2 )

E−

Z

We have the following lemma. Lemma 10.45. For p ≥ δ, we have Z  K≥R (τ1 , τ2 ) + E . Rp+2 E[Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 ) M≥R (τ1 ,τ2 )

 +O()B˙ δ ; 4m0 [ψ](τ1 , τ2 ) . R R Proof of Lemma 10.45. The terms Σ≤R (τ ) P · NΣ and M≤R (τ1 ,τ2 ) E on the right can be estimated as follows: Z p P · N Σ . R E[Ψ](τ1 ), Σ≤R (τ1 ) Z P · NΣ . Rp E[Ψ](τ2 ), Σ≤R (τ2 ) Z . Rp+2 Mor[Ψ](τ1 , τ2 ). E M≤R (τ1 ,τ2 )

Hence, K≤R (τ1 , τ2 ) −

Z M≤R (τ1 ,τ2 )

E

. Rp+2 (E[Ψ](τ1 ) + E[Ψ](τ2 ) + Mor[Ψ](τ1 , τ2 )) .

670

CHAPTER 10

In view of the improved Morawetz Theorem 10.1 we have, for fixed δ > 0, E[Ψ](τ2 ) + Morr[Ψ](τ1 , τ2 ) + F [Ψ](τ1 , τ2 ) .

E[Ψ](τ1 ) + Jδ [N, ψ](τ1 , τ2 ) +O()B˙ δ ; 4m [ψ](τ1 , τ2 ) 0

which implies Z K≥R (τ1 , τ2 ) + M≥R (τ1 ,τ2 )

.

E

  Rp+2 E[Ψ](τ1 ) + Jδ [N, ψ](τ1 , τ2 ) + O()B˙ δ ; 4m0 [ψ](τ1 , τ2 ) Z + fp eˇ4 Ψ · N . M(τ1 ,τ2 )

Together with the definition (5.3.7) of Jp and the fact that p ≥ δ, we infer Z K≥R (τ1 , τ2 ) + E M≥R (τ1 ,τ2 )



 . Rp+2 E[Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 ) + O()B˙ δ ; 4m0 [ψ](τ1 , τ2 ) which concludes the proof of Lemma 10.45. The proof of Theorem 10.37 now proceeds according to the following steps. (Bulk terms for r ≥ R) E. M≥R (τ1 ,τ2 )

Step 1. R

We prove the following lower bound for

Lemma 10.46. Given a fixed δ > 0 we have for all δ ≤ p ≤ 2 − δ and R  m δ ,   δ, Z Z   1 E ≥ rp−1 p|ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + r−2 |Ψ|2 ) 4 M≥R (τ1 ,τ2 ) M≥R (τ1 ,τ2 ) − O()Morr[Ψ](τ1 , τ2 ).

(10.2.15)

Proof of Lemma 10.46. We make use of Proposition 10.43 according to which    m  1 0 1 2fp 2 0 E[X, w, M ] = fp |ˇ e4 (Ψ)| + − fp Q34 + Err , ; fp [Ψ] 2 2 r r    m  p 1 = rp−1 |ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + V |Ψ|2 ) + Err , ; fp [Ψ] 2 2 r   p 1 ≥ rp−1 |ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + r−2 |Ψ|2 ) 2 2  m  +Err , ; fp [Ψ] r

671

REGGE-WHEELER TYPE EQUATIONS

where  m  Err , , fp (Ψ) r

m  |e4 Ψ|2 + r−2 Ψ|2 2 r
rp O()w1,1 |e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2   rp O()w2,1/2 |e3 Ψ|(|e4 Ψ| + r−1 |∇ / Ψ|) + |∇ / Ψ|2 + r−2 |Ψ|2 m Err + Err(),  mr   O rp−1 |ˇ e4 Ψ|2 + r−2 |Ψ|2 , r
O()rp−1 |ˇ e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 + r−2 |e3 Ψ|2 . rp O

= + + .

Err

m

r Err()

Thus, Z M≥R (τ1 ,τ2 )

E

≥

= =

Z

rp−1

M≥R (τ1 ,τ2 )

− O

m Z

R Z − O()

 1 |ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + r−2 |Ψ|2 ) 2 2   rp−1 |ˇ e4 Ψ|2 + r−2 |Ψ|2

M≥R (τ1 ,τ2 )

M≥R (τ1 ,τ2 )

− O()

p

Z M≥R (τ1 ,τ2 )

rp−1 |ˇ e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2




rp−3 |e3 Ψ|2 .

For δ ≤ p ≤ 2 − δ, R  m δ and   δ we can absorb all error terms except the last, i.e., Z Z   1 E ≥ rp−1 p|ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + r−2 |Ψ|2 ) 4 M≥R (τ1 ,τ2 ) M≥R (τ1 ,τ2 ) Z − O() rp−3 |e3 Ψ|2 . M≥R (τ1 ,τ2 )

Note also that for all δ ≤ p ≤ 2 − δ we have Z rp−3 |e3 Ψ|2 . Morr(τ1 , τ2 ). M≥R (τ1 ,τ2 )

Hence, for all δ ≤ p ≤ 2 − δ and R  m δ ,   δ, Z Z   1 E ≥ rp−1 p|ˇ e4 (Ψ)|2 + (2 − p)(|∇ / Ψ|2 + r−2 |Ψ|2 ) 4 M≥R (τ1 ,τ2 ) M≥R (τ1 ,τ2 ) − O()Morr[Ψ](τ1 , τ2 ) as desired. Combining (10.2.15) with Lemma 10.45, we deduce  K≥R (τ1 , τ2 ) + B˙ p,R [Ψ](τ1 , τ2 ) . Rp+2 E[Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 )  +O()B˙ δ ; 4m0 [ψ](τ1 , τ2 ) . (10.2.16)

672

CHAPTER 10

Step 2. (Boundary terms for r ≥ R.) Recall that, according to Proposition 10.43, 1 = fp |ˇ e4 Ψ|2 − r−2 e4 (rfp |Ψ|2 ) + O(r−1 )fp (|e4 Ψ|2 + r−2 |Ψ|2 ), 2

P · e4

and according to Lemma 10.44 P · NΣ∗

  1 1 1 f Q34 + f (ˇ e4 Ψ)2 + div Σ∗ r−1 f |Ψ|2 νΣ∗ 2 2 2 −1 +O(mr + )(|f | + r|f 0 |)(|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 ).

=

Recalling the definition of Z Z K≥R = P · e4 + Σ≥R (τ2 )

Σ∗ (τ1 ,τ2 )

P · NΣ∗ −

Z Σ≥R (τ1 )

P · e4

we write Z K≥R

= Σ≥R (τ2 )

1 2

Z

1 + 2 Z +

Z

−

fp |ˇ e4 Ψ|2 +

Σ≥R (τ2 )

Σ≥R (τ1 )

1 2

Z Σ∗ (τ1 ,τ2 )

r−2 e4 (rfp |Ψ|2 ) +

1 2

  Z rp Q34 + (ˇ e4 Ψ)2 −

fp (ˇ e4 Ψ)2

Σ≥R (τ1 )

Z div

 Σ∗

Σ∗ (τ1 ,τ2 )

r−1 f |Ψ|2 νΣ∗



r−2 e4 (rfp |Ψ|2 )

O(mr−1 + )rp−2 (|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 )

Σ∗ (τ1 ,τ2 )

Z + O() Σ≥R (τ2 )

−

rp−1 (|e4 Ψ|2 + r−2 |Ψ|2 ) !

Z r

p−1

2

(|e4 Ψ| + r

−2

Σ≥R (τ1 )

2

|Ψ| ) .

Now, the following integrations by parts hold true: Z r−2 e4 (rfp |Ψ|2 ) Σ≥R (τ )

 1 r e4 (rfp |Ψ| ) e4 (r) r≥R Sr Z  Z Z

1 e4 r−1 fp |Ψ|2 − e4 r−2 + κr−2 rfp |Ψ|2 r≥R e4 (r) Sr Σ≥R (τ )   Z Z Z 2e4 (r) p−1 2 p−1 2 r |Ψ| − r |Ψ| − κ− r−1 fp |Ψ|2 r S∗ (τ ) SR (τ ) Σ≥R (τ ) Z Z Z p−1 2 p−1 2 r |Ψ| − r |Ψ| + O() rp−3 |Ψ|2

Z = = = =

Z

−2

S∗ (τ )

2

SR (τ )

Σ≥R (τ )

and Z div Σ∗ (τ1 ,τ2 )

 Σ∗

r−1 f |Ψ|2 νΣ∗



Z = S∗ (τ2 )

rp−1 |Ψ|2 −

Z S∗ (τ1 )

rp−1 |Ψ|2

673

REGGE-WHEELER TYPE EQUATIONS

where S∗ (τ ) denotes the 2-sphere Σ∗ ∩ Σ(τ ). Note that the boundary terms cancel, except the one on r = R, and hence Z Z   Z 1 K≥R = fp |ˇ e4 Ψ|2 + rp Q34 + (ˇ e4 Ψ)2 − fp |ˇ e4 Ψ|2 2 Σ∗ (τ1 ,τ2 ) Σ≥R (τ2 ) Σ≥R (τ1 ) Z + O(mr−1 + )rp−2 (|e4 Ψ|2 + |∇ / Ψ|2 + r−2 |Ψ|2 ) Σ∗ (τ1 ,τ2 )

Z + O() Σ≥R (τ2 )

− +

1 2

Z

rp−1 (|e4 Ψ|2 + r−2 |Ψ|2 ) !

Z r

2

(|e4 Ψ| + r

−2

Σ≥R (τ1 )

rp−1 |Ψ|2 −

SR (τ2 )

p−1

1 2

Z

2

|Ψ| )

rp−1 |Ψ|2 .

SR (τ1 )

Using Q34 + |ˇ e4 Ψ|

2 4Υ 2 1 = |∇ / Ψ| + 2 |Ψ| + e4 Ψ + Ψ r r 4Υ 1 2 = |∇ / Ψ|2 + 2 |Ψ|2 + (e4 Ψ)2 + 2 |Ψ|2 + Ψ · e4 (Ψ) r r r 4Υ − 3 2 ≥ |∇ / Ψ|2 + |Ψ|2 + |e4 Ψ|2 r2 3

2

2

and the fact that 4Υ ≥ 3 + 2/3 for r ≥ R and R large enough, we infer K≥R

Z

1 2

≥

Σ≥R (τ2 )

rp |ˇ e4 Ψ|2 −

Z + O()

r

p−3

Σ≥R (τ2 )

1 2

+

Z r SR (τ2 )

p−1

Σ≥R (τ1 )

2

|Ψ| −

1 |Ψ| − 2 2

!

Z

rp |ˇ e4 Ψ|2 + F˙p [Ψ](τ1 , τ2 ) !

Z r

p−3

Σ≥R (τ1 )

Z SR (τ1 )

2

|Ψ|

rp−1 |Ψ|2 .

(10.2.17)

Next, recall that, according to Proposition 10.43, we have P · e4 ≥ We infer Z Σ≥R (τ2 )

P · e4

≥

1 p−2 p r (p − 1)2 |Ψ|2 − r−2 e4 (rfp |Ψ|2 ). 8 2

1 8

Z r Σ≥R (τ2 )

p−2

p (p − 1) |Ψ| − 2 2

2

Z Σ≥R (τ2 )

r−2 e4 (rfp |Ψ|2 ).

674

CHAPTER 10

Integrating by parts similarly as before, we infer Z Z Z 1 p P · e4 ≥ rp−2 (p − 1)2 |Ψ|2 − rp−1 |Ψ|2 8 Σ≥R (τ2 ) 2 S∗ (τ ) Σ≥R (τ2 ) Z Z p + rp−1 |Ψ|2 + O() rp−3 |Ψ|2 . 2 SR (τ ) Σ≥R (τ ) Arguing as for the proof of (10.2.17) except for the boundary term on Σ≥R (τ2 ) for which we use the above estimate, we deduce ! Z Z 1 1 p−2 2 2 p 2 K≥R ≥ r (p − 1) |Ψ| + − r |ˇ e4 Ψ| + F˙p [Ψ](τ1 , τ2 ) 8 Σ≥R (τ2 ) 2 Σ≥R (τ1 ) Z + O() rp−1 (|e4 Ψ|2 + r−2 |Ψ|2 ) Σ≥R (τ2 )

− +

1−p 2

!

Z r

p−1

2

(|e4 Ψ| + r

Σ≥R (τ1 )

Z r S∗ (τ2 )

p−1

p |Ψ| + 2 2

Z r SR (τ2 )

−2

2

|Ψ| )

p−1

1 |Ψ| − 2 2

Z SR (τ1 )

rp−1 |Ψ|2 .

We first focus on the case δ ≤ p ≤ 1 − δ, in which case the previous estimate yields ! Z Z δ2 1 p−2 2 p 2 K≥R ≥ r |Ψ| + − r |ˇ e4 Ψ| + F˙p [Ψ](τ1 , τ2 ) 8 Σ≥R (τ2 ) 2 Σ≥R (τ1 ) ! Z Z p−1 2 −2 2 p−3 2 + O() r (|e4 Ψ| + r |Ψ| ) − r |Ψ| Σ≥R (τ2 )

+

1 2

Z SR (τ2 )

rp−1 |Ψ|2 −

Σ≥R (τ1 )

1 2

Z SR (τ1 )

rp−1 |Ψ|2 .

Together with (10.2.17) and the fact that   δ 2 by assumption, we infer in view of the definition of E˙ p,R [Ψ] for δ ≤ p ≤ 1 − δ, Z ˙ ˙ ˙ Ep,R [Ψ](τ2 ) + Fp [Ψ](τ1 , τ2 ) . K≥R + Ep,R [Ψ](τ1 ) + rp−1 |Ψ|2 . SR (τ2 )

Together with (10.2.16), we deduce for δ ≤ p ≤ 1 − δ E˙ p,R [Ψ](τ2 ) + F˙p [Ψ](τ1 , τ2 ) + B˙ p,R [Ψ](τ1 , τ2 ) Z  ˙ . Ep,R [Ψ](τ1 ) + rp−1 |Ψ|2 + Rp+2 E[Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 ) SR (τ2 )

 +O()B˙ δ ; 4m0 [ψ](τ1 , τ2 ) . In view of the improved Morawetz Theorem 10.1, and thanks also to the term B˙ p,R [Ψ](τ1 , τ2 ) on the left-hand side, we may absorb the term O()B˙ δ ; 4m [ψ](τ1 , τ2 ) 0

675

REGGE-WHEELER TYPE EQUATIONS

and obtain

.

E˙ p,R [Ψ](τ2 ) + F˙p [Ψ](τ1 , τ2 ) + B˙ p,R [Ψ](τ1 , τ2 )   Rp+2 Ep [Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 )

(10.2.18)

which is the desired estimate in the case δ ≤ p ≤ 1 − δ. Finally, we focus on the remaining case, i.e., 1 − δ ≤ p ≤ 2 − δ. Combining (10.2.17) and (10.2.16), arguing as in the proof of (10.2.18), and in view of the definition of E˙ p,R [Ψ] for 1 − δ ≤ p ≤ 2 − δ, we obtain E˙ p,R [Ψ](τ2 ) + F˙p [Ψ](τ1 , τ2 ) + B˙ p,R [Ψ](τ1 , τ2 ) . Rp+2 (Ep [Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 )) Z Z +O() rp−3 |Ψ|2 + r−1−δ |Ψ|2 Σ≥R (τ2 )

.

R

p+2

Σ≥R (τ2 )

(Ep [Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 )) + E1−δ [Ψ](τ2 )

where we also used the fact that p ≤ 2 − δ so that p − 3 ≤ −1 − δ. Together with the fact that E˙ p,R [Ψ](τ ) ≥ E˙ 1−δ,R [Ψ](τ ) for p ≥ 1 − δ and (10.2.18), we infer E˙ p,R [Ψ](τ2 ) + F˙p [Ψ](τ1 , τ2 ) + B˙ p,R [Ψ](τ1 , τ2 ) . Rp+2 (Ep [Ψ](τ1 ) + Jp [N, ψ](τ1 , τ2 )) for all δ ≤ p ≤ 2 − δ as desired. This concludes the proof of Theorem 10.37. 10.3

HIGHER WEIGHTED ESTIMATES

We use a variation of the method of [5] to derive slightly stronger weighted estimates. This allows us to prove Theorem 5.18 for s = 0 in section 10.4.6. The proof for higher order derivatives s ≤ ksmall + 29 will be provided in section 10.4.6. As in the previous section we rely on the assumptions (10.2.7)–(10.2.10) to which we add: RP5. The assumptions RP0–RP4 hold true for one extra derivative with respect to d. RP6. e4 (m) satisfies the following improvement of RP4: |d≤2 e4 (m)| . w2,1 . 10.3.1

(10.3.1)

Wave equation for ψˇ

Proposition 10.47. Assume ψ verifies 2 ψ = −κκψ + N . Then ψˇ = f2 eˇ4 ψ verifies:

676

CHAPTER 10

1. In the region r ≥ 6m0 ,     3 2 3m (2 + κκ) ψˇ = r2 e4 (N ) + N + 1− e4 ψˇ + O(r−2 )d≤1 ψ r r r +rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ. 2. In the region 4m0 ≤ r ≤ 6m0 , (2 + κκ) ψˇ =

  3 f2 e4 (N ) + N + O(1)d≤2 ψ. r

The proof of Proposition 10.47 is postponed to section D.4. 10.3.2

The rp -weighted estimates for ψˇ

The goal of this section is to prove Theorem 5.18 in the case s = 0. The proof for higher order derivatives s ≤ ksmall + 29 will be provided in section 10.4.6. Proof of Theorem 5.18 in the case s = 0. We write, in accordance with Proposition 10.47,   3 ˇ ˇ ˇ  ψ − V ψ = N + f2 e4 + N r where

ˇ = N

    


1 − 3m e4 ψˇ + O(r−2 )d≤1 ψ + rΓb e4 dψ r +d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ,

   

O(1)d≤2 ψ,

2 r

r ≥ 6m0 ,

(10.3.2)

4m0 ≤ r ≤ 6m0 .

ˇ This yields, using We apply the first part of Proposition 10.43 to ψ replaced by ψ. also (10.1.13),   3 ˇ ˇ ˇ ˇ ˇ DivPq ( ψ) = Eq ( ψ) + fq eˇ4 ψ · N + fq eˇ4 ψ · f2 e4 + N, r where, with f = fq , Xq = fq e4 , wq =

2fq r , Mq

= 2r−1 fq0 e4 ,

  1 ˇ 2 + 1 2fq − f 0 Q34 ( ψ) ˇ + Errq ( ψ), ˇ = E[Xq , wq , Mq ] = fq0 |ˇ e4 ( ψ)| q 2 2 r   ˇ ˇ : = Err , m ; fq [ ψ] Errq ( ψ) r m   ˇ 2 + O m + w3,1 rq | ψ| ˇ2 = O 2 rq |e4 ψ| 4 r r
ˇ 2 + |∇ ˇ 2 + r−2 | ψ| ˇ2 + O()w1,1 rq |e4 ψ| / ψ|  
ˇ + r−1 |∇ ˇ |e3 ψ| ˇ + |∇ ˇ 2 + r−2 | ψ| ˇ2 , + O()w2,1/2 rq |e4 ψ| / ψ| / ψ| ˇ Eq ( ψ)

ˇ Pk ( ψ)

ˇ = P[Xq , wq , Mq ]( ψ).

677

REGGE-WHEELER TYPE EQUATIONS

We then integrate on the domain M(τ1 , τ2 ) to derive, exactly as in the proof of Theorem 10.37 (see section 10.2.3), Z Z Z
ˇ Pq · e4 + Pq · NΣ∗ + Eq + fq eˇ4 ψˇN Σ(τ2 )

Z = Σ(τ1 )

M(τ1 ,τ2 )

Σ∗ (τ1 ,τ2 )

Pq · e4 −



Z M(τ1 ,τ2 )

fq eˇ4 ψˇ · f2 e4 +

3 r

 N.

(10.3.3)

All terms can be treated exactly as in the proof of Theorem 10.37, except for the bulk term, i.e., we obtain the following analog of (10.2.1): Z ˇ 2) + ˇN ˇ 1 , τ2 ) ˇ ) + F˙q [ ψ](τ E˙ q ; R [ ψ](τ (Eq + rq eˇ4 ( ψ) M(τ1 ,τ2 )



   3 ˇ ˇ . Eq [ ψ](τ1 ) + Jq ψ, f2 e4 + N (τ1 , τ2 ). r Since all terms for r ≤ R can be controlled by one derivative of ψ, we infer Z ˇ 2 ) + Morr[ ψ](τ ˇ 1 , τ2 ) + ˇN ˇ 1 , τ2 ) ˇ ) + F˙q [ ψ](τ E˙ q [ ψ](τ (Eq + rq eˇ4 ( ψ) M≥R (τ1 ,τ2 )

.

ˇ 1 ) + Jˇq [ ψ, ˇ N ](τ1 , τ2 ) + Rq+3 (E 1 [ψ](τ2 ) + Morr1 [ψ](τ1 , τ2 )). (10.3.4) Eq [ ψ](τ

Also, since δ ≤ max(q, δ) ≤ 1 − δ, we have in view of Theorem 5.17 in the case s = 115 sup τ ∈[τ1 ,τ2 ]

.

1 1 Emax(q,δ) [ψ](τ ) + Bmax(q,δ) [ψ](τ1 , τ2 )

1 1 Emax(q,δ) [ψ](τ1 ) + Jmax(q,δ) [ψ, N ](τ1 , τ2 ).

(10.3.5)

In view of (10.3.4) and (10.3.5), it thus only remains to estimate the integral Z ˇN ˇ ), (Eq + rq eˇ4 ( ψ) M≥R (τ1 ,τ2 )

i.e., we need to derive the analog of (10.2.15) used in the proof of Theorem 10.37. This is achieved in Proposition 10.48 below, which together with (10.3.4) and (10.3.5) immediately yields the proof of Theorem 5.18 in the case s = 0. Proposition 10.48. The following estimate holds true: Z Z  ˇN ˇ 2 + (2 − q)|∇ ˇ2 ˇ) ≥ 1 (Eq + rq eˇ4 ( ψ) rq−1 (2 + q)|ˇ e4 ψ| / ψ| 8 M≥R (τ1 ,τ2 ) M≥R (τ1 ,τ2 )  −2 ˇ 2 ˇ ) + 2r | ψ| − O() sup E˙ q,R [ ψ](τ τ1 ≤τ ≤τ2



 1 1 − O(1) Emax(q,δ) [ψ](τ1 ) + Jmax(q,δ) [ψ, N ] . 15 The proof of Theorem 5.17 for higher derivatives s ≥ 1, even though proved later in section 10.4.5, is in fact independent of the proof of Theorem 5.18 and can thus be invoked here.

678

CHAPTER 10

ˇ, We now focus on the proof of Proposition 10.48. In view of the definition of N we have for r ≥ R ˇ N

=

A0

=

A1

=

A2

=

A0 + A1 + A2 , 2 ˇ 2 ˇ e4 ψ = (ˇ e4 ψˇ − r−1 ψ), r r 6m − 2 e4 ψˇ + O(r−2 )d≤1 ψ, r ˇ Err[g ψ],

ˇ Err[g ψ]

=

r2 Γg e4 dψ + rd≤1 (Γg )d≤1 ψ + d≤1 (Γg )d2 ψ.

Also, recall that we have for r ≥ R ˇ Eq ( ψ)

=

E[Xq , wq , Mq ] =

q q−1 ˇ 2 + 2 − q rq−1 Q34 ( ψ) ˇ + Errq ( ψ). ˇ r |ˇ e4 ( ψ)| 2 2

Consequently, we write ˇN ˇ ) = I0 + I1 + I2 , (Eq + rq eˇ4 ( ψ)
1 ˇ 2 + (2 − q)|∇ ˇ 2 + 4(2 − q)r−2 ψˇ2 I0 : = rq−1 q|ˇ e4 ψ| / ψ| 2 ˇ e4 ψˇ − r−1 ψ) ˇ + 2rq−1 eˇ4 ψ(ˇ  1 ˇ 2 + (2 − q)|∇ ˇ2 = rq−1 (q + 4)|ˇ e4 ψ| / ψ| (10.3.6) 2  + 4(2 − q)r−2 ψˇ2 − 4r−1 eˇ4 ψˇ ψˇ ,     ˇ −6me4 ψˇ + O(1)d≤1 ψ + O m rq−3 ( ψ) ˇ 2, I1 : = rq−2 eˇ4 ( ψ) r ˇ + rq eˇ4 ( ψ)A ˇ 2. I2 : = Errq ( ψ) We will rely on the following two lemmas. Lemma 10.49. The following lower bound estimate holds true for q ≤ 1 − δ and r ≥ R, where R is sufficiently large: I0 + I1

≥


1 q−1 ˇ 2 + (2 − q)|∇ ˇ 2 + 2r−2 | ψ| ˇ2 r (2 + q)|ˇ e4 ψ| / ψ| 4 −O(1)rq−3 (d≤1 ψ)2 .

(10.3.7)

Lemma 10.50. The following estimate holds true for the error term I2 : Z   ˇ ) + m0 +  B˙ q,R [ ψ](τ ˇ 1 , τ2 ) |I2 | .  sup E˙ q,R [ ψ](τ R τ1 ≤τ ≤τ2 M≥R (τ1 ,τ2 )   1 1 + sup Eq [ψ](τ ) + Bq [ψ](τ1 , τ2 ) + Jq [ψ, N ](τ1 , τ2 ) . τ1 ≤τ ≤τ2

We postpone the proof of Lemma 10.49 and Lemma 10.50 to finish the proof of Proposition 10.48.

679

REGGE-WHEELER TYPE EQUATIONS

Proof of Proposition 10.48. In view of Lemma 10.49 and Lemma 10.50, we have Z ˇN ˇ) (Eq + rq eˇ4 ( ψ) M≥R (τ1 ,τ2 )

Z

Z

=

(I0 + I1 ) + M≥R (τ1 ,τ2 )

≥

1 4

Z M≥R (τ1 ,τ2 )

− O(1)

ˇ 2 + (2 − q)|∇ ˇ 2 + 2r−2 | ψ| ˇ2 rq−1 (2 + q)|ˇ e4 ψ| / ψ|

Z




rq−3 (d≤1 ψ)2

M≥R (τ1 ,τ2 )

  ˇ ) + O m0 +  B˙ q,R [ ψ](τ ˇ 1 , τ2 ) E˙ q,R [ ψ](τ R τ1 ≤τ ≤τ2   1 1 O() sup Eq [ψ](τ ) + Bq [ψ](τ1 , τ2 ) + Jq [ψ, N ](τ1 , τ2 )

− O() −

I2 M≥R (τ1 ,τ2 )

sup

τ1 ≤τ ≤τ2

so that, since 1 − δ < q ≤ 1 − δ, and for R sufficiently large and small16 , Z ˇN ˇ) (Eq + rq eˇ4 ( ψ) M≥R (τ1 ,τ2 )

≥

1 8

Z M≥R (τ1 ,τ2 )

− O()

sup




ˇ ) E˙ q,R [ ψ](τ

τ1 ≤τ ≤τ2

 − O()

ˇ 2 + (2 − q)|∇ ˇ 2 + 2r−2 | ψ| ˇ2 rq−1 (2 + q)|ˇ e4 ψ| / ψ|

sup τ1 ≤τ ≤τ2

 Eq1 [ψ](τ ) + Bq1 [ψ](τ1 , τ2 ) + Jq [ψ, N ](τ1 , τ2 ) .

In view of (10.3.5), we infer Z Z  ˇN ˇ2 ˇ) ≥ 1 (Eq + rq eˇ4 ( ψ) rq−1 (2 + q)|ˇ e4 ψ| 8 M≥R (τ1 ,τ2 ) M≥R (τ1 ,τ2 )  ˇ 2 + 2r−2 | ψ| ˇ2 + (2 − q)|∇ / ψ| − O()

sup

ˇ ) E˙ q,R [ ψ](τ

τ1 ≤τ ≤τ2

  1 1 − O(1) Emax(q,δ) [ψ](τ1 ) + Jmax(q,δ) [ψ, N ] which concludes the proof. It finally remains to prove Lemma 10.49 and Lemma 10.50. 16 Using also the second bound on Morr from Theorem 10.1 and the bound on B ˙ δ,4m from the 0 rp estimates of Theorem 10.37. See also Remark 10.38.

680

CHAPTER 10

Proof of Lemma 10.49. Note that ˇ 2 − 4r−1 (ˇ ˇ ψˇ + 4(2 − q)r−2 | ψ| ˇ2 (q + 4)|ˇ e4 ψ| e4 ψ)
ˇ 2 + (6 − 4q)r−2 | ψ| ˇ 2 + 2 eˇ4 ψˇ − r−1 ψˇ 2 = (q + 2)|ˇ e4 ψ| ˇ 2 + (6 − 4q)r−2 | ψ| ˇ2 ≥ (q + 2)|ˇ e4 ψ| ˇ 2 + 2r−2 | ψ| ˇ 2, ≥ (q + 2)|ˇ e4 ψ|

where we used the fact that q ≤ 1 − δ. Hence, I0

≥


1 q−1 ˇ 2 + (2 − q)|∇ ˇ 2 + 2r−2 | ψ| ˇ2 . r (2 + q)|ˇ e4 ψ| / ψ| 2

We also have m

1
1 ˇ 2 + r−2 | ψ| ˇ 2 + O(1) rq−1 (ˇ ˇ 2 2 rq−3 (d≤1 ψ)2 2 . I1 ≤ O rq−1 |ˇ e4 ψ| e4 ψ) r Thus if m0 /R is sufficiently small, and since q ≤ 1 − δ, we deduce, for r ≥ R, I0 + I1 ≥


1 q−1 ˇ 2 + (2 − q)|∇ ˇ 2 + 2r−2 | ψ| ˇ 2 − O(1)rq−3 (d≤1 ψ)2 r (2 + q)|ˇ e4 ψ| / ψ| 4

as desired. Proof of Lemma 10.50. Recall that I2

=

A2 = ˇ = Err[g ψ]

ˇ + rq eˇ4 ( ψ)A ˇ 2, Errq ( ψ) ˇ Err[g ψ],

rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ,     ˇ = O m rq |e4 ψ| ˇ 2 + O m + w3,1 rq | ψ| ˇ2 Errq ( ψ) r2 r4
ˇ 2 + |∇ ˇ 2 + r−2 | ψ| ˇ2 + O()w1,1 rq |e4 ψ| / ψ|  
ˇ + r−1 |∇ ˇ |e3 ψ| ˇ + |∇ ˇ 2 + r−2 | ψ| ˇ2 . + O()w2,1/2 rq |e4 ψ| / ψ| / ψ|

Hence, 
ˇ τ −1−δdec |ˇ |I2 | . rq |ˇ e4 ( ψ)| e4 d≤1 ψ| + |d≤1 ψ|
 1 +r−1 τ − 2 −δdec |e3 ψ| + r−1 |d2 ψ| m 
ˇ 2 + |∇ ˇ 2 + r−2 | ψ| ˇ 2 + rq−2 τ − 12 −δdec |e4 ψ||e ˇ 3 ψ| ˇ + +  rq−1 |ˇ e4 ψ| / ψ| r ˇ 3 ψ|. ˇ +rq−3 |∇ / ψ||e

681

REGGE-WHEELER TYPE EQUATIONS

This yields, using q ≤ 1 − δ, Z |I2 | M≥R (τ1 ,τ2 )

 12  ˇ ) E˙ q,R [ ψ](τ sup

 .



sup τ1 ≤τ ≤τ2

+

m

0

R 

+

τ1 ≤τ ≤τ2

Eq1 [ψ](τ ) + Bq1 [ψ](τ1 , τ2 )



ˇ 1 , τ2 ) +  B˙ q,R [ ψ](τ sup

ˇ ) + B˙ q [ ψ](τ ˇ 1 , τ2 ) E˙ q,R [ ψ](τ

 12

τ1 ≤τ ≤τ2

.

! 12

Z M≥R (τ1 ,τ2 )

  ˇ ) + m0 +  B˙ q,R [ ψ](τ ˇ 1 , τ2 ) E˙ q,R [ ψ](τ R τ1 ≤τ ≤τ2   Z + sup Eq1 [ψ](τ ) + Bq1 [ψ](τ1 , τ2 ) +  

 12

ˇ2 rq−4 |e3 ψ|

sup

τ1 ≤τ ≤τ2

M≥R (τ1 ,τ2 )

ˇ For this Next, we estimate the term involving e3 ψ. formula in Lemma 5.20 which we recall below:   1 2 ψ = −e3 e4 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − 2

ˇ 2. rq−4 |e3 ψ|

we need to appeal to the 1 κe3 ψ + 2ηeθ ψ. 2

We have for r ≥ 6m0 e3 ψˇ = = =

e3 (re4 (rψ)) = re3 (re4 ψ) + e3 (r)e4 (rψ) r2 e3 e4 ψ + 2re3 (r)e4 ψ + e3 (r)e4 (r)ψ     1 1 r2 −2 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ 2 2 +2re3 (r)e4 ψ + e3 (r)e4 (r)ψ

so that ˇ . r2 |N | + r|e3 ψ| + |d≤2 ψ| |e3 ψ| and hence Z M≥R (τ1 ,τ2 )

ˇ2 rq−4 |e3 ψ|

Z .

Since q ≤ 1 − δ, we infer Z ˇ2 rq−4 |e3 ψ| M≥R (τ1 ,τ2 )

M≥R (τ1 ,τ2 )

  rq−4 r4 |N |2 + r2 |e3 ψ|2 + |d≤2 ψ|2 .

Z .



(trap)

M(τ1 ,τ2 )

r1−δ |N |2 + Bq1 [ψ](τ1 , τ2 )

. Jq [ψ, N ](τ1 , τ2 ) + Bq1 [ψ](τ1 , τ2 )

682

CHAPTER 10

and thus Z M≥R (τ1 ,τ2 )

  ˇ ) + m0 +  B˙ q,R [ ψ](τ ˇ 1 , τ2 ) E˙ q,R [ ψ](τ R τ1 ≤τ ≤τ2   1 1 + sup Eq [ψ](τ ) + Bq [ψ](τ1 , τ2 ) τ1 ≤τ ≤τ2 Z ˇ2 + rq−4 |e3 ψ|

|I2 | . 

sup

M≥R (τ1 ,τ2 )

  ˇ ) + m0 +  B˙ q,R [ ψ](τ ˇ 1 , τ2 ) E˙ q,R [ ψ](τ R τ1 ≤τ ≤τ2   + sup Eq1 [ψ](τ ) + Bq1 [ψ](τ1 , τ2 ) + Jq [ψ, N ](τ1 , τ2 )

. 

sup

τ1 ≤τ ≤τ2

which concludes the proof of Lemma 10.50.

10.4

HIGHER DERIVATIVE ESTIMATES

We have proved, respectively in section 10.2 and section 10.3.2, Theorem 5.17 on basic weighted estimates (see Remark 10.38) and Theorem 5.18 on higher weighted estimates only in the case s = 0. In this section, we conclude the proof of these theorems by recovering higher order derivatives17 one by one. 10.4.1

Basic assumptions

Recall that any Ricci coefficient either belongs to Γg or Γb , where Γg and Γb are defined in section 5.1.2. We make use of the following non-sharp consequence of the estimates of Lemma 5.1. We assume, concerning the Ricci coefficients, 

|dk (Γg )| .

dec −2δ0 r2 u1+δ trap

|dk (Γb )| .

dec −2δ0 ru1+δ trap

|dk (α, β, ρˇ)| . |dk α| + r|dk β| .

  dec −2δ0 r3 u1+δ trap

 dec −2δ0 ru1+δ trap

for k ≤ ksmall + 30, for k ≤ ksmall + 30, for k ≤ ksmall + 30, for k ≤ ksmall + 30,

where we recall that δdec and δ0 are such that we have in particular 0 < 2δ0 < δdec . 10.4.2

Strategy for recovering higher order derivatives

So far, we have proved Theorem 5.17 in the case s = 018 in section 10.2, and Theorem 5.18 on higher weighted estimates in the case s = 0 in section 10.3. We 17 Respectively s ≤ k small + 30 in the case of Theorem 5.17, and s ≤ ksmall + 29 in the case of Theorem 5.18. 18 Recall that Theorem 5.17 in the case s = 0 is obtained as a consequence of Theorem 10.1 on Morawetz and energy estimates, and Theorem 10.37 on rp -weighted estimates, see Remark 10.38.

683

REGGE-WHEELER TYPE EQUATIONS

now conclude the proof of these theorems by recovering higher order derivatives one by one. Since going from s = 0 to s = 1 is analogous to going from s to s + 1, we will in fact consider only the former. More precisely, we assume the following bounds proved respectively in section 10.2 and section 10.3: sup

Ep [ψ](τ ) + Bp [ψ](τ1 , τ2 ) + Fp [ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

. Ep [ψ](τ1 ) + Jp [ψ, N ](τ1 , τ2 ),

(10.4.1)

and ˇ ) + Bq [ ψ](τ ˇ 1 , τ2 ) . Eq [ ψ](τ ˇ 1 ) + Jˇq [ ψ, ˇ N ](τ1 , τ2 ) Eq [ ψ](τ

sup

(10.4.2)

τ ∈[τ1 ,τ2 ] 1 +E 1max(q,δ) [ψ](τ1 ) + Jmax(q,δ) [ψ, N ],

and our goal is to prove the corresponding estimates for s = 1. We will proceed as follows: 1. We first commute the wave equation for ψ and ψˇ with T and derive (10.4.1) for ˇ T ψ instead of ψ, and (10.4.2) for T ψˇ instead of ψ. ˇ 2. We then commute the wave equation for ψ and ψ with r /d2 and derive (10.4.1) ˇ for r /d2 ψ instead of ψ, and (10.4.2) for r /d2 ψˇ instead of ψ. 3. We then use the wave equation satisfied by ψ to derive an estimate for R2 ψ in r ≤ 6m0 19 with a degeneracy at r = 3m. 4. We then commute the wave equation for ψ with R and remove the degeneracy at r = 3m for R2 ψ. 5. We then commute the wave equation for ψ with the redshift vectorfield YH and derive (10.4.1) for YH ψ instead of ψ. 6. We then commute the wave equation for ψ and ψˇ with f1 e4 and derive (10.4.1) ˇ where f1 = r for for re4 ψ instead of ψ, and (10.4.2) for f1 e4 ψˇ instead of ψ, r ≥ 6m0 and f1 = 0 for r ≤ 4m0 . 7. We finally gather all estimates and conclude. We will follow the above strategy in section 10.4.5 to prove Theorem 5.17, and in section 10.4.6 to prove Theorem 5.18. To this end, we first derive several commutator identities and estimates. 10.4.3 10.4.3.1

Commutation formulas with the wave equation Commutation with T

Lemma 10.51. We have, schematically, the following commutator formulae: [T, e4 ] = Γg d,

[T, e3 ] = Γb d,

[T, /dk ] = Γb d + Γb ,

[T, /d?k ] = Γb d + Γb .

Proof. Recall that we have [e3 , e4 ] 19 Note

=

2ωe4 − 2ωe3 + 2(η − η)eθ .

that any finite region in r strictly containing the trapping region would suffice.

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CHAPTER 10

We infer 2[T, e4 ]

=

[e4 + Υe3 , e4 ]

=

Υ[e3 , e4 ] − e4 (Υ)e3

= =


Υ 2ωe4 − 2ωe3 + 2(η − η)eθ − e4 (Υ)e3     1 −1 Υ 2ωe4 − 2 ω + Υ e4 (Υ) e3 + 2(η − η)eθ 2

=

(r−1 Γb + Γg )d

=

Γg d,

and 2[T, e3 ]

=

[e4 + Υe3 , e3 ]

=

[e4 , e3 ] − e3 (Υ)e3

= −2ωe4 + 2ωe3 − 2(η − η)eθ − e3 (Υ)e3   1 = −2ωe4 + 2 ω − e3 (Υ) e3 − 2(η − η)eθ 2   1 1 = −2ωe4 + 2 ω + Υ−1 e4 (Υ) − T (Υ) e3 − 2(η − η)eθ 2 2Υ = (Γg + Γb )d =

Γb d.

Next, recall in view of Lemma 2.54, the following commutation formulae for reduced scalars. 1. If f ∈ sk , 1 κ /dk f + Comk (f ), 2 1 Comk (f ) = − ϑ /d?k+1 f + (ζ − η)e3 f − kηe3 Φf − ξ(e4 f + ke4 (Φ)f ) − kβf, 2 1 [ /dk , e4 ] = κ /dk f + Comk (f ), 2 1 Comk (f ) = − ϑ /d?k+1 f − (ζ + η)e4 f − kηe4 Φf − ξ(e3 f + ke3 (Φ)f ) − kβf. 2 [ /dk , e3 ]f =

2. If f ∈ sk−1 , 1 ? κ /dk f + Com∗k (f ), 2 1 Com∗k (f ) = − ϑ /dk−1 f − (ζ − η)e3 f − (k − 1)ηe3 Φf + ξ(e4 f − (k − 1)e4 (Φ)f ) 2 − (k − 1)βf, 1 [ /d?k , e4 ]f = κ /dk f + Com∗k (f ), 2 1 ∗ Comk (f ) = − ϑ /dk−1 f + (ζ + η)e4 f − (k − 1)ηe4 Φf + ξ(e3 f − (k − 1)e3 (Φ)f ) 2 − (k − 1)βf. [ /d?k , e3 ]f =

685

REGGE-WHEELER TYPE EQUATIONS

We infer, schematically, 2[T, /dk ]

=

[e4 + Υe3 , /dk ]

=

[e4 , /dk ] + Υ[e3 , /dk ] − eθ (Υ)e3 m 1 − (κ + Υκ) /dk + Γb d + r−1 Γb + 2eθ e3 2 r Γb d + Γb .

= =

The estimate for [T, /d?k ] is similar and left to the reader. This concludes the proof of the lemma. Lemma 10.52. We have T (κ) = d

≤1

 Γg ,

T

 1 2ω − κ = d≤1 Γb , 2

T (K) = d≤1 Γg .

Proof. We have 2T (κ)

=

(e4 + Υe3 )κ 1 1 = − κ2 − 2ωκ + 2 /d1 ξ + 2(η + η + 2ζ)ξ − ϑ2 2 2  1 1 +Υ − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2(ξξ + η 2 ) 2 2 1 2 1 = − κ − 2ωκ − κκΥ + 2ωκΥ + 2Υρ 2 2   1 1 +2 /d1 ξ + 2Υ /d1 η + 2(η + η + 2ζ)ξ − ϑ2 + Υ − ϑϑ + 2(ξξ + η 2 ) 2 2 = r−1 dΓb + r−1 Γb = d≤1 Γg .

Also, we have T (r) = e4 (r) + Υe3 (r) = e4 (r) − Υ + Υ(e3 (r) − 1) ∈ rΓb . We infer T

  1 2ω − κ 2

    1 1 2 + Γb − T κ + r 2 r T (r) = − 2 + dΓb r = d≤1 Γb = T

and 

   1 1 T (K) = T +T K − 2 r2 r 2T (r) = − 3 + r−1 Γb r = r−1 d(Γb ) + r−1 Γb = d≤1 Γg .

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CHAPTER 10

This concludes the proof of the lemma. Corollary 10.53. We have [T, 2 ]ψ

= d≤1 (Γg )d≤2 ψ.

Proof. Recall that we have   1 1 = −e3 e4 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ. 2 2

2 ψ We infer [T, 2 ]ψ

  1 −[T, e3 ]e4 ψ − e3 [T, e4 ]ψ + [T, 4 / 2 ]ψ + 2ω − κ [T, e4 ]ψ 2   1 1 1 +T 2ω − κ e4 ψ − κ[T, e3 ]ψ − T (κ)e3 ψ + 2η[T, eθ ]ψ 2 2 2 +2T (η)eθ ψ

=

and hence, using also 4 / 2 = − /d?2 /d2 + 2K, [T, 2 ]ψ −[T, e3 ](r−1 dψ) − d[T, e4 ]ψ − r−1 d[T, /d2 ]ψ − [T, /d?2 ]r−1 dψ + 2T (K)ψ     1 1 1 1 + 2ω − κ [T, e4 ]ψ + T 2ω − κ r−1 dψ − κ[T, e3 ]ψ − T (κ)dψ 2 2 2 2

=

+2η[T, eθ ]ψ + 2T (η)r−1 dψ. In view of [T, e4 ] = Γg d,

[T, e3 ] = Γb d,

[T, /dk ] = Γb d + Γb ,

[T, /d?k ] = Γb d + Γb

and ≤1

T (κ) = d

 Γg ,

T

 1 2ω − κ = d≤1 Γb , 2

T (K) = d≤1 Γg ,

we deduce, schematically, [T, 2 ]ψ

= d≤1 (Γg )d≤2 ψ + r−1 d≤1 (Γb )d≤2 ψ = d≤1 (Γg )d≤2 ψ.

This concludes the proof of the corollary. 10.4.3.2

Commutation with angular derivatives

Lemma 10.54. We have, schematically, [r /dk , e4 ]f, [r /d?k , e4 ]f = Γg d≤1 f, [r2 4 / k , e4 ]f = d≤1 (Γg )d≤2 f

[r /dk , e3 ]f = −rηe3 (f ) + Γb d≤1 f, [r /d?k , e3 ]f = rηe3 (f ) + Γb d≤1 f. Proof. Recall from Lemma 2.68 that the following commutation formulae holds

687

REGGE-WHEELER TYPE EQUATIONS

true: 1. If f ∈ sk ,   1 [r /dk , e4 ] = r Comk (f ) − A /dk f , 2   1 [r /dk , e3 ]f = r Comk (f ) − A /dk f . 2 2. If f ∈ sk−1 ,  1 ? =r − A /dk f , 2   1 ? ∗ ? [r /dk , e3 ]f = r Comk (f ) − A /dk f , 2 [r /d?k , e4 ]f



Com∗k (f )

where A = 2/re4 (r) − κ and A = 2/re3 (r) − κ. Now, we have Comk (f ) = r−1 Γg d≤1 f,

Com∗k (f ) = r−1 Γg d≤1 f, Com∗k (f ) = ηe3 (f ) + r−1 Γb d≤1 f,

Comk (f ) = −ηe3 (f ) + r−1 Γb d≤1 f,

which together with the fact that A ∈ Γg and A ∈ Γb implies, schematically, [r /dk , e4 ]f, [r /d?k , e4 ]f = Γg d≤1 f, [r /dk , e3 ]f = −rηe3 (f ) + Γb d≤1 f, [r /d?k , e3 ]f = rηe3 (f ) + Γb d≤1 f. Since 4 / k = − /d?k /dk + kK, we infer [r2 4 / k , e4 ]

=

[−r2 /d?k /dk + kr2 K, e4 ]

=

−[r /d?k , e4 ]r /dk − r /d?k [r /dk , e4 ] − ke4 (r2 K)

=

d≤1 (Γg )d≤2 f.

This concludes the proof of the lemma. Corollary 10.55. We have r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

=

−rη2 ψ + d≤1 (Γg )d≤2 ψ

and r /d?2 (1 φ) − (2 − 3K)(r /d?2 φ)

=

rη1 φ + d≤1 (Γg )d≤2 φ.

Proof. Recall that we have 2 ψ

=

  1 1 −e3 e4 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ 2 2

and 

1 φ

 1 1 = −e3 e4 φ + 4 / 2 φ + 2ω − κ e4 φ − κe3 φ + 2ηeθ φ. 2 2

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CHAPTER 10

We infer r /d2 (2 ψ) − 1 (r /d2 ψ)

= −[r /d2 , e3 ]e4 ψ − e3 [r /d2 , e4 ]ψ + r( /d2 4 / −4 / 1 /d2 )ψ + [r, 4 / 1 ] /d2 ψ     2 1 1 1 +reθ 2ω − κ e4 ψ + 2ω − κ [r /d2 , e4 ]ψ − reθ (κ)e3 ψ 2 2 2 1 − κ[r /d2 , e3 ]ψ + 2r /d2 (ηeθ ψ) − 2ηeθ (r /d2 ψ), 2 and r /d?2 (1 φ) − 2 (r /d?2 φ)

= −[r /d?2 , e3 ]e4 φ − e3 [r /d?2 , e4 ]φ + r( /d?2 4 / −4 / 2 /d?2 )φ + [r, 4 / 2 ] /d?2 φ     1 1 1 1 −reθ 2ω − κ e4 φ + 2ω − κ [r /d?2 , e4 ]φ + reθ (κ)e3 φ 2 2 2 1 − κ[r /d?2 , e3 ]φ + 2r /d2 (ηeθ φ) − 2ηeθ (r /d2 φ), 2 and hence, using also in particular the following identities from Proposition 2.28, /d2 4 /2 − 4 / 1 /d2

/d?2 4 /1 − 4 / 2 /d?2

= =

−K /d2 + 2eθ (K),

−3K /d?2 − eθ (K),

we infer

=

r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

−[r /d2 , e3 ]e4 ψ − d[r /d2 , e4 ]ψ + 2reθ (K)ψ + [r, 4 / 1 ](r−1 dψ)     1 1 1 +reθ 2ω − κ r−1 dψ + 2ω − κ [r /d2 , e4 ]ψ − reθ (κ)dψ 2 2 2 1 − κ[r /d2 , e3 ]ψ + 2d(r−1 ηdψ) − 2r−1 ηd(dψ), 2

and r /d?2 (1 φ) − (2 − 3K)(r /d?2 φ)

= −[r /d?2 , e3 ]e4 φ − d[r /d?2 , e4 ]φ − reθ (K)φ + [r, 4 / 2 ](r−1 dφ)     1 1 1 −reθ 2ω − κ r−1 dφ + 2ω − κ [r /d?2 , e4 ]φ + reθ (κ)dφ 2 2 2 1 − κ[r /d?2 , e3 ]φ + 2d(r−1 ηdφ) − 2r−1 ηd(dφ). 2 This yields, schematically, r /d2 (2 ψ) − (1 − K)(r /d2 ψ) =

1 −[r /d2 , e3 ]e4 ψ − d[r /d2 , e4 ]ψ + r−1 [r /d2 , e4 ]ψ − κ[r /d2 , e3 ]ψ + d≤1 (Γg )d≤2 ψ 2

689

REGGE-WHEELER TYPE EQUATIONS

and r /d?2 (1 φ) − (2 − 3K)(r /d?2 φ)

1 −[r /d?2 , e3 ]e4 φ − d[r /d?2 , e4 ]φ + r−1 [r /d?2 , e4 ]φ − κ[r /d?2 , e3 ]φ + d≤1 (Γg )d≤2 φ 2

=

where we used the fact that r−1d≤1 Γb is at least as good as d≤1 Γg and the fact that r−1 eθ (r) is Γg . Next, we rely on [r /dk , e4 ]f, [r /d?k , e4 ]f = Γg d≤1 f, [r /dk , e3 ]f = −rηe3 (f ) + Γb d≤1 f, [r /d?k , e3 ]f = rηe3 (f ) + Γb d≤1 f to infer r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

=

1 rηe3 e4 ψ + rκηe3 ψ + d≤1 (Γg )d≤2 ψ 2

r /d?2 (1 φ) − (2 − 3K)(r /d?2 φ)

=

1 −rηe3 e4 φ − rκηe3 φ + d≤1 (Γg )d≤2 φ. 2

and

This yields r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

=

    1 rη −2 ψ + 4 / 2 ψ + 2ω − κ e4 ψ + 2ηeθ ψ 2 +d≤1 (Γg )d≤2 ψ

=

−rη2 ψ + d≤1 (Γg )d≤2 ψ

and

= =

r /d?2 (1 φ) − (2 − 3K)(r /d?2 φ)     1 −rη −1 φ + 4 / 2 φ + 2ω − κ e4 φ + 2ηeθ φ + d≤1 (Γg )d≤2 φ 2 rη1 φ + d≤1 (Γg )d≤2 φ,

where we used the fact that r−1d≤1 Γb is at least as good as d≤1 Γg . This concludes the proof of the corollary. 10.4.3.3

Commutation with R in the region r ≤ r0

We derive in the following lemma commutator identities that are non-sharp as far as decay in r is concerned. This is sufficient for our needs since we will commute the wave equation with R only in the region r ≤ r0 for a fixed r0 ≥ 4m0 large enough. We will use in particular the following estimate: max

k≤ksmall +30

|d(Γg )| .

 dec −2δ0 r2 u1+δ trap

.

(10.4.3)

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CHAPTER 10

Lemma 10.56. We have [R, e4 ] = O

!

 dec −2δ0 u1+δ trap

[r /dk , R]f, [r /d?k , R]f = O

d, 

dec −2δ0 u1+δ trap

[r 4 / k , R]f = O

!

 dec −2δ0 u1+δ trap

d≤1 f,

!



2

2m [R, e3 ] = − 2 e3 + O r !

dec −2δ0 u1+δ trap

d≤2 f.

Proof. Recall that R is defined by R=

1 (e4 − Υe3 ) 2

and that we have [e3 , e4 ]

=

2ωe4 − 2ωe3 + 2(η − η)eθ .

We infer [R, e4 ]

= =

1 Υ 1 [−Υe3 , e4 ] = − [e3 , e4 ] + e4 (Υ)e3 2 2 2 !   m  Υωe3 + 2 e4 (r) e3 + O d, dec −2δ0 r u1+δ trap

and 1 1 1 [e4 − Υe3 , e3 ] = [e4 , e3 ] + e3 (Υ)e3 2 2 2 !   m  = ωe3 + 2 e3 (r) e3 + O d, dec −2δ0 r u1+δ trap

[R, e3 ]

=

and hence, [R, e4 ] = O



!

dec −2δ0 u1+δ trap

d,

2m [R, e3 ] = − 2 e3 + O r



!

dec −2δ0 u1+δ trap

d.

Also, recall that we have [r /dk , e4 ]f, [r /d?k , e4 ]f = Γg d≤1 f, [r2 4 / k , e4 ]f = d≤1 (Γg )d≤2 f,

[r /dk , e3 ]f = −rηe3 (f ) + Γb d≤1 f, [r /d?k , e3 ]f = rηe3 (f ) + Γb d≤1 f.

d,

691

REGGE-WHEELER TYPE EQUATIONS

We infer [r /d?k , e4 ]f

[r /dk , e4 ]f,

=O

[r 4 / k , e4 ]f = O [r /dk , e3 ]f = O

d≤1 f,

dec −2δ0 u1+δ trap



2

!

 !

dec −2δ0 u1+δ trap !  dec −2δ0 u1+δ trap

d≤2 f, ≤1

d

[r /d?k , e3 ]f

f,



=O

dec −2δ0 u1+δ trap

! d≤1 f.

Together with the definition for R, we deduce [r /dk , R]f, [r /d?k , R]f = O [r2 4 / k , R]f = O



 dec −2δ0 u1+δ trap !

! d≤1 f,

d≤2 f.

dec −2δ0 u1+δ trap

This concludes the proof of the lemma. Corollary 10.57. We have in the region r ≤ r0  2 (Rψ)

=

3m 1− r



2

d ψ+O

 dec −2δ0 u1+δ trap

! d2 ψ + O(1)d≤1 ψ + O(1)d≤1 N.

Proof. Recall that we have 2 ψ

  1 1 = −e4 (e3 (ψ)) + 4 / 2 ψ − κe4 ψ + − κ + 2ω e3 ψ + 2ηeθ ψ. 2 2

Multiplying by r2 , we infer 2

r 2 ψ

=

  1 2 1 2 −r e4 (e3 (ψ)) + r 4 / 2 ψ − r κe4 ψ + r − κ + 2ω e3 ψ + 2rηreθ ψ 2 2 2

2

and hence R(r2 2 ψ)

1 = r2 2 (Rψ) − [R, r2 e4 e3 ]ψ + [R, r2 4 / 2 ]ψ − R(r2 κ)e4 ψ    2 1 2 1 − r κ[R, e4 ]ψ + R r2 − κ + 2ω e3 ψ 2 2   1 +r2 − κ + 2ω [R, e3 ]ψ + 2R(rη)reθ ψ + 2rη[R, reθ ]ψ. 2

Using the commutation identities of the previous lemma, we infer in the region r ≤ r0 !  2 2 2 R(r 2 ψ) = r 2 (Rψ) − [R, r e4 e3 ]ψ + O d2 ψ + O(1)dψ. dec −2δ0 u1+δ trap

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CHAPTER 10

Also, since ψ satisfies 2 ψ = V ψ + N , we infer in the region r ≤ r0 !  2 2 r 2 (Rψ) = [R, r e4 e3 ]ψ + O d2 ψ + O(1)d≤1 ψ + O(1)d≤1 N. dec −2δ0 u1+δ trap Next, recall that we have !



[R, e4 ] = O

dec −2δ0 u1+δ trap

d,

[R, e3 ] = −

2m e3 + O()d. r2

We infer [R, r2 e4 e3 ]ψ

= =

=

R(r2 )e4 e3 + r2 [R, e4 ]e3 + r2 e4 [R, e3 ]    1 2m e4 (r2 ) − Υe3 (r2 ) e4 e3 ψ + r2 e4 − 2 e3 ψ 2 r !  +O d2 ψ dec −2δ0 u1+δ trap !    2 r − 3m e4 e3 ψ + O d2 ψ + O(1)dψ dec −2δ0 u1+δ trap

and thus, in the region r ≤ r0 ,  2 (Rψ)

=

3m 1− r





2

d ψ+O

dec −2δ0 u1+δ trap

! d2 ψ + O(1)d≤1 ψ + O(1)d≤1 N

as desired. 10.4.3.4

Commutation with the redshift vectorfield

Let a positive bump function κ = κ(r), supported in the region in [−2, 2] and equal to 1 for [−1, 1]. Recall that the redshift vectorfield is given by   Υ YH = κH Y(0) , κH := κ δH where Y(0) is defined by Y(0) = ae3 + be4 + 2T,

a=1+

5 (r − 2m), 4m

b=

5 (r − 2m). 4m

Lemma 10.58. We have [2 , e3 ]ψ

=

−2ωe3 (e3 ψ) + κe4 (e3 ψ) + κ2 ψ + d≤1 (Γg )d2 ψ + r−2 d≤1 ψ.

Proof. Recall that we have 2 ψ

  1 1 = −e4 (e3 (ψ)) + 4 / 2 ψ − κe4 ψ + − κ + 2ω e3 ψ + 2ηeθ ψ. 2 2

693

REGGE-WHEELER TYPE EQUATIONS

Since we have [e4 , e3 ] = 2ωe3 + r−1 Γb d/, [ /d?k , e3 ] =

[ /dk , e3 ] =

1 κ /dk + Γb d + r−1 Γb , 2

1 ? κ /dk + Γb d + r−1 Γb , 2

we infer [2 , e3 ]ψ

1 = −[e4 , e3 ](e3 (ψ)) + [4 / 2 , e3 ] − κ[e4 , e3 ](ψ) 2   1 1 + e3 (κ)e4 (ψ) − e3 − κ + 2ω e3 (ψ) + 2η[eθ , e3 ]ψ − 2e3 (η)eθ (ψ) 2 2 = −2ωe3 (e3 ψ) + κ4 / 2 ψ + d≤1 (Γg )d2 ψ + r−2 d≤1 ψ.

Using again 2 ψ

  1 1 = −e4 (e3 (ψ)) + 4 / 2 ψ − κe4 ψ + − κ + 2ω e3 ψ + 2ηeθ ψ, 2 2

we deduce [2 , e3 ]ψ

  = −2ωe3 (e3 ψ) + κ 2 ψ + e4 (e3 ψ) + d≤1 (Γg )d2 ψ + r−2 d≤1 ψ = −2ωe3 (e3 ψ) + κe4 (e3 ψ) + κ2 ψ + d≤1 (Γg )d2 ψ + r−2 d≤1 ψ.

This concludes the proof of the lemma. Lemma 10.59. There exists a scalar function d0 satisfying the bound d0

=

1 + O(δH ) on the support of κH , 2m0

such that we have, schematically,  [2 , YH ]ψ

= d0 Y(0) (YH ψ) + 1Υ≤2δH +

2 ψ + dT ψ + d≤1 (Γg )d2 ψ +

1 ≤1 ψ 2 d δH

1 1δ ≤Υ≤2δH d≤2 ψ. δH H

Proof. We have Y(0) = ae3 + be4 + 2T

= ae3 + b(2T − Υe3 ) + 2T =

(a − Υb)e3 + 2(1 + b)T.

Thus, in view of the commutator identities [T, 2 ]ψ

=

d≤1 (Γg )d≤2 ψ,

[2 , e3 ]ψ

=

−2ωe3 (e3 ψ) + κe4 (e3 ψ) + κ2 ψ + d≤1 (Γg )d2 ψ + r−2 d≤1 ψ,



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CHAPTER 10

we deduce, schematically, [2 , Y(0) ]ψ

= =

[2 , (a − Υb)e3 ]ψ + [2 , 2(1 + b)T ]ψ

(a − Υb)[2 , e3 ]ψ + gαβ Dα (a)Dβ e3 ψ + 2(1 + b)[2 , T ]ψ

+2gαβ Dα (b)Dβ T ψ + d≤1 ψ   1 = (a − Υb) − 2ωe3 (e3 ψ) + κe4 (e3 ψ) + κ2 ψ − e3 (a)e4 (e3 ψ) 2 1 − e4 (a)e3 (e3 ψ) + dT ψ + d≤1 (Γg )d2 ψ + d≤1 ψ. 2 Since e4 = −Υe3 + 2T , we infer schematically   Υ 1 [2 , Y(0) ]ψ = (a − Υb)(−2ω − Υκ) + e3 (a) − e4 (a) e3 (e3 ψ) 2 2 +2 ψ + dT ψ + d≤1 (Γg )d2 ψ + d≤1 ψ. We deduce [2 , YH ]ψ

=

[2 , κH Y(0) ]ψ

=

κH [2 , Y(0) ]ψ + κ0H d≤2 ψ + κ00H d≤1 ψ   Υ 1 κH (a − Υb)(−2ω − Υκ) + e3 (a) − e4 (a) e3 (e3 ψ) 2 2   1 ≤1 ≤1 2 +1Υ≤2δH 2 ψ + dT ψ + d (Γg )d ψ + 2 d ψ δH 1 + 1δH ≤Υ≤2δH d≤2 ψ. δH

=

Now, we have κH e3 (e3 ψ)

= = =

1 κH Y(0) (e3 ψ) + T dψ a − Υb 1 κH Y(0) (Y(0) ψ) + dT ψ + d≤1 ψ (a − Υb)2 1 1 ≤1 Y(0) (YH ψ) + dT ψ + d ψ (a − Υb)2 δH

and hence [2 , YH ]ψ

=

1 (a − Υb)(−2ω − Υκ) + Υ 2 e3 (a) − 2 e4 (a) Y(0) (YH ψ) (a − Υb)2   1 ≤1 ≤1 2 +1Υ≤2δH 2 ψ + dT ψ + d (Γg )d ψ + 2 d ψ δH 1 + 1δH ≤Υ≤2δH d≤2 ψ. δH

695

REGGE-WHEELER TYPE EQUATIONS

Now, we have, in view of the definition of a and b, 1 (a − Υb)(−2ω − Υκ) + Υ 2 e3 (a) − 2 e4 (a) (a − Υb)2

= = =

(1 + O(Υ))(−2ω + O(Υ)) + O(Υ) (1 + O(Υ))2 1 + O() + O(Υ) 2m 1 + O() + O(Υ) 2m0

where we also used our assumptions on ω and m. Thus, we have on the support of κH 1 (a − Υb)(−2ω − Υκ) + Υ 2 e3 (a) − 2 e4 (a) (a − Υb)2

= =

1 + O( + δH ) 2m0 1 + O(δH ) 2m0

where we used the fact that   δH by assumption. Setting d0

:=

1 (a − Υb)(−2ω − Υκ) + Υ 2 e3 (a) − 2 e4 (a) , 2 (a − Υb)

this concludes the proof of the lemma. 10.4.3.5

Commutation with re4

Lemma 10.60. We have, schematically,   Υ 2m 1 [2 , re4 ]ψ = 1+ eˇ4 (re4 ψ) + 2 ψ + Γg d2 ψ + r−2 dT ψ 2 r rΥ Υ +r−2 d/2 ψ + r−2 dψ. Proof. Recall that we have 2 ψ

  1 1 = −e3 (e4 (ψ)) + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ. 2 2

Since we have r r [re4 , e3 ] = 2rωe3 − κe4 + Γb d, [re4 , e4 ] = − κe4 + Γg d, 2 2 1 1 ? [ /dk , re4 ] = rκ /dk + Γg d + Γg , [ /dk , re4 ] = rκ /d?k + Γg d + Γg , 2 2

696

CHAPTER 10

we infer, schematically, [2 , re4 ]ψ

=

=

=

  1 −[e3 , re4 ]e4 ψ − e3 [e4 , re4 ]ψ + [4 / 2 , re4 ]ψ + 2ω − κ [e4 , re4 ]ψ 2   1 1 1 −re4 2ω − κ e4 ψ − κ[e3 , re4 ]ψ + re4 (κ)e3 ψ 2 2 2 +2η[eθ , re4 ]ψ − 2re4 (η)eθ ψ   r 1 1 2rωe3 − κe4 e4 ψ − e3 (rκe4 ψ) + rκ4 / 2 ψ + re4 (κ)e3 ψ 2 2 2 +Γg d2 ψ + r−2 dψ   r 1 1 2rωe3 − κe4 e4 ψ − rκe3 e4 ψ − rκ2 e3 ψ + Γg d2 ψ + r−2 d/2 ψ 2 2 4 +r−2 dψ.

Using again 2 ψ

  1 1 = −e3 (e4 (ψ)) + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ, 2 2

we have 1 1 − rκe3 e4 ψ − rκ2 e3 ψ 2 4

=

1 rκ2 ψ + r−2 d/2 ψ + r−2 dψ 2

and hence [2 , re4 ]ψ

 r 2rωe3 − κe4 e4 ψ + 2 ψ + Γg d2 ψ + r−2 d/2 ψ + r−2 dψ 2   1 r = 2rω (2T − e4 ) − κe4 e4 ψ + 2 ψ + Γg d2 ψ + r−2 d/2 ψ Υ 2 =



+r−2 dψ   1 r 1 = −2rω − κ e4 (e4 ψ) + 2 ψ + Γg d2 ψ + r−2 dT ψ + r−2 d/2 ψ Υ 2 Υ +r−2 dψ   2m 1 = Υ+ e4 (e4 ψ) + 2 ψ + Γg d2 ψ + r−2 dT ψ + r−2 d/2 ψ rΥ Υ +r−2 dψ   Υ 2m 1 = 1+ eˇ4 (re4 ψ) + 2 ψ + Γg d2 ψ + r−2 dT ψ + r−2 d/2 ψ 2 r rΥ Υ +r−2 dψ. This concludes the proof of the lemma. 10.4.4

Some weighted estimates for wave equations

Recall from Corollary 10.55 that we have the following commutator identity: r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

= −rη2 ψ + d≤1 (Γg )d≤2 ψ.

697

REGGE-WHEELER TYPE EQUATIONS

In particular, to derive weighted estimates for r /d2 , we need to derive weighted estimates for solutions φ to wave equations of the type (1 − V1 )φ

=

N,

where φ is a reduced 1-scalar and the potential V1 is given by V1 = V +K = −κκ+K. This is done in the following theorem. Theorem 10.61. Let φ a reduced 1-scalar solution to (1 − V1 )φ =

N,

V1 = −κκ + K.

Then, φ satisfies, for all δ ≤ p ≤ 2 − δ, sup

Ep [φ](τ ) + Bp [φ](τ1 , τ2 ) + Fp [φ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

Z 1 − 3m |φ|(|φ| + |Rφ|) . Ep [φ](τ1 ) + Jp [φ, N ](τ1 , τ2 ) + r (trap) M(τ ,τ ) 1 2 Z  + rp−3 |φ|(|φ| + |dφ|), (10.4.4) (trap)

M(τ1 ,τ2 )

and φˇ = f2 eˇ4 φ satisfies for all 1 − δ < q ≤ 1 − δ, sup

ˇ ) + Bq [φ](τ ˇ 1 , τ2 ) Eq [φ](τ

τ ∈[τ1 ,τ2 ] 1 ˇ 1 ) + Jˇq [φ, ˇ N ](τ1 , τ2 ) + E 1 . Eq [φ](τ max(q,δ) [φ](τ1 ) + Jmax(q,δ) [φ, N ] Z ˇ φ| ˇ + |dφ|). ˇ + rq−3 |φ|(| (10.4.5) M(τ1 ,τ2 )

Remark 10.62. Although we will not need it, we expect that the last two terms in the right-hand side of (10.4.4) and the last term in the right-hand side of (10.4.5) could be removed. Proof. We start with the following observations. • (10.4.5) is the analog of (10.4.1), i.e., of Theorem 5.18 in the case s = 0, with V replaced by V1 , and with the reduced 2-scalar ψ replaced by the reduced 1-scalar φ. The proof is in fact significantly easier in view of the presence of the term Z ˇ φ| ˇ + |dφ|) ˇ rq−3 |φ|(| M(τ1 ,τ2 )

on the right-hand side of (10.4.5). • (10.4.4) is the analog of (10.4.2), i.e., of Theorem 5.17 in the case s = 0, with V replaced by V1 , and with the reduced 2-scalar ψ replaced by the reduced 1-scalar φ. The proof is in fact significantly easier in view of the presence of the terms Z Z  1 − 3m |φ|(|φ| + |Rφ|) + rp−3 |φ|(|φ| + |dφ|) (trap) r (trap) M(τ ,τ ) M(τ ,τ ) 1 2 1 2 on the right-hand side of (10.4.4).

698

CHAPTER 10

• The boundary terms can be treated as in the proof of (10.4.1) and (10.4.2) in view of the fact that V1 is a positive potential.20 • The only place where there might be a potential difficulty concerns the proof of (10.4.5) in (trap) M where the second-to-last term on the right-hand side is required to have a more precise structure. In view of the above observations, and in particular of the last one, we focus on recovering the bulk term leading to (10.4.5) in (trap) M. To this end, we choose f and w as in Proposition 10.16. This yields21  3m 
1− r ˙ R, w](Ψ) ≥ f 0 |R(Ψ)|2 + r−1 1 − 3m f |∇ E[f / Ψ|2 + O |Ψ|2 . r r3 We infer ˙ R, w, M = 2hR](Ψ) ≥ E[f

0

2

f |R(Ψ)| + r  +O

1 − 3m r r3

−1





3m 1− r



f |∇ / Ψ|2

1 |Ψ|2 + r−2 (Υr2 h)0 |Ψ|2 + hΨR(Ψ). 2

We now choose a smooth h, compactly supported in [5/2m0 , 7/2m0 ], such that h(3m) = 0 and h0 (3m) = 1.22 We infer r−2 (Υr2 h)0 (3m) = 1/3 > 0 and hence   Υ ˙ R, w, M ](Ψ) ≥ f 0 |R(Ψ)|2 + r−1 1 − 3m f |∇ E[f / Ψ|2 + 3 |Ψ|2 r r  3m  1− r +O |Ψ|(|Ψ| + |R(Ψ)|). r3 In view of the choice of f in Proposition 10.16, we have    2 3m 3m 1− f ≥ 1− , r r

1 f & 3, r 0

and hence, there exist two constants c0 > 0 and C0 > 0 such that !  2 1 3m Υ 2 −1 2 2 |R(Ψ)| + r 1− |∇ / Ψ| + 3 |Ψ| r3 r r 3m 1 − r −C0 |Ψ|(|Ψ| + |R(Ψ)|). 3 r

˙ R, w0 , M ](Ψ) ≥ c0 E[f

The last term above is responsible for the second-to-last term on the right-hand 20 We

have V1 = −κκ + K =

4Υ + 1 + O() r2

in view of the assumptions so that V1 is indeed a positive potential. 21 Note that Proposition 10.16 does not use the particular form of the potential and the type of the reduced scalar φ and hence holds in our more general case. 22 This differs from the choice of h in the proof of (10.4.1) in order to avoid using a Poincar´ e inequality (which depends on the type of the reduced scalar) and the particular form of the potential V1 .

699

REGGE-WHEELER TYPE EQUATIONS

side of (10.4.5). Next, we have the following consequence of (10.4.1) and Theorem 10.61. Corollary 10.63. Let φ be a reduced k-scalar for k = 1, 2 such that φ satisfies23 !  (k − W )φ = O dφ1 + φ2 dec −2δ0 r2 u1+δ trap where φ1 and φ2 are given reduced scalars, and where W = V in the case k = 2 and W = V1 in the case k = 1. Then, φ satisfies, for all δ ≤ p ≤ 2 − δ, sup

Ep [φ](τ ) + Bp [φ](τ1 , τ2 ) + Fp [φ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! .

Ep [φ](τ1 ) + 

2

sup E[φ1 ](τ ) + Bp [φ1 ](τ1 , τ2 )

+ Jp [φ, φ2 ](τ1 , τ2 )

[τ1 ,τ2 ]

Z 3m  + 1 − |φ|(|φ| + |Rφ|) + rp−3 |φ|(|φ| + |dφ|). (trap) r (trap) M(τ ,τ ) M(τ ,τ ) 1 2 1 2 Z

Proof. The wave equation for φ satisfies the assumptions of (10.4.1) and Theorem 10.61 with !  N = O dφ1 + φ2 . dec −2δ0 r2 u1+δ trap We deduce sup

Ep [φ](τ ) + Bp [φ](τ1 , τ2 ) + Fp [φ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

" .



Ep [φ](τ1 ) + Jp φ, O Z + (trap) M(τ

1 ,τ2 )

!

#

dφ1 + φ2 (τ1 , τ2 ) dec −2δ0 r2 u1+δ trap Z  1 − 3m |φ|(|φ| + |Rφ|) + rp−3 |φ|(|φ| + |dφ|). (trap) r M(τ1 ,τ2 )

Now, in view of the definition Z p Jp,R [ψ, N ](τ1 , τ2 ) = r eˇ4 ψN , M≥R (τ1 ,τ2 )

Z

2

τ2

Jp [ψ, N ](τ1 , τ2 ) = τ1

dτ kN kL2 ( (trap) Σ(τ ))

Z +



(trap)

M(τ1 ,τ2 )

r1+δ |N |2

+ Jp,4m0 [ψ, N ](τ1 , τ2 ), 23 Recall

that we have δ0  δdec in view of (5.1.1), and hence δdec − 2δ0 > 0.

700

CHAPTER 10

we have " Jp φ, O Jp φ, O

# dφ1 + φ2 (τ1 , τ2 )

dec −2δ0 r2 u1+δ trap

" .

!



!



# dφ1 (τ1 , τ2 ) + Jp [φ, φ2 ] (τ1 , τ2 )

dec −2δ0 r2 u1+δ trap

and, for δ ≤ p ≤ 2 − δ, using also δdec − 2δ0 > 0, we have " ! #  Jp φ, O dφ1 (τ1 , τ2 ) dec −2δ0 r2 u1+δ trap  Z τ2 2 Z dτ  . 2 kdφ1 kL2 ( (trap) Σ(τ )) 1+δ −2δ0 + 2 rδ−3 |dφ1 |2 dec (trap) τ τ M(τ1 ,τ2 ) Z 1 + rp−2 eˇ4 (ψ)dφ1 M≥4m0 (τ1 ,τ2 ) Z  . 2 sup kdφ1 k2L2 ( (trap) Σ(τ )) + 2 rp−3 |d≤2 ψ|2 (trap)

[τ1 ,τ2 ]

! 12

Z +

r



(trap)

p−3

M(τ1 ,τ2 )

M(τ1 ,τ2 )

! 21

Z

2

|dφ|

(trap)

! . 2

r



sup E 1 [φ1 ](τ ) + Bp1 [φ1 ](τ1 , τ2 )

p−3

M(τ1 ,τ2 )

|dφ1 |

2


1
1 +  Bp1 [φ1 ](τ1 , τ2 ) 2 Bp1 [φ](τ1 , τ2 ) 2 .

[τ1 ,τ2 ]

We immediately deduce sup

Ep [φ](τ ) + Bp [φ](τ1 , τ2 ) + Fp [φ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! .

Ep [φ](τ1 ) + 

2

sup E[φ1 ](τ ) + Bp [φ1 ](τ1 , τ2 )

+ Jp [φ, φ2 ](τ1 , τ2 )

[τ1 ,τ2 ]

Z 3m  + rp−3 |φ|(|φ| + |dφ|). 1 − r |φ|(|φ| + |Rφ|) + (trap) (trap) M(τ ,τ ) M(τ1 ,τ2 ) 1 2 Z

This concludes the proof of the corollary. Finally, we end this section with the following lemma. Lemma 10.64. Let φ be a reduced k-scalar for k = 1, 2, and let X a vectorfield. We have, for all δ ≤ p ≤ 2 − δ, Z 1 − 3m |dφ|(|Xφ| + |R(Xφ)|) r (trap) M(τ ,τ ) 1 2 Z  + rp−3 |dφ|(|Xφ| + |d(Xφ)|) (trap)

M(τ1 ,τ2 )

1

1

. (Bp [Xφ](τ1 , τ2 )) 2 (Bp [φ](τ1 , τ2 )) 2 . Proof. The proof follows immediately from the definition of Bp [φ](τ1 , τ2 ).

701

REGGE-WHEELER TYPE EQUATIONS

10.4.5

Proof of Theorem 5.17

We now conclude the proof of Theorem 5.17 for all 0 ≤ s ≤ ksmall +30 by recovering higher derivatives s ≥ 1 one by one starting from the estimate s = 0 provided by (10.4.1). As explained in section 10.4.2, it suffices to recover the estimates for s = 1 from the one for s = 0 as the procedure to recover the estimate for s + 1 from the one for s is completely analogous. We now follow the strategy outlined in section 10.4.2. 10.4.5.1

Recovering estimates for T ψ

Recall that ψ satisfies V = −κκ,

2 ψ = V ψ + N,

and recall also from Corollary 10.53 that we have = d≤1 (Γg )d≤2 ψ.

[T, 2 ]ψ We infer 2 (T ψ) + V T (ψ)

T (N ) + d≤1 (Γg )d≤2 ψ.

=

In view of Corollary 10.63 with φ = T (ψ), φ1 = d≤1 ψ and φ2 = T (N ), and in view of (10.4.3), we deduce sup

Ep [T ψ](τ ) + Bp [T ψ](τ1 , τ2 ) + Fp [T ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! . Ep [T ψ](τ1 ) + 

2

≤1

sup E[d

≤1

ψ](τ ) + Bp [d

ψ](τ1 , τ2 )

[τ1 ,τ2 ]

Z 1 − 3m |T φ|(|T φ| + |R(T φ)|) +Jp [T (ψ), T (N )](τ1 , τ2 ) + r (trap) M(τ ,τ ) 1 2 Z  + rp−3 |T φ|(|T φ| + |d(T φ)|), (trap)

M(τ1 ,τ2 )

and hence, using Lemma 10.64 with X = T , we infer for any δ ≤ p ≤ 2 − δ, sup

Ep [T ψ](τ ) + Bp [T ψ](τ1 , τ2 ) + Fp [T ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! . Ep [T ψ](τ1 ) +

Jp1 [ψ, N ](τ1 , τ2 )

2

+

1

sup E [ψ](τ ) +

Bp1 [ψ](τ1 , τ2 )

[τ1 ,τ2 ]

+Bp [ψ](τ1 , τ2 ). 10.4.5.2

(10.4.6)

Recovering estimates for r /d2 ψ

Recall that ψ satisfies 2 ψ = V ψ + N,

V = −κκ,

702

CHAPTER 10

and recall also from Corollary 10.55 that we have r /d2 (2 ψ) − (1 − K)(r /d2 ψ)

= −rη2 ψ + d≤1 (Γg )d≤2 ψ.

We infer = rη2 ψ + r /d2 (N ) + d≤1 (Γg )d≤2 ψ

1 (r /d2 ψ) + (V − K)r /d2 ψ

=

−rηN + r /d2 (N ) + d≤1 (Γg )d≤2 ψ,

and hence = −rηN + r /d2 (N ) + d≤1 (Γg )d≤2 ψ.

1 (r /d2 ψ) + (V − K)r /d2 ψ

In view of Corollary 10.63 with φ = r /d2 ψ, φ1 = d≤1 ψ and φ2 = −rηN + r /d2 (N ), and in view of (10.4.3), we deduce sup

Ep [r /d2 ψ](τ ) + Bp [r /d2 ψ](τ1 , τ2 ) + Fp [r /d2 ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! .

sup E[d≤1 ψ](τ ) + Bp [d≤1 ψ](τ1 , τ2 )

Ep [r /d2 ψ](τ1 ) + 2

[τ1 ,τ2 ]

h i +Jp r /d2 ψ, −rηN + r /d2 (N ) (τ1 , τ2 ) Z 3m + 1 − r |r /d2 φ|(|r /d2 φ| + |R(r /d2 φ)|) (trap) M(τ ,τ ) 1 2 Z  + rp−3 |r /d2 φ|(|r /d2 φ| + |d(r /d2 φ)|), (trap)

M(τ1 ,τ2 )

and hence, using Lemma 10.64 with X = r /d2 , we infer for any δ ≤ p ≤ 2 − δ, sup

Ep [r /d2 ψ](τ ) + Bp [r /d2 ψ](τ1 , τ2 ) + Fp [r /d2 ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ]

! .

Ep [r /d2 ψ](τ1 ) +

Jp1 [ψ, N ](τ1 , τ2 )

+

2

1

sup E [ψ](τ ) +

Bp1 [ψ](τ1 , τ2 )

[τ1 ,τ2 ]

+Bp [ψ](τ1 , τ2 ). 10.4.5.3

(10.4.7)

Recovering estimates for Rψ in r ≤ r0

We start with the following lemma. Lemma 10.65. Let ψ satisfy 2 ψ = V ψ + N,

V = −κκ.

Then, R2 ψ satisfies R2 ψ

= −ΥN + T 2 ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ + O(r−2 )ψ.

703

REGGE-WHEELER TYPE EQUATIONS

Proof. Recall that we have 2 ψ

  1 1 −e3 (e4 (ψ)) + 4 / 2 ψ − κe3 ψ + − κ + 2ω e4 ψ + 2ηeθ ψ 2 2

=

and e4 = T + R,

Υe3 = (T − R).

We infer Υ2 ψ

=

−(T − R)(T + R)ψ + Υ4 / 2ψ −

  Υ 1 κe3 ψ + Υ − κ + 2ω e4 ψ 2 2

+2Υηeθ ψ =

−T 2 ψ + R2 ψ − [T, R]ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ

and hence R2 ψ

= −Υ2 ψ + T 2 ψ − [T, R]ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ

= −ΥN + T 2 ψ − [T, R]ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ + O(r−2 )ψ where we used the fact that 2 ψ = V ψ + N and V = κκ = O(r−2 ). Also, we have [T, R]ψ

= = =

1 [e4 + Υe3 , e4 − Υe3 ]ψ 4  1 − e4 (Υ)e3 + Υ[e3 , e4 ] ψ 2 O(r−2 )dψ

and thus R2 ψ

= −ΥN + T 2 ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ + O(r−2 )ψ.

This concludes the proof of the lemma. We now estimate Rψ in r ≤ r0 for a fixed r0 ≥ 4m0 that will be chosen large enough. First, in view of the identity of the previous lemma, i.e., R2 ψ

= −ΥN + T 2 ψ + O(r−2 ) d/2 ψ + O(r−1 )dψ + O(r−2 )ψ,

we infer  2 3m 1− (R2 ψ)2 r Σr≤r0 (τ ) Mr≤r0 (τ1 ,τ2 ) ! Z 2 E[T ψ] + E[r /d2 ψ] + E[ψ] + N

Z sup [τ1 ,τ2 ]

.

sup

|R2 ψ|2 +

[τ1 ,τ2 ]

Z +

Z

Σr≤r0 (τ )

N 2 + Morr[T ψ](τ1 , τ2 )

Mr≤r0 (τ1 ,τ2 )

+Morr[r /d2 ψ](τ1 , τ2 ) + Morr[ψ](τ1 , τ2 ).

(10.4.8)

704

CHAPTER 10

Next, we remove the degeneracy of the above estimate at r = 3m. Recall from Corollary 10.57 that we have in the region r ≤ 4m0 !   3m  2 2 (Rψ) = 1− d ψ+O d2 ψ + O(1)d≤1 ψ + O(1)d≤1 N. dec −2δ0 r u1+δ trap Then, 1. multiplying Rψ with a cut-off function equal to one on [5/2m0 , 7/2m0 ] and vanishing on [9/4m0 , 4m0 ] and inferring the corresponding wave equation from the above one for Rψ, 2. relying on the Morawetz estimate of Proposition 10.12 with the particular choice f (r) = r − 3m, 3. adding a large multiple of the energy estimate, 4. and using Proposition 10.32 for the boundary terms, we easily infer the following estimate: Z (R2 ψ)2 (trap) M(τ

1 ,τ2 )



Z . Mr≤4m0 (τ1 ,τ2 )

3m 1− r

2

! 2

2

≤1

2

(d ψ) + (dψ) + (d

N)

2

+E[Rψ](τ1 ) +  sup E 1 [ψ](τ ). [τ1 ,τ2 ]

Together with (10.4.8), we infer Z Z 2 2 sup |R ψ| + [τ1 ,τ2 ]

.

Mr≤r0 (τ1 ,τ2 )

Σr≤r0 (τ )



E[Rψ](τ1 ) + sup

|R2 ψ|2

(10.4.9)

E 1 [ψ](τ ) + E[T ψ](τ ) + E[r /d2 ψ](τ ) + E[ψ](τ )



[τ1 ,τ2 ]

+Jp1 [ψ, N ](τ1 , τ2 ) 10.4.5.4

+ Morr[T ψ](τ1 , τ2 ) + Morr[r /d2 ψ](τ1 , τ2 ) + Morr[ψ](τ1 , τ2 ).

Recovering estimates for YH ψ

Recall that ψ satisfies V = −κκ,

2 ψ = V ψ + N, and recall also from Lemma 10.59  [2 , YH ]ψ

= d0 Y(0) (YH ψ) + 1Υ≤2δH +

≤1

2 ψ + dT ψ + d

1 (Γg )d ψ + 2 d≤1 ψ δH

1 1δ ≤Υ≤2δH d≤2 ψ δH H

where the scalar function d0 satisfies the bound d0

=

1 + O(δH ) on the support of κH . 2m0

2



705

REGGE-WHEELER TYPE EQUATIONS

We infer 2 (YH ψ) − V YH (ψ)

  1 ≤1 2 = d0 Y(0) (YH ψ) + 1Υ≤2δH N + dT ψ + d ψ + 2 d ψ δH 1 + 1δH ≤Υ≤2δH d≤2 ψ + YH (N ). δH

Then, 1. 2. 3. 4. 5.

we use the redshift vectorfield YH as a multiplier, we rely on Proposition 10.29, we use the fact that d0 ≥ 0, we add a large multiple of the energy, and we use Proposition 10.32 for the boundary terms.

We easily infer sup E[YH ψ] + Morr[YH ψ](τ1 , τ2 ) [τ1 ,τ2 ]

. E[YH ψ](τ1 ) + Jp1 [ψ, N ](τ1 , τ2 ) + Morr[dψ](τ1 , τ2 ) + Morr[Rψ](τ1 , τ2 ) +Morr[T ψ](τ1 , τ2 ) + Morr[r /d2 ψ](τ1 , τ2 ) + Morr[ψ](τ1 , τ2 ). 10.4.5.5

(10.4.10)

Recovering estimates for re4 ψ in r ≥ r0

Recall that ψ satisfies 2 ψ = V ψ + N,

V = −κκ,

and recall also from Lemma 10.60   Υ 2m 1 [2 , re4 ]ψ = 1+ eˇ4 (re4 ψ) + 2 ψ + Γg d2 ψ + r−2 dT ψ + r−2 d/2 ψ r rΥ2 Υ +r−2 dψ. We infer 2 (re4 ψ) − V re4 (ψ)

=

Υ r

   2m 1 1+ e ˇ (re ψ) + O d2 ψ + r−2 dT ψ 4 4 2 2 rΥ r Υ

+r−2 d/2 ψ + r−2 d≤1 ψ + N + re4 (N ). Then, 1. as in section 10.2.3, we use the vectorfield fp e4 as a multiplier, where we have fp = θr0 (r)rp e4 and the cut-off θr0 (r) is equal to one in the region r ≥ r0 and vanishes in the region r ≤ r0 /2, 2. we rely on Proposition 10.43 to control the bulk and the boundary terms, 3. and we use the fact that the prefactor of the term eˇ4 (re4 ψ) on the right-hand side is positive for r ≥ 4m0 , i.e.,   Υ 2m 1+ ≥ 0 for r ≥ 4m0 . r rΥ2

706

CHAPTER 10

We easily infer sup

Ep,r≥r0 [re4 ψ](τ ) + Bp,r≥r0 [re4 ψ](τ1 , τ2 ) + Fp,r≥r0 [re4 ψ](τ1 , τ2 )

τ ∈[τ1 ,τ2 ] 1 Ep1 [re4 ψ](τ1 ) + Jp1 [ψ, N ](τ1 , τ2 ) + Bp,r [ψ](τ1 , τ2 ) + Bp1 [ψ](τ1 , τ2 ) 0 /2≤r 0 is chosen small enough so that the term λBδ [ψ](τ1 , τ2 ) can be absorbed by the LHS in (10.5.3). Concerning the boundary terms on A(τ1 , τ2 ) ∪ Σ(τ2 ) ∪ Σ∗ (τ1 , τ2 ) appearing in the right-hand side of (10.5.4), the potential V0 does not appear in the boundary term of the rp -weighted estimates, but it does appear in the boundary term of the energy estimates.28 More precisely, it appears in Z Z   Q34 = |∇ / φ|2 + V0 φ2 . A(τ1 ,τ2 )∪Σ(τ2 )∪Σ∗ (τ1 ,τ2 )

A(τ1 ,τ2 )∪Σ(τ2 )∪Σ∗ (τ1 ,τ2 )

Now, we have in view of the definition of V0 Z Z Q34 ≥ A(τ1 ,τ2 )∪Σ(τ2 )∪Σ∗ (τ1 ,τ2 )

A(τ1 ,τ2 )∪Σ(τ2 )∪Σ∗ (τ1 ,τ2 )

−O(1)

|∇ / φ|2

Z A(τ1 ,τ2 )∪Σ(τ2 )∪Σ∗ (τ1 ,τ2 )

φ2 r3

and the control of the boundary terms follows. This concludes the proof of 10.68. We are now in position to prove Theorem 10.67. Note first that we have Z (d≤s φ)2 . Fδs−1 [φ](τ1 , τ2 ) 3 r A(τ1 ,τ2 )∪Σ∗ (τ1 ,τ2 ) R ≤s 2 which explains why the term A(τ1 ,τ2 )∪Σ∗ (τ1 ,τ2 ) (d r3φ) , that one would a priori expect in view of (10.5.4), is not present on the right-hand side of (10.5.2). Also, the estimates for ψ and φ are similar, so we focus on the estimate for ψ. Proof of Theorem 10.67. The proof of Theorem 10.67 follows along the same lines as the one of Theorem 5.17. More precisely, following the strategy in section 10.4.2, we recover derivatives one by one starting from Theorem 10.68 and use it iteratively in conjonction with the commutator estimates of section 10.4.3. The only difference is the treatment of the derivatives for s ≥ ksmall +1 as we assume that the estimates of section 10.4.1 for the Ricci coefficients and curvature components only hold for k ≤ ksmall derivatives. Thus, to conclude, we need to consider the terms for which at least ksmall +1 derivatives fall on the Ricci coefficients and curvature components. Since on the other hand we have s ≤ klarge − 1, in view of the definition (3.3.7) of 2 28 The boundary term of the r p -weighted estimates involves only Q 44 = (e4 φ) , while the one of the energy estimate involves also Q34 = |∇ / φ|2 + V0 φ2 .

715

REGGE-WHEELER TYPE EQUATIONS

ksmall in terms of klarge , and in view of the commutator estimates of section 10.4.3, one easily checks that these terms are bounded in absolute value from above by   |d≤s (Γg )| + r−1 |d≤s (Γb )| |d≤ksmall ψ|. We thus need, in view of Theorem 10.68, to estimate Z  2 r1+δ |d≤s (Γg )| + r−1 |d≤s (Γb )| |d≤ksmall ψ|2 M(τ1 ,τ2 ) Z ≤ksmall ˇ + |T ds ψ||d≤s (Γ)||d ψ| (trap) M(τ



sup

.

M(τ1 ,τ2 )

1 ,τ2 )

r |d≤ksmall ψ|2 2

Z M(τ1 ,τ2 )

 2 r−1+δ |d≤s (Γg )| + r−1 |d≤s (Γb )| ! 12

! +

sup

u

(trap) M(τ

×

1 ,τ2 )

(trap) M(τ

1 ,τ2 )

1 2 +δdec

M(τ1 ,τ2 )

+

≤ksmall

|d

ψ|

sup τ ∈[τ1 ,τ2 ]

Eδs [ψ](τ )

! 12

Z

sup

.

1 2 +δdec

rutrap

ˇ2 |d≤s Γ| !2 ≤ksmall

|d

1 2 +δdec

sup M(τ1 ,τ2 )

rutrap

ψ|

Ds [Γ] ! 12

! ≤ksmall

|d

ψ|

sup τ ∈[τ1 ,τ2 ]

Eδs [ψ](τ )

p

Ds [Γ]

where we have used the definition of Ds [Γ]. We infer Z  2 r1+δ |d≤s (Γg )| + r−1 |d≤s (Γb )| |d≤ksmall ψ|2 M(τ1 ,τ2 ) Z ≤ksmall ˇ + |T ds ψ||d≤s (Γ)||d ψ| (trap) M(τ

.

−1

λ

1 ,τ2 )

1 2 +δdec

sup M(τ1 ,τ2 )

rutrap

!2 ≤ksmall

|d

ψ|

Ds [Γ] + λ

sup τ ∈[τ1 ,τ2 ]

Eδs [ψ](τ )

for any λ > 0 and the last term is then absorbed from the left-hand side of the desired estimate by choosing λ > 0 small enough which concludes the proof of Theorem 10.67. Proposition 10.69. Let ψ a reduced 2-scalar satisfying e2 , 2 ψ = f2 (r, m)Y(0) ψ + N where the function f2 is smooth and positive, and where the vectorfield Y(0) has been introduced in Proposition 10.29 in connection with the redshift vectorfield and is given by     5 5 Y(0) := 1 + (r − 2m) + Υ e3 + 1 + (r − 2m) e4 . 4m 4m

716

CHAPTER 10

Also, assume that the Ricci coefficients and curvature components associated to the global null frame we are using satisfy the estimates of section 10.4.1 for k ≤ ksmall derivatives. Then, for any 1 ≤ s ≤ klarge − 1, we have Z Z (ds+1 ψ)2 . Eδs [ψ](τ1 ) + (ds+1 ψ)2 (int) M(τ

(ext) M

1 ,τ2 )

r≤ 5 m0 2

(τ1 ,τ2 )

2

 +Ds [Γ] 

r|d≤ksmall ψ|

sup (int) M(τ

1 ,τ2

)∪ (ext) M

Z

r≤ 5 m0 2



+ (int) M(τ

1 ,τ2 )∪

(ext) M

 e2 )2 . (d≤s ψ)2 + (d≤s+1 N

r≤ 5 m0 2

Proof. Recall from Proposition 10.29 that the redshift vectorfield is given by ! Υ YH := κH Y(0) , κH := κ , 1 10 δH where κ is a positive bump function κ = κ(r), supported in the region in [−2, 2] and equal to 1 for [−1, 1]. To estimate ψ in (int) M, we consider   r − 2m0 (1 + 2δH ) ψe := κ e 2m0 δH where κ e is a positive bump function κ = κ(r), supported in the region in (−∞, 1] and equal to 1 for (−∞, 0]. Since (int) M is included in r ≤ 2m0 (1 + 2δH ), we infer in particular ψe = ψ on

(int)

M,

e ⊂ supp(ψ)

(int)

M(τ1 , τ2 ) ∪

(ext)

Mr≤2m0 (1+3δH ) .

Also, we have, in view of the wave equation for ψ, e20 2 ψe = f2 (r, m)Y(0) ψe + N e 0 satisfies where N 2 Z (int) M(τ

1 ,τ2

)∪ (ext) M

e20 )2 (d≤s+1 N

Z

(d≤s+1 ψ)2

. (ext) M

r≤ 5 m0 2

r≤ 5 m0 2

(τ1 ,τ2 )

Z

e2 )2 . (d≤s+1 N

+ (int) M(τ

(ext) M 1 ,τ2 )∪ r≤ 5 m0 2

717

REGGE-WHEELER TYPE EQUATIONS

Since ψe = ψ on (int) M, it thus suffices to prove for ψe the following estimate: Z e 2 . E s [ψ](τ e 1) (ds+1 ψ) δ M(τ1 ,τ2 )

2

 +Ds [Γ] 

sup (int) M(τ

(ext) M 1 ,τ2 )∪ r≤ 5 m0 2

Z

  e 2 + (d≤s+1 N e20 )2 . (d≤s ψ)

+ (int) M(τ

1 ,τ2 )∪

(ext) M

e r|d≤ksmall ψ|

r≤ 5 m0 2

This estimate follows from first deriving the corresponding estimate for s = 0 by using the redshift as a multiplier, and then by recovering derivatives one by one using commutation with T , d/ and the redshift vectorfield. Note that • ψe is supported on r ≤ 2m0 (1 + 2δH ) and hence is estimated on 2m0 (1 − 2δH ) ≤ r ≤ 2m0 (1 + 2δH ) so that the redshift vectorfield YH has good properties, both as a multiplier and e as a commutator, on the support of ψ; • and the term f2 (r, m)Y(0) yields a good sign when using YH as a multiplier since the function f2 (r, m) is positive, and since YH = κH Y(0) . This concludes the proof of the proposition.

Appendix A Appendix to Chapter 2 A.1

PROOF OF PROPOSITION 2.64

In a neighborhood of a given sphere S, we consider a (u, s, θ, ϕ) coordinates system, where θ is such that e4 (θ) = 0. Then, in this coordinates system, we have ∂s = e4 . Since we have Z ∂s

 f

=

Z 

S

 ∂s f + g(Deθ ∂s , eθ )f + g(Deϕ ∂s , eϕ )f ,

S

we infer Z e4

 f

S

Z =

(e4 (f ) + κf ). S

In particular, choosing f = 1, we deduce 1 e4 (|S|) |S|

= κ

and since |S| = 4πr2 , e4 (r) =

rκ . 2

Next, let ∂u the coordinates vectorfield in the (u, s, θ, ϕ) coordinates system. We have Z  Z   ∂u f = ∂u f + g(Deθ ∂u , eθ )f + g(Deϕ ∂u , eϕ )f S ZS   = ∂u f + g(Deθ ∂u , eθ )f − g(∂u , Deϕ eϕ )f ZS   = ∂u f + g(Deθ ∂u , eθ )f + g(∂u , Da (Φ)ea )f . S

On the other hand, we have, see (2.2.42),   1 1√ 1 ∂u = ς e3 − Ωe4 − γbeθ . 2 2 2 We infer g(Deθ ∂u , eθ ) + g(∂u , Da (Φ)ea )

=

1 1 1 √ ςκ − ςΩκ − /d1 (ς γb) 2 2 2

720

APPENDIX A

and thus   Z  Z 1 1 ς e3 − Ωe4 f = 2 2 S S

 1 1 1√ 1 ς e3 − Ωe4 − γbeθ f + ςκf 2 2 2 2 ! 1 1 √ − ςΩκf − /d1 ( γb)f . 2 2

We deduce Z  Z  Z −1 e3 f = Ωe4 f +ς S

S



√

S

!

ςe3 f − ςΩe4 f + ςκf − ςΩκf − /d1 (ς γbf ) .

Next, we use Z



e4

f

Z =

(e4 (f ) + κf )

S

S

Z

√ /d1 (ς γbf )

and =

0.

S

We infer Z  e3 f = S

Z Ω

(e4 (f ) + κf ) + ς S

=

−1

!

Z

ςe3 f − ςΩe4 f + ςκf − ςΩκf

S

Z
ˇ + ς −1 Ωˇ ς(e3 f + κf ) + Ω ς (e4 f + κf ) S S Z Z ˇ 4 f + κf ). −ς −1 Ω ςˇ(e4 f + κf ) − ς −1 Ως(e ς −1

Z

S

We further write Z ς −1 ς(e3 f + κf ) S

S

Z Z = ς −1 ς (e3 f + κf ) + ς −1 ςˇ (e3 f + κf ) S S Z Z = (e3 f + κf ) + (ς −1 ς − 1) (e3 f + κf ) S S Z −1 +ς ςˇ (e3 f + κf ) S Z Z Z −1 −1 = (e3 f + κf ) − ς ςˇ (e3 f + κf ) + ς ςˇ (e3 f + κf ). S

S

S

721

APPENDIX TO CHAPTER 2

Hence, Z  e3 f = S  Z  Err e3 f = S

+ −

 Z  (e3 f + κf ) + Err e3 f , S S Z Z −ς −1 ςˇ (e3 f + κf ) + ς −1 ςˇ (e3 f + κf ) S S Z Z
ˇ + ς −1 Ωˇ Ω ς (e4 f + κf ) − ς −1 Ω ςˇ(e4 f + κf ) S S Z ˇ 4 f + κf ) ς −1 Ως(e Z

S

as desired. In particular, choosing f = 1, we infer 1 e3 (|S|) |S|

=


ˇ + ς −1 Ωˇ ˇςκ κ − ς −1 ςˇ κ + ς −1 ςˇκ + Ω ς κ − ς −1 Ω ςˇ κ − ς −1 Ω

=


ˇ + ς −1 Ωˇ ˇ ς κ. κ − ς −1 ςˇ κ + ς −1 ςˇκ ˇ+ Ω ς κ − ς −1 Ω ςˇ κ ˇ − ς −1 Ω

Hence, since |S| = 4πr2 , recalling the definition of A, 2e3 (r) r

=


ˇ + ς −1 Ωˇ ˇςκ κ − ς −1 ςˇ κ + ς −1 ςˇκ ˇ+ Ω ς κ − ς −1 Ω ςˇ κ ˇ − ς −1 Ω

=

κ + A.

This concludes the proof of Proposition 2.64.

A.2

PROOF OF PROPOSITION 2.71

We start with the proof for R e4 (m). Recall that the Hawking mass m is given by 1 the formula 2m = 1 + r 16π S κκ. Differentiating in the e4 direction, we deduce Z  Z   2e4 (m) 2me4 (r) 1 1 − = e κκ = e4 (κκ) + κκ2 . 4 2 r r 16π 16π S S Now, making use of the e4 transport equations of Proposition 2.63,     1 1 1 1 e4 (κκ) = κ − κ2 − ϑ2 + κ − κκ + 2ρ − 2 /d1 ζ − ϑϑ + 2ζ 2 2 2 2 2 1 1 = −κκ2 + 2κρ − 2κ /d1 ζ − κϑ2 − κϑϑ + 2κζ 2 . 2 2 We infer 2e4 (m) m − κ = r r = =

 Z  1 1 1 2κρ − 2κ /d1 ζ − κϑ2 − κϑϑ + 2κζ 2 16π S 2 2  Z  1 1 1 1 |S|κ ρ + 2ˇ κρˇ + 2eθ (κ)ζ − κϑ2 − κϑϑ + 2κζ 2 8π 16π S 2 2  Z  2 r 1 1 1 κρ + 2ˇ κρˇ + 2eθ (κ)ζ − κϑ2 − κϑϑ + 2κζ 2 2 16π S 2 2

722

APPENDIX A

and hence    Z  r3 2m r 1 2 1 2 e4 (m) = κ ρ + 3 + − κϑ − κϑϑ + 2ˇ κρˇ + 2eθ (κ)ζ + 2κζ . 4 r 32π S 2 2 R 1 Using the identity ρ = − 2m r 3 + 16πr 2 S ϑϑ (see (2.2.12) of Proposition 2.59), we deduce  Z  r 1 1 e4 (m) = − κϑ2 − (κ − κ)ϑϑ + 2ˇ κρˇ + 2eθ (κ)ζ + 2κζ 2 32π S 2 2  Z  r 1 1 = − κϑ2 − κ ˇ ϑϑ + 2ˇ κρˇ + 2eθ (κ)ζ + 2κζ 2 32π S 2 2 Z r = Err1 32π S as desired. In the same vein, 2e3 (m) 2me3 (r) − r r2

=

1 e3 16π

Z

 Z   1 κκ = e3 (κκ) + κ2 κ + E1 , 16π S S

with E1 the error term defined in Proposition 2.64  Z  1 E1 = Err e3 κκ . 16π S We make use of the e3 transport equations of Proposition 2.63,   1 1 e3 (κκ) = κ − κ κ + 2ωκ + 2 /d1 η + 2ρ − ϑ ϑ + 2η 2 2 2   1 2 1 + κ − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2 2  
1 2 2 = −κκ + 2κ /d1 η + 2κ /d1 ξ + 2ρκ + κ 2η − ϑϑ + 2κ η − 3ζ ξ 2 1 2 − κϑ . 2

723

APPENDIX TO CHAPTER 2

Therefore, setting E2 =

= = = = +

1 16π

R  S



κ 2η 2 − 12 ϑϑ + 2κ η − 3ζ ξ − 12 κϑ2 ,

2e3 (m) m − (κ + A) rZ r   1 2κ /d1 η + 2κ /d1 ξ + 2ρκ + E1 + E2 16π S Z   1 − 2eθ (κ)η − 2eθ (κ)ξ + 2(ρ + ρˇ)(κ + κ ˇ ) + E1 + E2 16π S Z   1 2 1 r ρκ + − 2eθ (κ)η − 2eθ (κ)ξ + 2ˇ ρκ ˇ + E1 + E2 2 16π S Z  1 2  2m 1 r κ − 3 + ϑϑ 2 r 16πr2 S Z   1 − 2eθ (κ)η − 2eθ (κ)ξ + 2ˇ ρκ ˇ + E1 + E2 . 16π S

We deduce 2e3 (m) r

= + = +

Z   1 1 − 2eθ (κ)η − 2eθ (κ)ξ + 2ˇ ρκ ˇ + κϑϑ + E1 16π S 2   Z  
1 1 1 m κ 2η 2 − ϑϑ + 2κ η − 3ζ ξ − κϑ2 + A 16π S 2 2 r Z   1 1 − 2eθ (κ)η + 2κη 2 − 2eθ (κ)ξ + 2κηξ − κϑ2 16π S 2 Z   1 1 m 2ˇ ρκ ˇ − 6κ ζ ξ − κ ˇ ϑ ϑ + E1 + A, 16π S 2 r

i.e., e3 (m)

= +

Z   r 1 − 2eθ (κ)η + 2κη 2 − 2eθ (κ)ξ + 2κηξ − κϑ2 32π S 2 Z   r r 1 m  2ˇ ρκ ˇ − 6κ ζ ξ − κ ˇϑ ϑ + E1 + A . 32π S 2 2 r

It remains to calculate E1 + m r A. Using the definitions of E1 and A and grouping similar terms appropriately we find   Z m 1 m E1 + A = −ς −1 ςˇ (e3 (κκ) + κκ2 ) + κ r 16π S r   Z 1 m +ς −1 ςˇ (e3 (κκ) + κκ2 ) + ςˇκ ˇ 16π S r   Z
1 m ˇ + ς −1 Ωˇ + Ω ς (e4 (κκ) + κ2 κ) + κ 16π S r   Z 1 m −ς −1 Ω ςˇ(e4 (κκ) + κ2 κ) + ςˇκ ˇ 16π S r   Z 1 ˇ 4 (κκ) + κ2 κ) + m Ωςκ ˇ −ς −1 Ως(e . 16π S r

724

APPENDIX A

Now, we have from the above calculations e4 (κκ) + κκ2

2κρ − 2κ /d1 ζ + Err[e4 (κκ)], 1 1 Err[e4 (κκ)] = − κϑ2 − κϑϑ + 2κζ 2 , 2 2 e3 (κκ) + κκ2 = 2ρκ + 2κ /d1 η + 2κ /d1 ξ + Err[e3 (κκ)],  
1 1 Err[e3 (κκ)] = κ 2η 2 − ϑϑ + 2κ η − 3ζ ξ − κϑ2 . 2 2 =

We infer E1 +

m A = r

  Z 1 m −ς −1 ςˇ (2ρκ + 2κ /d1 η + 2κ /d1 ξ + Err[e3 (κκ)]) + κ 16π S r   Z 1 m +ς −1 ςˇ (2ρκ + 2κ /d1 η + 2κ /d1 ξ + Err[e3 (κκ)]) + ςˇκ ˇ 16π S r   Z
1 m ˇ + ς −1 Ωˇ + Ω ς (2κρ − 2κ /d1 ζ + Err[e4 (κκ)]) + κ 16π S r   Z 1 m −ς −1 Ω ςˇ(2κρ − 2κ /d1 ζ + Err[e4 (κκ)]) + ςˇκ ˇ 16π S r   Z 1 mˇ ˇ −ς −1 Ως(2κρ − 2κ /d1 ζ + Err[e4 (κκ)]) + Ωςκ 16π S r

and hence m A r  Z  1 1 −1 −ς ςˇ 2ˇ ρκ ˇ − 2eθ (κ)η − 2eθ (κ)ξ + κϑϑ + Err[e3 (κκ)] 16π S 2   Z 1 m −1 +ς ςˇ (2ρˇ κ + 2ˇ ρκ + 2ˇ ρκ ˇ + 2κ /d1 η + 2κ /d1 ξ + Err[e3 (κκ)]) + ςˇκ ˇ 16π S r    Z
1 1 ˇ + ς −1 Ωˇ + Ω ς 2ˇ κρˇ + 2eθ (κ)ζ + κϑϑ + Err[e4 (κκ)] 16π S 2   Z 1 m −1 −ς Ω ςˇ(2ρˇ κ + 2ˇ ρκ + 2ˇ ρκ ˇ − 2κ /d1 ζ + Err[e4 (κκ)]) + ςˇκ ˇ 16π S r   Z 1 mˇ −1 ˇ −ς Ως(2ρˇ κ + 2ˇ ρκ + 2ˇ ρκ ˇ − 2κ /d1 ζ + Err[e4 (κκ)]) + Ωςκ . 16π S r E1 + =

We deduce e3 (m)

Z Z
r
r −1 ˇ = 1 − ς ςˇ Err1 + Ω + ς Ωˇ ς Err1 32π S 32π S Z
r ςˇ 2ρˇ +ς −1 κ + 2ˇ ρκ + 2κ /d1 η + 2κ /d1 ξ + Err2 32π S Z r ˇ (2ρˇ −ς −1 (Ωˇ ς + Ως) κ + 2ˇ ρκ − 2κ /d1 ζ + Err2 ) 32π S h i m ˇ − ς −1 −ˇ ςκ ˇ + Ω ςˇκ ˇ + Ωςκ , r −1

725

APPENDIX TO CHAPTER 2

where we have introduced Err1

=

Err1

=

Err2

=

1 2ˇ κρˇ + 2eθ (κ)ζ + κϑϑ + Err[e4 (κκ)], 2 1 2ˇ ρκ ˇ − 2eθ (κ)η − 2eθ (κ)ξ + κϑϑ + Err[e3 (κκ)], 2 2ˇ ρκ ˇ + Err[e4 (κκ)],

Err2

=

2ˇ ρκ ˇ + Err[e3 (κκ)].

In view of the definition of Err[e4 (κκ)] and Err[e3 (κκ)], this concludes the proof of Proposition 2.71.

A.3

PROOF OF LEMMA 2.72

Recall that we have e4 (κ)

1 1 = − κ2 − ϑ2 . 2 4

We infer e4 (κ)

= =

and hence   2 e4 κ − = r =

1 1 e4 (κ) + κ ˇ 2 = − κ2 − ϑ2 + κ ˇ2 2 4 1 1 1 2 − κ2 − ϑ2 + κ ˇ 2 4 2

1 1 1 2 2 e4 (r) 1 1 1 1 2 − κ2 − ϑ2 + κ ˇ + = − κ2 + κ − ϑ2 + κ ˇ 2 4 2 r r 2 r 4 2   1 2 1 1 2 − κ κ− − ϑ2 + κ ˇ . 2 r 4 2

Next, using e4 (ω)

= ρ + ζ(2η + ζ)

we infer that e4 (ω) and hence  m e4 ω − 2 r

= e4 (ω) + κ ˇω ˇ = ρ + ζ(2η + ζ) + κ ˇω ˇ,

2me4 (r) e4 (m) − 2 r3 r 2m m 2 e4 (m) = ρ+ 3 + 2 κ− − + 3ζ(2η + ζ) + κ ˇω ˇ r r r r2 = e4 (ω) +

as stated. Next, using 1 1 e3 (κ) + κκ − 2ωκ = 2 /d1 η + 2ρ − ϑϑ + 2η 2 2 2

726

APPENDIX A

we deduce e3 (κ)

= =

1 1 − κκ + 2ωκ + 2ρ − ϑϑ + 2η 2 2 2 1 1 1 − κ κ + 2ω κ + 2ρ + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + 2η 2 . 2 2 2

Making use of Corollary 2.66 e3 (κ)

= =

e3 (κ) + Err[e3 κ] 1 1 1 − κ κ + 2ω κ + 2ρ + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + 2η 2 + Err[e3 κ] 2 2 2

and 

= = =

 2 e3 κ − r 2 r e3 (κ) + 2 (κ + A) r 2 1 1 1 1 1 − κ κ + 2ω κ + 2ρ + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + 2η 2 + κ + A + Err[e3 κ] 2  2 2 r r  1 2 1 1 1 − κ κ− + 2ω κ + 2ρ + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + 2η 2 + A + Err[e3 κ]. 2 r 2 2 r

Now, 2ω κ + 2ρ = =

 κ−  2ω κ − 2ω

Hence,     2 1 2 e3 κ − + κ κ− r 2 r

 2 + r  2 + r



= +

4 ω + 2ρ r   4 m 2m ω− 2 +2 ρ+ 3 . r r r

  4 m 2m 2ω + ω− 2 +2 ρ+ 3 r r r 1 1 1 2η 2 + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + A + Err[e3 κ]. 2 2 r 2 κ− r



In view of Corollary 2.66 the error term Err[e3 (κ)] is given by   Err[e3 (κ)] = −ς −1 ςˇ (e3 κ + κκ − κκ) + ς −1 ςˇ(e3 κ + κκ) − ςˇκ ˇκ   
 ˇ + ς −1 Ωˇ + Ω ς e4 κ + κ2 − κ2 − ς −1 Ω ςˇ(e4 κ + κ2 ) − ςˇκ ˇκ   ˇ 4 κ + κ2 ) − Ως ˇ κκ + κ − ς −1 Ως(e ˇκ ˇ.

APPENDIX TO CHAPTER 2

727

Together with the null structure equations for e3 (κ) and e4 (κ), we infer   1 1 −1 2 Err[e3 (κ)] = −ς ςˇ κκ + 2ωκ + 2ρ + 2 /d1 η − ϑϑ + 2η − κ κ 2 2 !   1 1 −1 +ς ςˇ κκ + 2ωκ + 2ρ + 2 /d1 η − ϑϑ + 2η 2 − ςˇκ ˇκ 2 2  
1 1 ˇ + ς −1 Ωˇ + Ω ς κ2 − ϑ2 − κ2 2 4 !   1 2 1 2 −1 −ς Ω ςˇ κ − ϑ − ςˇκ ˇκ 2 4 !   1 1 −1 ˇ ˇ κκ + κ −ς Ως κ2 − ϑ2 − Ως ˇκ ˇ, 2 4 and hence Err[e3 (κ)]

 
1 1 ˇ + ς −1 Ωˇ = −ς −1 − κ κ + 2ω κ + 2ρ ςˇ − κ2 Ω ς 2 2   1 1 −ς −1 ςˇ κ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 2 2 !   1 1 −1 2 +ς ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η − ςˇκ ˇκ 2 2 !    
1 1 1 1 ˇ + ς −1 Ωˇ + Ω ς κ ˇ 2 − ϑ2 − ς −1 Ω ςˇ κ2 − ϑ2 − ςˇκ ˇκ 2 4 2 4 !   1 1 −1 2 2 ˇ ˇ κκ + κ −ς Ως κ − ϑ − Ως ˇκ ˇ (A.3.1) 2 4

so that, in view of the definition of A, we obtain     2 1 2 e3 κ − + κ κ− r 2 r        2 4 m 2m 1 −1 = 2ω κ − + ω− 2 +2 ρ+ 3 −ς − κ κ + 2ω κ + 2ρ ςˇ r r r r 2   

1 2 ˇ 1 −1 1 ˇ 2 −1 −1 − κ Ω + ς Ωˇ ς − ς κˇ ς + κ Ω + ς Ωˇ ς + Err e3 κ − , 2 r r r

728

APPENDIX A

with    2 Err e3 κ − r 1 1 1 1 1 ˇ = 2η 2 + 2ˇ ωκ ˇ− κ ˇκ ˇ − ϑϑ + ς −1 ςˇκ ˇ − ς −1 Ω ςˇκ ˇ − ς −1 Ωςκ 2 2 r r r   1 1 −ς −1 ςˇ κ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 2 2 !   1 1 +ς −1 ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η 2 − ςˇκ ˇκ 2 2 !    
1 1 2 1 2 1 2 −1 −1 2 ˇ + Ω + ς Ωˇ ς κ ˇ − ϑ − ς Ω ςˇ κ − ϑ − ςˇκ ˇκ 2 4 2 4 !   1 1 ˇ ˇ κκ + κ −ς −1 Ως κ2 − ϑ2 − Ως ˇκ ˇ. 2 4 This concludes the proof of Lemma 2.72.

A.4

PROOF OF PROPOSITION 2.73

In view of Corollary 2.66 applied to 1 1 e4 (κ) + κ2 = − ϑ2 , 2 2 we deduce 1 2 1 2 1 2 e4 κ ˇ + κˇ κ=− κ ˇ − κ ˇ − (ϑ − ϑ2 ). 2 2 2 In view of Corollary 2.66 applied to 1 1 e4 (κ) + κκ = −2 /d1 ζ + 2ρ − ϑϑ + 2ζ 2 2 2 we deduce 1 1 1 1 e4 κ ˇ + κˇ κ+ κ ˇκ = − κ ˇκ ˇ− κ ˇκ ˇ+F −F 2 2 2 2 where F −F

= =

    1 1 2 2 −2 /d1 ζ + 2ρ − ϑϑ + 2ζ − −2 /d1 ζ + 2ρ − ϑϑ + 2ζ 2 2     1 1 −2 /d1 ζ + 2ˇ ρ + − ϑϑ + 2ζ 2 − − ϑϑ + 2ζ 2 . 2 2

729

APPENDIX TO CHAPTER 2

Hence, 1 1 e4 κ ˇ + κˇ κ+ κ ˇκ = 2 2 Err[e4 κ ˇ] :

−2 /d1 ζ + 2ˇ ρ + Err[e4 κ ˇ]     1 1 1 1 2 2 = − κ ˇκ ˇ− κ ˇκ ˇ + − ϑϑ + 2ζ − − ϑϑ + 2ζ . 2 2 2 2

In view of Corollary 2.66 applied to e4 (ω) = ρ + 3ζ 2 we deduce e4 ω ˇ

= −ˇ κω ˇ + (ρ + 3ζ 2 ) − (ρ + 3ζ 2 ) = ρˇ − κ ˇω ˇ + 3(ζ 2 − ζ 2 ).

In view of Corollary 2.66 applied to 3 1 e4 (ρ) + κρ = /d1 β − ϑα − ζβ 2 2 we deduce 3 3 3 1 e4 ρˇ + κˇ ρ + ρˇ κ=− κ ˇ ρˇ + κ ˇ ρˇ + /d1 β − 2 2 2 2



1 ϑα + ζβ 2



 +

 1 ϑα + ζβ . 2

In view of Corollary 2.66 applied to 3 e4 µ + κµ = Err[e4 µ], 2 we deduce 3 3 3 1 e4 µ ˇ + κˇ µ + µˇ κ = − κ ˇµ ˇ+ κ ˇµ ˇ + Err[e4 µ] − Err[e4 µ]. 2 2 2 2 In view of Corollary 2.66 applied to e4 (Ω) = −2ω we deduce ˇ −e4 (Ω)

=

ˇ 2ˇ ω−κ ˇΩ

as stated. In view of Corollary 2.66 applied to the equation 1 1 e3 (κ) + κκ = 2 /d1 η + 2ρ + 2η 2 + 2ωκ − ϑϑ 2 2 we deduce e3 (ˇ κ)

= e3 (κ) − e3 (κ) − Err[e3 (κ)] 1 1 = − κκ + 2 /d1 η + 2ρ + 2η 2 + 2ωκ − ϑϑ 2 2 1 1 2 + κκ − 2ρ − 2η − 2ωκ + ϑϑ − Err[e3 κ] 2 2 1 = 2 /d1 η + 2ˇ ρ − (κˇ κ + κˇ κ) + 2 (ωˇ κ + κˇ ω) 2   1
1 +2 η 2 − η 2 − κ ˇκ ˇ + 2ˇ ωκ ˇ− ϑϑ − ϑϑ − Err[e3 κ]. 2 2

730

APPENDIX A

Now, recall that we have, in view of (A.3.1),  
1 1 ˇ + ς −1 Ωˇ Err[e3 (κ)] = −ς −1 − κ κ + 2ω κ + 2ρ ςˇ − κ2 Ω ς 2 2   1 1 −1 2 −ς ςˇ κ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 2 !   1 1 −1 +ς ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η 2 − ςˇκ ˇκ 2 2 !    
1 1 1 1 ˇ + ς −1 Ωˇ + Ω ς κ ˇ 2 − ϑ2 − ς −1 Ω ςˇ κ2 − ϑ2 − ςˇκ ˇκ 2 4 2 4 !   1 1 −1 2 2 ˇ ˇ κκ + κ −ς Ως κ − ϑ − Ως ˇκ ˇ. 2 4 We deduce e3 (ˇ κ)

=

1 2 /d1 η + 2ˇ ρ − (κˇ κ + κˇ κ) + 2 (ωˇ κ + κˇ ω) 2  
1 1 ˇ + ς −1 Ωˇ +ς −1 − κ κ + 2ω κ + 2ρ ςˇ + κ2 Ω ς 2 2   1
1 +2 η 2 − η 2 − κ ˇκ ˇ + 2ˇ ωκ ˇ− ϑϑ − ϑϑ 2 2   1 1 −1 +ς ςˇ κ ˇκ ˇ + 2ˇ ωκ ˇ − ϑϑ + 2η 2 2 2 !   1 1 −1 2 −ς ςˇ κκ + 2ωκ + 2ˇ ρ + 2 /d1 η − ϑϑ + 2η − ςˇκ ˇκ 2 2 !    
1 1 1 1 −1 −1 ˇ + ς Ωˇ − Ω ς κ ˇ 2 − ϑ2 + ς Ω ςˇ κ2 − ϑ2 − ςˇκ ˇκ 2 4 2 4 !   1 2 1 2 −1 ˇ ˇ +ς Ως κ − ϑ − Ως κ κ − κ ˇκ ˇ 2 4

as desired. In view of Corollary 2.66 applied to the equation 1 1 e3 (κ) + κ2 = 2 /d1 ξ − 2ω κ + 2(η − 3ζ)ξ − ϑ2 2 2 we deduce e3 (ˇ κ) + κ κ ˇ =

2 /d1 ξ − 2 (ˇ ωκ+ωκ ˇ)  1 2 1 2 − κ ˇ − 2ˇ ωκ ˇ + 2(η − 3ζ)ξ − 2(η − 3ζ)ξ − ϑ − ϑ2 − Err[e3 κ] 2 2

APPENDIX TO CHAPTER 2

731

where     ˇκ −Err[e3 κ] = ς −1 ςˇ e3 κ + κ2 − κκ − ς −1 ςˇ(e3 κ + κ2 ) − ςˇκ  
ˇ + ς −1 Ωˇ − Ω ς (e4 κ + κκ − κ κ) + ς −1 Ω ςˇ(e4 κ + κκ) − ςˇκ ˇκ   ˇ 4 κ + κκ) − Ως ˇ κκ − κ + ς −1 Ως(e ˇ2 . In view of the null structure equations for e3 (κ) and e4 (κ), we infer    
1 1 ˇ + ς −1 Ωˇ −Err[e3 κ] = ς −1 ςˇ − κ2 − 2ω κ − Ω ς − κ κ + 2ρ 2 2   1 2 1 + ς −1 ςˇ κ ˇ − 2ˇ ωκ ˇ + 2(η − 3ζ)ξ − ϑ2 2 2  
1 1 ˇ + ς −1 Ωˇ − Ω ς κ ˇκ ˇ − ϑ ϑ + 2ζ 2 2 2 !   1 2 1 2 −1 −ς ςˇ κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ − ςˇκ ˇκ 2 2 !   1 1 + ς −1 Ω ςˇ κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 − ςˇκ ˇκ 2 2 !   1 1 −1 2 ˇ ˇ κκ − κ +ς Ως κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ − Ως ˇ2 2 2 and hence   1 2 /d1 ξ − 2 (ˇ ωκ+ωκ ˇ ) + ς −1 ςˇ − κ2 − 2ω κ 2  
1 ˇ + ς −1 Ωˇ − Ω ς − κ κ + 2ρ + Err[e3 (ˇ κ)], 2  1 2 1 2 = − κ ˇ − 2ˇ ωκ ˇ + 2(η − 3ζ)ξ − 2(η − 3ζ)ξ − ϑ − ϑ2 2 2 !   1 2 1 2 −1 −ς ςˇ κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ − ςˇκ ˇκ 2 2 !   1 1 −1 +ς Ω ςˇ κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 − ςˇκ ˇκ 2 2 !   1 1 ˇ ˇ κκ − κ +ς −1 Ως κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 − Ως ˇ2 2 2

e3 (ˇ κ) + κ κ ˇ =

Err[e3 (ˇ κ)]

as desired. In view of Corollary 2.66 applied to the equation 3 1 e3 (ρ) + κρ = /d1 β − ϑα − ζβ + 2ηβ + 2ξβ 2 2

732

APPENDIX A

we deduce 3 3 e3 ρˇ + κˇ ρ+ κ ˇρ 2 2



 1 = /d1 β − ϑα + ζβ − 2ηβ − 2ξβ 2   1 3 + ϑα + ζβ − 2ηβ − 2ξβ − κ ˇ ρˇ − Err[e3 ρ] 2 2

where   −Err[e3 (ρ)] = ς −1 ςˇ (e3 ρ + κρ − κρ) − ς −1 ςˇ(e3 ρ + κρ) − ςˇκ ˇρ  
ˇ + ς −1 Ωˇ − Ω ς (e4 ρ + κρ − κ ρ) + ς −1 Ω ςˇ(e4 ρ + κρ) − ςˇκ ˇρ   ˇ 4 ρ + κρ) − Ως ˇ κ −κ + ς −1 Ως(e ˇ ρˇ. In view of the null structure equations for e3 (ρ) and e4 (ρ), we infer
3 3 ˇ + ς −1 Ωˇ −Err[e3 (ρ)] = − κ ρς −1 ςˇ + κ ρ Ω ς 2 2   1 1 + ς −1 ςˇ − κ ˇ ρˇ − ϑ α − ζ β + 2(η β + ξ β) 2 2 !   1 1 − ς −1 ςˇ − κρ + /d1 β − ϑ α − ζ β + 2(η β + ξ β) − ςˇκ ˇρ 2 2  
1 1 −1 ˇ − Ω + ς Ωˇ ς − κ ˇ ρˇ − ϑ α − ζβ 2 2 !   1 1 −1 + ς Ω ςˇ − κρ + /d1 β − ϑ α − ζβ − ςˇκ ˇρ 2 2 !   1 1 −1 ˇ ˇ +ς Ως − κρ + /d1 β − ϑ α − ζβ − Ως κ − κ ˇ ρˇ 2 2

733

APPENDIX TO CHAPTER 2

and hence
3 3 3 ˇ + ς −1 Ωˇ ς + Err[e3 ρˇ], − ρˇ κ + /d1 β − κ ρς −1 ςˇ + κ ρ Ω 2 2 2     1 1 3 − ϑα + ζβ − 2ηβ − 2ξβ + ϑα + ζβ − 2ηβ − 2ξβ − κ ˇ ρˇ 2 2 2   1 1 +ς −1 ςˇ − κ ˇ ρˇ − ϑ α − ζ β + 2(η β + ξ β) 2 2 !   1 1 −1 −ς ςˇ − κρ + /d1 β − ϑ α − ζ β + 2(η β + ξ β) − ςˇκ ˇρ 2 2  
1 1 ˇ + ς −1 Ωˇ − Ω ς − κ ˇ ρˇ − ϑ α − ζβ 2 2 !   1 1 −1 +ς Ω ςˇ − κρ + /d1 β − ϑ α − ζβ − ςˇκ ˇρ 2 2 !   1 1 −1 ˇ − κρ + /d1 β − ϑ α − ζβ − Ως ˇ κ −κ +ς Ως ˇ ρˇ, 2 2

3 e3 ρˇ + κˇ ρ = 2 Err[e3 ρˇ]

=

which ends the proof of Proposition 2.73.

A.5

PROOF OF PROPOSITION 2.74

In view of the null structure equation for e3 (ζ), we have 1 κξ + 2 /d?1 ω 2

1 1 1 = e3 (ζ) + κ(ζ + η) − 2ω(ζ − η) − β + ϑ(ζ + η) − ϑξ 2 2 2

and hence 1 κξ + 2 /d?1 ω 2

 =

 1 1 κ + 2ω + ϑ η + e3 (ζ) − β 2 2 1 1 1 + κζ − 2ωζ + ϑζ − ϑξ 2 2 2

which is the first desired identity. To prove the second identity we start with 1 e3 (κ) + κ κ − 2ωκ = 2 /d1 η + 2ρ − 12 ϑ ϑ + 2η 2 . 2 Applying eθ , 1 1 e3 (eθ (κ)) + [eθ , e3 ]κ + κeθ (κ) + κeθ (κ) − 2ωeθ (κ) − 2κeθ (ω) 2 2 1 = 2eθ ( /d1 η) + 2eθ (ρ) − eθ (ϑ ϑ) + 2eθ (η 2 ). 2

734

APPENDIX A

Since [eθ , e3 ]κ = 12 (κ + ϑ)eθ κ + (ζ − η)e3 κ − ξe4 κ we deduce 2eθ ( /d1 η) + ηe3 (κ) + 2eθ (η 2 )

−ξe4 (κ) − 2κeθ (ω) + e3 (eθ (κ) 1 1 1 + (κ + ϑ)eθ (κ) + ζe3 (κ) + κeθ (κ) + κeθ (κ) 2 2 2 1 − 2ωeθ (κ) − 2eθ (ρ) + eθ (ϑ ϑ), 2

=

or, making use of the equations for e3 κ and e4 κ in Proposition 2.63,   1 1 2 2eθ ( /d1 η) + − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η η + 2eθ (η 2 ) 2 2   1 1 = − − κ2 − ϑ2 ξ − 2κeθ (ω) + e3 (eθ (κ)) 2 2   1 1 1 2 + (κ + ϑ)eθ (κ) + − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η ζ 2 2 2 1 1 1 + κeθ (κ) + κeθ (κ) − 2ωeθ (κ) − 2eθ (ρ) + eθ (ϑ ϑ). 2 2 2 Since eθ = − /d?1 , /d?1 /d1 = /d2 /d?2 + 2K and K = −ρ − 14 κκ + 14 ϑϑ, we infer that  1 3 − 2 /d2 /d?2 + κκ + 2ωκ + 2 /d1 η + 6ρ − ϑϑ + 2η 2 η + 2eθ (η 2 ) 2  2  1 = κ κξ + 2 /d?1 ω + e3 (eθ (κ)) 2   1 1 1 2 + (κ + ϑ)eθ (κ) + − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η ζ 2 2 2 1 1 1 1 + κeθ (κ) + κeθ (κ) − 2ωeθ (κ) − 2eθ (ρ) + eθ (ϑ ϑ) + ϑ2 ξ. 2 2 2 2 

Making use of the previously derived identity,   1 1 1 ? 2 /d1 ω + κξ = κ + 2ω + ϑ η + e3 (ζ) − β 2 2 2 1 1 1 + κζ − 2ωζ + ϑζ − ϑξ, 2 2 2 we infer that  1 3 − 2 /d2 /d?2 + κκ + 2ωκ + 2 /d1 η + 6ρ − ϑϑ + 2η 2 η + 2eθ (η 2 ) 2 2    1 1 = κ κ + 2ω + ϑ η + e3 (ζ) − β 2 2   1 1 1 +κ κζ − 2ωζ + ϑζ − ϑξ + e3 (eθ (κ)) 2 2 2   1 1 1 2 + (κ + ϑ)eθ (κ) + − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η ζ 2 2 2 1 1 1 1 + κeθ (κ) + κeθ (κ) − 2ωeθ (κ) − 2eθ (ρ) + eθ (ϑ ϑ) + ϑ2 ξ, 2 2 2 2 

735

APPENDIX TO CHAPTER 2

or

=

  1 3 ? 2 −2 /d2 /d2 + 6ρ + 2 /d1 η − κϑ − ϑϑ + 2η η + 2eθ (η 2 ) 2 2
κ e3 (ζ) − β + e3 (eθ (κ))   1 1 1 +κ κζ − 2ωζ + ϑζ − ϑξ 2 2 2   1 1 1 2 + (κ + ϑ)eθ (κ) + − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η ζ 2 2 2 1 1 1 1 + κeθ (κ) + κeθ (κ) − 2ωeθ (κ) − 2eθ (ρ) + eθ (ϑ ϑ) + ϑ2 ξ 2 2 2 2

and hence

=

  1 ? 2 2 /d2 /d2 − 2 /d1 η + κϑ − 2η η − 2eθ (η 2 ) 2
κ −e3 (ζ) + β − e3 (eθ (κ))   1 1 1 3 −κ κζ − 2ωζ + ϑζ − ϑξ + 6ρη − ϑϑη 2 2 2 2   1 1 1 − (κ + ϑ)eθ (κ) − − κκ + 2ωκ + 2 /d1 η + 2ρ − ϑϑ + 2η 2 ζ 2 2 2 1 1 1 1 − κeθ (κ) − κeθ (κ) + 2ωeθ (κ) + 2eθ (ρ) − eθ (ϑ ϑ) − ϑ2 ξ 2 2 2 2

which is the second desired identity. To prove the third identity we start with 1 e3 (κ) + κ2 + 2ω κ = 2

1 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 . 2

Taking eθ = − /d?1 and using /d?1 /d1 = /d2 /d?2 + 2K as before,

= =

e3 (eθ (κ)) + [eθ , e3 ]κ + κeθ (κ) + 2ωeθ (κ) + 2κeθ (ω)   1 −2 /d?1 /d1 ξ + 2eθ (η − 3ζ)ξ − eθ (ϑ2 ) 2   1 ? −2( /d2 /d2 + 2K)ξ + 2eθ (η − 3ζ)ξ − eθ (ϑ2 ). 2

Thus, since [eθ , e3 ]κ = 12 (κ + ϑ)eθ κ + (ζ − η)e3 κ − ξe4 κ, −2( /d2 /d?2 + 2K)ξ

= +

1 e3 (eθ (κ)) + 2κeθ (ω) + (κ + ϑ)eθ κ + (ζ − η)e3 κ − ξe4 κ 2   1 κeθ (κ) + 2ωeθ (κ) − 2eθ (η − 3ζ)ξ + eθ (ϑ2 ). 2

736

APPENDIX A

Making use of the equations for e3 κ, e4 κ in Proposition 2.63 2( /d2 /d?2 + 2K)ξ

1 2κ /d?1 ω − e3 (eθ (κ)) − (κ + ϑ)eθ κ 2   1 2 1 − (ζ − η) − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2 2   1 1 + ξ − κ κ − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 2 2   1 − κeθ (κ) − 2ωeθ (κ) + 2eθ (η − 3ζ)ξ − eθ (ϑ2 ). 2 =

We deduce

=

2( /d2 /d?2 + K)ξ     1 1 2κ /d?1 ω + η − κ2 − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 + 2eθ (η − 3ζ)ξ 2 2 1 1 −e3 (eθ (κ)) − eθ (ϑ2 ) − (κ + ϑ)eθ (κ) 2 2   1 2 1 −ζ − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2 2   1 1 +ξ − κ κ − 2K − 2 /d1 ζ + 2ρ − ϑ ϑ + 2ζ 2 − κeθ (κ) − 2ωeθ (κ) 2 2 −6ηζξ − 6eθ (ζξ).

Making use of K = −ρ − 14 κκ + 14 ϑϑ and reorganizing we deduce 2( /d2 /d?2 + K)ξ   1 1 = 2κ /d?1 ω + η − κ2 − 2ω κ + 2 /d1 ξ + 2ηξ − ϑ2 + 2eθ (ηξ) − e3 (eθ (κ)) 2 2 1 1 − eθ (ϑ2 ) − (κ + ϑ)eθ (κ) 2 2  1 2 1 −ζ − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2 2   2 +ξ 4ρ − ϑϑ − 2 /d1 ζ + 2ζ − κeθ (κ) − 2ωeθ (κ) − 6ηζξ − 6eθ (ζξ). We make use again of the identity   1 1 1 1 1 1 2 /d?1 ω + κξ = κ + 2ω + ϑ η + e3 (ζ) − β + κζ − 2ωζ + ϑζ − ϑξ, 2 2 2 2 2 2

APPENDIX TO CHAPTER 2

737

to derive

=

2( /d2 /d?2 + K)ξ     1 1 1 κ − κξ + κ + 2ω + ϑ η + e3 (ζ) − β 2 2 2     1 1 1 1 1 +κ κζ − 2ωζ + ϑζ − ϑξ + η − κ2 − 2ω κ + 2 /d1 ξ + 2ηξ − ϑ2 2 2 2 2 2 1 1 +2eθ (ηξ) − e3 (eθ (κ)) − eθ (ϑ2 ) − (κ + ϑ)eθ (κ) 2 2   1 2 1 −ζ − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ2 2 2   +ξ 4ρ − ϑϑ − 2 /d1 ζ + 2ζ 2 − κeθ (κ) − 2ωeθ (κ) − 6ηζξ − 6eθ (ζξ).

Grouping terms and using once more the identity K = −ρ − 14 κκ + 14 ϑϑ we deduce 2 /d2 /d?2 ξ  
1 1 = −e3 (eθ (κ)) + κ e3 (ζ) − β + 2 /d1 ξ + κ ϑ + 2ηξ − ϑ2 η + 2eθ (ηξ) 2 2   1 1 1 1 1 1 − eθ (ϑ2 ) + κ κζ − 2ωζ + ϑζ − ϑξ − (κ + ϑ)eθ (κ) − ϑϑξ 2 2 2 2 2 2   1 2 1 2 −ζ − κ − 2ω κ + 2 /d1 ξ + 2(η − 3ζ)ξ − ϑ 2 2   2 +ξ 6ρ − ϑϑ − 2 /d1 ζ + 2ζ − κeθ (κ) − 2ωeθ (κ) − 6ηζξ − 6eθ (ζξ) which is the third desired identity. This concludes the proof of Proposition 2.74.

A.6

PROOF OF PROPOSITION 2.90

The proof follows by straightforward calculations using the definition of Ricci coefficients and curvature components with respect to the two frames. Recall the transformation (2.3.3)   1 e04 = λ e4 + f eθ + f 2 e3 , 4     1 1 1 1 e0θ = 1 + f f eθ + f e4 + f 1 + f f e3 , 2 2 2 4      1 1 1 1 e03 = λ−1 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 . 2 16 4 4

738

APPENDIX A

We first derive the transformation formulae for κ. We have, under a transformation of type (2.3.3), χ0

= = = =

=

g(De0θ e04 , eθ0 )      1 2 0 0 g Deθ λ e4 + f eθ + f e3 , eθ 4     1 λg De0θ e4 + f eθ + f 2 e3 , e0θ 4     λ λg De0θ e4 , e0θ + λe0θ (f )g(eθ , e0θ ) + e0θ (f 2 )g(e3 , e0θ ) + λf g De0θ eθ , e0θ 4 λ 2 0 + f g(De0θ e3 , eθ ) 4       1 λ 0 λg De0θ e4 , eθ + λ 1 + f f e0θ (f ) − f e0θ (f 2 ) + λf g De0θ eθ , e0θ 2 4 λ + f 2 χ + l.o.t. 4

We recall that the lower order terms we denote by l.o.t., here and throughout the proof, are linear with respect to Γ = {ξ, ξ, ϑ, κ ˇ , η, η, ζ, κ ˇ , ϑ} and quadratic or higher order in f, f , and do not contain derivatives of these latter. We also recall that χ = 12 (κ + ϑ), χ = 12 (κ + ϑ). Next, we compute       1 1 0 g De0θ e4 , eθ = g De0θ e4 , 1 + f f eθ + f e3 + l.o.t. 2 2     1 = 1 + f f g D(1+ 1 f f )eθ + 1 f e4 + 1 f e3 e4 , eθ 2 2 2 2   1 + f g Deθ + 12 f e4 + 12 f e3 e4 , e3 + l.o.t. 2
= 1 + f f χ + f ξ + f η + f ζ + f f ω − f 2 ω + l.o.t., and   f g De0θ eθ , e0θ

= =

  1   1 f f g De0θ eθ , e4 + f 2 g De0θ eθ , e3 2 2 1 1 2 − f f χ − f χ + l.o.t. 2 2

This yields 0

χ

=

λg



De0θ e4 , e0θ



    1 λ + λ 1 + f f e0θ (f ) − f e0θ (f 2 ) + λf g De0θ eθ , e0θ 2 4

λ + f 2 χ + l.o.t. 4   1 1 1 1 = λ χ + 1 + f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ − f 2 χ + f f χ 2 4 4 2 ! +f f ω − f 2 ω + l.o.t. .

APPENDIX TO CHAPTER 2

739

Hence,   1 = χ0 + e04 Φ = χ0 + λ e4 + f eθ + f 2 e3 Φ 4

κ0

1 1 1 = λ κ + e0θ (f ) + eθ (Φ)f + (κ − ϑ)f 2 + f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) 8 2 4 ! 1 1 +f ξ − f 2 χ + f f χ + f f ω − f 2 ω + l.o.t. 4 2 1 1 1 = λ κ + /d1 0 (f ) + f f e0θ (f ) + f e0θ (f 2 ) − f 2 κ + f (ζ + η) + f ξ + f f ω 2 4 4 ! −f 2 ω + l.o.t. and   1 = χ0 − e04 Φ = χ0 − λ e4 + f eθ + f 2 e3 Φ 4

ϑ0

1 1 1 = λ ϑ + e0θ (f ) − eθ (Φ)f − (κ − ϑ)f 2 + f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) 8 2 4 ! 1 1 +f ξ − f 2 χ + f f χ + f f ω − f 2 ω + l.o.t. 4 2 1 1 1 = λ ϑ − /d?2 0 (f ) + f f e0θ (f ) − f e0θ (f 2 ) + f f κ + f (ζ + η) + f ξ + f f ω 2 4 4 ! −f 2 ω + l.o.t. . This yields κ0

= λ (κ + /d1 0 (f )) + λErr(κ, κ0 ), 1 1 1 Err(κ, κ0 ) = f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ + f 2 κ + f f ω − f 2 ω 2 4 4 +l.o.t. 1 = f (ζ + η) + f ξ + f 2 κ + f f ω − f 2 ω + l.o.t. 4 and ϑ0

= λ (ϑ − /d?2 0 (f )) + λErr(ϑ, ϑ0 ), 1 1 1 Err(ϑ, ϑ0 ) = f f e0θ (f ) − f e0θ (f 2 ) + f (ζ + η) + f ξ + f f κ + f f ω − f 2 ω 2 4 4 +l.o.t. 1 = f (ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t. 4 Next, we derive the transformation formula for κ and ϑ. We have, under a

740

APPENDIX A

transformation of type (2.3.3), χ0 = g(De0θ e03 , eθ0 )         1 1 1 1 = g De0θ λ−1 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 , e0θ 2 16 4 4        1 1 1 1 = λ−1 g De0θ 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 , e0θ 2 16 4 4      −1 λ 1 1 e0 f f + f 2 f 2 g(e3 , e0θ ) + λ−1 e0θ f 1 + f f g(eθ , e0θ ) = 2 θ 8 4     λ−1 0 2
1 1 + eθ f g(e4 , e0θ ) + λ−1 1 + f f + f 2 f 2 g De0θ e3 , e0θ 4 2 16       1 1 +λ−1 f 1 + f f g De0θ eθ , e0θ + λ−1 f 2 g De0θ e4 , e0θ 4 4        −1 λ 1 1 1 = − f e0θ f f + f 2 f 2 + λ−1 1 + f f e0θ f 1 + f f 2 8 2 4       −1
λ 1 1 − f 1 + f f e0θ f 2 + λ−1 1 + f f g De0θ e3 , e0θ 4 4 2   1 +λ−1 f g De0θ eθ , e0θ + λ−1 f 2 χ + l.o.t. 4 Then, we easily derive by symmetry from the formula for κ and ϑ
κ0 = λ−1 κ + /d1 0 (f ) + λ−1 Err(κ, κ0 ),     1 1 3 1 Err(κ, κ0 ) = − f e0θ f f + f 2 f 2 + f f + (f f )2 e0θ (f ) 2 8 4 8    

1 1 1 1 + 1 + f f f e0θ f f − f 1 + f f e0θ f 2 + f (−ζ + η) 4 2 4 4 1 2 +f ξ − f κ + f f ω − f 2 ω + l.o.t. 4 1 2 0 1 = − f eθ (f ) + f (−ζ + η) + f ξ − f 2 κ + f f ω − f 2 ω + l.o.t. 4 4 and ϑ0


= λ ϑ − /d?2 0 (f ) + λ−1 Err(ϑ, ϑ0 ),     1 1 3 1 Err(ϑ, ϑ0 ) = − f e0θ f f + f 2 f 2 + f f + (f f )2 e0θ (f ) 2 8 4 8    

1 1 1 1 + 1 + f f f e0θ f f − f 1 + f f e0θ f 2 + f (−ζ + η) 4 2 4 4 1 +f ξ + f f κ + f f ω − f 2 ω + l.o.t. 4 1 2 0 1 = − f eθ (f ) + f (−ζ + η) + f ξ + f f κ + f f ω − f 2 ω + l.o.t. 4 4 Next, we derive the transformation formula for ζ. We have, under a transfor-

741

APPENDIX TO CHAPTER 2

mation of type (2.3.3), 2ζ 0

= g(De0θ e04 , e03 )      1 2 0 0 = g Deθ λ e4 + f eθ + f e3 , e3 4     1 2 0 0 0 = −2eθ (log(λ)) + λg Deθ e4 + f eθ + f e3 , e3 4   1 = −2e0θ (log(λ)) + λe0θ (f )g (eθ , e03 ) + λe0θ (f 2 )g (e3 , e03 ) + λg De0θ e4 , e03 4   1   0 2 +λf g De0θ eθ , e3 + λf g De0θ e3 , e03  4    1 1 0 = −2eθ (log(λ)) + f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + λg De0θ e4 , e03 4 8   +λf g De0θ eθ , e03 + l.o.t.

We compute   λg De0θ e4 , e03 = = =

  g De0θ e4 , e3 + f eθ + l.o.t.     g Deθ + 12 f e4 + 12 f e3 e4 , e3 + f g Deθ + 12 f e4 + 12 f e3 e4 , eθ + l.o.t. 2ζ + 2ωf − 2ωf + f χ + l.o.t.

and   λf g De0θ eθ , e03 = = =

  f g De0θ eθ , e3 + l.o.t.   f g Deθ + 12 f e4 + 12 f e3 eθ , e3 + l.o.t. −f χ + l.o.t.

This yields 2ζ

0

    1 1 = + f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + λg De0θ e4 , e03 4 8   +λf g De0θ eθ , e03 + l.o.t.   1 1 = 2ζ − 2e0θ (log(λ)) + f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + 2ωf − 2ωf + f χ 4 8 −f χ + l.o.t. −2e0θ (log(λ))

and hence 1 = ζ − e0θ (log(λ)) + (−f κ + f κ) + f ω − f ω + Err(ζ, ζ 0 ),   4 1 1 1 1 Err(ζ, ζ 0 ) = f 1 + f f e0θ (f ) − f 2 e0θ (f 2 ) + (−f ϑ + f ϑ) + l.o.t. 2 4 16 4 1 0 1 = f e (f ) + (−f ϑ + f ϑ) + l.o.t. 2 θ 4 ζ0

742

APPENDIX A

Next, we derive the transformation formulae for η. We have, under a transformation of type (2.3.3),
2η 0 = g De03 e04 , e0θ      1 = g De03 λ e4 + f eθ + f 2 e3 , e0θ 4     1 = λg De03 e4 + f eθ + f 2 e3 , e0θ 4
1
= λg De03 e4 , e0θ + λe03 (f )g (eθ , e0θ ) + λf g De03 eθ , e0θ + λe03 (f 2 )g (e3 , e0θ ) 4 1 2 0 + λf g(De03 e3 , eθ ) 4  

1 1 = λ 1 + f f e03 (f ) − λf e03 (f 2 ) + λg De03 e4 , e0θ + λf g De03 eθ , e0θ 2 4 +l.o.t. We compute λg

De03 e4 , e0θ






 1 = λg De03 e4 , eθ + f e3 + l.o.t. 2   1 = g De3 +f eθ e4 , eθ + f g (De3 e4 , e3 ) + l.o.t. 2 = 2η + f χ − 2ωf + l.o.t.

and λf g De03 eθ , e0θ




=

l.o.t.

This yields 2η 0

  1 = λ 1 + f f e03 (f ) − 2 +l.o.t.   1 = λ 1 + f f e03 (f ) − 2



1 λf e03 (f 2 ) + λg De03 e4 , e0θ + λf g De03 eθ , e0θ 4 1 λf e03 (f 2 ) + 2η + f χ − 2ωf + l.o.t. 4

and hence η0 Err(η, η 0 )

1 1 = η + λe03 (f ) + κf − f ω + Err(η, η 0 ), 2 4 1 1 1 0 = λf f e3 (f ) − λf e03 (f 2 ) + f ϑ + l.o.t. 4 8 4 1 = f ϑ + l.o.t. 4

Next, we derive the transformation formulae for η. We have, under a transfor-

743

APPENDIX TO CHAPTER 2

mation of type (2.3.3), 2η 0
= g De04 e03 , e0θ         1 1 2 2 1 1 2 −1 0 0 = g De4 λ 1 + f f + f f e3 + f 1 + f f eθ + f e4 , eθ 2 16 4 4        1 1 1 1 = λ−1 g De04 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 , e0θ 2 16 4 4    
1 −1 0 1 2 2 1 1 2 2 0 −1 = λ e4 f f + f f g (e3 , eθ ) + λ 1 + f f + f f g De04 e3 , e0θ 2 8 2 16     
1 1 −1 0 0 −1 +λ e4 f 1 + f f g (eθ , eθ ) + λ f 1 + f f g De04 eθ , e0θ 4 4
1 −1 0 2 1 + λ e4 (f )g (e4 , e0θ ) + λ−1 f 2 g De04 e4 , e0θ 4 4        1 −1 0 1 2 2 1 1 = − λ f e4 f f + f f + λ−1 1 + f f e04 f 1 + f f 2 8 2 4  

1 1 − λ−1 f 1 + f f e04 (f 2 ) + λ−1 g De04 e3 , e0θ + λ−1 f g De04 eθ , e0θ + l.o.t. 4 4 We compute λ−1 g De04 e3 , e0θ




  1 = λ−1 g De04 e3 , eθ + f e4 + l.o.t. 2 = 2η + f χ − 2f ω + l.o.t.

and λ−1 f g De04 eθ , e0θ




=

l.o.t.

This yields 2η 0

       1 1 1 1 = − λ−1 f e04 f f + f 2 f 2 + λ−1 1 + f f e04 f 1 + f f 2 8 2 4  

1 1 − λ−1 f 1 + f f e04 (f 2 ) + λ−1 g De04 e3 , e0θ + λ−1 f g De04 eθ , e0θ 4 4 +l.o.t.        1 1 1 1 = − λ−1 f e04 f f + f 2 f 2 + λ−1 1 + f f e04 f 1 + f f 2 8 2 4   1 1 − λ−1 f 1 + f f e04 (f 2 ) + 2η + f χ − 2f ω + l.o.t. 4 4

744

APPENDIX A

and hence 1 1 = η + λ−1 e04 (f ) + κf − f ω + Err(η, η 0 ), 2 4   1 1 1 2 2 0 −1 0 −1 0 Err(η, η ) = λ f f e4 (f ) − λ f e4 f f + f f 4 4 8    
1 −1 1 1 1 2 −1 1 0 +λ 1 + f f e4 f f − λ f 1 + f f e04 (f 2 ) + f ϑ 8 2 8 4 4 +l.o.t. 1 1 = − f 2 λ−1 e04 (f ) + f ϑ + l.o.t. 8 4 η0

Next, we derive the transformation formulae for ξ. We have, under a transformation of type (2.3.3),
2ξ 0 = g De04 e04 , e0θ      1 2 0 0 = g De4 λ e4 + f eθ + f e3 , eθ 4     1 = λg De04 e4 + f eθ + f 2 e3 , e0θ 4

1 = λg De04 e4 , e0θ + λe04 (f )g (eθ , e0θ ) + λf g De04 eθ , e0θ + λe04 (f 2 )g (e3 , e0θ ) 4
1 2 0 + λf g De04 e3 , eθ 4  

1 1 = λ 1 + f f e04 (f ) − λf e04 (f 2 ) + λg De04 e4 , e0θ + λf g De04 eθ , e0θ 2 4 +l.o.t. We compute λg

De04 e4 , e0θ




= = =

  1 λg De04 e4 , eθ + f e3 + l.o.t. 2 1 λ2 g (De4 +f eθ e4 , eθ ) + λ2 f g (De4 e4 , e3 ) + l.o.t. 2 2λ2 ξ + λ2 f χ + 2λ2 f ω + l.o.t.

and λf g De04 eθ , e0θ




=

l.o.t.

This yields 2ξ

0

=

=

  1 λ 1 + f f e04 (f ) − 2 +l.o.t.   1 λ 1 + f f e04 (f ) − 2



1 λf e04 (f 2 ) + λg De04 e4 , e0θ + λf g De04 eθ , e0θ 4 1 λf e04 (f 2 ) + 2λ2 ξ + λ2 f χ + 2λ2 f ω + l.o.t. 4

745

APPENDIX TO CHAPTER 2

and hence 0

=

Err(ξ, ξ 0 )

=

ξ

=

 1 −1 0 1 λ ξ + λ e4 (f ) + ωf + f κ + λ2 Err(ξ, ξ 0 ), 2 4 1 −1 1 1 λ f f e04 (f ) − λ−1 f e04 (f 2 ) + f ϑ + l.o.t. 4 8 4 1 f ϑ + l.o.t. 4 2



In the particular case when λ = 1, f = 0, see Remark 2.91, the error term takes the form  
1 3 1 1 2 1 1 0 Err(ξ, ξ ) = f ϑ + f η + 2ζ − η − f ω + χ − f 4 ξ. 4 4 4 2 16 Next, we derive the transformation formulae for ξ. We have, under a transformation of type (2.3.3), 2ξ 0
= g De03 e03 , e0θ         1 1 2 2 1 1 2 −1 0 = g De03 λ 1 + f f + f f e3 + f 1 + f f eθ + f e4 , eθ 2 16 4 4        1 1 1 1 = λ−1 g De03 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 , e0θ 2 16 4 4  
1 −1 0 1 2 2 = λ e3 f f + f f g (e3 , e0θ ) + λ−1 g De03 e3 , e0θ 2 8   
1 −1 0 +λ e3 f 1 + f f g (eθ , e0θ ) + λ−1 f g De03 eθ , e0θ 4 1 −1 0 2 + λ e3 (f )g (e4 , e0θ ) + l.o.t. 4        1 −1 0 1 2 2 1 1 −1 0 = − λ f e3 f f + f f +λ 1 + f f e3 f 1 + f f 2 8 2 4  

1 −1 1 − λ f 1 + f f e03 (f 2 ) + λ−1 g De03 e3 , e0θ + λ−1 f g De03 eθ , e0θ + l.o.t. 4 4 We compute λ−1 g De03 e3 , e0θ




  1 = λ−1 g De03 e3 , eθ + f e4 + l.o.t. 2   1 = λ−2 g De3 +f eθ e3 , eθ + λ−2 f g (De3 e3 , e4 ) + l.o.t. 2 = 2λ−2 ξ + λ−2 f χ + 2λ−2 f ω + l.o.t.

and λ−1 f g De03 eθ , e0θ




=

l.o.t.

746

APPENDIX A

This yields 2ξ 0

=

=

       1 1 1 1 − λ−1 f e03 f f + f 2 f 2 + λ−1 1 + f f e03 f 1 + f f 2 8 2 4  

1 1 − λ−1 f 1 + f f e03 (f 2 ) + λ−1 g De03 e3 , e0θ + λ−1 f g De03 eθ , e0θ 4 4 +l.o.t.        1 1 1 1 − λ−1 f e03 f f + f 2 f 2 + λ−1 1 + f f e03 f 1 + f f 2 8 2 4   1 −1 1 − λ f 1 + f f e03 (f 2 ) + 2λ−2 ξ + λ−2 f χ + 2λ−2 f ω + l.o.t. 4 4

and hence ξ0

=

Err(ξ, ξ 0 )

=

=

  1 1 λ−2 ξ + λe03 (f ) + ω f + f κ + λ−2 Err(ξ, ξ 0 ), 2 4    
1 1 1 1 1 − λf e03 f f + f 2 f 2 + λf f e03 (f ) + λ 1 + f f e03 f f 2 4 8 4 8 2   1 1 1 − λf 1 + f f e03 (f 2 ) + f ϑ + l.o.t. 8 4 4 1 2 0 1 − λf e3 (f ) + f ϑ + l.o.t. 8 4

Next, we derive the transformation formulae for ω. We have, under a transformation of type (2.3.3),
4ω 0 = g De04 e04 , e03      1 = g De04 λ e4 + f eθ + f 2 e3 , e03 4     1 = −2e04 (log λ) + λg De04 e4 + f eθ + f 2 e3 , e03 4

0 0 0 0 = −2e4 (log λ) + λg De4 e4 , e3 + λe4 (f )g (eθ , e03 ) + λf g De04 eθ , e03
1 1 + λe04 (f 2 )g (e3 , e03 ) + λf 2 g De04 e3 , e03 4 4  
1 1 0 = −2e4 (log λ) + f 1 + f f e04 (f ) − f 2 e04 (f 2 ) + λg De04 e4 , e03 4 8
0 0 +λf g De4 eθ , e3 + l.o.t. We compute λg

De04 e4 , e03




    1 = g De04 e4 , 1 + f f e3 + f eθ + l.o.t. 2     1 = λ 1 + f f g De4 +f eθ + 14 f 2 e3 e4 , e3 2 +λf g (De4 +f eθ e4 , eθ ) + l.o.t.   1 = 4λ 1 + f f ω + 2λf ζ − λf 2 ω + 2λf ξ + λf f χ + l.o.t. 2

747

APPENDIX TO CHAPTER 2

and λf g De04 eθ , e03




=


f g De04 eθ , e3 + l.o.t.

=

λf g (De4 +f eθ eθ , e3 ) + l.o.t.

=

−2λf η − λf 2 χ + l.o.t.

This yields 4ω

0

=

=

−2e04 (log λ)

 +f

1 1 + ff 4




1 e04 (f ) − f 2 e04 (f 2 ) + λg De04 e4 , e03 8


+λf g De04 eθ , e03 + l.o.t.     1 1 2 0 2 1 0 0 −2e4 (log λ) + f 1 + f f e4 (f ) − f e4 (f ) + 4λ 1 + f f ω 4 8 2 +2λf ζ − λf 2 ω + 2λf ξ + λf f χ − 2λf η − λf 2 χ + l.o.t.

and hence   1 = λ ω − λ−1 e04 (log(λ)) + λErr(ω, ω 0 ), 2   1 1 1 1 1 1 1 Err(ω, ω 0 ) = f 1 + f f e04 (f ) − f 2 e04 (f 2 ) + ωf f − f η + f ξ + f ζ 4 4 32 2 2 2 2 1 2 1 1 2 − κf + f f κ − ωf + l.o.t. 8 8 4 1 0 1 1 1 1 1 1 1 = f e (f ) + ωf f − f η + f ξ + f ζ − κf 2 + f f κ − ωf 2 4 4 2 2 2 2 8 8 4 +l.o.t. ω0

In the particular case, see Remark 2.91, when λ = 1, f = 0 we have the more precise formula ω0

=


1 1 ω + f (ζ − η) − f 2 2ω + κ + ϑ + f ξ . 2 8

Next, we derive the transformation formulae for ω. We have, under a transfor-

748

APPENDIX A

mation of type (2.3.3), 4ω 0
= g De03 e03 , e04         1 1 1 1 = g De03 λ−1 1 + f f + f 2 f 2 e3 + f 1 + f f eθ + f 2 e4 , e04 2 16 4 4 0 = 2e3 (log(λ))        1 1 2 2 1 1 2 −1 0 +λ g De03 1 + f f + f f e3 + f 1 + f f eθ + f e4 , e4 2 16 4 4   1 1 = 2e03 (log(λ)) + λ−1 e03 f f + f 2 f 2 g (e3 , e04 ) 2 8     
1 1 −1 0 −1 0 +λ 1 + f f g De03 e3 , e4 + λ e3 f 1 + f f g (eθ , e04 ) 2 4
1 +λ−1 f g De03 eθ , e04 + λ−1 e03 (f 2 )g (e4 , e04 ) + l.o.t. 4      1 2 2 1 1 0 0 0 = 2e3 (log(λ)) − e3 f f + f f + f e3 f 1 + f f − f 2 e03 (f 2 ) 8 4 8  

1 +λ−1 1 + f f g De03 e3 , e04 + λ−1 f g De03 eθ , e04 + l.o.t. 2 We compute  
1 λ−1 1 + f f g De03 e3 , e04 2  
1 = 1 + f f g De03 e3 , e4 + f eθ 2     1 = λ−1 1 + f f g D(1+ 1 f f )e3 +f eθ + 1 f 2 e4 e3 , e4 + f eθ + l.o.t. 2 4 2   1 = 4λ−1 1 + f f ω − 2λ−1 f ζ − λ−1 f 2 ω + 2λ−1 f f ω 2 +2λ−1 f ξ + λ−1 f f χ + l.o.t. and λ−1 f g De03 eθ , e04





= f g De03 eθ , e4 + l.o.t.   = λ−1 f g De3 +f eθ eθ , e4 + l.o.t. = −2λ−1 f η − λ−1 f 2 χ + l.o.t.

This yields 4ω

0

=

    1 2 2 1 1 0 − ff + f f + f e3 f 1 + f f − f 2 e03 (f 2 ) 8 4 8   1 +4λ−1 1 + f f ω − 2λ−1 f ζ − λ−1 f 2 ω + 2λ−1 f f ω 2

2e03 (log(λ))

e03



+2λ−1 f ξ + λ−1 f f χ − 2λ−1 f η − λ−1 f 2 χ + l.o.t.

749

APPENDIX TO CHAPTER 2

and hence  1 0 ω = λ ω + λe3 (log(λ)) + λ−1 Err(ω, ω 0 ), 2      1 0 1 2 2 1 0 1 1 0 Err(ω, ω ) = − e3 f f + f f + f e3 f 1 + f f − f 2 e03 (f 2 ) 4 8 4 4 32 1 1 1 1 1 1 +ωf f − f η + f ξ − f ζ − κf 2 + f f κ − ωf 2 + l.o.t. 2 2 2 8 8 4 1 0 1 1 1 1 2 1 1 = − f e3 (f ) + ωf f − f η + f ξ − f ζ − κf + f f κ − ωf 2 4 2 2 2 8 8 4 +l.o.t. 0

−1



Next we derive the formula for α. We have   1 2 1 2 0 0 0 2 α = R(e4 , e4 ) = λ R e4 + f eθ + f e3 , e4 + f eθ + f e3 4 4   1 = λ2 R44 + 2f R4θ + f 2 Rθθ + f 2 R34 + l.o.t. 2   3 = λ2 α + 2f β + f 2 ρ + l.o.t. 2 and hence α0

=

Err(α, α0 )

=

λ2 α + λ2 Err(α, α0 ), 3 2f β + f 2 ρ + l.o.t. 2

The formula for α is easily derived by symmetry from the one on α. Next we derive the formula for β. We have β0

= = = =

R(e04 , e0θ )     1 2 1 1 λR e4 + f eθ + f e3 , 1 + f f eθ + (f e4 + f e3 ) + l.o.t. 4 2 2   1 1 λ R4θ + f Rθθ + f R44 + f R43 + l.o.t. 2 2   3 1 λ β + f ρ + f α + l.o.t. 2 2

and hence  3 β = λ β + f ρ + λErr(β, β 0 ), 2 1 0 Err(β, β ) = f α + l.o.t. 2 0



The formula for β is easily derived by symmetry from the one on β.

750

APPENDIX A

Finally, we derive the formula for ρ. We have     1 1 1 ρ0 = R(e04 , e03 ) = R e4 + f eθ + f 2 e3 , 1 + f f e3 + f eθ + f 2 e4 + l.o.t. 4 2 4 1 = R43 + f f R43 + f R4θ + f Rθ3 + f f Rθθ + l.o.t. 2 3 = ρ + ρf f + f β + f β + l.o.t. 2 and hence ρ0

= ρ + Err(ρ, ρ0 ), 3 Err(ρ, ρ0 ) = ρf f + f β + f β + l.o.t. 2 This concludes the proof of Proposition 2.90.

A.7

PROOF OF LEMMA 2.92

For ξ 0 and ω 0 , we need a more precise transformation formula than the ones of Proposition 2.90. We have 2ξ 0

=

g(De04 e04 , e0θ )

λ2 g(Dλ−1 e04 (λ−1 e04 ), e0θ )   1 2 −1 0 = λ g Dλ−1 e04 (λ e4 ), eθ + f e3 2   f 1 2 2 −1 0 +λ g Dλ−1 e04 (λ e4 ), e4 + f eθ + f e3 2 4  
f 1 = λ2 g Dλ−1 e04 (λ−1 e04 ), eθ + f e3 + λ2 g Dλ−1 e04 (λ−1 e04 ), λ−1 e04 2 2   1 = λ2 g Dλ−1 e04 (λ−1 e04 ), eθ + f e3 . 2 =

751

APPENDIX TO CHAPTER 2

Also, we have 4ω 0 =

g(De04 e04 , e03 )

=

−2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), λe03 )

=

=

 −2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), e3 ) + λf g Dλ−1 e04 (λ−1 e04 ), eθ +   1 1 + λf 2 g Dλ−1 e04 (λ−1 e04 ), e4 + f eθ + f 2 e3 4 4  −2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), e3 ) + λf g Dλ−1 e04 (λ−1 e04 ), eθ +

1 f e3 2



1 f e3 2




1 + λf 2 g Dλ−1 e04 (λ−1 e04 ), λ−1 e04 4 =

  1 −2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), e3 ) + λf g Dλ−1 e04 (λ−1 e04 ), eθ + f e3 . 2

In view of the change of frame formula for ξ 0 , we infer 4ω 0

= −2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), e3 ) + λ−1 f ξ 0 .

Next, we compute g Dλ−1 e04 (λ−1 e04 ), eθ




= = =

=

    1 g Dλ−1 e04 e4 + f eθ + f 2 e3 , eθ 4

1 g Dλ−1 e04 e4 , eθ + λ−1 e04 (f ) + f 2 g Dλ−1 e04 e3 , eθ 4     1 g De4 +f eθ + 14 f 2 e3 e4 , eθ + e4 + f eθ + f 2 e3 f 4  1 2  + f g De4 +f eθ + 14 f 2 e3 e3 , eθ 4   1 1 1 2ξ + f χ + f 2 η + e4 + f eθ + f 2 e3 f + f 2 η 2 4 2 1 3 1 4 + f χ + f ξ. 4 8

Also, we have g

Dλ−1 e04 (λ−1 e04 ), e3






   1 2 = g Dλ−1 e04 e4 + f eθ + f e3 , e3 4

= g Dλ−1 e04 e4 , e3 + f g Dλ−1 e04 eθ , e3     = g De4 +f eθ + 14 f 2 e3 e4 , e3 + f g De4 +f eθ + 14 f 2 e3 eθ , e3 =

1 4ω + 2f ζ − f 2 ω − 2ηf − f 2 χ − f 3 ξ. 2

752

APPENDIX A

We deduce 2ξ

0

 1 = λ g + f e3 2 (   1 2 1 2 1 1 2 = λ 2ξ + f χ + f η + e4 + f eθ + f e3 f + f 2 η + f 3 χ 2 4 2 4  ) 1 1 1 + f 4 ξ + f 4ω + 2f ζ − f 2 ω − 2ηf − f 2 χ − f 3 ξ 8 2 2 (   1 1 1 = λ2 2ξ + e4 + f eθ + f 2 e3 f + f χ + 2f ω + f 2 η − f 2 η + f 2 ζ 4 2 2 ) 1 1 1 − f 3χ − f 3ω − f 4ξ 4 2 8 2



Dλ−1 e04 (λ−1 e04 ), eθ

and 4ω 0

= −2e04 (log(λ)) + λg(Dλ−1 e04 (λ−1 e04 ), e3 ) + λ−1 f ξ 0 (   1 = λ 4ω − 2 e4 + f eθ + f 2 e3 log(λ) + 2f ζ − f 2 ω − 2ηf − f 2 χ 4 ) 1 − f 3 ξ + λ−1 f ξ 0 . 2

If ξ 0 = 0, we infer   1 1 1 1 2ξ + e4 + f eθ + f 2 e3 f + f χ + 2f ω + f 2 η − f 2 η + f 2 ζ − f 3 χ 4 2 2 4 1 3 1 − f ω − f 4ξ = 2 8

0

and hence λ−1 e04 (f ) +

κ 2

 + 2ω f

=

1 1 1 1 1 −2ξ − ϑf − f 2 η + f 2 η − f 2 ζ + f 3 κ + f 3 ω 2 2 2 8 2 1 3 1 4 + f ϑ+ f ξ 8 8

which yields the desired transport equation for f κ  λ−1 e04 (f ) + + 2ω f = −2ξ + E1 (f, Γ), 2 1 1 1 1 1 E1 (f, Γ) = − ϑf − f 2 η + f 2 η − f 2 ζ + f 3 κ + f 3 ω 2 2 2 8 2 1 3 1 4 + f ϑ + f ξ. 8 8 Also, if ξ 0 = 0 and ω 0 = 0, we infer   1 2 1 0 = 4ω − 2 e4 + f eθ + f e3 log(λ) + 2f ζ − f 2 ω − 2ηf − f 2 χ − f 3 ξ 4 2

753

APPENDIX TO CHAPTER 2

and hence λ−1 e04 (log(λ))

1 1 1 1 2ω + f ζ − f 2 ω − ηf − f 2 κ − f 2 ϑ − f 3 ξ 2 4 4 4

=

which yields the desired transport equation for log(λ) λ−1 e04 (log(λ))

=

2ω + E2 (f, Γ), 1 1 1 1 = f ζ − f 2 ω − ηf − f 2 κ − f 2 ϑ − f 3 ξ. 2 4 4 4

E2 (f, Γ)

Finally, we derive the transport equation for f . In view of the transformation formulas of Proposition 2.90 for ζ 0 and η 0 , and the fact that we assume ζ 0 + η 0 = 0, we have 1 −1 0 λ e4 (f ) 2

1 1 1 = −(ζ + η) + e0θ (log(λ)) − f κ + f ω − f e0θ (f ) + f 2 λ−1 e04 (f ) 4 2 8 1 − f ϑ + l.o.t. 4

Together with the above identity for λ−1 e04 (f ), we infer κ f 2

=

−2(ζ + η) + 2e0θ (log(λ)) + 2f ω + E3 (f, f , Γ),

E3 (f, f , Γ)

=

1 −f e0θ (f ) − f ϑ + l.o.t., 2

λ−1 e04 (f ) +

which yields the third identity of the statement. This concludes the proof of Lemma 2.92.

A.8

PROOF OF COROLLARY 2.93

In view of Lemma 2.92 and the fact that (e3 , e4 , eθ ) emanates from an outgoing geodesic foliation and hence ξ = 0,

ω = 0,

ζ + η = 0,

we have κ f = E1 (f, Γ), 2 −1 0 λ e4 (log(λ)) = E2 (f, Γ), κ λ−1 e04 (f ) + f = 2e0θ (log(λ)) + 2f ω + E3 (f, f , Γ). 2

λ−1 e04 (f ) +

The second equation is the desired identity for log(λ). We still need to derive the first and the third identities. We start with the first one. We have  κ  λ−1 e04 (rf ) = r − f + E1 (f, Γ) + λ−1 e04 (r)f 2  r 2λ−1 e04 (r) = − κ− f + rE1 (f, Γ). 2 r

754

APPENDIX A

Since λ−1 e04

= e4 + f eθ +

f2 e3 , 4

we infer λ−1 e04 (r)

=

r f2 κ+ e3 (r) 2 4

and hence λ−1 e04 (rf )

= −

r 2

  e3 (r) 2 κ ˇ− f f + rE1 (f, Γ) 2r

as desired. Next, we have   λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω  κ  = r − f + 2e0θ (log(λ)) + 2f ω + E3 (f, f , Γ) − 2r2 e0θ (E2 (f, Γ)) 2   κ +rΩ − f + E1 (f, Γ) + λ−1 e04 (r)f − 2r2 [λ−1 e04 , e0θ ] log(λ) 2 −4rλ−1 e04 (r)e0θ (log(λ)) + rλ−1 e04 (Ω)f + λ−1 e04 (r)f Ω     r 2λ−1 e04 (r) = − κ− f + 2r 1 − 2λ−1 e04 (r) e0θ (log(λ)) 2 r     r 2λ−1 e04 (r) −2r2 λ−1 [e04 , e0θ ] log(λ) + r λ−1 e04 (Ω) + 2ω f − κ− Ωf 2 r −2r2 e0θ (log(λ))λ−1 e04 (log(λ)) + rE3 (f, f , Γ) − 2r2 e0θ (E2 (f, Γ)) + rΩE1 (f, Γ). Since we have λ−1 e04

= e4 + f eθ +

f2 e3 , 4

we infer λ−1 e04 (r)

=

r f2 κ+ e3 (r), 2 4

λ−1 e04 (Ω)

=

e4 (Ω) + f eθ (Ω) +

=

f2 e3 (Ω) 4 f2 −2ω + f eθ (Ω) + e3 (Ω). 4

Together with the transport equation for log(λ) and the commutator identity for

APPENDIX TO CHAPTER 2

755

[e04 , e0θ ], we infer   λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω     r e3 (r) 2 e3 (r) 2 0 = − κ ˇ− f f + 2r 1 − rκ − f eθ (log(λ)) 2 2r 2     f r e3 (r) 2 2 −1 0 0 0 2 +r λ (κ + ϑ )eθ (log(λ)) + r eθ (Ω) + e3 (Ω) f − κ ˇ− f Ωf 4 2 2r −2r2 e0θ (log(λ))E2 (f, Γ) + rE3 (f, f , Γ) − 2r2 e0θ (E2 (f, Γ)) + rΩE1 (f, Γ). Now, recall the following transformation formulas λ−1 κ0

= κ + /d01 (f ) + Err(κ, κ0 ), 1 Err(κ, κ0 ) = f (ζ + η) − f 2 κ − f 2 ω + l.o.t. 4 We infer

=

  λ−1 e04 rf − 2r2 e0θ (log(λ)) + rf Ω       r e3 (r) 2 2 e3 (r) 2 0 − κ ˇ− f f + r2 κ ˇ− κ− − f eθ (log(λ)) 2 2r r r   2 0 0 −1 0 0 +r /d1 (f ) + Err(κ, κ ) + λ ϑ eθ (log(λ))     f r e3 (r) 2 2 κ ˇ− f Ωf +r eθ (Ω) + e3 (Ω) f − 4 2 2r −2r2 e0θ (log(λ))E2 (f, Γ) + rE3 (f, f , Γ) − 2r2 e0θ (E2 (f, Γ)) + rΩE1 (f, Γ).

This concludes the proof of Corollary 2.93.

A.9

PROOF OF LEMMA 2.91

Recall that we have obtained in section A.7 (     1 2 1 1 1 1 0 2 2ξ = λ 2ξ + e4 + f eθ + f e3 f + κ + 2ω f + ϑf + f 2 η − f 2 η 4 2 2 2 2 ) 1 1 1 +f 2 ζ − f 3 χ − f 3 ω − f 4 ξ , 4 2 8 (   1 2 0 4ω = λ 4ω − 2 e4 + f eθ + f e3 log(λ) + 2f ζ − f 2 ω − 2ηf − f 2 χ 4 ) 1 − f 3 ξ + λ−1 f ξ 0 . 2

756

APPENDIX A

In the case where λ = 1 and f = 0, we immediately infer 2ξ

0

4ω 0



   1 2 1 1 1 1 = 2ξ + e4 + f eθ + f e3 f + κ + 2ω f + ϑf + f 2 η − f 2 η 4 2 2 2 2 1 1 1 +f 2 ζ − f 3 χ − f 3 ω − f 4 ξ, 4 2 8 1 2 = 4ω + 2f ζ − f ω − 2ηf − f 2 χ − f 3 ξ, 2

and hence ξ

0

ω0

  1 0 1 1 1 1 1 1 = ξ + e4 (f ) + κ + ω f + ϑf + f 2 η − f 2 η + f 2 ζ − f 3 χ 2 4 4 4 4 2 8 1 3 1 4 − f ω − f ξ, 4 16 1 1 1 1 1 1 = ω + f ζ − f 2 ω − ηf − f 2 κ − f 2 ϑ − f 3 ξ. 2 4 2 8 8 8

Finally, we compute the change of frame formula for ζ 0 and η 0 when λ = 1, f = 0. We have in this case e04 e0θ e03

1 = e4 + f eθ + f 2 e3 , 4 1 = eθ + f e3 , 2 = e3 ,

and hence 2ζ 0

= = = = = =

  g De0θ e04 , e03     1 2 g De0θ e4 + f eθ + f e3 , e3 4     g De0θ e4 , e3 + f g De0θ eθ , e3     g Deθ + 12 f e3 e4 , e3 + f g Deθ + 12 f e3 eθ , e3 2ζ − 2ωf − χf − ξf 2     1 1 2ζ − κ + 2ω f − f ϑ + fξ 2 2

757

APPENDIX TO CHAPTER 2

and 2η 0


= g De03 e04 , e0θ     1 = g De03 e4 + f eθ + f 2 e3 , e0θ 4

1 = g De03 e4 , e0θ + e03 (f )g (eθ , e0θ ) + f g De03 eθ , e0θ + e03 (f 2 )g (e3 , e0θ ) 4 1 2 0 + f g(De03 e3 , eθ ) 4     1 1 1 0 = e3 (f ) + g De3 e4 , eθ + f e3 + f g De3 eθ , f e3 + f 2 g(De3 e3 , eθ ) 2 2 4 1 = 2η + e03 (f ) − 2f ω − f 2 ξ 2

which yields the desired change of frame formula for ζ 0 and η 0 . This concludes the proof of Lemma 2.91.

A.10

PROOF OF PROPOSITION 2.99

Recall that we have     1 2 4 2 q = r e3 (e3 (α)) + (2κ − 6ω)e3 (α) + −4e3 (ω) + 8ω − 8ω κ + κ α , 2 which we write in the form q = r4 J where J

=

V

=

e3 (e3 (α)) + (2κ − 6ω)e3 (α) + V α, 1 −4e3 (ω) + 8ω 2 − 8ω κ + κ2 . 2

We make use of the general1 Bianchi equations, see Proposition 2.57, 1 3 e3 α + κα = − /d?2 β + 4ωα − ϑρ + Err[e3 (α)], 2 2 e3 β + κβ = − /d?1 ρ + 2ωβ + 3ηρ + Err[e3 (β)], 3 e3 ρ + κρ = /d1 β + Err[e3 (ρ)], 2 as well as the null structure equations (see Proposition 2.56) 1 1 e3 ϑ + κ ϑ − 2ωϑ = −2 /d?2 η − κ ϑ + Err[e3 (ϑ)], 2 2 1 2 e3 (κ) + κ + 2ω κ = 2 /d1 ξ + Err[e3 (κ)], 2 where Err[e3 (α)], Err[e3 (β)], Err[e3 (ρ)], Err[e3 (ϑ)], Err[e3 (κ)] denote the corresponding quadratic terms in each equation. We also make use of the commutation 1 In

an arbitrary Z-invariant frame.

758

APPENDIX A

formula (see Lemma 2.54) [e3 , /d?2 ]β

=

Com∗2 (β)

=

1 − κ /d?2 β − Com∗2 (β), 2 1 − ϑ /d1 β − (ζ − η)e3 β − ηe3 Φβ + ξ(e4 β − e4 (Φ)β) − β · β. 2

Thus, J

  1 3 = e3 − κα − /d?2 β + 4ωα − ϑρ + Err[e3 (α)] + (2κ − 6ω)e3 (α) + V α 2 2  
3 1 3 = κ − 2ω e3 α + − e3 κ + 4e3 (ω) + V α − /d?2 e3 β − [e3 , /d?2 ]β − ϑe3 ρ 2 2 2 3 − ρe3 ϑ + e3 Err[e3 (α)] 2    3 1 3 = κ − 2ω − κα − /d?2 β + 4ωα − ϑρ + Err[e3 (α)] 2 2 2   1 1 3 3 + − e3 κ + 4e3 (ω) + V α − /d?2 e3 β + κ /d?2 β − ϑe3 ρ − ρe3 ϑ 2 2 2 2 + e3 Err[e3 (α)] + Com∗2 (β) 1 − e3 κ + 4e3 (ω) + V 2   !    3 1 3 3 + κ − 2ω − κ + 4ω α− ϑe3 ρ + ρe3 ϑ + ϑρ κ − 2ω 2 2 2 2   3 + e3 Err[e3 (α)] + κ − 2ω Err[e3 (α)] + Com∗2 (β). 2 = − /d?2 e3 β + (−κ + 2ω) /d?2 β +

Hence, J = − /d?2 e3 β + (−κ + 2ω) /d?2 β −

3 2



3 ϑe3 ρ + ρe3 ϑ + ϑρ( κ − 2ω) 2



+ W α + E,    1 3 1 W : = − e3 κ + 4e3 (ω) + V + κ − 2ω − κ + 4ω , 2 2 2   3 E : = e3 Err[e3 (α)] + κ − 2ω Err[e3 (α)] + Com∗2 (β). 2

(A.10.1)

Now, ignoring cubic and higher order terms, − /d?2 e3 β + (−κ + 2ω) /d?2 β

= − /d?2 (−κβ − /d?1 ρ + 2ωβ + 3ηρ + Err[e3 (β)]) + (−κ + 2ω) /d?2 β =

/d?2 /d?1 ρ − 3ρ /d?2 η + β /d?1 (κ − 2ω) − 3η /d?1 ρ − /d?2 Err[e3 (β)].

APPENDIX TO CHAPTER 2

759

Also,

=

=

  3 ϑe3 ρ + ρe3 ϑ + ϑρ κ − 2ω 2     3 3 ϑ − κρ + /d1 β + ϑρ κ − 2ω 2 2   1 1 +ρ − κ ϑ + 2ωϑ − 2 /d?2 η − κ ϑ + Err[e3 (ϑ)] 2 2 1 1 − κρϑ − κρϑ − 2ρ /d?2 η + ϑ /d1 β + ρErr[e3 (ρ)] 2 2

and W

=

= = = =

  1 1 − e3 κ + 4e3 (ω) + −4e3 (ω) + 8ω 2 − 8ω κ + κ2 2 2    3 1 + κ − 2ω − κ + 4ω 2 2     1 1 3 − e3 κ + 8ω 2 − 8ω κ + κ2 + − κ2 − 8ω 2 + 7ωκ 2 2 4 1 1 2 − e3 κ − κ − ωκ 2 4  1 1 2 1 − − κ − 2ω κ + 2 /d1 ξ + Err[e3 (κ)] − κ2 − ωκ 2 2 4 1 − /d1 ξ − Err[e3 (κ)]. 2

Thus, back to (A.10.1), J

/d?2 /d?1 ρ − 3ρ /d?2 η + β /d?1 (κ − 2ω) − 3η /d?1 ρ − /d?2 Err[e3 (β)]   3 1 1 ? − − κρϑ − κρϑ − 2ρ /d2 η + ϑ /d1 β + ρErr[e3 (ρ)] − /d1 ξα + E 2 2 2 3 3 = /d?2 /d?1 ρ + ρ(κϑ + κϑ) + β /d?1 (κ − 2ω) − 3η /d?1 ρ − ϑ /d1 β − /d1 ξα 4 2 3 − ρErr[e3 (ρ)] + E. 2 =

In other words, 3 /d?2 /d?1 ρ + ρ(κϑ + κϑ) + Err 4 3 ? Err := β /d1 (κ − 2ω) − 3η /d?1 ρ − ϑ /d1 β − /d1 ξα 2   3 3 − ρErr[e3 (ρ)] + e3 Err[e3 (α)] + κ − 2ω Err[e3 (α)] + Com∗2 (β) + l.o.t. 2 2 J

=

It remains to analyze the lower order terms according to our convention in Definition

760

APPENDIX A

2.94. Note that we can write the first line in the expression of Err Err1

= r−1 Γb · β + r−2 Γg · d/Γg + r−2 Γg d/Γb + r−1 d/Γb · α = r−1 d/≤1 Γb · β + r−2 Γg d/Γb + l.o.t.

On the other hand, Err[e3 (ρ)] Err[e3 (α)] e3 Err[e3 (α)]

1 = − ϑ α − ζ β + 2(η β + ξ β) = Γg · Γb + Γb · β, 2 = (ζ + 4η)β = Γg · β, = e3 (ζ + 4η)β + (ζ + 4η)e3 (β) = =

Com∗2 (β)

= = + =

e3 (ζ + 4η) · β + (ζ + 4η)(−κβ − /d?1 ρ + 2ωβ + 3ηρ) e3 (ζ + 4η) · β + r−1 Γg β + r−2 Γg d/Γg + r−3 Γg · Γg ,

1 − ϑ /d1 β − (ζ − η)e3 β − ηe3 Φβ + ξ(e4 β − e4 (Φ)β) − β · β 2 1 − ϑ /d1 β − (ζ − η) (−κβ − /d?1 ρ + 2ω β + 3ηρ) − ηe3 (Φ)β 2 ξ (−2κβ + /d?2 α − 2ωβ) − ξe4 Φβ − β · β + l.o.t. r−1 Γb · d/≤1 β + r−2 Γg · d/Γb + l.o.t.

Therefore, schematically, Err

= e3 (ζ + 4η) · β + r−1 Γb d/≤1 β + r−2 Γg d/Γb + l.o.t.

and therefore,
Err[q] = r4 Err = r4 e3 (ζ + 4η) · β + r−1 Γb · d/≤1 β + r−2 Γg · d/Γb + l.o.t. Since e3 ζ ∈ r−1 dΓb and β ∈ r−1 Γg we rewrite in the form Err[q]

= r4 e3 η · β + r2 d≤1 Γb · Γg ).

This concludes the proof of Proposition 2.99.

A.11

PROOF OF PROPOSITION 2.100

We start with the formula (2.3.11)   3 3 5 ? ? rq = r /d2 /d1 ρ + κρϑ + κρϑ + rErr[q] 4 4 with Err[q] given by (2.3.12). Taking the e3 derivative we deduce e3 (rq) = r5 L + 5e3 (r) q + e3 (rErr[q]) − 5e3 (r)Err[q],   3 3 ? ? L : = e3 /d2 /d1 ρ + e3 (κρϑ) + e3 (κρϑ) . 4 4

(A.11.1)

761

APPENDIX TO CHAPTER 2

We calculate L as follows: 3 3 L = e3 /d?2 /d?1 ρ + e3 (κρϑ) + e3 (κρϑ) 4 4 3 3 ? ? ? ? = /d2 /d1 e3 (ρ) + [e3 , /d2 /d1 ]ρ + e3 (κρϑ) + e3 (κρϑ). 4 4 Ignoring cubic and higher order terms e3 (κρϑ)

= κρe3 (ϑ) + e3 (κ)ρϑ + κe3 (ρ)ϑ   1 1 = κρ − κ ϑ + 2ωϑ − 2 /d?2 η − κ ϑ + Err[e3 ϑ] 2 2   1 + − κ2 − 2ω κ + 2 /d1 ξ + Err[e3 κ] ρϑ 2   3 1 +κ − κρ + /d1 β − ϑ α + Err[e3 (ρ)] ϑ 2 2   5 1 = κρ − κ ϑ − 2 /d?2 η − κ ϑ + κρErr[e3 ϑ] + 2 /d1 ξ ρϑ + κ( /d1 β)ϑ. 2 2

We deduce   5 1 e3 (κρϑ) = κρ − κ ϑ − 2 /d?2 η − κ ϑ + E1 , 2 2

(A.11.2)

E1 = 2 /d1 ξ ρϑ + κ( /d1 β)ϑ + κρErr[e3 (ϑ)], where Err[e3 (ϑ)], Err[e3 (κ)], Err[e3 (ρ)] denote the quadratic error terms in the corresponding equations. Also, e3 (κρϑ)

= =

κρe3 (ϑ) + e3 (κ)ρϑ + κe3 (ρ)ϑ   κρ − κ ϑ − 2ω ϑ − 2α − 2 /d?2 ξ + Err[e3 (ϑ)]   1 + − κ κ + 2ωκ + 2 /d1 η + 2ρ + Err[e3 (κ)] ρϑ 2   3 + − κρ + /d1 β + Err[e3 (ρ)] κϑ. 2

Hence, ignoring the higher order terms,   e3 (κρϑ) = κρ − 3κ ϑ − 2α − 2 /d?2 ξ + 2ρ2 ϑ + E2 , E2 : = 2ρ /d1 ηϑ + κ /d1 βϑ + κρ Err[e3 (ϑ)]. Also, we have /d?2 /d?1 e3 (ρ)

= =

E3

=

  3 /d?2 /d?1 − κρ + /d1 β + Err[e3 (ρ)] 2 3 3 /d?2 /d?1 /d1 β − ρ /d?2 /d?1 κ − κ /d?2 /d?1 ρ + E3 , 2 2 /d?2 /d?1 Err[e3 (ρ)] − 3 /d?1 κ · /d?1 ρ,

(A.11.3)

762

APPENDIX A

i.e.,   5 1 e3 (κρϑ) = κρ − κ ϑ − 2 /d?2 η − κ ϑ + E1 , 2 2   ? e3 (κρϑ) = κρ − 3κ ϑ − 2α − 2 /d2 ξ + 2ρ2 ϑ + E2 ,

(A.11.4)

3 3 /d?2 /d?1 e3 (ρ) = /d?2 /d?1 /d1 β − ρ /d?2 /d?1 κ − κ /d?2 /d?1 ρ + E3 . 2 2 Now, in view of Lemma 2.54 we have (for f = /d?1 ρ ∈ s2−1 ), [e3 , /d?2 ] /d?1 ρ =

1 − κ /d?2 /d?1 ρ − Com∗2 ( /d?1 ρ) 2

and [e3 , /d?1 ]ρ = = − = E41

=

1 1 −[e3 , eθ ]ρ = − κ /d?1 ρ − ϑeθ ρ + (ζ − η)e3 ρ − ξe4 ρ 2 2   1 ? 1 3 − κ /d1 ρ − ϑeθ ρ + (ζ − η) − κρ + /d1 β + Err[e3 (ρ)] 2 2 2   3 ξ − κρ + /d1 β + Err[e4 (ρ)] 2  1 ? 3  − κ /d1 ρ − ρ (ζ − η)κ − ξκ + E41 , 2 2 1 (ζ − η) /d1 β − ξ /d1 β + (ζ − η)e3 [(ρ)] − ϑeθ ρ − ξErr[e4 (ρ)]. 2

We deduce /d?2 [e3 , /d?1 ]ρ

E4

 1   3  /d?2 − κ /d?1 ρ − ρ κ(ζ − η) − κξ + E41 2 2  1 ? ? 3  ? = − κ /d2 /d1 ρ − ρ κ /d2 (ζ − η) − κ /d?2 ξ + E4 , 2 2 1 3 3 = /d?2 E41 − /d?1 κ · /d?1 ρ − (ζ − η) /d?1 (κρ) + ξ /d?1 (κρ). 2 2 2 =

Hence, since [e3 , /d?2 /d?1 ]ρ = [e3 , /d?2 ] /d?1 ρ + /d?2 [e3 , /d?1 ]ρ, [e3 , /d?2 /d?1 ]ρ = −κ /d?2 /d?1 ρ −


3 ? /d2 (ζ − η)κρ − /d?2 ξκρ − Com∗2 ( /d?1 ρ) + E4 .(A.11.5) 2

763

APPENDIX TO CHAPTER 2

We deduce, recalling (A.11.4), L = = − + + = + +

3 3 /d?2 /d?1 e3 (ρ) + [e3 , /d?2 /d?1 ]ρ + e3 (κρϑ) + e3 (κρϑ) 4 4 3 ? ? 3 ? ? ? ? /d2 /d1 /d1 β − ρ /d2 /d1 κ − κ /d2 /d1 ρ + E3 2 2  3 ? ? ? κ /d2 /d1 ρ − /d2 (ζ − η)κρ − /d?2 ξκρ − Com∗2 ( /d?1 ρ) + E4   2  3 5 1 κρ − κ ϑ − 2 /d?2 η − κ ϑ + E1 4 2 2   3  κρ − 3κ ϑ − 2α − 2 /d?2 ξ + 2ρ2 ϑ + Err1 4 5 3 3 /d?2 /d?1 /d1 β − κ /d?2 /d?1 ρ − ρ /d?2 /d?1 κ − ρκ /d?2 ζ 2 2   2  3 3 5 1 3  κρ − κ ϑ − κ ϑ + κρ − 3κ ϑ − 2α + ρ2 ϑ 4 2 2 4 2 ∗ ? E1 + E2 + E3 + E4 − Com2 ( /d1 ρ),

i.e., 5 3 3 L = /d?2 /d?1 /d1 β − κ /d?2 /d?1 ρ − ρ /d?2 /d?1 κ − ρκ /d?2 ζ 2 2 2   3 5 1 3 3 − κρ κ ϑ + κ ϑ − κρ (3κ ϑ + 2α) + ρ2 ϑ + E, 4 2 2 4 2

(A.11.6)

E = E1 + E2 + E3 + E4 − Com∗2 ( /d?1 ρ). On the other hand, in view of (2.3.11), writing e3 r = 5e3 (r)q = =

r 2

(κ + A),

5 r κq + 5rAq 2   5 5 3 3 r κ /d?2 /d?1 ρ + κρϑ + κρϑ + Err + 5rAq. 2 4 4

Hence, in view of (A.11.1) and (A.11.6), e3 (rq)

= r5 L + 5e3 (r)q + e3 (rErr) − 5e3 (r)Err (  5 ? ? 3 ? ? 3 5 ? ? ? = r /d2 /d1 /d1 β − κ /d2 /d1 ρ − ρ /d2 /d1 κ − ρκ /d2 ζ 2 2 2 )   3 5 1 3 3 2 − κρ κ ϑ + κ ϑ − κρ (3κ ϑ + 2α) + ρ ϑ 4 2 2 4 2   5 5 3 3 e3 r 5 + r κ /d?2 /d?1 ρ + κρϑ + κρϑ + r5 E + e3 (r5 Err) − 5 r Err 2 4 4 r + 5rAq   3 ? ? 3 3 3 5 ? ? ? 2 = r /d2 /d1 /d1 β − ρ /d2 /d1 κ − κρ /d2 ζ − κρα + (2ρ − κκρ)ϑ 2 2 2 4 + Err[e3 (rq)]

764

APPENDIX A

where = e3 (rErr[q]) − 5e3 rErr[q] + 5rAq + r5 E

Err[e3 (rq)]

= re3 (Err[q]) + Err[q] + 5rAq + r5 E

(A.11.7)

and E

=

E1 + E2 + E3 + E4 − Com∗2 ( /d?1 ρ)

with E1 = 2 /d1 ξ ρϑ + κ( /d1 β)ϑ + κρErr[e3 (κ)], E2 = 2ρ /d1 ηϑ + κ /d1 βϑ + κρ Err[e3 (ϑ)], E3 = /d?2 /d?1 Err[e3 (ρ)] − 3 /d?1 κ · /d?1 ρ, 1 3 3 E4 = /d?2 E41 − /d?1 κ · /d?1 ρ − (ζ − η) /d?1 (κρ) + ξ /d?1 (κρ), 2 2 2 1 E41 = (ζ − η) /d1 β − ξ /d1 β − ϑeθ ρ, 2 1 ∗ ? ? Com2 ( /d1 ρ) = − ϑ /d1 /d1 ρ − (ζ − η)e3 /d?1 ρ − ηe3 Φ /d?1 ρ + ξ(e4 /d?1 ρ − e4 (Φ) /d?1 ρ) 2 − β /d?1 ρ. Note also that (ζ − η)e3 /d?1 ρ =

(ζ − η) · /d?1 e3 ρ   1 ? 1 +(ζ − η) − κ /d1 ρ − ϑeθ ρ + (ζ − η)e3 ρ − ξe4 ρˇ . 2 2

Using our schematic notation Err[e3 (κ)]

=

Err[e3 (ϑ)]

=

Err[e3 (ρ)]

=

Γb · Γb + l.o.t., Γb · Γb + l.o.t.,

Γg · α + l.o.t. = Γg · Γb + l.o.t.,

and E1 E2 E3 E41 E4 Com∗2 ( /d?1 ρ)

= r−4 Γb · d/≤1 Γb + l.o.t.,

= r−4 Γb · d/≤1 Γb + r−2 Γb · β + l.o.t.,

= r−2 d/2 (Γg · Γb ) + r−3 ( d/Γg ) · ( d/Γg ),

= r−2 Γg · ( d/Γb ) + r−1 Γb · d/β + r−2 Γb d/ · Γg = r−2 d/(Γg · Γb ) + l.o.t.,

= r−3 d/2 (Γg · Γb ) + l.o.t.,

= r−3 Γb · d≤2 Γg + r−2 d/Γb · Γg + l.o.t.,

and, since r−1 Γb can be replaced by Γg and d/β can be replaced by r−1 Γg , E

= r−3 d/2 (Γg · Γb ) + l.o.t.

765

APPENDIX TO CHAPTER 2

Taking into account the expression of Err[q] in Proposition 2.99 we write h i re3 (Err[q]) + Err[q] = re3 r4 e3 η · β + r2 d≤1 Γb · Γg ) + r4 e3 η · β +r2 d≤1 Γb · Γg )

= r5 d≤1 e3 η · β + r3 d≤2 Γb · Γg and therefore, back to (A.11.7), Err[e3 (rq)]

= e3 (rErr[q]) + Err[q] + rAq + r5 E

= r5 d≤1 e3 η · β + r3 d≤2 Γb · Γg + rΓb q + r2 d/2 (Γg · Γb ) + l.o.t.

= rΓb q + r5 d≤1 e3 η · β + r3 d≤2 Γb · Γg .

This concludes the proof of Proposition 2.100.

A.12

PROOF OF THE TEUKOLSKY-STAROBINSKY IDENTITY

According to Proposition 2.100 we have ( ) 3 ? ? 3 3 3 5 ? ? ? 2 e3 (rq) = r /d2 /d1 /d1 β − ρ /d2 /d1 κ − κρ /d2 ζ − κρα + (2ρ − κκρ)ϑ 2 2 2 4 +Err[e3 q]. We infer that (

2

3 3 3 e3 /d?2 /d?1 /d1 β − e3 (ρ /d?2 /d?1 κ) − e3 (κρ /d?2 ζ) − e3 (κρα) 2 2 2 )  (A.12.1) 3  + e3 (2ρ2 − κκρ)ϑ + 7re3 (r)e3 (rq) 4
+ r2 e3 Err[e3 q] + rErr[e3 q] + l.o.t.

e3 (r e3 (rq)) = r

7

We first compute e3 /d?2 /d?1 /d1 β = =

/d?2 /d?1 /d1 (e3 β) + [e3 , /d?2 ] /d?1 /d1 β + /d?2 [e3 , /d?1 ] /d1 β + /d?2 /d?1 [e3 , /d1 ]β 1 1 − κ /d?2 + ϑ /d1 + (ζ − η)e3 + ηe3 Φ − ξ(e4 − e4 (Φ)) 2 2 !    1 1 +β /d?1 /d1 β + /d?2 − κ /d?1 + ϑ /d0 + (ζ − η)e3 − ξe4 /d1 β 2 2    1 1 ? ? ? + /d2 /d1 − κ /d1 + ϑ /d2 − (ζ − η)e3 + ηe3 Φ + ξ(e4 + e4 (Φ)) + β β . 2 2 /d?2 /d?1 /d1 (e3 β) +

In view of our general commutation formulas in Lemma 2.54 and our notation

766

APPENDIX A

convention for error terms,2 we have3

= =

/d?2 [e3 ,

[e3 , /d?2 ] /d?1 /d1 β   1 1 − κ /d?2 + ϑ /d1 + (ζ − η)e3 + ηe3 Φ − ξ(e4 − e4 (Φ)) + β /d?1 /d1 β 2 2 1 ? ? − κ /d2 /d1 /d1 β + r−4 Γb · d/≤3 Γb + l.o.t., 2

/d?1 ] /d1 β

  1 ? 1 − κ /d1 + ϑ /d0 + (ζ − η)e3 − ξe4 /d1 β 2 2   1 − κ /d?2 /d?1 /d1 β + r−4 Γb · d/≤3 Γb + Γ≤1 /≤2 Γb + l.o.t., b · d 2 1 1 /d?2 /d?1 − κ /d1 + ϑ /d?2 − (ζ − η)e3 + ηe3 Φ + ξ(e4 + e4 (Φ)) 2 2 ! ! /d?2

= =

/d?2 /d?1 [e3 , /d1 ]β

=



+β β   1 − κ /d?2 /d?1 /d1 β + r−4 Γb · d/≤3 Γb + Γ≤1 /≤2 Γb + l.o.t. b · d 2

=

Hence, schematically, e3 /d?2 /d?1 /d1 β

=

3 /d?2 /d?1 /d1 (e3 β) − κ /d?2 /d?1 /d1 β + r−4 d/≤3 (Γb · Γb ). 2

Using the Bianchi identity e3 β = /d2 α − 2(κ + ω) β + (−2ζ + η)α + 3ξρ, we further deduce /d?2 /d?1 /d1 (e3 β) = /d?2 /d?1 /d1 /d2 α − 2(κ + ω) /d?2 /d?1 /d1 β + 3ρ /d?2 /d?1 /d1 ξ + r−4 d/≤3 (Γb · Γb ), i.e., e3 /d?2 /d?1 /d1 β = /d?2 /d?1 /d1 /d2 α − 2(κ + ω) /d?2 /d?1 /d1 β + 3ρ /d?2 /d?1 /d1 ξ 3 − κ /d?2 /d?1 /d1 β + r−4 d/≤3 (Γb · Γb ). 2

(A.12.2)

We next calculate the second term e3 (ρ /d?2 /d?1 κ) on the right-hand side of (A.12.1) e3 (ρ /d?2 /d?1 κ)

= ρ /d?2 /d?1 e3 κ + ρ[e3 , /d?2 ] /d?1 κ + ρ /d?2 [e3 , /d?1 ]κ + e3 (ρ) /d?2 /d?1 κ.

Using the equation for e3 κ in Proposition 2.56 we derive   1 2 ? ? ? ? ρ /d2 /d1 e3 κ = ρ /d2 /d1 − κ − 2ω κ + 2 /d1 ξ + Γb · Γb , 2 = −ρ (κ + 2ω) /d?2 /d?1 κ − 2ρκ /d?2 /d?1 ω + 2ρ /d?2 /d?1 /d1 ξ + r−5 d/≤2 Γb · Γb . Also, 2 In

particular we write β ∈ r−1 Γb . also commute once more e3 and e4 with / d?1 , /d1 , / d?2 , /d2 and use Bianchi.

3 We

767

APPENDIX TO CHAPTER 2

[e3 , /d?2 ] /d?1 κ

 = =

/d?2 [e3 , /d?1 ]κ = =

1 1 − κ /d?2 + ϑ /d1 + (ζ − η)e3 + ηe3 Φ − ξ(e4 − e4 (Φ)) + β 2 2 1 − κ /d?2 /d?1 κ + r−2 Γb · d/≤2 Γg , 2   1 ? 1 ? ? /d2 − κ /d1 − ϑ /d1 + (ζ − η)e3 − ξe4 κ 2 2 1 − κ /d?2 /d?1 κ + /d?2 (ζ − η)e3 κ − /d?2 ξe4 κ + r−2 d/≤2 (Γb · Γg ). 2



/d?1 κ

Using also the Bianchi equation e3 ρ

3 1 = − κρ + /d1 β − ϑ α − ζ β + 2(η β + ξ β), 2 2

we deduce e3 (ρ /d?2 /d?1 κ)

= −ρ (κ + 2ω) /d?2 /d?1 κ − 2ρκ /d?2 /d?1 ω + 2ρ /d?2 /d?1 /d1 ξ
3 − ρκ /d?2 /d?1 κ + ρ /d?2 (ζ − η)e3 κ − /d?2 (ξ)e4 κ − ρκ /d?2 /d?1 κ 2 + r−5 d/≤2 (Γb · Γg ),

i.e., 7 e3 (ρ /d?2 /d?1 κ) = − ρκ /d?2 /d?1 (κ) − 2ρω /d?2 /d?1 κ − 2ρκ /d?2 /d?1 ω + 2ρ /d?2 /d?1 /d1 ξ 2 (A.12.3)
+ ρ /d?2 (ζ − η)e3 κ − /d?2 (ξ)e4 κ + r−5 d/≤2 (Γb · Γg ). Now, /d?2 (ζ − η)e3 κ − /d?2 (ξ)e4 κ   1 = /d?2 (ζ − η) − κ2 − 2ω κ + 2 /d1 ξ + Γb · Γb 2   1 − /d?2 ξ − κκ + 2ωκ + 2 /d1 η + 2ρ + Γg · Γb 2     1 1 = /d?2 (ζ − η) − κ2 − 2ω κ − /d?2 ξ − κκ + 2ωκ + 2ρ + r−2 d/≤1 Γb · d/≤1 Γb . 2 2 Therefore, back to (A.12.3), 7 e3 (ρ /d?2 /d?1 κ) = − ρκ /d?2 /d?1 (κ) − 2ρω /d?2 /d?1 κ − 2ρκ /d?2 /d?1 ω + 2ρ /d?2 /d?1 /d1 ξ 2     1 1 + ρ /d?2 (ζ − η) − κ2 − 2ω κ − ρ /d?2 ξ − κκ + 2ωκ + 2ρ (A.12.4) 2 2 + r−5 d/≤2 (Γb · Γg ). We next estimate the third term e3 (κρ /d?2 ζ) on the right-hand side of (A.12.1), e3 (κρ /d?2 ζ)

= κρ /d?2 (e3 ζ) + κρ[e3 , /d?2 ]ζ + e3 (κ)ρ /d?2 ζ + κe3 (ρ) /d?2 ζ.

768

APPENDIX A

Using again the equations e3 (κ)

=

e3 ρ =

1 − κ2 − 2ω κ + 2 /d1 ξ + Γb · Γb , 2 3 − κρ + /d1 β + Γg · Γb , 2

which yields e3 (κ)ρ /d?2 ζ κe3 ρ /d?2 ζ



 1 2 = − κ − 2ω κ ρ /d?2 ζ + r−5 d/Γb · d/Γg , 2 3 2 = − κ ρ + r−4 d/Γb · d/Γg , 2

the equation e3 ζ

=

1 1 − κ(ζ + η) + 2ω(ζ − η) + β + 2 /d?1 ω + 2ωξ + κξ + Γb · Γb , 2 2

and the commutator formula   1 ? 1 ? [e3 , /d2 ]ζ = − κ /d2 + ϑ /d1 + (ζ − η)e3 + ηe3 Φ − ξ(e4 − e4 (Φ)) + β ζ 2 2 1 ? = − κ /d2 ζ + r−1 Γb · d/≤1 Γg , 2 we deduce

= − − − = +

e3 (κρ /d?2 ζ)   1 1 κρ /d?2 − κ(ζ + η) + 2ω(ζ − η) + β + 2 /d?1 ω + 2ωξ + κξ + Γb · Γb 2 2 1 2 ? κ ρ /d2 ζ + r−5 Γb · d/≤1 Γg 2 1 2 ? κ ρ /d2 ζ − 2ω κρ /d?2 ζ + r−5 d/≤1 Γb · d/≤1 Γg 2 3 2 ? κ ρ /d2 ζ + r−4 d/Γb · d/Γg 2   1 1 κρ −3κ /d?2 ζ − κ /d?2 η − 2ω /d?2 η + /d?2 β + 2 /d?2 /d?1 ω + 2ω /d?2 ξ + κ /d?2 ξ 2 2 r−4 d/≤1 Γb · d/≤1 Γg + r−5 d/≤1 Γb · d/≤1 Γb ,

i.e., e3 (κρ /d?2 ζ)

= κρ

1 − 3κ /d?2 ζ − κ /d?2 η − 2ω /d?2 η + /d?2 β + 2 /d?2 /d?1 ω + 2ω /d?2 ξ 2 !

1 + κ /d?2 ξ 2

+ r−4 d/≤1 Γb · d/≤1 Γg + r−5 d/≤1 Γb · d/≤1 Γb . (A.12.5)

769

APPENDIX TO CHAPTER 2

For the fourth term on the right-hand side of (A.12.1) we have e3 (κρα)

= κρe3 (α) + e3 (κ)ρα + κe3 (ρ)α   1 1 = κρe3 (α) + − κ κ + 2ωκ + 2 /d1 η + 2ρ − ϑ ϑ + 2(ξ ξ + η η) ρα 2 2   3 1 +κ − κρ + /d1 β − ϑ α − ζ β + 2(η β + ξ β) α 2 2 = κρe3 (α) + (−2κκ + 2ωκ + 2ρ) ρα + r−3 d/Γb + r−2 Γb · Γb ) · α,

i.e., e3 (κρα)

= κρe3 (α) + (−2κκ + 2ωκ + 2ρ) ρα + r−3 d/Γb + r−2 Γb · Γb ) · α.

(A.12.6)

Finally, for the fifth term on the right-hand side of (A.12.1), using the e3 equations for ϑ, ρ, κ, κ,   e3 (2ρ2 − κκρ)ϑ = (2ρ2 − κκρ)e3 ϑ + 4ρe3 (ρ)ϑ − e3 (κ)κρϑ − κe3 (κ)ρϑ −κκe3 (ρ)ϑ

=

i.e.,   e3 (2ρ2 − κκρ)ϑ

  (2ρ2 − κκρ) − κ ϑ − 2ω ϑ − 2α − 2 /d?2 ξ + Γb · Γb   3 +4ρ − κρ + /d1 β + Γg · Γb ϑ 2   1 − − κ κ + 2ωκ + 2ρ + 2 /d1 η + Γb · Γb κρϑ 2   1 −κ − κ2 − 2ω κ + 2 /d1 ξ + Γb · Γb ρϑ 2   3 −κκ − κρ + /d1 β + Γg · Γb ϑ, 2

  (2ρ2 − κκρ) − κ ϑ − 2ω ϑ − 2α − 2 /d?2 ξ − 6κρ2 ϑ     1 1 2 3 − − κ κ + 2ωκ + 2ρ κρϑ + κ κ + 2ω κ ρϑ + κ2 κρ 2 2 2 =

+ r−5 d/≤1 Γb · Γb , from which   e3 (2ρ2 − κκρ)ϑ    7 2 ? 2 2 2 = (2ρ − κκρ) − 2α − 2 /d2 ξ + κκ ρ − 10κρ + 2κκρω − 4ωρ ϑ 2 +r−5 d/≤1 Γb · Γb .

(A.12.7)

770

APPENDIX A

Recalling (A.12.1) 3 3 3 = e3 /d?2 /d?1 /d1 β − e3 (ρ /d?2 /d?1 κ) − e3 (κρ /d?2 ζ) − e3 (κρα) 2 2 2  3  2 −6 + e3 (2ρ − κκρ)ϑ + 7r e3 (r)e3 (rq) 4
+ r2 e3 Err[e3 q] + rErr[e3 q] + l.o.t.

r−7 e3 (r2 e3 (rq))

and making use of (A.12.4)–(A.12.7) we deduce r−7 e3 (r2 e3 (rq)) = −

− − + +

3 /d?2 /d?1 /d1 /d2 α − 2(κ + ω) /d?2 /d?1 /d1 β + 3ρ /d?2 /d?1 /d1 ξ − κ /d?2 /d?1 /d1 β 2 ( 3 7 − ρκ /d?2 /d?1 (κ) − 2ρω /d?2 /d?1 κ − 2ρκ /d?2 /d?1 ω + 2ρ /d?2 /d?1 /d1 ξ 2 2    ) 1 2 1 ? ? +ρ /d2 (ζ − η) − κ − 2ω κ − ρ /d2 ξ − κκ + 2ωκ + 2ρ 2 2 (  ) 3 1 ? 1 ? ? ? ? ? ? ? κρ −3κ /d2 ζ − κ /d2 η − 2ω /d2 η + /d2 β + 2 /d2 /d1 ω + 2ω /d2 ξ + κ /d2 ξ 2 2 2 ( ) 3 −3 −2 κρe3 (α) + (−2κκ + 2ωκ + 2ρ) ρα + r d/Γb + r Γb · Γb ) · α 2 (  )   7 3 (2ρ2 − κκρ) − 2α − 2 /d?2 ξ + κκ2 ρ − 10κρ2 + 2κκρω − 4ωρ2 ϑ 4 2  
7r−6 e3 (r)e3 (rq) + Err + r−7 r2 e3 Err[e3 q] + rErr[e3 q] + l.o.t.,

where the error term Err is given by Err

=

r−3 d/Γb + r−2 Γb · Γb ) · α + r−4 d/≤1 Γb · d/≤1 Γg

+r−5 d/≤1 Γb · d/≤1 Γb .

(A.12.8)

Denoting the expression of the left-hand side of the identity (2.3.15) by I, i.e., I := e3 (r2 e3 (rq)) + 2ωr2 e3 (rq), we deduce r−7 I

= + − + +

  7 /d?2 /d?1 /d1 /d2 α − κ + 2ω /d?2 /d?1 /d1 β 2     3 7 3 7 3 κ + 2ω ρ /d?2 /d?1 κ + κ + 2ω κρ /d?2 ζ − κρ /d?2 β 2 2 2 2 2  3 3 κρe3 (α) − − 3κκ + 2ωκ + 4ρ ρα 2 2  3 7 2 κκ ρ − 10κρ2 + 2κκρω − 4ωρ2 ϑ 4 2   g r−7 7re3 (r)e3 (rq) + 2ωr2 e3 (rq) + Err,

771

APPENDIX TO CHAPTER 2

g is given by where the new error term Err  
g = Err + r−7 r2 e3 Err[e3 q] + rErr[e3 q] + 2ωr−5 Err[e3 q]. Err To calculate the term J := 7re3 (r)e3 (rq) + 2ωr2 e3 (rq) in the last row we make use once more of the identity of Lemma 2.100 to derive J    7 r  2 r κ + 2ω e3 (rq) + 7r e3 (r) − κ e3 (rq) 2 2   7 r2 κ + 2ω e3 (rq) + r2 Γb e3 (rq) 2    7 3 ? ? 3 3 3 7 ? ? ? 2 r κ + 2ω /d2 /d1 /d1 β − ρ /d2 /d1 κ − κρ /d2 ζ − κρα + (2ρ − κκρ)ϑ 2 2 2 2 4   7 r2 κ + 2ω Err[e3 (rq)] + r2 Γb e3 (rq), 2

= = = + i.e.,

= +

r−7 J   7 3 ? ? 3 3 3 ? ? ? 2 κ + 2ω /d2 /d1 /d1 β − ρ /d2 /d1 κ − κρ /d2 ζ − κρα + (2ρ − κκρ)ϑ 2 2 2 2 4   7 r−5 κ + 2ω Err[e3 (rq)] + r−5 Γb e3 (rq). 2

Combining and simplifying, r−7 I

=

3 3 3 /d?2 /d?1 /d1 /d2 α − κρ /d?2 β − κρe3 (α) − 2 2 2 9 2 g g − κρ ϑ + Err, 4



 1 κκ + 4ωκ + 4ρ ρα 2

where g g Err

g + r−5 = Err



 7 κ + 2ω Err[e3 (rq)] + r−5 Γb e3 (rq). 2

Using Bianchi to replace /d?2 β, we deduce r

−7

I

= − =

  3 1 3 −1 − κρ −e4 α − κα + 4ωα − ρϑ + r Γg · Γb 2 2 2   3 3 1 9 2 κρe3 (α) − κκ + 4ωκ + 4ρ ρα − κρ ϑ + r−5 Γg · Γb + l.o.t. 2 2 2 4   3 3 ? ? /d2 /d1 /d1 /d2 α + κρe4 α − κρe3 (α) − 6 κω + ωκ + ρ ρα + r−5 Γg · Γb . 2 2 /d?2 /d?1 /d1 /d2 α

772

APPENDIX A

Hence, in view of the definition of I, we infer I

= e3 (r2 e3 (rq)) + 2ωr2 e3 (rq) ( o 3 3 7 = r /d?2 /d?1 /d1 /d2 α + κρe4 α − κρe3 (α) + Err[ST ] 2 2

where  7 Err[ST ] = r Err + r κ + 2ω Err[e3 (rq)] + r2 Γb e3 (rq) + r2 Γg · Γb 2  
+ r7 Err + r2 e3 Err[e3 q] + rErr[e3 q] + 2ωr2 Err[e3 q] + r2 Γb e3 (rq) 7g

2



+ r2 Γg · Γb . Recall that, see (A.12.8), Err

r−3 d/Γb + r−2 Γb · Γb ) · α + r−4 d/≤1 Γb · d/≤1 Γg + r−5 d/≤1 Γb · d/≤1 Γb .

=

Hence, Err[ST ]

= r4 d/Γb + rΓb · Γb ) · α + r3 d/≤1 Γb · d/≤1 Γg + r2 d/≤1 Γb · d/≤1 Γb
+ r2 Γb e3 (rq) + r2 e3 Err[e3 q] + rErr[e3 q] + 2ωr2 Err[e3 q].

Recall that, see Proposition 2.100, Err[e3 (rq)]



= rΓb q + r5 d≤1 e3 η · β + r3 d≤2 Γb · Γg .

Therefore, E


= r2 e3 Err[e3 q] + rErr[e3 q] + 2ωr2 Err[e3 q] 

= r2 Γb e3 (rq) + e3 (Γb )rq + r7 d≤2 e3 η · β + r5 d≤3 Γb · Γg

+ r2 Γb q + r6 d≤1 e3 η · β + r4 d≤2 Γb · Γg 

= r2 Γb e3 (rq) + (d≤1 Γb )rq + r7 d≤2 e3 η · β + r5 d≤3 Γb · Γg .

Thus, Err[T S]

= r4 d/Γb + rΓb · Γb ) · α + r2 Γb e3 (rq) + (d≤1 Γb )rq

+ r7 d≤2 e3 η · β + r5 d≤3 Γb · Γg

which ends the proof of Proposition 2.101.



773

APPENDIX TO CHAPTER 2

A.13

PROOF OF PROPOSITION 2.107

In this section we give a proof of Proposition 2.107, i.e., we derive the wave equation for the extreme curvature component α, g α = −4ωe4 (α) + (2κ + 4ω) e3 (α) + V α + Err(g α),   (A.13.1) 1 V : = −4e4 (ω) + κκ − 10κω + 2κω − 8ωω − 4ρ + 4eθ (Φ)2 α, 2 where Err(g α)

=

1 3 1 ϑe3 (α) + ϑ2 ρ + eθ (Φ)ϑβ − κ(ζ + 4η)β − (ζ + η)e4 (β) 2 4 2 −ξe3 (β) + eθ (Φ)(2ζ + η)α + β 2 + e4 (Φ)ηβ + e3 (Φ)ξβ −(ζ + 4η)e4 (β) − (e4 (ζ) + 4e4 (η))β − 2(κ + ω)(ζ + 4η)β

+2eθ (κ + ω)β − eθ ((2ζ + η)α) − 3ξeθ (ρ) + 2ηeθ (α) 3 + ϑ(eθ (β) + eθ (Φ)β) + 3ρ(η + η + 2ζ)ξ + (eθ (η) + eθ (Φ)η)α 2 1 1 3 + κϑα − 2ωϑα − ϑϑα + ξξα + η 2 α + ϑζβ + 3ϑ(ηβ + ξβ) 4 2 2 1 − ϑ(ζ + 4η)β. 2 The equation for α can then be easily inferred by symmetry. Proof. We make use of the Bianchi identities eθ (β) − eθ (Φ)β e4 (β) + 2(κ + ω)β

 3 − 4ω α + ϑ ρ − (ζ + 4η)β, 2 2 = eθ (α) + 2eθ (Φ)α + (2ζ + η)α + 3ξρ, = e3 (α) +

κ

to infer that e4 (e3 (α)) κ  = e4 (eθ (β)) − eθ (Φ)e4 (β) − e4 (eθ (Φ))β − − 4ω e4 (α) 2   e4 (κ) 3 3 − − 4e4 (ω) α − ϑe4 (ρ) − e4 (ϑ)ρ + (ζ + 4η)e4 (β) 2 2 2 +(e4 (ζ) + 4e4 (η))β   = e4 (eθ (β)) − eθ (Φ) eθ (α) + 2eθ (Φ)α − 2(κ + ω)β + (2ζ + η)α + 3ξρ   κ  e4 (κ) −(D4 Dθ Φ + DD4 eθ Φ)β − − 4ω e4 (α) − − 4e4 (ω) α 2 2 3 3 − ϑe4 (ρ) − e4 (ϑ)ρ + (ζ + 4η)e4 (β) + (e4 (ζ) + 4e4 (η))β. 2 2

774

APPENDIX A

Hence, e4 (e3 (α))

= e4 (eθ (β)) − eθ (Φ)(eθ (α) + 2eθ (Φ)α) + 2eθ (Φ)(κ + ω)β − 3eθ (Φ)ξρ    κ e4 (κ) − 4ω e4 (α) − − 4e4 (ω) α +e4 (Φ)eθ (Φ)β − 2 2 3 3 − ϑe4 (ρ) − e4 (ϑ)ρ − eθ (Φ)(2ζ + η)α − β 2 − e4 (Φ)ηβ − e3 (Φ)ξβ 2 2 +(ζ + 4η)e4 (β) + (e4 (ζ) + 4e4 (η))β,

and eθ (eθ (α)) = eθ (e4 (β)) + 2(κ + ω)eθ (β) + 2eθ (κ + ω)β − 2eθ (Φ)eθ (α) − 2eθ (eθ (Φ))α

−eθ ((2ζ + η)α) − 3eθ (ξρ)  κ   3 = eθ (e4 (β)) + 2(κ + ω) eθ (Φ)β + e3 (α) + − 4ω α + ϑ ρ − (ζ + 4η)β 2 2 +2eθ (κ + ω)β − 2eθ (Φ)eθ (α) − 2(Dθ Dθ Φ + DDθ eθ Φ)α − eθ ((2ζ + η)α) −3eθ (ξρ)

κ  = eθ (e4 (β)) + 2(κ + ω)eθ (Φ)β + 2(κ + ω)e3 (α) + 2(κ + ω) − 4ω α 2   1 1 2 +3(κ + ω)ϑ ρ − 2eθ (Φ)eθ (α) − 2 ρ − eθ (Φ) + χe4 (Φ) + χe3 (Φ) α 2 2 −3eθ (ξ)ρ − 2(κ + ω)(ζ + 4η)β + 2eθ (κ + ω)β − eθ ((2ζ + η)α) − 3ξeθ (ρ). In view of Lemma 2.102, we have g f

  1 1 = −e4 (e3 (f )) + eθ (eθ (f )) − κe4 (f ) + − κ + 2ω e3 (f ) + eθ (Φ)eθ (f ) 2 2 +2ηeθ (f ).

We infer g α

  1 1 = −e4 (e3 (α)) + eθ (eθ (α)) − κe4 (α) + − κ + 2ω e3 (α) + eθ (Φ)eθ (α) 2 2 +2ηeθ (α) 3 3 = [eθ , e4 ](β) − e4 (Φ)eθ (Φ)β + ϑe4 (ρ) + e4 (ϑ)ρ + 3(κ + ω)ϑ ρ 2  2  3 −3(eθ (ξ) − eθ (Φ)ξ)ρ − 4ωe4 (α) + κ + 4ω e3 (α) 2 +

e4 (κ) − 4e4 (ω) + κκ − 8κω + κω − 8ωω − 2ρ + 4eθ (Φ)2 − χe4 (Φ) 2 !

−χe3 (Φ) α + eθ (Φ)(2ζ + η)α + β 2 + e4 (Φ)ηβ + e3 (Φ)ξβ −(ζ + 4η)e4 (β) − (e4 (ζ) + 4e4 (η))β − 2(κ + ω)(ζ + 4η)β +2eθ (κ + ω)β − eθ ((2ζ + η)α) − 3ξeθ (ρ) + 2ηeθ (α).

775

APPENDIX TO CHAPTER 2

Next, we have [eθ , e4 ](β)

= = −

χeθ (β) − (ζ + η)e4 (β) − ξe3 (β)   κ  3 χ eθ (Φ)β + e3 (α) + − 4ω α + ϑ ρ − (ζ + 4η)β 2 2 (ζ + η)e4 (β) − ξe3 (β)

and hence  = −4ωe4 (α) +

g α

+ +

 3 κ + χ + 4ω e3 (α) + V1 α 2

3 3 3 ϑe4 (ρ) + e4 (ϑ)ρ + 3(κ + ω)ϑ ρ − 3(eθ (ξ) − eθ (Φ)ξ)ρ + χϑ ρ 2 2 2 Err1

where V1

:= −

e4 (κ) − 4e4 (ω) + κκ − 8κω + κω − 8ωω − 2ρ + 4eθ (Φ)2 − χe4 (Φ) 2 κ χe3 (Φ) + χ − 4χω, 2

:= eθ (Φ)ϑβ − χ(ζ + 4η)β − (ζ + η)e4 (β) − ξe3 (β) + eθ (Φ)(2ζ + η)α

Err1

+ −

β 2 + e4 (Φ)ηβ + e3 (Φ)ξβ − (ζ + 4η)e4 (β) − (e4 (ζ) + 4e4 (η))β

2(κ + ω)(ζ + 4η)β + 2eθ (κ + ω)β − eθ ((2ζ + η)α) − 3ξeθ (ρ) + 2ηeθ (α).

Next, we make use of e4 (ϑ) + κϑ + 2ωϑ = −2α + 2(eθ (ξ) − eθ (Φ)ξ) + 2(η + η + 2ζ)ξ, 3 1 e4 (ρ) + κρ = eθ (β) + eθ (Φ)β − ϑα + ζβ + 2(ηβ + ξβ), 2 2 to calculate the term I

:= = + =

3 3 3 ϑe4 (ρ) + e4 (ϑ)ρ + 3(κ + ω)ϑ ρ − 3(eθ (ξ) − eθ (Φ)ξ)ρ + χϑ ρ 2  2 2  3 3 3 ϑ − κρ + /d1 β + ρ (−κϑ − 2ωϑ − 2α + 2(eθ (ξ) − eθ (Φ)ξ)) 2 2 2 3κ+ϑ 3(κ + ω)ϑ ρ − 3(eθ (ξ) − eθ (Φ)ξ)ρ + ϑ ρ + l.o.t. 2 2 3 3 −3ρα + ϑ /d1 β + ϑ2 ρ. 2 4

Hence, 

g α

 3 = −4ωe4 (α) + κ + χ + 4ω e3 (α) + (V1 − 3ρ) α 2 3 3 + Err1 + ϑ /d1 β + ϑ2 ρ. 2 4

776

APPENDIX A

Using, also, 1 e4 (κ) + κκ − 2ωκ = 2

1 2(eθ (η) + eθ (Φ)η) + 2ρ − ϑϑ + 2(ξξ + η 2 ) 2

and the identities, 2χ = κ + ϑ, as well as 2χ = κ + ϑ, we finally obtain g α

=

−4ωe4 (α) + (2κ + 4ω) e3 (α) + V α + Err[α]

as desired. We write schematically the error term Err[α] 1 3 1 = ϑe3 (α) + ϑ2 ρ + eθ (Φ)ϑβ − κ(ζ + 4η)β − (ζ + η)e4 (β) − ξe3 (β) 2 4 2 + eθ (Φ)(2ζ + η)α + β 2 + e4 (Φ)ηβ + e3 (Φ)ξβ − (ζ + 4η)e4 (β)

− (e4 (ζ) + 4e4 (η))β − 2(κ + ω)(ζ + 4η)β + 2eθ (κ + ω)β − eθ ((2ζ + η)α) 3 − 3ξeθ (ρ) + 2ηeθ (α) + ϑ(eθ (β) + eθ (Φ)β) + 3ρ(η + η + 2ζ)ξ 2 1 1 3 + (eθ (η) + eθ (Φ)η)α + κϑα − 2ωϑα − ϑϑα + ξξα + η 2 α + ϑζβ 4 2 2 1 + 3ϑ(ηβ + ξβ) − ϑ(ζ + 4η)β 2 as follows: 

 1 1 1 Err[g α] = Γg + d/Γg α + Γg e3 (α) + Γg d/α r r r   1 1 1 + β + Γg + dΓg β + Γg d(β) + Γg e3 (β) r r r 1 2 + (Γg ) ρ + Γg d/(ρ) r ≤1 = Γg e3 (α, β) + r−1 Γ≤1 (α, β, ρˇ) + β 2 + Γ2g ρ. g ·d This concludes the proof of Proposition 2.107.

A.14

PROOF OF THEOREM 2.108

Recall the symbolic notation used in the statement of the theorem. n o n o Γg = ϑ, η, η, ζ, A , Γb = ϑ, ξ, A , n o n o dΓg = dϑ, reθ (κ), dη, dη, dζ, dA , dΓb = dϑ, eθ (κ), dξ, dA , where A = 2r e4 (r) − κ,

A = 2r e3 (r) − κ. We also denote, for s ≥ 2,

ds Γg

= ds−1 dΓg ,

ds Γb = ds−1 dΓb ,

777

APPENDIX TO CHAPTER 2

for higher derivatives with respect to d = (e3 , re4 , d/) (see definition 2.39 for the notation d/ and d/s ). We also recall Remark 2.95. Remark A.1. According to the main bootstrap assumptions BA-E, BA-D (see section 3.4.1.) the terms Γb behave worse in powers of r than the terms in Γg . Thus, in the symbolic expressions below, we replace the terms of the form Γg + Γb by Γb . We also replace r−1 Γb by Γg . We will denote l.o.t. all cubic and higher error ˇ R. ˇ We also include in l.o.t. terms which decay faster in powers of r terms in Γ, than those taken into account by the main quadratic terms. Recall that q = r4 Q(α),

(A.14.1)

where Q is the operator 1 W := −4e3 (ω) + 8ω 2 − 8ω κ + κ2 . (A.14.2) 2

Q := e3 e3 + (2κ − 6ω)e3 + W,

Lemma A.2. The quantity q is fully invariant with respect to the conformal frame transformations e03 = λ−1 e3 ,

e4 = λe4 ,

e0θ = eθ .

Proof. The proof is an immediate consequence of Definition A.3 and Lemmas A.5, A.4 below. We recall that under the above mentioned frame transformation we have α0 ω0

= λ2 α, β 0 = λβ, ρ0 = ρ, κ0 = λ−1 κ, κ0 = λκ, η 0 = η, η 0 = η,     1 1 −1 0 = λ ω + e3 (log λ) , ω = λ ω − e4 (log λ) , ζ 0 = ζ − eθ (log λ). 2 2

Definition A.3. We say that a reduced tensor is conformal invariant of type4 a, i.e., a-conformal invariant, if under the conformal change of frames e03 = λ−1 e3 , e04 = λe4 , it transforms by f 0 = λa f. Lemma A.4. Let f be an a-conformal invariant tensor. 1. The tensor ∇3 f

:= e3 f − 2aωf

(A.14.3)

:= e4 f + 2aωf

(A.14.4)

is a − 1 conformal invariant. 2. The tensor ∇4 f 4 Note

that for a given Ricci or curvature coefficient a coincides with the signature of the component.

778

APPENDIX A

is a + 1 conformal invariant. 3. The tensor (c)

∇ /A f

=

∇ / A f + aζA f

(A.14.5)

is a-conformal invariant. Proof. Immediate verification. Lemma A.5. We have Q(α)

=

1 ∇3 (∇3 α) + 2κ∇3 α + κ2 α. 2

Proof. We have ∇3 (∇3 α)

= =

∇3 (e3 α − 4ωα) = e3 (e3 α − 4ωα) − 2ω(e3 α − 4ωα) e3 e3 α − 4e3 ωα − 4ωe3 α − 2ωe3 α + 8ω 2 α.

Hence, Q(α)

1 = ∇3 (∇3 α) + 2κ∇3 α + κα 2 1 = e3 e3 α − 4e3 ωα − 4ωe3 α − 2ωe3 α + 8ω 2 α + 2κ(e3 α − 4ωα) + κ2 α 2   1 = e3 e3 α + (2κ − 6ω)e3 α + −4e3 ω + 8ω 2 − 8κ ω + κ2 2

as stated. Remark A.6. Using the definitions of ∇3 , ∇4 the null structure equations for κ, κ take the form 1 ∇3 κ + κ2 = 2 /d1 ξ + Γb · Γb = r−1 d/Γb + l.o.t., 2 1 ∇4 κ + κ κ = 2 /d1 η + 2ρ + Γg · Γb = 2ρ + r−1 d/Γg , 2 1 ∇3 κ + κ κ = 2 /d1 η + 2ρ + Γg · Γb = 2ρ + r−1 d/Γg , 2 1 ∇4 κ + κ2 = 2 /d1 ξ + Γg · Γg = r−1 d/Γg . 2

(A.14.6)

Also, since ρ is 0-conformal 3 ∇3 ρ + κρ = /d1 β + Γg · Γb = r−1 d/Γg . 2

(A.14.7)

Definition A.7. Given f an a-conformal S-tangent tensor we define its a-conformal Laplacian to be (c)

4 /f =

(c)

∇ / A (c) ∇ / A f.

779

APPENDIX TO CHAPTER 2

Lemma A.8. The following formula holds true for a 2-conformal tensor f
(c) 4 /f = 4 / 2 f + 4ζ∇ / f + 2 div ζ + 2|ζ|2 f. In particular, we have (c)

4 /f

= 4 / 2 f + r−1 d/≤1 (Γg · f ).

Proof. Immediate verification. The goal of this section is to prove Theorem 2.108 which we recall below for the convenience of the reader. Theorem A.9. The invariant scalar quantity q defined in (2.3.10) verifies the equation 2 q + κκ q = Err[2 q]

(A.14.8)

where, schematically, Err[2 q]

  := r2 d≤2 (Γg · (α, β)) + e3 r3 d≤2 (Γg · (α, β)) + d≤1 (Γg · q) + l.o.t.

Definition A.10. Given a quadratic or higher order E we say the following: 1. E ∈ Good if r4 E can be expressed in the form (2.4.8). 2. E ∈ Good1 if after applying r4 e3 or r3 it can be expressed in the form (2.4.8). 3. E ∈ Good2 if after applying r4 e3 e3 , r4 e3 or r3 it can be expressed in the form (2.4.8). In view of the definition we note that (e3 + r−1 )Good1 = Good,

QGood2 = Good.

To prove the theorem we have to check that Err[2 q] = r4 Good. A.14.1

The Teukolsky equation for α

We recall below Proposition 2.107. Lemma A.11. We have 2 α V

= −4ωe4 (α) + (4ω + 2κ)e3 (α) + V α + Err[g α], 1 = −4ρ − 4e4 (ω) − 8ωω + 2ω κ − 10κ ω + κ κ, 2

where Err[g α] is given schematically by Err(g α)

:=

  Γg e3 (α) + r−1 d≤1 (η, Γg )(α, β) + ξ(e3 (β), r−1 dˇ ρ).

Remark A.12. Since ξ vanishes for r ≥ 4m0 , η ∈ Γg and e3 α = r−1 dα, we deduce Err(g α) ∈ Good2 .

780

APPENDIX A

Lemma A.13. The Teukolsky equation for α can be written in the form L(α)

= Good2

(A.14.9)

where L is the operator L

:= −∇4 ∇3 +

  5 1 1 4 / 2 − κ∇3 − κ∇4 − −4ρ + κκ . (A.14.10) 2 2 2

(c)

We also note that, for a 0-conformal tensor f , 2 f

=

−∇4 ∇3 f +

1 1 4 / 2 f − κ∇3 f − κ∇4 f + r−1 Γg · d/f. (A.14.11) 2 2

(c)

Proof. Recall that we have (see Definition 2.103)   1 1 2 α = −e4 (e3 (α)) + 4 / 2 α − κe4 (α) + − κ + 2ω e3 (α) + 2ηeθ (α). 2 2 Therefore, L(α)

  1 1 = −e4 (e3 (α)) + 4 / 2 α − κe4 (α) + − κ + 2ω e3 (α) + 2ηeθ (α) 2 2 + 4ωe4 (α) − (4ω + 2κ)e3 (α) − V α     1 5 = −e4 (e3 (α)) + 4 / 2α − κ − 4ω e4 α − κ + 2ω e3 α + 2ηeθ (α) − V α 2 2     1 5 (c) = −e4 (e3 (α)) + 4 / 2α − κ − 4ω e4 α − κ + 2ω e3 α − V α + Good2 . 2 2 On the other hand, ∇4 (∇3 (α))




= ∇4 e3 α − 4ωα = e4 e3 α − 4ωα + 2ω e3 α − 4ωα = e4 e3 α − 4ωe4 α − 4e4 ωα + 2ωe3 α − 8ωωα.

Hence, 5 1 −∇4 ∇3 α − κ∇3 α − κ∇4 α 2 2

= −e4 e3 α + 4ωe4 α + 4e4 ωα − 2ωe3 α + 8ωωα 5 1 κ e3 α − 4ωα) − κ(e4 α + 4ωα) 2 2   1 5 = −e4 e3 α − (κ − 4ω)e4 α − κ + 2ω e3 α 2 2
+ 4e4 ω + 8ωω + 10κω − 2ωκ α. −

781

APPENDIX TO CHAPTER 2

We deduce, with V 0 = −4ρ + 12 κκ,

= + =

5 1 −∇4 ∇3 α − κ∇3 α − κ∇4 − V 0 α 2 2   1 5 −e4 e3 α − (κ − 4ω)e4 α − κ + 2ω e3 α 2 2   1 4e4 ω + 8ωω + 10κω − 2ωκ + 4ρ − κκ α 2   1 5 −e4 e3 α − (κ − 4ω)e4 α − κ + 2ω e3 α − V α. 2 2

Hence, Lα

−∇4 ∇3 α +

=

  5 1 1 4 / 2 α − κ∇3 α − κ∇4 α − −4ρ + κκ α ∈ Good2 2 2 2

(c)

as desired. The proof of the second part of the lemma follows in the same manner.

A.14.2

Commutation lemmas

The goal of the following lemmas is to calculate the commutator of Q with L. Lemma A.14. Given f an a-conformal tensor we have [∇3 , ∇4 ]f

=

2aρf + r−1 Γg d/≤1 f.

(A.14.12)

Proof. We have [∇3 , ∇4 ]f

= ∇ 3 ∇4 f − ∇ 4 ∇3 f



= e3 − 2(a + 1)ω e4 f + 2aωf − e4 + 2(a − 1)ω (e3 f − 2aωf = e3 e4 f − 2(a + 1)ωe4 f + 2ae3 (ωf ) − 4a(a + 1)ωωf

− e4 e3 f − 2(a − 1)ωe3 f + 2ae4 (ωf ) + 4a(a − 1)ωω   = [e3 , e4 ]f − 2ωe4 f + 2ωe3 (f ) + 2a e3 ω + e4 ω − 4ωω f. Recall that [e3 , e4 ] e3 ω + e4 ω − 4ωω

= −2ωe3 + 2ωe4 + 2(η − η)eθ ,

= ρ + Γ g · Γb .

We deduce5 [∇3 , ∇4 ]f

=

2aρ + r−1 Γg d/≤1 f

as stated. Lemma A.15. Assume f a-conformal and g is b-conformal. Then f g is a + b5 Recall

that η ∈ Γg in the frame we are using.

782

APPENDIX A

conformal and ∇3 (f g) ∇4 (f g)

= f ∇3 g + g∇3 f, = f ∇4 g + g∇4 f.

Proof. Indeed ∇3 (f g) = e3 (f g) − 2(a + b)ωf g = f e3 g + ge3 f − 2(a + b)ωf g = f ∇3 g + g∇3 f as stated. Lemma A.16. We have 1 [Q, ∇3 ]α = κ2 ∇3 α + κ3 α + Good1 , 2
1 [Q, ∇4 ]α = 2ρ + κκ ∇3 α + κκ2 α + Good1 . 2

(A.14.13)

Also,
1 [Q, ∇4 ∇3 ]α = − 2ρ + κκ ∇3 ∇3 α + κ2 ∇4 ∇3 α + κ3 ∇4 α 2     (A.14.14) 1 2 3 2 1 + 3ρκ − κκ ∇3 α + κ − κκ + 2ρ α + Good. 2 2 2 Proof. We have6      1 1 [Q, ∇3 ]α = ∇3 ∇3 + 2κ∇3 + κ2 ∇3 α − ∇3 ∇3 ∇3 + 2κ∇3 + κ2 α 2 2 = −2∇3 (κ)∇3 α − κ(∇3 κ)α     1 2 1 2 −1 −1 = −2 − κ + r Γb ∇3 α − κ − κ + r Γb α 2 2 1 = κ2 ∇3 α + κ3 α + r−1 Γg d≤1 α 2 and [Q, ∇4 ]α      1 1 = ∇3 ∇3 + 2κ∇3 + κ2 ∇4 α − ∇4 ∇3 ∇3 + 2κ∇3 + κ2 α 2 2

= ∇3 ∇3 ∇4 − ∇4 ∇3 ∇3 α + 2κ ∇3 ∇4 − ∇4 ∇3 α − 2∇4 κ∇3 α − κ(∇4 κ)α   = ∇3 ∇3 , ∇4 ]α + [∇3 , ∇4 ]∇3 α + 2κ[∇3 , ∇4 ]α − 2∇4 κ∇3 α − κ(∇4 κ)α. In view of Lemma A.14 we have  ∇3 , ∇4 ]α =  ∇3 , ∇4 ]∇3 α = 6 Recall

that r−1 Γb = Γg .

4ρα + r−1 Γg · d/≤1 α,

2ρ∇3 α + r−1 Γg · d/≤1 ∇3 α.

783

APPENDIX TO CHAPTER 2

Hence, [Q, ∇4 ]α

  = ∇3 4ρα + r−1 Γg d/≤1 α     + 2ρ + r−1 Γg d/≤1 ∇3 α + 2κ 4ρα + r−1 Γg d/≤1 α − 2∇4 κ∇3 α − κ(∇4 κ)α =



6ρ − 2∇4 κ ∇3 α + 4∇3 ρ + 8κρ − κ∇4 κ α + Good1 .

We now note, using the equations for ∇4 ρ and ∇4 κ,   3 −1 4∇3 ρ + 8κρ − κ∇4 κ = 4 − κρ + r d/Γg + 8κρ 2   1 −1 −κ − κκ + 2ρ + r d/Γg 2 1 2 = κκ + r−1 d/Γg , 2   1 6ρ − 2∇4 κ = 6ρ − 2 − κκ + 2ρ + r−1 d/Γg 2 =

2ρ + κκ + r−1 d/Γg .

Hence, [Q, ∇4 ]α

=


1 2ρ + κκ ∇3 α + κκ2 α + Good1 2

as stated. Also, [Q, ∇4 ∇3 ]α

=

  [Q, ∇4 ]∇3 α + ∇4 [Q, ∇3 ]α .

(A.14.15)

We first calculate, as above, for f = ∇3 α [Q, ∇4 ]f      1 1 = ∇3 ∇3 + 2κ∇3 + κ2 ∇4 f − ∇4 ∇3 ∇3 + 2κ∇3 + κ2 f 2 2

= ∇3 ∇3 ∇4 − ∇4 ∇3 ∇3 f + 2κ ∇3 ∇4 − ∇4 ∇3 f − 2∇4 κ∇3 α − κ(∇4 κ)f   = ∇3 ∇3 , ∇4 ]f + [∇3 , ∇4 ]∇3 f + 2κ[∇3 , ∇4 ]f − 2∇4 κ∇3 f − κ(∇4 κ)f. In view of Lemma A.14, since f = ∇3 α is 1-conformal and ∇3 f is 0-conformal, we have  ∇3 , ∇4 ]f = 2ρf + r−1 Γg d/≤1 f,  ∇3 , ∇4 ]∇3 f = r−1 Γg d/≤1 ∇3 f.

784

APPENDIX A

Hence [Q, ∇4 ]f

    = ∇3 2ρf + r−1 Γg d/≤1 f + r−1 Γg d/≤1 ∇3 f   + 2κ 2ρf + r−1 Γg d/≤1 f − 2∇4 κ∇3 f − κ(∇4 κ)f

= 2ρ − 2∇4 κ ∇3 f + 2∇3 ρ + 4κρ − κ∇4 κ f + r−1 Γg d/≤1 ∇3 f + r−2 Γg d/≤1 f.

Therefore,

[Q, ∇4 ]∇3 α = 2ρ − 2∇4 κ ∇3 ∇3 α + 2∇3 ρ + 4κρ − κ∇4 κ ∇3 α + r−2 Γg d≤2 α. As above, 2ρ − 2∇4 κ = 2∇3 ρ + 4κρ − κ∇4 κ = =

  1 2ρ − 2 − κκ + 2ρ + r−1 Γg = −2ρ + κκ + r−1 Γg , 2     3 1 2 − κρ + r−1 Γg + 4κρ − κ − κκ + 2ρ + r−1 Γg 2 2 1 2 κκ − ρκ + r−1 Γg . 2

Hence, since r−1 Γg (∇3 ∇3 α, ∇3 α) = r−2 Γg · d≤2 α = Good,  
1 2 [Q, ∇4 ]∇3 α = − 2ρ + κκ ∇3 ∇3 α + κκ − ρκ ∇3 α + Good. (A.14.16) 2 We deduce [Q, ∇4 ∇3 ]α

= = + = +

  [Q, ∇4 ]∇3 α + ∇4 [Q, ∇3 ]α  
1 2 − 2ρ + κκ ∇3 ∇3 α + κκ − ρκ ∇3 α 2   1 ∇4 κ2 ∇3 α + κ3 α + Good1 + Good 2
1 − 2ρ + κκ ∇3 ∇3 α + κ2 ∇4 ∇3 α + κ3 ∇4 α 2   1 2 3 2 2 ∇4 (κ ) + κκ − ρκ ∇3 α + κ ∇4 κα + Good. 2 2

Note that 1 ∇4 (κ2 ) + κκ2 − ρκ 2

= =

3 2 κ ∇4 κ = 2

  1 1 2κ − κκ + 2ρ + r−1 d/Γg + κκ2 − ρκ 2 2 1 2 3ρκ − κκ + r−2 d/Γg , 2  3 2 1 κ − κκ + 2ρ + r−1 d/Γg . 2 2

785

APPENDIX TO CHAPTER 2

Hence,
1 − 2ρ + κκ ∇3 ∇3 α + κ2 ∇4 ∇3 α + κ3 ∇4 α 2     1 2 3 2 1 + 3ρκ − κκ ∇3 α + κ − κκ + 2ρ α + Good 2 2 2

[Q, ∇4 ∇3 ]α

=

as stated. Lemma A.17. Given f a 2-conformal tensor in s2 we have [∇3 ,

4 / ]f = −κ (c)4 / f + r−1 d≤2 (Γb · f ).

(c)

Proof. Recall that for a 2-conformal spacetime tensor f we have (c)

4 /f

=

4 / f + r−1 d/≤1 (Γg · f ).

Hence, [∇3 ,

(c)

4 / ]f


[∇3 , 4 / ]f + ∇3 r−1 d/≤1 (Γg · f ) + r−1 d/≤1 (Γg · ∇3 f ).

=

On the other hand, since ∇ / ω = r−1 d/Γb , ∇ / 2 ω = r−2 d/2 Γb , [∇3 , 4 / ]f = [e3 − 4ω, 4 / ]f = [e3 , 4 / ]f + r−2 d/≤2 (Γb · f ). We deduce [∇3 ,

(c)

4 / ]f

=


[e3 , 4 / ]f + r−2 d≤2 (Γb · f ) + e3 r−1 d/≤1 (Γg · f ) .

In the reduced form, for an s2 tensor f , [∇3 ,

(c)

4 / ]f

=


[e3 , 4 / 2 ]f + r−2 d≤2 (Γb · f ) + e3 r−1 d/≤1 (Γg · f ) .

We now recall that 4 / 2 = − /d?2 /d2 + 2K. Hence, applying the commutation Lemma7 2.54, [4 / 2 , e3 ]f

[− /d?2 /d2 + 2K, e3 ]f = − /d?2 [ /d2 , e3 ]f − [ /d?2 , e3 ] /d2 f − 2e3 (K)f     1 1 ? ∗ ? = − /d2 κ /d2 + Com2 (f ) − κ /d2 + Com2 ( /d2 f ) − 2e3 (K) 2 2 = −κ /d?2 /d2 f − 2e3 (K)f + eθ (κ) /d2 f − /d?2 (Com2 (f )) − Com∗2 ( /d2 f ) =

= −κ /d?2 /d2 f − 2e3 (K)f + r−2 d≤2 (Γb · f ) + r−1 d≤1 (Γg · e3 f )

= κ4 / 2 f − 2(e3 K + κK)f + r−2 d≤2 (Γb · f ) + r−1 d≤1 (Γg · e3 f ). 7 Recall

that we have

Com2 (f )

=

Com∗2 (f )

=

1 ? − ϑ/ d3 f + (ζ − η)e3 f − 2ηe3 Φf − ξ(e4 f + ke4 (Φ)f ) − 2βf, 2 1 − ϑ /d1 f − (ζ − η)e3 f − ηe3 Φf + ξ(e4 f − e4 (Φ)f ) − βf. 2

786

APPENDIX A

Note that, ignoring the quadratic terms,     1 1 e3 K + κK = −e3 ρ + κκ − κ ρ + κκ 4 4  1 = −e3 ρ − κρ − e3 (κκ) + κκ2 4 1 = κρ − /d1 β 2      1 1 1 − κ − κ2 − 2ω κ + κ − κκ + 2ωκ + 2 /d1 η + 2ρ + κκ2 4 2 2 1 = − /d1 β − κ /d1 η. 2 We deduce = −κ4 / 2 f + r−1 d≤2 (Γb · f ).

[e3 , 4 / 2 ] = −[4 / 2 , e3 ]f Consequently, [∇3 ,

/ f + r−1 d≤2 (Γb · f ) 4 / ]f = −κ (c)4

(c)

as stated. Lemma A.18. We have [Q,

(c)

4 / ]α

5 = −2κ∇3 (c)4 / α − κ2 (c)4 / α + Good. 2

(A.14.17)

Proof. We have [Q,

(c)

4 / 2 ]α

  1 2 = ∇3 ∇3 + 2κ∇3 + κ 2

  1 2 4 /α − 4 / ∇3 ∇3 α + 2κ∇3 α + κ α 2   1 (c) (c) (c) 2 (c) = ∇3 [∇3 , 4 / ]α + [∇3 , 4 / ]e3 α + [2κ∇3 , 4 / ]α + κ , 4 / 2 α. 2 (c)

(c)

Note that [2κ∇3 , (c)4 / ]α   1 2 (c) κ , 4 /2 α 2

=

2κ[∇3 ,

=

Good.

(c)

4 / ]α + Good,

787

APPENDIX TO CHAPTER 2

Hence, using the previous commutation lemma, [Q,

(c)

4 / 2 ]α

∇3 [∇3 , (c)4 / ]α + [∇3 , (c)4 / ]e3 α + 2κ[∇3 , (c)4 / ]α + Good   (c) −1 ≤2 ∇3 − κ 4 / α + r d (Γb · α)   − κ (c)4 / ∇3 α + r−1 d≤2 (Γb · ∇3 α)   2κ − κ (c)4 / α + r−1 d≤2 (Γb · α) + Good

−κ ∇3 (c)4 / α + (c)4 / ∇3 α − ∇3 κ + 2κ2 (c)4 / α + Good

−κ 2∇3 (c)4 / α − [∇3 , (c)4 / ]α − ∇3 κ + 2κ2 (c)4 / α + Good
(c) 2 (c) −2κ∇3 4 / α − ∇3 κ + 3κ 4 / α + Good.

= = + + = = =

Note that 

2 (c)

(∇3 κ + 3κ )

4 /α

=

5 2 κ + r−1 d/Γb 2



(c)

4 /α =

5 2 κ + r−2 d/Γg · d/≤2 α. 2

Hence, [Q,

(c)

4 / 2 ]α

5 = −2κ∇3 (c)4 / α − κ2 (c)4 / α + Good 2

as stated. Lemma A.19. We have Q(f g) = Q(f )g + f Q(g) + 2∇3 f ∇3 g − 12 κ2 f g. Also, [Q, f e4 ]g

=

[Q, f ∇3 ]g

=

1 Q(f )∇4 g + f [Q, e4 ]g + 2∇3 f ∇3 ∇4 g − κ2 f ∇4 g, 2 1 Q(f )∇3 g + f [Q, ∇3 ]g + 2∇3 f ∇3 ∇3 g − κ2 f ∇3 g. 2

Proof. Recall that Q =

1 ∇3 ∇3 + 2κ∇3 + κ2 . 2

Hence, Q(f g)

=

  1 ∇3 ∇3 + 2κ∇3 + κ2 (f g) 2

1 (∇3 ∇3 f )g + f (∇3 ∇3 g) + 2∇3 f ∇3 g + 2κ(∇3 f g + f ∇3 g) + κ2 f g 2   = ∇3 ∇3 f + 2κ∇3 f g + 2∇3 f ∇3 g + f Q(g) =

1 = Q(f )g + f Q(g) + 2∇3 f ∇3 g − κ2 f g. 2

788

APPENDIX A

Also, [Q, f ∇4 ]g

= Q(f ∇4 g) − f ∇4 Q(g) = Q(f )∇4 g + f Q∇4 (g) + 2∇3 f ∇3 ∇4 g 1 2 − κ f ∇4 g − f ∇4 Q(g) 2   1 = Q(f ) − κf ∇4 g + f [Q, ∇4 ]g + 2∇3 f ∇3 ∇4 g. 2

Similarly,  [Q, f ∇3 ]g

=

1 Q(f ) − κ2 f 2

 ∇3 g + f [Q, ∇3 ]g + 2∇3 f ∇3 ∇3 g

as stated. A.14.3

Main commutation

Proposition A.20. The following identity holds true. [Q, L]α = −2κ∇4 Q(α) + CQ Q(α) + Good,

(A.14.18)

where CQ

=

7 −8ρ − κκ. 2

Proof. In view of Lemma A.13, we have Lα

=

−∇4 ∇3 α +

  5 1 1 4 / 2 α − κ∇3 α − κ∇4 α − −4ρ + κκ α = Good2 . 2 2 2

(c)

Hence, we infer [Q, L]α

1 5 = −[Q, ∇4 ∇3 ]α + [Q, 4 / 2 ]α − [Q, κ∇4 ]α − [Q, κ∇3 ]α 2 2   1 + Q, 4ρ − κκ α 2 = I +J +K +L+M (A.14.19)

with I, J, K, L, M , denoting each of the commutators on the left of (A.14.19). A.14.3.1

Expression for I

In view of Lemma A.16 we have, for I = −[Q, ∇4 ∇3 ]α,  
1 3 1 2 2 I = (2ρ − κκ ∇3 ∇3 α − κ ∇4 ∇3 α − κ ∇4 α − − κκ + 3ρκ ∇3 α 2 2   (A.14.20) 3 2 1 − κ − κκ + 2ρ α + Good. 2 2

789

APPENDIX TO CHAPTER 2

A.14.3.2

Expression for J

Using Lemma A.18, J

=

[Q,

5 4 / ]α = −2κ∇3 (c)4 / α − κ2 (c)4 / α. 2

(c)

Recalling the definition of L and the fact that Lα = Good1 we write   5 1 1 4 / 2 α = ∇4 ∇3 α + κ∇3 α + κ∇4 α + −4ρ + κκ α + Good1 . 2 2 2 Hence,     5 1 1 = −2κ∇3 ∇4 ∇3 α + κ∇3 α + κ∇4 α + −4ρ + κκ α 2 2 2     5 2 5 1 1 − κ ∇4 ∇3 α + κ∇3 α + κ∇4 α + −4ρ + κκ α 2 2 2 2   1 = −2κ∇3 ∇4 ∇3 α − 5κκ∇3 ∇3 α − κ2 ∇3 ∇4 α − 2κ −4ρ + κκ ∇3 α 2     5 1 1 − 2κ ∇3 κ∇3 α + ∇3 κ∇4 α + ∇3 −4ρ + κκ α 2 2 2     5 2 5 1 1 − κ ∇4 ∇3 α + κ∇3 α + κ∇4 α + −4ρ + κκ α . 2 2 2 2

J

According to Lemma A.14 ∇3 ∇ 4 ∇3 α

= = =

∇ 3 ∇4 α

=

∇4 ∇3 ∇3 α + [∇3 , ∇4 ]∇3 α

∇4 ∇3 ∇3 α + 2ρ∇3 α + r−1 Γg d/≤1 ∇3 α ∇4 ∇3 ∇3 α + 2ρ∇3 α + Good, ∇4 ∇3 α + 4ρα + Good1 .

We deduce, modulo Good error terms, J

= − − −



−2κ ∇4 ∇3 ∇3 α + 2ρ∇3 α − 5κκ∇3 ∇3 α − κ2 ∇4 ∇3 α + 4ρα   1 2κ −4ρ + κκ ∇3 α − 5κ∇3 κ∇3 α − κ∇3 κ∇4 α 2   1 2κ∇3 −4ρ + κκ α 2     5 2 5 1 1 κ ∇4 ∇3 α + κ∇3 α + κ∇4 α + −4ρ + κκ α . 2 2 2 2

Grouping terms, we rewrite in the form J

= −2κ∇4 ∇3 ∇3 α − 5κκ∇3 ∇3 α + J43 ∇4 ∇3 α + J4 ∇4 α + J3 ∇3 α + J0 α.

790

APPENDIX A

We calculate the coefficients J43 , J4 , J3 , J0 as follows. J43

=

J4

=

J3

= = =

J0

= = = − =

5 7 −κ2 − κ2 = − κ2 , 2 2   5 3 1 5 3 −κ∇3 κ − κ = −κ − κ2 + r−1 d/Γb − κ3 = − κ3 + r−2 d/Γb , 4 2 4 4   1 25 −4κρ − 2κ −4ρ + κκ − 5κ∇3 κ − κ3 2 4   29 1 4κρ − κ3 − 5κ − κκ + 2ρ + r−1 d/Γg 4 2 19 3 −6κρ − κ + r−2 d/Γg , 4     1 5 1 2 −4ρκ − 2κ∇3 −4ρ + κκ − κ2 −4ρ + κκ 2 2 2 5 6ρκ2 − κκ3 + 8κ∇3 ρ − κ (κ∇3 κ + κ∇3 κ) 4   5 3 2 6ρκ − κκ3 + 8κ − κρ + r−1 d/Γg 4 2      1 1 κ κ − κκ + 2ρ + r−1 Γg + κ − κ2 + r−1 d/Γb 2 2 5 −8κ2 ρ − κκ3 + κκ3 + r−3 d/Γb + r−2 Γg . 4

Hence J4 ∇4 α J3 ∇3 α J0 α

3 = − κ3 + Good, 4  19 = −6κρ − κ3 ∇3 α + Good, 4 1 = −8κ2 ρ − κκ3 + Good. 4

We finally derive 7 J = −2κ∇4 ∇3 ∇3 α − 5κκ∇3 ∇3 α − κ2 ∇4 ∇3 α 2     (A.14.21) 3 3 19 2 1 3 2 − κ ∇4 α − 6κρ + κκ ∇3 α − 8κ ρ + κκ α + Good. 4 4 4 A.14.3.3

Expression for K

Also, using Lemma A.19
and Lemma A.16 (according to which we have the identity [Q, ∇4 ]α = 2ρ + κκ ∇3 α + 12 κκ2 α + Good1 )   i 1h 1 1 K = − Q, κ∇4 α = − Q(κ)∇4 α + κ[Q, ∇4 ]α + 2∇3 κ∇3 ∇4 α − κ3 ∇4 α 2 2 2     1 1 1 1 = − Q(κ) − κ3 ∇4 α − κ (2ρ + κκ) ∇3 α + κκ2 α 2 2 2 2 − ∇3 κ∇3 ∇4 α + Good.

791

APPENDIX TO CHAPTER 2

Hence, K = −∇3 κ∇3 ∇4 α −

1 2




1 1 1 Q(κ) − κ3 ∇4 α − κ 2ρ + κκ ∇3 α − κκ3 α + Good. 2 2 4

We calculate the expression, 1 Q(κ) − κ3 2

=

= =

  1 2 −1 ∇3 ∇3 κ + 2κ∇3 κ = ∇3 − κ + r d/Γb 2   1 2 +2κ − κ + r−1 d/Γb 2

−κ ∇3 κ + κ2 + ∇3 r−1 d/Γb + r−2 d/Γb
1 − κ3 + ∇3 r−1 d/Γb + r−2 d/Γb . 2

Hence, K


1 1 1 = −∇3 κ∇3 ∇4 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α − κκ3 α 4 2 4
+∇3 r−1 d/Γb ∇4 α + Good.

We note that
∇3 r−1 d/Γb ∇4 α

= =


∇3 r−1 d/Γb ∇4 α − r−1 d/Γb ∇3 ∇4 α
∇3 r−1 d/Γg d≤1 α − r−2 d/Γg d≤2 α = Good.

We deduce K


1 1 1 = −∇3 κ∇3 ∇4 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α − κκ3 α + Good 4 2 4  
1 2 1 1 1 = − − κ + r−1 d/Γb ∇3 ∇4 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α − κκ3 α 2 4 2 4 +Good
1 2 1 1 1 = κ ∇3 ∇4 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α − κκ3 α + Good. 2 4 2 4

In view of Lemma A.14, we have [∇3 , ∇4 ]α = 4ρα + r−1 Γg d/≤1 α. Hence K


1
1 2 1 κ ∇4 ∇3 α + 4ρα + r−1 Γg d/≤1 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α 2 4 2 1 3 − κκ α + Good 4   1 2 1 1 1 = κ ∇4 ∇3 α + κ3 ∇4 α − κ (2ρ + κκ) ∇3 α + κ2 2ρ − κκ α + Good. 2 4 2 4 =

We have thus derived K

=


1 2 1 1 κ ∇4 ∇3 α + κ3 ∇4 α − κ 2ρ + κκ ∇3 α 2  4 2 1 2 +κ 2ρ − κκ α + Good. 4

(A.14.22)

792

APPENDIX A

A.14.3.4

Expression for L

According to Lemma A.19 and Lemma A.16 (according to which we have the identity [Q, ∇3 ]α = κ2 ∇3 α + 12 κ3 α + Good1 )   i 5h 5 1 L = − Q, κe3 α = − Q(κ)∇3 α + κ[Q, ∇3 ]α + 2∇3 κ∇3 ∇3 α − κκ2 ∇3 α 2 2 2     5 1 1 = − Q(κ)∇3 α + κ κ2 ∇3 α + κ3 α + 2∇3 κ∇3 ∇3 α − κκ2 ∇3 α 2 2 2 +Good   5 1 5 = −5∇3 κ∇3 ∇3 α − Q(κ) + κκ ∇3 α − κκ3 α + Good. 2 2 4 Note that Q(κ)

1 = ∇3 ∇3 κ + 2κ∇3 κ + κκ2 2     1 1 1 = ∇3 − κκ + 2ρ + r−1 d/Γg + 2κ − κκ + 2ρ + r−1 d/Γg + κκ2 2 2 2  
1 3 1 = − κ∇3 κ + κ∇3 κ + 2 − κρ + r−1 d/Γg − κκ2 + 4ρκ + κκ2 2 2 2
−1 −2 + e3 r d/Γg + r d/Γg
1 1 = − κ∇3 κ + κ∇3 κ + ρκ − κκ. 2 2

Therefore, Q(κ)

=

    1 1 2 1 1 1 −1 −1 − κ − κ + r d/Γb − κ − κκ + 2ρ + r d/Γg + κρ − κκ2 2 2 2 2 2

+

r−1 d≤2 Γg = r−1 d≤2 Γg .

We deduce 5 5 L = −5∇3 κ∇3 ∇3 α − κκ2 ∇3 α + κκ3 α + Good 4  4 1 5 2 5 = −5 − κκ + 2ρ ∇3 ∇3 α − κκ ∇3 α + κκ3 α + Good. 2 4 4 Therefore, 

 1 5 5 L = −5 − κκ + 2ρ ∇3 ∇3 α − κκ2 ∇3 α + κκ3 α + Good. 2 4 4

(A.14.23)

793

APPENDIX TO CHAPTER 2

A.14.3.5

Expression for M

Similarly, according to Lemma A.19,   1 M = Q, 4ρ − κκ α 2       1 1 1 2 1 = Q 4ρ − κκ α + 2∇3 4ρ − κκ ∇3 α − κ 4ρ − κκ α, 2 2 2 2 i.e., M

      1 1 1 1 = Q 4ρ − κκ α + 2∇3 4ρ − κκ ∇3 α − κ2 4ρ − κκ α. 2 2 2 2

We calculate   1 ∇3 4ρ − κκ = 2 = − =

1 1 4∇3 ρ − κ∇3 κ − κ∇3 κ 2 2     3 1 1 4 − κρ + r−1 d/Γg − κ − κ2 + r−1 d/Γb 2 2 2   1 1 κ − κκ + 2ρ + r−1 d/Γg 2 2 1 −7κρ + κκ2 + r−1 d/Γg . 2

We deduce        1 1 2 1 1 2 M = Q 4ρ − κκ − κ 4ρ − κκ α + −7κρ + κκ ∇3 α + Good. 2 2 2 2 It remains to calculate     1 1 2 1 M0 = Q 4ρ − κκ − κ 4ρ − κκ 2 2 2     1 1 = ∇3 ∇3 4ρ − κκ + 2κ∇3 4ρ − κκ 2 2     1 2 1 2 −1 −1 = ∇3 −7κρ + κκ + r d/Γg + 2κ −7κρ + κκ + r d/Γg 2 2   1 2 1 2 = −7ρ∇3 κ − 7κ∇3 ρ + κ ∇3 κ + κκ∇3 κ + 2κ −7κρ + κκ 2 2 +

∇3 (r−1 d/Γg ) + r−2 d/Γg .

794

APPENDIX A

Hence,     1 3 = −7ρ − κ2 + r−1 d/Γb − 7κ − κρ + r−1 d/Γg 2 2     1 2 1 1 + κ − κκ + 2ρ + r−1 d/Γg + κκ − κ2 + r−1 d/Γb 2 2 2   1 + 2κ −7κρ + κκ2 + ∇3 (r−1 d/Γg ) + r−2 d/Γg 2 1 = κ2 ρ + κκ3 + ∇3 (r−1 d/Γg ) + r−2 d/Γg . 4

M0

We conclude M

=

    1 1 κ2 ρ + κκ3 α + 2 −7κρ + κκ2 ∇3 α + Good. (A.14.24) 4 2

Indeed note that ∇3 (r−1 d/Γg )α A.14.3.6


= ∇3 r−1 d/Γg α − r−1 d/Γg ∇3 α = Good.

End of the proof of Proposition A.20

Using the equations (A.14.20)–(A.14.24) we deduce, in view of (A.14.19), [Q, L]α = I + J + K + L + M  
1 3 1 2 2 = 2ρ − κκ ∇3 ∇3 α − κ ∇4 ∇3 α − κ ∇4 α − − κκ + 3ρκ ∇3 α 2 2   3 1 7 − κ2 − κκ + 2ρ α − 2κ∇4 ∇3 ∇3 α − 5κκ∇3 ∇3 α − κ2 ∇4 ∇3 α 2 2 2     3 19 1 − κ3 ∇4 α − 6κρ + κκ2 ∇3 α − 8κ2 ρ + κκ3 α 4 4 4  
1 2 1 3 1 1 2 + κ ∇4 ∇3 α + κ ∇4 α − κ(2ρ + κκ ∇3 α + κ 2ρ − κκ α 2 4 2 4   1 5 5 − 5 − κκ + 2ρ ∇3 ∇3 α − κκ2 ∇3 α − κκ3 α 2 4 4     1 1 + κ2 ρ + κκ3 α + 2 −7κρ + κκ2 ∇3 α + Good. 4 2 We deduce 0 0 [Q, L]α = −2κ∇4 ∇3 ∇3 α + C33 ∇3 ∇3 α + C43 ∇4 ∇3 α + C40 ∇4 α + C30 ∇3 α + C00 α,

795

APPENDIX TO CHAPTER 2

with 0 C33

=

0 C43

=

C40

=

C30

=

C00

=

 
1 7 2ρ − κκ − 5κκ − 5 − κκ + 2ρ = −8ρ − κκ, 2 2 7 1 −κ2 − κ2 + κ2 = −4κ2 , 2 2 1 3 3 3 1 3 − κ − κ + κ = −κ3 , 2 4 4 
5 1 2 19 2 1 κκ − 3ρκ − 6κρ + κκ − κ 2ρ + κκ − κκ2 2 4 2 4   1 +2 −7κρ + κκ2 = −24κρ − 5κκ2 , 2       3 2 1 1 3 1 5 2 2 − κ − κκ + 2ρ − 8κ ρ + κκ + κ 2ρ − κκ − κκ3 2 2 4 4 4   1 3 + κ2 ρ + κκ3 = −8κ2 ρ − κκ3 . 4 4

Finally we write, recalling the definition of Q = ∇3 ∇3 + 2κ∇3 + 12 κ2 , ∇3 ∇3 α

1 = Q(α) − 2κ∇3 α − κ2 α 2

and ∇ 4 ∇3 ∇3 α

1 = ∇4 Q(α) − 2κ∇4 ∇3 α − κ2 ∇4 α − 2∇4 κ∇3 α − κ∇4 κα. 2

Hence, 0 −2κ∇4 ∇3 ∇3 α + C33 ∇3 ∇3 α

= −2κ∇4 Q(α) + 4κ2 ∇4 ∇3 α + κ3 ∇4 α + 4κ∇4 κ∇3 α   1 2 2 0 + 2κ ∇4 κα + C33 Q(α) − 2κ∇3 α − κ α . 2 We deduce [Q, L]α

= + +

0 −2κ∇4 Q(α) + C33 Q(α) + 4κ2 ∇4 ∇3 α + κ3 ∇4 α  
1 0 0 0 4κ∇4 κ − 2κC33 ∇3 α + 2κ2 ∇4 κ − κ2 C33 α + C43 ∇ 4 ∇3 α 2 C40 ∇4 α + C30 ∇3 α + C00 α.

0 Thus, setting CQ = C33 , we deduce

[Q, L]α

= −2κ∇4 Q(α) + CQ Q(α) + C43 ∇4 ∇3 α + C4 ∇4 α + C3 ∇3 α + C0 α +Good,

796

APPENDIX A

where 7 0 = C33 = −8ρ − κκ, 2 0 = 4κ2 + C43 = 4κ2 − 4κ2 = 0,

CQ C43

= κ3 + C40 = κ3 − κ3 = 0.

C4 Also, C3

= = = =

C0

= = = =


0 2κ 2∇4 κ − C33 + C30   7 2κ −κκ + 4ρ + r−1 d/Γg + 8ρ + κκ + C30 2  
5 2κ 12ρ + κκ + − 24κρ − 5κκ2 + r−2 d/Γg 2 r−2 d/Γg , 1 0 2κ2 ∇4 κ − κ2 C33 + C00 2     1 1 7 3 2κ2 − κκ + 2ρ + r−1 d/Γg + κ2 8ρ + κκ − 8κ2 ρ − κκ3 2 2 2 4 3 3 8κ2 ρ + κκ3 − 8κ2 ρ − κκ3 + r−3 d/Γg 4 4 r−3 d/Γg .

We have therefore checked that [Q, L]α

= −2κ∇4 Q(α) + CQ Q(α) + Good,

7 CQ = −8ρ − κκ, 2

as stated in Proposition A.20. A.14.4

Proof of Theorem 2.108

We start with the following: Lemma A.21. We have 2 (f r4 )



= r4 2 f − 2r4 κe4 f + κe3 f + r4 − 5κκ − 4ρ f + O(r4 d≤1 Γg · f ).

We postpone the proof of the lemma to the end of the section and continue below the proof of the theorem. According to Lemma A.13 L(α)

=

Good2

where L is the operator Lα

= −∇4 ∇3 α +

  5 1 1 4 / 2 α − κ∇3 α − κ∇4 α − −4ρ + κκ α. 2 2 2

(c)

Applying Q and recalling the definition of the error terms Good we derive L(Qα)

= −[Q, L]α + Good.

797

APPENDIX TO CHAPTER 2

Thus, in view of Proposition A.20, [Q, L]α

= −2κ∇4 Q(α) + CQ Q(α),

CQ = −8ρ − 72 κκ.

We deduce L(Qα)

=

2κ∇4 Q(α) − CQ Q(α).

Therefore, modulo Good terms, 2κ∇4 Q(α) − CQ Q(α)

5 1 −∇4 ∇3 (Qα) + (c)4 / 2 (Qα) − κ∇3 Q(α) − κ∇4 Q(α) 2 2   1 −4ρ + κκ Q(α). 2

= −

We deduce −∇4 ∇3 (Qα) +

5 5 4 / 2 (Qα) − κ∇3 Q(α) − κ∇4 Q(α) 2   2  1 + CQ − −4ρ + κκ Q(α) 2

(c)

=

Good.

In view of the expression for 2 in the second part of Lemma A.13, we rewrite in the form
2 Q(f ) − 2κ∇3 Q(α) − 2κ∇4 Q(α) − 4ρ + 4κκ Q(α) = Good + r−1 Γg · d/Q(α). Finally, making use of Lemma A.21 and recalling that q = r4 Q(α),

2 q = r4 2 (Qα) − 2r4 κe4 (Qα) + κe3 (Qα) + r4 − 5κκ − 4ρ Qf

+O(r4 d≤1 Γg · Q(α))  
= r4 2κ∇3 Q(α) + 2κ∇4 Q(α) + 4ρ + 4κκ Q(α) + Good

− 2r4 κe4 (Qα) + κe3 (Qα) + r4 − 5κκ − 4ρ Qf + O(d≤1 Γg · q) =

−κκq + r4 Good.

This ends the proof of Theorem 2.108. A.14.4.1

Proof of Lemma A.21

We have 2 (f r4 )

= Dα Dα (f r4 ) = Dα (Dα f r4 + f Dα r4 ) = r4 2 f + 2Dα (r4 )Dα f + f (r)
= r4 2 f − e3 (r4 )e4 f + e4 (r4 )e3 f + f (r4 ) + r4 Γg df
= r4 2 f − 4r3 e3 (r)e4 f + e4 (r)e3 f + f (r4 ) + r4 Γg df   = r4 2 f − 2r4 (κ + Γb )e4 f + (κ + Γg )e3 f + f (r4 ) + r4 Γg · df
= r4 2 f − 2r4 κe4 f + κe3 f + f (r4 ) + r4 Γg · f.

798

APPENDIX A

Also, (r4 )

  1 1 = −e4 (e3 (r4 )) − κe4 (r4 ) + − κ + 2ω e3 (r4 ) + 4 / (r4 ) + 2ηeθ (r4 ) 2 2   r 1 r = −4e4 (r3 e3 (r)) − 2r3 κ (κ + Γg ) + 4r3 − κ + 2ω (κ + Γb ) 2 2 2 +4 / (r4 ) + 2ηeθ (r4 )   1 = −12r2 (e4 r)(e3 r) − 4r3 e4 e3 r − r4 κκ + 2r4 − κ + 2ω κ + O(r3 Γb ) 2 r  = −3r4 (κ + Γb )(κ + Γg ) − 4r3 e4 (κ + Γb ) − 2r4 κκ + 4r4 ωκ 2 +O(r3 Γb ).

Hence, (r4 )

= −5r4 κκ + 4r4 ωκ − 2r3 e4 (rκ) + O(r4 d≤1 Γg ).

Note that e4 (rκ)

=

  r 1 r re4 (κ) + κ(κ + Γg ) = r − κ κ + 2ωκ + 2 /d1 η + 2ρ + κ(κ + Γg ) 2 2 2

=

2rρ + 2rωκ + O(d≤1 Γg ).

Hence, (r4 )

= =

−5r4 κκ + 4r4 ωκ − 2r3 (2rρ + 2rωκ) + O(r4 d≤1 Γg )
r4 − 5κκ − 4ρ + O(r4 d≤1 Γg ).

We conclude 2 (f r4 ) as stated.

=



r4 2 f − 2r4 κe4 f + κe3 f + r4 − 5κκ − 4ρ f + O(r4 d≤1 Γg )

Appendix B Appendix to Chapter 8 B.1

PROOF OF PROPOSITION 8.14

Proposition B.1. The following wave equations hold true. 1. The null curvature component ρ verifies the identity g ρ := κe4 ρ + κe3 ρ +

 3 κ κ + 2ρ ρ + Err[g ρ], 2

where Err[g ρ]

=

  3 1 ρ − ϑ ϑ + 2(ξ ξ + η η) 2 2    3 1 + κ − 2ω ϑ α − ζ β − 2(η β + ξ β) 2 2 1 − ϑ /d?2 β + (ζ − η)e3 β − ηe3 (Φ)β − ξ(e4 β + e4 (Φ)β) − ββ 2   1 −e3 − ϑ α + ζ β + 2(η β + ξ β) 2   − /d?1 (κ)β + 2 /d?1 (ω)β + 3η /d?1 (ρ) − /d1 − ϑβ + ξα − 2ηeθ ρ.

2. The small curvature quantity ρ˜ := r

2



2m ρ+ 3 r



verifies the wave equation 8m g (˜ ρ) + 3 ρ˜ = r


  g (r) − 2r − 2m 3m 4Υ r2 −6m − κκ + 2 r2 r r 3m − (Aκ + Aκ) + Err[g ρ˜], r

800

APPENDIX B

where ! 6m 3 2 3 4 e3 (r) 4 e4 (r) Err[g ρ˜] := − AA + 2 ρ˜ + A + A ρ˜ r r 2 3 r 3 r    8m  3 8m 2 + κκ − 3 + 2 g (r2 ) + 3 ρ˜ 2 r 3r r 2 2 −Ae3 (˜ ρ) − Ae4 (˜ ρ) + Ae3 (m) + Ae4 (m) r   r 1 2 a +4D (m)Da + g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. r r Proof. We prove the result in the following steps. Step 1. We start by deriving the wave equation for ρ. From Bianchi, ρ satisfies 3 1 e4 ρ + κρ = /d1 β − ϑ α + ζ β + 2(η β + ξ β). 2 2 Differentiating with respect to e3 , we obtain   3 3 1 e3 (e4 (ρ)) + κe3 (ρ) + e3 (κ)ρ = e3 ( /d1 β) + e3 − ϑ α + ζ β + 2(η β + ξ β) . 2 2 2 Also, β satisfies from Bianchi e3 β + κβ

= − /d?1 ρ + 2ωβ + 3ηρ − ϑβ + ξα.

Differentiating with respect to /d1 , we infer /d1 (e3 β) + κ /d1 β − /d?1 (κ)β

= − /d1 /d?1 ρ + 2ω /d1 β − 2 /d?1 (ω)β + 3ρ /d1 η − 3η /d?1 (ρ)   + /d1 − ϑβ + ξα

and hence /d1 /d?1 ρ =

− /d1 (e3 β) − κ /d1 β + 2ω /d1 β + 3ρ /d1 η

  + /d?1 (κ)β − 2 /d?1 (ω)β − 3η /d?1 (ρ) + /d1 − ϑβ + ξα . Next, we add the equation for /d1 /d?1 ρ from the one for e3 (e4 (ρ)). This yields 3 3 e3 (e4 (ρ)) + /d1 /d?1 ρ + κe3 (ρ) + e3 (κ)ρ 2 2 

=

 1 [e3 , /d1 ]β − κ /d1 β + 2ω /d1 β + 3ρ /d1 η + e3 − ϑ α + ζ β + 2(η β + ξ β) 2   + /d?1 (κ)β − 2 /d?1 (ω)β − 3η /d?1 (ρ) + /d1 − ϑβ + ξα .

Next, we recall the following commutator identity 1 1 [e3 , /d1 ]β = − κ /d1 β + ϑ /d?2 β − (ζ − η)e3 β + ηe3 (Φ)β + ξ(e4 β + e4 (Φ)β) + ββ. 2 2

801

APPENDIX TO CHAPTER 8

We infer e3 (e4 (ρ)) + =

/d1 /d?1 ρ

3 3 + κe3 (ρ) + e3 (κ)ρ + 2 2



3 κ − 2ω 2

 /d1 β − 3ρ /d1 η

1 ? ϑ /d2 β − (ζ − η)e3 β + ηe3 (Φ)β + ξ(e4 β + e4 (Φ)β) + ββ 2   1 +e3 − ϑ α + ζ β + 2(η β + ξ β) 2   + /d?1 (κ)β − 2 /d?1 (ω)β − 3η /d?1 (ρ) + /d1 − ϑβ + ξα .

Next, we make use of the Bianchi identities and the null structure equations to compute   3 3 e3 (κ)ρ + κ − 2ω /d1 β − 3ρ /d1 η 2 2   3 1 1 = ρ − κ κ + 2ωκ + 2 /d1 η + 2ρ − ϑ ϑ + 2(ξ ξ + η η) 2 2 2    3 3 1 + κ − 2ω e4 ρ + κρ + ϑ α − ζ β − 2(η β + ξ β) − 3ρ /d1 η 2 2 2    3 3  = κ − 2ω e4 ρ + ρ κ κ + 2ρ 2 2      1 3 3 1 + ρ − ϑ ϑ + 2(ξ ξ + η η) + κ − 2ω ϑ α − ζ β − 2(η β + ξ β) . 2 2 2 2 This yields

=

   3 3 3  e3 (e4 (ρ)) − 4 / ρ + κe3 (ρ) + κ − 2ω e4 ρ + ρ κ κ + 2ρ 2 2 2      3 1 3 1 − ρ − ϑ ϑ + 2(ξ ξ + η η) − κ − 2ω ϑ α − ζ β − 2(η β + ξ β) 2 2 2 2 1 ? + ϑ /d2 β − (ζ − η)e3 β + ηe3 (Φ)β + ξ(e4 β + e4 (Φ)β) + ββ 2   1 +e3 − ϑ α + ζ β + 2(η β + ξ β) 2   + /d?1 (κ)β − 2 /d?1 (ω)β − 3η /d?1 (ρ) + /d1 − ϑβ + ξα ,

where we used the fact that /d1 /d?1 = −4 /. Next, recall the formula for the wave operator acting on a scalar ψ   1 1 g ψ = −e3 e4 ψ + 4 / ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ. 2 2

802

APPENDIX B

We infer    3 3 3  e3 (e4 (ρ)) − 4 / ρ + κe3 (ρ) + κ − 2ω e4 ρ + ρ κκ + 2ρ 2 2 2   1 1 = −g ρ + 2ω − κ e4 ρ − κe3 ρ + 2ηeθ ρ 2 2    3 3 3  + κe3 (ρ) + κ − 2ω e4 ρ + ρ κ κ + 2ρ 2 2 2 and hence g ρ =

 3 κe4 ρ + κe3 ρ + κ κ + 2ρ ρ 2      3 1 3 1 + ρ − ϑ ϑ + 2(ξ ξ + η η) + κ − 2ω ϑ α − ζ β − 2(η β + ξ β) 2 2 2 2 1 − ϑ /d?2 β + (ζ − η)e3 β − ηe3 (Φ)β − ξ(e4 β + e4 (Φ)β) − ββ 2   1 −e3 − ϑ α + ζ β + 2(η β + ξ β) 2   ? − /d1 (κ)β + 2 /d?1 (ω)β + 3η /d?1 (ρ) − /d1 − ϑβ + ξα − 2ηeθ ρ.

Step 2. We derive the following identity g (r2 ρ) = −Ae3 (r2 ρ) − Ae4 (r2 ρ) 3 + 2

! 4 e3 (r) 4 e4 (r) 2 2 A + A + κκ + 2ρ + 2 g (r ) r2 ρ 3 r 3 r 3r

(B.1.1)

+ 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. Proof. r2 ρ satisfies the following wave equation g (r2 ρ)

= r2 g ρ + 2Da (r2 )Da (ρ) + ρg (r2 ).

On the other hand, recall that we have g ρ =

κe4 ρ + κe3 ρ +

 3 κ κ + 2ρ ρ + Err[g ρ]. 2

We deduce g (r2 ρ)

=



r2 κ − e4 (r2 ) e3 ρ + r2 κ − e3 (r2 ) e4 ρ   3 2 + κκ + 2ρ + 2 g (r2 ) r2 ρ + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ] 2 3r

= −Ae3 (r2 ρ) − Ae4 (r2 ρ) 3 + 2

! 4 e3 (r) 4 e4 (r) 2 2 A + A + κκ + 2ρ + 2 g (r ) r2 ρ 3 r 3 r 3r

+4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ] as desired.

APPENDIX TO CHAPTER 8

803

Step 3. We now derive the desired formula for g ρ˜. In view of the definition of ρ˜, we have   2m 2 g (˜ ρ) = g (r ρ) + g r     1 1 2 2 a = g (r ρ) + 2mg + 4D (m)Da + g (m). r r r Together with B.1.1 we deduce g (˜ ρ)

=

−Ae3 (r2 ρ) − Ae4 (r2 ρ)

! 4 e3 (r) 4 e4 (r) 2 2 A + A + κκ + 2ρ + 2 g (r ) r2 ρ 3 r 3 r 3r     1 1 2 ? ? a +4r /d1 (r) /d1 (ρ) + 2mg + 4D (m)Da + g (m) r r r 3 + 2

+4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. Next, we use r2 ρ = ρ˜ − 2mr−1 . This yields ! 3 8m 2 2 g (˜ ρ) − κκ − 3 + 2 g (r ) ρ˜ 2 r 3r   1 3m 12m2 2m e3 (r) e4 (r) = 2mg − κκ + − 3 g (r2 ) − 6mA 2 − 6mA 2 r r r4 r r r ! 3 3 4 e3 (r) 4 e4 (r) 2 + 2 ρ˜2 + A + A ρ˜ − Ae3 (˜ ρ) − Ae4 (˜ ρ) + Ae3 (m) r 2 3 r 3 r r   2 1 2 + Ae4 (m) + 4Da (m)Da + g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. r r r Note that in Schwarzschild   3 8m 2 8m 2 κκ − 3 + 2 g (r ) = − 3 2 r 3r r

804

APPENDIX B

and hence 8m g (˜ ρ) + 3 ρ˜  r 1 3m 12m2 2m e3 (r) e4 (r) = 2mg − κκ + − 3 g (r2 ) − 6m 2 − 6mA 2 4 r r r r r r ! 3 3 4 e3 (r) 4 e4 (r) + 2 ρ˜2 + A + A ρ˜ r 2 3 r 3 r ! ! 3 8m 2 8m 2 + κκ − 3 + 2 g (r ) + 3 ρ˜ 2 r 3r r −Ae3 (˜ ρ) − Ae4 (˜ ρ) +   1 +4Da (m)Da + r Also, we have   1 1 g − 3 g (r2 ) r r

2 2 Ae3 (m) + Ae4 (m) r r 2 g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. r

g (r) Dα (r)Dα (r) g (r) Dα (r)Dα (r) +2 −2 2 −2 2 3 r r r r3 g (r) = −3 2 r = −

and hence 8m g (r) 3m 12m2 ρ ˜ − 6m − κκ + r3 r2 r r4 ! e3 (r) e4 (r) 3 2 3 4 e3 (r) 4 e4 (r) −6m 2 − 6mA 2 + 2 ρ˜ + A + A ρ˜ r r r 2 3 r 3 r ! ! 3 8m 2 8m + κκ − 3 + 2 g (r2 ) + 3 ρ˜ 2 r 3r r

g (˜ ρ) +

−Ae3 (˜ ρ) − Ae4 (˜ ρ) +   1 +4Da (m)Da + r

2 2 Ae3 (m) + Ae4 (m) r r 2 g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. r

Finally, since −6m

e3 (r) e4 (r) − 6mA 2 2 r r

= −3mA

κ κ 6m − 3mA − AA r r r

and g (r) 3m 12m2 −6m 2 − κκ + r r r4

g (r) − 2r − = −6m r2

2m r2




3m − r



4Υ κκ + 2 r

 ,

805

APPENDIX TO CHAPTER 8

we obtain
  g (r) − 2r − 2m 8m 3m 4Υ r2 g (˜ ρ) + 3 ρ˜ = −6m − κκ + r r2 r r2 κ 6m κ AA −3mA − 3mA − r r r ! 3 2 3 4 e3 (r) 4 e4 (r) + 2 ρ˜ + A + A ρ˜ r 2 3 r 3 r ! ! 3 8m 2 8m 2 + κκ − 3 + 2 g (r ) + 3 ρ˜ 2 r 3r r −Ae3 (˜ ρ) − Ae4 (˜ ρ) +   1 +4Da (m)Da + r

2 2 Ae3 (m) + Ae4 (m) r r 2 g (m) + 4r /d?1 (r) /d?1 (ρ) + r2 Err[g ρ]. r

This concludes the proof of the proposition.

Appendix C Appendix to Chapter 9 C.1

PROOF OF LEMMA 9.11

We start with the following: Lemma C.1. Let k ≥ 0 an integer and let f ∈ sk (S). Then, we have Z 1 √ (◦  #λ γ k √ S # ( /dk f ) = p /dk (f # ) + U γ(eθ (κ) − eθ (ϑ)) dλ 2 γS # 0 Z 1  #λ k √ + S γeθ κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ dλ 4 0 ! ) # k k 1/2 0 # 0 + κ − ϑ − Ω(κ − ϑ) − 2bγ eθ Φ U + (κ + ϑ) S f # 4 2 where for 0 ≤ λ ≤ 1, #λ denotes the pullback by ◦ ◦

ψλ (u, s, θ)

◦

:=

◦

(u + λU (θ), s + λS(θ), θ).

◦

Proof. For p ∈ S and f a Z-invariant scalar function on S, we have by definition of the pushforward of a vectorfield [Ψ# (∂θ )f ]Ψ(p)

=

[∂θ (f ◦ Ψ)]p .

We infer # ( /dS kf)

=

1 p

γS #

  ∂θ (f # ) + k∂θ (Φ# )f #

and hence # ( /dS kf)

= =

√   γ p eθ (f # ) + keθ (Φ)f # + k(eθ (Φ# ) − eθ (Φ))f # γS # √ ◦  γ p /dk (f # ) + k(eθ (Φ# ) − eθ (Φ))f # . γS #

Next, we have eθ (Φ# ) − eθ (Φ)

=

√

γ

−1



∂θ (Φ# ) − ∂θ Φ



807

APPENDIX TO CHAPTER 9

and   ◦ ◦ ∂θ (Φ# ) − ∂θ Φ (u, s, θ)

◦

◦

◦ ◦

∂θ [Φ(u + U (θ), s + S(θ), θ)] − ∂θ Φ(u, s, θ)

=

◦

◦

◦ ◦

(∂θ Φ)(u + U (θ), s + S(θ), θ) − ∂θ Φ(u, s, θ) h i ◦ ◦ + (∂u Φ)# U 0 + (∂s Φ)# S 0 (u, s, θ) Z 1 i d h ◦ ◦ = (∂θ Φ)(u + λU (θ), s + λS(θ), θ) dλ 0 dλ h i ◦ ◦ + (∂u Φ)# U 0 + (∂s Φ)# S 0 (u, s, θ) Z 1 ◦ ◦ = U (θ) (∂u ∂θ Φ)(u + λU (θ), s + λS(θ), θ)dλ =

0

Z 1 ◦ ◦ +S(θ) (∂s ∂θ Φ)(u + λU (θ), s + λS(θ), θ)dλ 0 h i ◦ ◦ + (∂u Φ)# U 0 + (∂s Φ)# S 0 (u, s, θ) which we rewrite #

∂θ (Φ ) − ∂θ Φ = U

Z

1

(∂u ∂θ Φ)

#λ

Z dλ + S

0

1

(∂s ∂θ Φ)#λ dλ + (∂u Φ)# U 0 + (∂s Φ)# S 0

0 ◦ ◦

◦

◦

where #λ denotes the pullback by the map ψλ (u, s, θ) = (u + λU (θ), s + λS(θ), θ). Next, recall that ∂s = e4 ,

 1 e3 − Ωe4 − bγ 1/2 eθ , 2

∂u =

∂θ =

√

γeθ .

Hence, ∂θ ∂s Φ

=

∂θ ∂u Φ

=

√

γeθ e4 (Φ),   1√ γeθ e3 Φ − Ωe4 Φ − bγ 1/2 eθ Φ , 2

which yields #

∂θ (Φ ) − ∂θ Φ

1

Z = U

√

γeθ e4 (Φ)

 #λ

dλ

0

   #λ 1√ γeθ e3 Φ − Ωe4 Φ − bγ 1/2 eθ Φ dλ 2 0 # 1 + e3 Φ − Ωe4 Φ − bγ 1/2 eθ Φ U 0 + (e4 Φ)# S 0 2 Z 1  #λ 1 √ = U γ(eθ (κ) − eθ (ϑ)) dλ 2 0 Z 1  #λ 1 √ + S γeθ κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ dλ 4 0 # 1 1 + κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ U 0 + (κ + ϑ)# S 0 . 4 2 Z

+S

1



808

APPENDIX C

We deduce # ( /dS kf)

=

Z 1 √ (◦  #λ γ k √ # p /dk (f ) + U γ(eθ (κ) − eθ (ϑ)) dλ 2 γS # 0 Z 1  #λ k √ + S γeθ κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ dλ 4 0 ! ) # k k 1/2 0 # 0 + κ − ϑ − Ω(κ − ϑ) − 2bγ eθ Φ U + (κ + ϑ) S f # . 4 2

This concludes the proof of the lemma. We are ready to prove the higher derivative comparison Lemma 9.11 which we recall below. ◦ ◦

◦

Lemma C.2. Let S ⊂ R = R(, δ) as in Definition 9.1 verifying the assumptions ◦

A1–A3. Let Ψ : S → S be Z-invariant deformation. Assume the bound k(U 0 , S 0 )k

◦ −1

◦

L∞ 1 (S)

+r

max

0≤s≤smax −1

k(U 0 , S 0 )k

◦ ◦ ◦

hs (S,g /)

. δ.

(C.1.1)

Then, we have for any reduced scalar h defined on R khkhs (S) . sup |d≤k h| for 0 ≤ s ≤ smax . R

Also, if f ∈ hs (S) and f # is its pullback by ψ, we have kf khs (S) = kf # k

◦

hs (S, g / S,# )

= kf # k

◦

◦ ◦

hs (S,g /)

(1 + O()) for 0 ≤ s ≤ smax − 1.

Remark C.3. Note that the estimates of the lemma are independent of the size ◦

◦

◦ ◦

r of the sphere S = S(u, s) ⊂ R, see Definition 9.1. To simplify the argument ◦ below we assume r ≈ 1. The general case can be easily deduced by a simple scaling argument or making obvious adjustments in the inequalities below. Proof. We argue by iteration. We consider the following iteration assumptions: If (9.2.14) holds, then we have khkhs (S) . sup |d≤s h|,

(C.1.2)

R

and if (9.2.14) holds, then we have kf # k

◦

hs (S, g / S,# )

= kf # k

◦ ◦ ◦

hs (S,g /)

(1 + O(δ)). (C.1.3)

First, note that (C.1.2) holds trivially for s = 0 and (C.1.3) holds for s = 0 by Lemma 9.8. Thus, from now on, we assume that (C.1.2) and (C.1.3) hold for some s with 0 ≤ s ≤ smax − 2, and our goal is to prove that it also holds for s replaced by s + 1. We start with (C.1.2). We have S S /dS k h = eθ h + eθ (Φ)h.

809

APPENDIX TO CHAPTER 9

Now, recall that we have 1 S S eS θ = p S ∂θ , ∂θ |Ψ(p) = γ



    1 1 1 √ S 0 − ΩU 0 e4 + U 0 e3 + γ 1 − bU 0 eθ . 2 2 2 Ψ(p)

This yields ( ( /dS k h)|Ψ(p)

=

    1 1 S 0 − ΩU 0 e4 (h) + S 0 − ΩU 0 e4 (Φ)h 2 2 !)   1 0 1 0 1 0 √ + U e3 (h) + U e3 (Φ)h + γ 1 − bU /dk (h) 2 2 2 1 p γS

.

|Ψ(p)

Together with the iteration assumption (C.1.3), we infer k /dS k hkhs (S) = .

◦

# k( /dS ◦ ◦ (1 + O(δ)) k h) k hs (S,g /)

   

1 1 1

S 0 − Ω# U 0 (e4 (h))# + S 0 − Ω# U 0 (e4 (Φ)h)#

p

γ S,# 2 2 !  

p 1 0 1 1

+ U (e3 (h))# + U 0 (e3 (Φ)h)# + γ # 1 − b# U 0 ( /dk (h))#

2 2 2

, ◦ ◦

hs (S,g /)

i.e., k /dS k hkhs (S)

p

!

γ#

# + p − 1 ( /dk (h)) ◦ ◦

◦ ◦ hs (S,g /) γ S,# hs (S,g /)

   

1 1 1

+ p S 0 − Ω# U 0 (e4 (h))# + S 0 − Ω# U 0 (e4 (Φ)h)#

γ S,# 2 2 !

1 0 1 0 1p # # 0 # # # + U (e3 (h)) + U (e3 (Φ)h) − γ b U ( /dk (h)) .

◦ ◦ 2 2 2

. ( /dk (h))#

hs (S,g /)

◦

◦

Together with a non-sharp product rule in hs (S, g/ ) and the repeated use of the iteration assumptions (C.1.2), (C.1.3), we can bound the right-hand side of the

810

APPENDIX C

above inequality by  

√

√

#  . 1 + k γkhs (S) + ( γ)

 



1

1



 p  + p

◦

γS

γ S,# ∞ ◦ h∞ 1 (S) hs (S) h1 (S)  



1

1



 p  × k /dk (h)khs (S) + k(U 0 , S 0 )k ◦ ◦ + p ◦

γS

γ S,# ∞ ◦ hs (S,g / )∩h∞ 1 (S) hs (S) h1 (S)  

√ √ × 1 + k(Ω, b γ)k + (Ω, b γ)# ◦ 

hs (S)

h∞ 1 (S)

 

× (e3 , e4 , /dk )h, e3 (Φ)h, e4 (Φ)h

.

hs (S)

Therefore k /dS k hkhs (S) can be bounded by   

 

√

1

1 √



#  . 1 + k γkhs (S) + ( γ) ∞ ◦  p + p

γS

γ S,# ∞ ◦ h1 (S) hs (S) h1 (S)  



1

1 ≤s



 × sup d /dk h +  p + p

γS

γ S,# ∞ ◦ R hs (S) h1 (S)  

√ √ #

× 1 + k(Ω, b γ)khs (S) + (Ω, b γ) ◦ h∞ 1 (S) ≤s × sup d (dh, e3 (Φ)h, e4 (Φ)h) , R

where we used in the last inequality the assumption (9.2.14) on (U 0 , S 0 ). Together with (9.1.12) and (9.1.15), we infer k /dS k hkhs (S) (  .

 



1

1



 p  + p

◦

γS

γ S,# ∞ ◦ h∞ 1 (S) hs (S) h1 (S) 

 )

1

√

 1 + (Ω, b γ)# + p ◦

h∞

γ S,# ∞ ◦ 1 (S)

√

# 1 + 1 +

( γ)





1

+ 1 + p

γS hs (S) ≤s+1 × sup d h .



h1 (S)

R

Also, for a reduced scalar v defined on R, we have in view of the assumption (9.2.14)

811

APPENDIX TO CHAPTER 9

on (U 0 , S 0 ) kv # k

◦

h∞ 1 (S)

= .

kv ◦ ψk ∞ ◦ h1 (S)   0 1 + sup |ψ (θ)| sup |d≤1 v| R

0≤θ≤π

.

 1 + k(U 0 , S 0 )k

 ◦

h∞ 1 (S)

sup |d≤1 v| R

◦

.

(1 + δ) sup |d≤1 v|.

(C.1.4)

R

Together with (9.1.12) and (9.1.15), we infer



( )

1

1



S k /dk hkhs (S) . 1 + p + p sup d≤s+1 h .

γS

γ S,# ∞ ◦ R h (S) h (S) s

1

Now, recall that γ S (ψ(θ))

  1 = γ(ψ(θ)) + Ω(ψ(θ)) + (b(ψ(θ)))2 γ(ψ(θ)) (U 0 (θ))2 4 0 0 − 2U (θ)S (θ) − γ(ψ(θ))b(ψ(θ))U 0 (θ).

Together with a repeated application of the iteration assumptions and a non-sharp ◦

◦

product rule in hs (S, g/ ) and (C.1.4), this yields

S

γ + γ S,# ∞ ◦ hs (S) h1 (S)   . 1 + sup |d≤1 (γ, Ω, b2 γ, bγ)| + sup |d≤s (γ, Ω, b2 γ, bγ)| R R   0 0 0 0 × 1 + k(U , S )k ∞ ◦ + k(U , S )k ◦ ◦ h1 (S)

hs (S,g /)

. 1 where we used in the last estimate the assumption (9.2.14) on (U 0 , S 0 ) and (9.1.15). We infer



1

1



+ p . 1

p

γS

γ S,# ∞ ◦ hs (S)

h1 (S)

and hence k /dS k hkhs (S)

.

sup d≤s+1 h R

which corresponds to the first iteration assumption (C.1.2) with s replaced with s + 1 for s ≤ smax − 2. Next, we focus on recovering the second iteration assumption (C.1.3) with s replaced with s + 1 for s ≤ smax − 2. Recall from Lemma C.1 that we have for

812

APPENDIX C

f ∈ sk (S) # ( /dS kf)

=

Z 1 √ (◦  #λ γ k √ p /dk (f # ) + U γ(eθ (κ) − eθ (ϑ)) dλ 2 γS # 0 Z 1  #λ k √ + S γeθ κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ dλ 4 0 ! ) # k k 1/2 0 # 0 + κ − ϑ − Ω(κ − ϑ) − 2bγ eθ Φ U + (κ + ϑ) S f # 4 2

where for 0 ≤ λ ≤ 1, #λ denotes the pullback by ◦ ◦

ψλ (u, s, θ)

=

◦

◦

(u + λU (θ), s + λS(θ), θ).

For convenience, we rewrite some of the terms as follows: eθ (κ) − eθ (ϑ)

=

bγ 1/2 eθ Φ

=

1 − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ), 2 1 1/2 γ ( /d1 b + /d?2 b), 2

and   eθ κ − ϑ − Ω(κ − ϑ) − 2bγ 1/2 eθ Φ 1 1 = − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ + Ω( /d1 ϑ − /d?2 ϑ) 2 2 ? 1/2 ? 1/2 + /d1 (γ )( /d1 ϑ + /d2 ϑ)b − 2γ eθ (beθ Φ) 1 1 = − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ + Ω( /d1 ϑ − /d?2 ϑ) 2 2 ? 1/2 ? 1/2 ? + /d1 (γ )( /d1 ϑ + /d2 ϑ)b − 2γ (−eθ (Φ) /d2 b − Kb) 1 1 = − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ + Ω( /d1 ϑ − /d?2 ϑ) 2 2 1 + /d?1 (γ 1/2 )( /d1 ϑ + /d?2 ϑ)b + γ 1/2 ( /d2 /d?2 b + /d?3 /d?2 b) + 2γ 1/2 Kb 2 where we used the identities = −(eθ (Φ))2 − K, 1 1/2 2γ 1/2 eθ Φ /d?2 b = γ ( /d2 /d?2 b + /d?3 /d?2 b). 2 eθ (eθ (Φ))

813

APPENDIX TO CHAPTER 9

This yields # ( /dS kf)  #λ Z 1 √ (◦ γ k 1 √ # ? ? = p /dk (f ) + U γ − /d1 (κ) − ( /d1 ϑ − /d2 ϑ) dλ 2 2 γS # 0 Z 1 k 1 √ + S γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ 4 2 0

1 + Ω( /d1 ϑ − /d?2 ϑ) + /d?1 (γ 1/2 )( /d1 ϑ + /d?2 ϑ)b 2 !!#λ 1 1/2 ? ? ? 1/2 + γ ( /d2 /d2 b + /d3 /d2 b) + 2γ Kb dλ 2 ! ) # k k 1/2 ? 0 # 0 + κ − ϑ − Ω(κ − ϑ) − γ ( /d1 ϑ + /d2 ϑ)b U + (κ + ϑ) S f # . 4 2 ◦

◦

Next, we take the hs (S, g/ )-norm of this identity, and we use the iteration ◦

assumption to replace the norm on the left-hand side with the hs (S, g/ S,# )-norm. We infer ◦

# k( /dS (1 + O(δ)) ◦ kf) k h (S, g / S,# )

√ (s  #λ Z 1

◦ γ k 1 √

# ? ? = p /dk (f ) + U γ − /d1 (κ) − ( /d1 ϑ − /d2 ϑ) dλ

γS # 2 2 0 Z 1 k 1 √ + S γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ 4 2 0

1 + Ω( /d1 ϑ − /d?2 ϑ) + /d?1 (γ 1/2 )( /d1 ϑ + /d?2 ϑ)b 2 !!#λ 1 1/2 ? ? ? 1/2 + γ ( /d2 /d2 b + /d3 /d2 b) + 2γ Kb dλ 2 # k + κ − ϑ − Ω(κ − ϑ) − γ 1/2 ( /d1 ϑ + /d?2 ϑ)b U 0 4 ! )

k

# 0 + (κ + ϑ) S f # .

◦ ◦ 2 hs (S, g /)

814

APPENDIX C ◦

◦

Next, we use a non-sharp product rule in hs (S, g/ ) to infer ◦

# k( /dS (1 + O(δ)) ◦ kf) k hs (S, g / S,# )  (

√



◦ # γ



 = 1 + O(1) p − 1

/dk (f ) ◦ ◦

◦ ◦

γS # ◦ hs (S, g /) h (S, g / )∩h∞ (S) s

+O(1) kU k

◦

1

◦

hs+1 (S, g /)

 #λ

√ 

1

× γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) dλ



◦ ◦ ◦ 2 0 hs (S, g / )∩h∞ ( S) 1 Z 1

√ 1

+kSk γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ ◦ ◦

hs+1 (S, g /) 0 2 Z

1

1 + Ω( /d1 ϑ − /d?2 ϑ) + /d?1 (γ 1/2 )( /d1 ϑ + /d?2 ϑ)b 2 !!#λ

1 1/2

? ? ? 1/2 + γ ( /d2 /d2 b + /d3 /d2 b) + 2γ Kb dλ

◦ ◦ ◦ 2 hs (S, g / )∩h∞ ( S) 1

 #

1/2 ?

+ kU 0 k ◦ ◦

κ − ϑ − Ω(κ − ϑ) − γ ( /d1 ϑ + /d2 ϑ)b ◦ ◦ ◦ hs+1 (S, g /) ∞ hs (S, g / )∩h1 (S) ! )

# 0

f + (κ + ϑ)# ◦ ◦ . ◦ kS k ◦ ◦ ◦ ◦ hs (S, g / )∩h∞ 1 (S)

hs+1 (S, g /)

hs+1 (S, g /)

815

APPENDIX TO CHAPTER 9

Since s+1 ≤ smax −1, we infer in view of (9.2.14) and the fact that U (0) = S(0) = 0, ◦

# k( /dS (1 + O(δ)) ◦ kf) k hs (S, g / S,# )  (

√



◦

γ

 /dk (f # ) = 1 + O(1) p − 1

◦ ◦

γS #

◦ ◦ ◦ hs (S, g /) hs (S, g / )∩h∞ 1 (S)



  Z # 1 λ ◦ 1

√

+O(δ) γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) dλ



◦ ◦ ◦ 2 0 ∞ hs (S, g / )∩h1 (S) Z 1

√ 1

+ γ − /d?1 (κ) − ( /d1 ϑ − /d?2 ϑ) + /d?1 (Ωκ) − /d?1 (Ω)ϑ

2 0

1 + Ω( /d1 ϑ − /d?2 ϑ) + /d?1 (γ 1/2 )( /d1 ϑ + /d?2 ϑ)b 2 !!#λ

1 1/2

dλ + γ ( /d2 /d?2 b + /d?3 /d?2 b) + 2γ 1/2 Kb

◦ ◦ ◦ 2 hs (S, g / )∩h∞ 1 (S)

 #

1/2 ?

+ κ − ϑ − Ω(κ − ϑ) − γ ( /d1 ϑ + /d2 ϑ)b

◦ ◦ ◦

hs (S, g / )∩h∞ 1 (S)

!

+ (κ + ϑ)#

◦

◦

hs (S, g /

)

#

f

◦ )∩h∞ 1 (S)

◦

◦

hs+1 (S, g /)

Next, we have by the iteration assumption (C.1.3)

 #

≤2 

−2 ˇ

/d

Γ, r γ − 1, b, Ω + Υ

◦

.

◦

h (S, g /)

.

s

 #λ

≤2 

−2 ˇ

+ sup /d Γ, r γ − 1, b, Ω + Υ

◦ ◦ 0≤λ≤1 hs (S, g /)



#

≤2  ˇ r−2 γ − 1, b, Ω + Υ

/d

Γ,

◦

h (S, g / S,# )

.

s

 #λ

≤2 

−2 ˇ

+ sup /d Γ, r γ − 1, b, Ω + Υ

◦ S,# 0≤λ≤1 λ) hs (S, g /

 

≤2 ˇ −2

Γ, r γ − 1, b, Ω + Υ

/d hs (S)

 

≤2 ˇ −2

+ sup /d Γ, r γ − 1, b, Ω + Υ

hs (Sλ )

0≤λ≤1

◦

where the surface Sλ is the image of S by ψλ . Since s ≤ smax − 2, we infer in view of our iteration assumption (C.1.2) and our assumptions (9.1.12), (9.1.15) on the

816

APPENDIX C

(u, s) foliation

 #

≤2 

ˇ r−2 γ − 1, b, Ω + Υ

/d

Γ,

◦ ◦ hs (S, g /)

 #λ

≤2 

ˇ r−2 γ − 1, b, Ω + Υ

+ sup Γ,

/d

0≤λ≤1

◦

◦

hs (S, g /)

  ˇ r−2 γ − 1, b, Ω + Υ Γ ˇ . sup d≤s /d≤2 Γ, R   ◦ ˇ r−2 γ − 1, b, Ω + Υ Γ ˇ . δ. . sup d≤s+2 Γ,

(C.1.5)

R

Also, we have

 #

≤2 

−2 ˇ

/d

Γ, r γ − 1, b, Ω + Υ

∞ ◦ h1 (S)

  #λ

≤2

−2 ˇ

+ sup /d Γ, r γ − 1, b, Ω + Υ

0≤λ≤1

◦

h∞ (S)

1



 

≤2 ˇ −2

= /d Γ, r γ − 1, b, Ω + Υ ◦ ψ ∞ ◦ h1 (S)



 

≤2 ˇ −2

+ sup /d Γ, r γ − 1, b, Ω + Υ ◦ ψλ

◦

h∞ 1 (S)

0≤λ≤1



     ≤3 ˇ −2 0 . sup d Γ, r γ − 1, b, Ω + Υ 1 + sup |ψ (θ)| R 0≤θ≤π       ˇ r−2 γ − 1, b, Ω + Υ . sup d≤3 Γ, 1 + k(U 0 , S 0 )k ∞ ◦ h1 (S)

R

◦



.

where we used our assumptions (9.1.12), (9.1.15) on the (u, s) foliation and our assumption (9.2.14) on (U 0 , S 0 ). Therefore,

 # ◦

≤2 

−2 ˇ

/d

Γ, r γ − 1, b, Ω + Υ . δ,

◦ h∞ 1 (S)

 #λ

≤2 

ˇ r−2 γ − 1, b, Ω + Υ

sup / d Γ,

0≤λ≤1

◦ ◦

(C.1.6)

. δ.

h∞ 1 (S)

We deduce ◦

# k( /dS (1 + O(δ)) ◦ kf) k hs (S, g / S,# )  

√

γ

 = 1 + O(1) p − 1

γS #

◦ ◦ ◦ hs (S, g / )∩h∞ 1 (S) ( )

◦

◦ #

#

× . ◦ ◦

/dk (f ) ◦ ◦ + O(δ) f hs+1 (S, g /) hs (S, g /)

Next, we estimate the term in the RHS involving γ and γ S # . From the proof of

817

APPENDIX TO CHAPTER 9

Lemma 9.8, we have γ

S,#

Z 1  Z 1   #λ 1 # 1/2 = U e3 − Ωe4 − bγ eθ γ dλ + S (e4 γ) λ dλ 2 0 0  # 1 2 # + Ω+ b γ (U 0 )2 − 2U 0 S 0 − (γb) U 0 . 4

−γ

Using a non-sharp product rule, we infer

S,#

γ − γ ◦ ◦ ◦ hs (S, g / )∩h∞ 1 (S) Z 1    #λ

1/2

. kU k ◦ ◦ e − Ωe − bγ e dλ ◦ 3 4 θ γ

◦ ◦ ◦ ∞ hs (S, g / )∩h1 (S) 0 hs (S, g / )∩h∞ 1 (S) Z 1

#λ kSk ◦ ◦ γ) ◦

(e

◦ ◦ ◦ dλ 4 hs (S, g / )∩h∞ hs (S, g / )∩h∞ 1 (S) 0 1 (S)

 #

1 2

kU 0 k ◦ ◦ + Ω + b γ ◦

◦ ◦ hs (S, g / )∩h∞ ◦ 4 1 (S) ∞ hs (S, g / )∩h1 (S) 0

0

.

+kU k ◦ ◦ ◦ kS k ◦ ◦ ◦ hs (S, g / )∩h∞ hs (S, g / )∩h∞ 1 (S) 1 (S)

# 0 + (γb) ◦ ◦ ◦ ◦ ◦ ◦ kU k hs (S, g / )∩h∞ hs (S, g / )∩h∞ 1 (S) 1 (S)

Z 1  #λ ◦

≤1  −2

/d

Ω + Υ δ r γ − 1, b,

◦ ◦

dλ

◦

hs (S, g / )∩h∞ 1 (S) ◦

0

◦
#

+δ r−2 γ − 1, b, Ω + Υ

◦

◦

◦

hs (S, g / )∩h∞ 1 (S)

+δ

where we used our assumption (9.2.14) on (U 0 , S 0 ) and U (0) = S(0) = 0. Using the estimates (C.1.5), (C.1.6) for (r−2 γ − 1, b, Ω + Υ), we infer ◦

S,#

γ − γ

◦

◦

hs (S, g /

Together with (9.1.15) for γ, we infer

√

γ

− 1

p

γS #

h

◦ )∩h∞ 1 (S)

.

δ.

◦

◦ ◦ / s (S, g

◦

.

δ

)∩h∞ 1 (S)

and hence # k( /dS kf) k

◦

(1 + O(δ)) hs (S, g / S,# )

  ( ◦ ◦

◦ #



# = 1 + O(δ)

/dk (f ) ◦ ◦ + O(δ) f ◦

hs (S, g /)

) ◦ ◦

hs+1 (S,g /)

.

818

APPENDIX C

Now, we have kf # k

◦

hs+1 (S, g / S,# )

kf # k

◦

◦

hs+1 (S, g /)

= kf # k

◦

L2 (S, g / S,# )

◦

hs (S, g / S,# )

,

◦

kf # k

=

# + k( /dS kf) k

◦ ◦ L2 (S, g /

)

+ k /dk (f # )k

◦

◦

hs (S, g /)

.

Together with Lemma 9.8, this yields kf # k

◦

hs+1

(S, g / S,# )

= kf # k

◦ ◦ ◦

hs+1 (S,g /)

(1 + O(δ)).

This corresponds to our iteration assumption (C.1.3) with s replaced with s + 1 for s ≤ smax − 2. Thus, we have finally derived both iteration assumption (C.1.3) and (C.1.2) with s replaced with s + 1 respectively for s ≤ smax − 2. Hence, we deduce that they hold for 0 ≤ s ≤ smax − 1, i.e., khkhs (S) . sup |d≤k h| for 0 ≤ s ≤ smax − 1

(C.1.7)

R

and kf # k

◦

hs

(S, g / S,# )

= kf # k

◦

◦ ◦

hs (S,g /)

(1 + O(δ)) for 0 ≤ s ≤ smax − 1.

Together with Lemma 9.7, we deduce kf khs (S) = kf # k

◦

hs

(S, g / S,# )

= kf # k

◦

◦ ◦

hs (S,g /)

(1 + O(δ)) for 0 ≤ s ≤ smax − 1.(C.1.8)

Finally, notice that the restriction s ≤ smax − 2 for the iteration assumptions (C.1.2), (C.1.3) was only necessary to replace s with s + 1 in (C.1.3). Indeed, a direct inspection of the proof reveals that to replace s with s + 1 in (C.1.2), we only need the restriction s ≤ smax − 1. Thus, running the iteration again, now with s = smax − 1, we deduce khkhs (S) . sup |d≤k h| for 0 ≤ s ≤ smax . R

This concludes the proof of the lemma.

Appendix D Appendix to Chapter 10 D.1

HORIZONTAL S-TENSORS

Consider a null pair e3 , e4 on (M, g) and, at every point p ∈ M, the horizontal space S = {e3 , e4 }⊥ . Let γ the metric induced on S. By definition, for all X, Y ∈ TS M, i.e., vectors in M tangent to S, h(X, Y ) = g(X, Y ). For any Y ∈ T (M) we define its horizontal projection, 1 1 Y ⊥ = Y + g(Y, e3 )e4 + g(Y, e3 )e4 . 2 2

(D.1.1)

Definition D.1. A k-covariant tensor U is said to be S-horizontal, U ∈ TkS (M), if for any X1 , . . . , Xk , we have U (Y1 , . . . , Yk ) = U (Y1⊥ , . . . , Yk⊥ ). We define the projection operator, 1 1 Πνµ := δµν − (e3 )µ (e4 )ν − (e4 )µ (e3 )ν . 2 2 Clearly Πµα Πβµ = Πβα . An arbitrary tensor Uα1 ...αm is said to be an S-horizontal tensor, or simply S-tensor, if Πβα11 . . . Πβαm Uβ1 ...βm = Uα1 ...αm . m Definition D.2. Given X ∈ T(M) and Y ∈ TS (M) we define ˙ XY D

:=

(DX Y )⊥ .

Remark D.3. In the particular case when S is integrable and both X, Y ∈ TS M ˙ X Y is the standard induced covariant differentiation on S. then D Definition D.4. Given a general, covariant, S-horizontal tensorfield U we define its horizontal covariant derivative according to the formula ˙ X U (Y1 , . . . , Yk ) = X(U (Y1 , . . . , Yk )) − U (D ˙ X Y1 , . . . , Yk ) − . . . − U (Y1 , . . . , D ˙ X Yk ), D where X ∈ TM and Y1 , . . . Yk ∈ TS M. Proposition D.5. For all X ∈ TM and Y1 , Y2 ∈ TS M, ˙ X Y1 , Y2 ) + h(Y1 , D ˙ X Y2 ). Xh(Y1 , Y2 ) = h(D

820

APPENDIX D

Proof. Indeed, we have Xh(Y1 , Y2 )

= Xg(Y1 , Y2 ) = g(DX Y1 , Y2 ) + g(Y1 , DX Y2 ) ˙ X Y1 , Y2 ) + g(Y1 , D ˙ X Y2 ) = g(D ˙ X Y1 , Y2 ) + h(Y1 , D ˙ X Y2 ) = h(D

as desired. Given an orthonormal frame e1 , e2 on S we have X ˙ µ eA = D (Λµ )AB eB A, B = 1, 2, B=1,2

where (Λµ )αβ := g(Dµ eβ , eα ). D.1.1

Mixed tensors

We consider tensors Tk M ⊗ TlS M, i.e., tensors of the form Uµ1 ...µk ,A1 ...AL for which we define ˙ µ Uν ...ν ,A ...A D 1 1 L k

= eµ Uν1 ...νk ,A1 ...Al −UDµ ν1 ...νk ,A1 ...Al − . . . − Uν1 ...Dµ νk ,A1 ...Al

−Uν1 ...νk ,D˙ µ A1 ...Al − . . . − Uν1 ...νk ,A1 ...D˙ µ Al . We are now ready to prove the following: Proposition D.6. We have the curvature formula ˙ µD ˙ ν −D ˙ νD ˙ µ )ΨA = RA B µν ΨB . (D More generally, ˙ µD ˙ ν −D ˙ νD ˙ µ )ΨλA = Rλ σ µν ΨσA + RA B µν ΨλB . (D Proof. Straightforward verification. D.1.2

Invariant Lagrangian

We introduce L

˙ µ ΨA D ˙ µ ΨB + W hAB ΨA ΨB . = g µν hAB D

Proposition D.7. The Euler-Lagrange equations are given by ˙ A = W ΨA Ψ ˙ µD ˙ ν ΨA . ˙ A := gµν D where Ψ

821

APPENDIX TO CHAPTER 10

Proof. The variation of the action is given by Z   ˙ µ ΨA D ˙ ν (δΨ)B + W ΨA δΨB dvg 0 = 2 hAB gµν D ZM   ˙ µ ΨA (δΨ)B dvg = 2 Dν gµν hAB D M Z   ˙ νD ˙ µ ΨA (δΨ)B − W ΨA δΨB dvg −2 hAB gµν D = −2

ZM M

  ˙ νD ˙ µ ΨA (δΨ)B − W ΨA δΨB dvg hAB gµν D

from which the proposition follows. D.1.3

Comparison of the Lagrangians

Let Ψ ∈ S2 (M) and ψ ∈ s2 its reduced form. Note that the Lagrangian of the scalar equation g ψ = V ψ + 4(eθ Φ)2 ψ is given by L(ψ)

:= gµν ∂µ ψ∂ν ψ + (V + 4(eθ Φ)2 )ψ 2

while the Lagrangian for ˙ gΨ = V Ψ  is given by ˙ µΨ · D ˙ ν Ψ + V Ψ · Ψ. = gµν D

L(Ψ) Proposition D.8. We have

L(Ψ) = 2L(ψ). Proof. Observe that ˙ µ ΨD ˙ νΨ gµν D

˙ 3Ψ · D ˙ 4Ψ + D ˙ θΨ · D ˙ θΨ + D ˙ ϕΨ · D ˙ ϕ Ψ. = −D

Now, recalling that ∇ / ϕ eϕ ∇ / θ eθ

= −eθ Φeθ , =

0,

∇ / ϕ eθ = eθ (Φ)eϕ , ∇ / θ eϕ = 0,

(D.1.2)

822

APPENDIX D

we deduce ˙ 3Ψ · D ˙ 4Ψ D ˙ θΨ · D ˙ θΨ D

= e3 Ψ · e4 Ψ = 2e3 ψe4 ψ ˙ θ Ψθθ D ˙ θ Ψθθ + 2D ˙ θ Ψθϕ D ˙ θ Ψθϕ + D ˙ θ Ψϕϕ D ˙ θ Ψϕϕ = D 2(eθ ψ)2 ˙ ϕ Ψθθ D ˙ ϕ Ψθθ + 2D ˙ ϕ Ψθϕ D ˙ ϕ Ψθϕ + D ˙ ϕ Ψϕϕ D ˙ ϕ Ψϕϕ = D

=

˙ ϕΨ · D ˙ ϕΨ D

= = =

2(eϕ ψ)2 + 2(−ΨD˙ ϕ θϕ − ΨθD˙ ϕ ϕ ) · (−ΨD˙ ϕ θϕ − ΨθD˙ ϕ ϕ )

2(eϕ ψ)2 + 2(−eθ (Φ)Ψϕϕ + eθ (Φ)Ψθθ ) · (−eθ (Φ)Ψϕϕ + eθ (Φ)Ψθθ ) 2(eϕ ψ)2 + 8(eθ Φ)2 ψ 2 .

Hence, ˙ µ ΨD ˙ νΨ gµν D

= −2e3 ψe4 ψ + 2(eθ ψ)2 + 2(eϕ ψ)2 + 4(eθ Φ)2 ψ 2

and L(Ψ)

D.1.4

= −2e3 Ψe4 ψ + 2(eθ ψ)2 + 2(eϕ ψ)2 + 8(eθ Φ)2 ψ 2 + 2V ψ 2 .

Energy-momentum tensor

Consider the energy-momentum tensor   ˙ µΨ · D ˙ ν Ψ − 1 gµν D ˙ λΨ · D ˙ λΨ + V Ψ · Ψ . Qµν := D 2 Lemma D.9. We have Dν Qµν

=


˙ µ Ψ · Ψ ˙ ν ΨA RABνµ ΨB − 1 Dµ V Ψ · Ψ. ˙ −Vψ +D D 2

Proof. We have Dν Qµν

=

  ˙ νD ˙ νΨ · D ˙ µΨ + D ˙ νΨ · D ˙ νD ˙ µ−D ˙ µD ˙ ν Ψ − V Dµ Ψ · Ψ D

=

1 − Dµ V Ψ · Ψ 2 ˙ µΨ · D ˙ νD ˙ νΨ + D ˙ ν ΨA RABνµ ΨB − V Dµ ΨΨ − 1 Dµ V Ψ · Ψ D 2
1 ν A B ˙ µ Ψ Ψ ˙ Ψ RABνµ Ψ − Dµ V Ψ · Ψ. ˙ −VΨ +D D 2

=

Lemma D.10. Relative to an arbitrary Z-polarized frame e3 , e4 , eθ , eϕ we have Q33

= |e3 Ψ|2 ,

Q34

= |∇ / Ψ|2 + V |Ψ|2 .

Q44

= |e4 Ψ|2 ,

823

APPENDIX TO CHAPTER 10

If ψ is the reduced form of Ψ, Q33

=

2(e3 ψ)2 ,

=

2(e4 ψ)2 ,

Q34

=

2(eθ ψ)2 + 2(eϕ ψ)2 + 2V |ψ|2 + 8(eθ Φ)2 ψ 2 .

Q44 Also,

gµν Qµν

= −L(Ψ) − V |Ψ|2 ,

|L(Ψ)| . |e3 Ψ| |e4 Ψ| + |∇ / Ψ|2 + V |Ψ|2 , and |QAB |

≤

|e3 Ψ||e4 Ψ| + |∇ / Ψ|2 + |V ||Ψ|2 ,

|QA4 |

≤

|e4 Ψ||∇ / Ψ|.

|QA3 |

D.2

≤

|e3 Ψ||∇ / Ψ|,

STANDARD CALCULATION

Proposition D.11. Consider an admissible spacetime M and Ψ ∈ S2 (M) and X a vectorfield of the form X = ae3 + be4 . 1. The 1-form Pµ = Qµν X ν verifies
˙ µ Ψ · Ψ ˙ − V Ψ − X(V )Ψ · Ψ. = X µD

D µ Pµ

2. Let X as above, w a scalar and M a 1-form. Define Pµ

=

1 ˙ µ Ψ − 1 |Ψ|2 ∂µ w + 1 |Ψ|2 Mµ . Pµ [X, w, M ] = Qµν X ν + wΨ · D 2 4 4

Then, with |Ψ|2 := Ψ · Ψ, Dµ Pµ [X, w, M ]

1 1 1 1 Q · (X) π − X(V )Ψ · Ψ + wL[Ψ] − |Ψ|2 g w 2 2 2 4 1 2 1 ˙ µΨ M µ + |Ψ| DivM + Ψ · D 4 2  
1 ˙ −VΨ . + X(Ψ) + wΨ · Ψ 2 =

Proof. Let Pµ [X, 0, 0] = Qµν X ν . Then, Dµ Pµ

= =

  ˙ µΨ · D ˙ νD ˙ ν Ψ − V Ψ + X µD ˙ ν ΨA RABνµ ΨB − 1 X µ Dµ V Ψ · Ψ X µD 2
1 ˙ µ Ψ · Ψ ˙ − V Ψ − X(V )|Ψ|2 . X µD 2

824

APPENDIX D

Assume X = ae3 + be4 . Then, since only the middle components of R are relevant, and recalling that RAB43 = − ?ρ ∈AB = 0, we derive ˙ ν ΨA RABν3 ΨB = aD ˙ 4 ΨA RAB43 ΨB + bD ˙ 3 ΨA RAB434 ΨB = 0. X µD ˙ − V Ψ, To prove the second part of the proposition we write with N [Ψ] := Ψ Dµ Pµ [X, w, M ]

= + + = + +

1 1 1 ˙ µΨ Q · (X) π + X(Ψ) · N [Ψ] − X(V )Ψ · Ψ + Dµ w Ψ · D 2 2 2 1 ˙µ ˙ µ Ψ + 1 wΨ ˙ µ Ψ∂µ w − 1 |Ψ|2 g w ˙ gΨ − 1 Ψ · D wD Ψ·D 2 2 2 4 1 2 1 µ ˙ |Ψ| DivM + Ψ · Dµ Ψ M 4 2 1 1 1 ˙µ (X) ˙ µΨ Q· π − X(V )Ψ · Ψ + w D Ψ·D 2 2 2 1 1 1 wΨ (V Ψ + N [Ψ]) − |Ψ|2 g w + |Ψ|2 DivM 2 4 4 1 µ ˙ Ψ · Dµ Ψ M + X(Ψ) · Ψ · N [Ψ]. 2

Hence, Dµ Pµ [X, w, M ]

= +

1 1 1 1 Q · (X) π − X(V )Ψ · Ψ + wL[Ψ] − |Ψ|2 g w 2 2 2  4  1 2 1 1 µ ˙ |Ψ| DivM + Ψ · Dµ Ψ M + X(Ψ) + wΨ · N [Ψ] 4 2 2

as desired. Remark D.12. As a consequence of the proposition above we deduce that every time we use vectorfields of the form ae3 +be4 as multipliers, the equation Ψ−V Ψ = N is treated exactly in the same manner as the scalar equation ψ − V ψ = N . Remark D.13. Note that in Schwarzschild, our potential V = −κκ = 4Υr−2 verifies      1 2m 2m 2m −2 −3 ∂r V = ∂r r 1− = −2r 1− + 4 4 r r r r − 3m = −2 . r4 D.3

VECTORFIELD Xf

Lemma D.14. Let Xf := f e4 ,

(X)

Λ :=

2f , r

(X)

π e :=

(X)

π−

(X)

Λg =

(X)

π−

2f g. r

825

APPENDIX TO CHAPTER 10

• We have (X)

π e44 = 0,

(X)

π e43

(X)

π e4θ = 2f ξ,

(X)

(X)

π4ϕ = 0,

(X)

π3ϕ = 0,   4f 2f = −2e4 f + 4f ω + = −2 e4 (f ) − + 4f ω, r r

π eAB = 2f (1+3)χAB −

2f g/ AB = 2f r

(X)

π e3θ = 2f (η + ζ),

(X)

π e33 = −8f ω − 4e3 (f ).



 1 χAB − δAB , r

(D.3.1)

(1+3)

• In particular, we have (X) (X) (X)

4f + O() min{w1,1 , w2,1/2 } (|f | + r|f 0 |) , r =  min{w2,1 , w3,1/2 },

π e43 = −2f 0 +

π e4A

π eAB = O() min{w1,1 , w2,1/2 }|f |,

(X)

(D.3.2)

π e3A = O()w1,1 |f |,

(X)

π e33 = 4f 0 Υ − 4Υ0 + O()w1,1 (|f | + r|f 0 |).

• We have  (X) Λ Proof. We calculate

= (X)

m 
2 00 f + O 4 + w3,1 |f | + r|f 0 | + r2 |f 00 | . r r

παβ = g(Deα X, eβ ) + g(Deβ X, eα ), (X)

π44

=

(X)

π43

=

(X)

π4θ

=

2f ξ,

πAB

=

2f (1+3)χAB ,

(X)

We deduce, for

(X)

π e=

(X)

π3θ

=

2f (η + ζ),

π33

=

−8f ω − 4e3 (f ).

π−

(X)

Λg =

=

0,

(X)

π e43

=

−2e4 f + 4f ω +

(X)

π e4θ

=

2f ξ,

=

2f

−2e4 f + 4f ω,

(X)

π e44

π eAB

0,

(X)

(X)

(X)

(D.3.3)

(1+3)

χAB

(X)

π−

2f r g,

  4f 2f = −2 e4 (f ) − + 4f ω, r r

2f − g/ AB = 2f r

(X)

π e3θ

=

2f (η + ζ),

(X)

π e33

=

−8f ω − 4e3 (f ).



(1+3)

χAB

 1 − δAB , r

Under the assumptions (10.2.7)–(10.2.8) on the Ricci coefficients (with respect to

826

APPENDIX D

the frame (e03 , e04 )), we deduce (X)

(X) (X)

π e43

=

−2e4 f + 4f ω = −2f 0 +

=

−2f 0 +

π e4A

=

π eAB

=

(X)

π e3A

(X)

π e33

4f +  min{w1,1 , w2,1/2 } (|f | + r|f 0 |) , r  min{w2,1 , w3,1/2 },  min{w1,1 , w2,1/2 }|f |,

=

min{w1,1 , w2,1/2 }|f |,

=

−8f ω − 4e3 (f ) = −8f

=

4f − 2f 0 (e4 (r) − 1) + 4f (ω − 1) r

m

 + w1,1 − 4f 0 (−Υ + w0,1 )

r2 4f 0 Υ − 4Υ0 + w1,1 (|f | + r|f 0 |).

To prove formula (D.3.3) we make use of the following (see also Lemma 10.11). Lemma D.15. If h = h(r) then  
2 2m h = Υh00 (r) + − 2 h0 + O()w2,1 |h| + r|h0 | + r2 |h00 | . r r Proof. For a general scalar h, h =

  1 1 (1+3) (1+3) − (e3 e4 + e4 e3 )h + 4 /h + ω− trχ e4 h 2 2   1 + (1+3)ω − (1+3)trχ e3 h 2

with 4 / h = eθ eθ h + (eθ Φ)2 eθ h = 0 if h is radial. Thus,   1 1 (1+3) (1+3) h = − (e3 e4 + e4 e3 )h + ω− trχ e4 h 2 2   1 + (1+3)ω − (1+3)trχ e3 h 2 1 = −f 00 (e3 r)(e4 r) − h0 (e3 e4 + e4 e3 )r 2      1 1 (1+3) 0 (1+3) (1+3) (1+3) + h ω− trχ e4 r + ω− trχ e3 r 2 2 m  = −h00 (−Υ + O()w0,1 )(1 + O()w1,1 ) + 2 + O()w1,1 h0 r "  m Υ + h0 + + O()w1,1 (1 + O()w1,1 ) r2 r #   1 + − + O()w1,1 (−Υ + O()w0,1 ) r  
2 2m = Υh00 + − 2 h0 + O()w2,1 |h| + r|h0 | + r2 |h00 | r r which concludes the proof of Lemma D.15.

827

APPENDIX TO CHAPTER 10

In view of Lemma D.15,   2f  (X) Λ =  r  00    0
2f 2 2m 2f = Υ + − 2 + O()w3,1 |f | + r|f 0 | + r2 |f 00 | . r r r r Note that 00    0 2f 2 2m 2f + − 2 r r r r  00    0  2f 4f 0 4f 2 2m 2f 2f Υ − 2 + 3 + − 2 − 2 r r r r r r r   2Υ 00 4f 0 4f 2m 2f 0 2f f − (Υ − 1) 2 + (Υ − 1) 3 − 2 − 2 r r r r r r m
2 + O 4 |f | + r|f 0 | + r2 |f 00 | . r r 

Υ = = = Hence,

 (X) Λ

=

m 
2 00 f + O 4 w3,1 |f | + r|f 0 | + r2 |f 00 | r r

as desired. This concludes the proof of Lemma D.14.

D.4

PROOF OF PROPOSITION 10.47

In view of the following Leibniz rule which holds for any scalar f , −4 / 2 (f ψ)

=

/d?2 /d2 (f ψ) + 2Kf ψ

=

/d?2 (f /d2 ψ + eθ (f )ψ) + 2Kf ψ

=

−f 4 / 2 ψ − eθ (f ) /d2 ψ + eθ (f ) /d?3 ψ − 4 / 0 (f )ψ,

we have the following computation e4 (2 (rψ))

= e4 (r2 ψ) − e4 (e3 (r)e4 ψ) − e4 (e4 (r)e3 ψ) − 2e4 (eθ (r) /d2 ψ)

+2e4 (eθ (r) /d?3 ψ) + e4 (0 (r)ψ) r  r  = e4 (r2 ψ) − e4 (κ + A)e4 ψ − e4 (κ + A)e3 ψ 2 2 −1 ≤1 ≤2 +e4 (0 (r)ψ) + r d (Γg )d ψ 1 1 = e4 (r2 ψ) − e4 (rκe4 ψ) − e4 (rκe3 ψ) + e4 (0 (r)ψ) + r−1 Err, 2 2

where we have introduced the notation, used throughout the proof of Proposition 10.47, Err

:= r2 Γg e4 e3 ψ + rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ.

828

APPENDIX D

Next, recall that we have 2 ψ

  1 1 = −e4 e3 ψ + 4 / 2 ψ + 2ω − κ e3 ψ − κe4 ψ + 2ηeθ ψ. 2 2

We infer 0 (r)

= = = =

  1 1 −e4 (e3 (r)) + 4 / 2 (r) + 2ω − κ e3 (r) − κe4 (r) + 2ηeθ (r) 2 2  r   1 r 1 r −e4 (κ + A) + 2ω − κ (κ + A) − κ (κ + A) + r−1 d≤1 Γg 2 2 2 2 2   1 1 1 1 − e4 (rκ) + 2ω − κ rκ − rκκ + rd≤1 Γg 2 2 2 4 2 + O(r−2 ) + d≤1 Γb r

and hence e4 (2 (rψ))

1 = e4 (r2 ψ) − e4 (rκe4 ψ) − 2 +d≤1 (Γg )d≤2 ψ 1 = e4 (r2 ψ) − e4 (rκe4 ψ) − 2

1 e4 (rκe3 ψ) + e4 (0 (r)ψ) 2 1 e4 (rκe3 ψ) + e4 2



2 ψ r



+O(r−3 )d≤1 ψ + r−1 Err so that e4 (r2 ψ)

=

1 1 e4 (2 (rψ)) + e4 (rκe4 ψ) + e4 (rκe3 ψ) − e4 2 2



2 ψ r



+O(r−3 )d≤1 ψ + r−1 Err. We infer 2 (e4 (rψ)) − e4 (r2 ψ)

=

1 1 [2 , e4 ](rψ) − e4 (rκe4 ψ) − e4 (rκe3 ψ) 2 2   2 −3 ≤1 +e4 ψ + O(r )d ψ + r−1 Err. r

Next, using again 

2 ψ

 1 1 = −e4 e3 ψ + 4 / 2 ψ + 2ω − κ e3 ψ − κe4 ψ + 2ηeθ ψ, 2 2

829

APPENDIX TO CHAPTER 10

we infer [2 , e4 ]ψ

=

=

  1 −e4 [e3 , e4 ]ψ + [4 / 2 , e4 ]ψ + 2ω − κ [e3 , e4 ]ψ 2   1 1 −e4 2ω − κ e3 ψ + e4 (κ)e4 ψ + 2η[eθ , e4 ]ψ − 2e4 (η)eθ ψ 2 2   1 −e4 [e3 , e4 ]ψ + [4 / 2 , e4 ]ψ + 2ω − κ [e3 , e4 ]ψ 2    1 1 − 2e4 (ω) − − κ2 − 2ωκ e3 ψ 2 2   1 1 + − κκ + 2ωκ + 2ρ e4 ψ + 2η[eθ , e4 ]ψ + r−2 d≤1 (Γg )dψ. 2 2

Now, recall [e3 , e4 ]

=

2ωe4 − 2ωe3 + 2(η − η)eθ ,

and, in view of Lemma 2.54, the following commutation formulae for reduced scalars: 1. If f ∈ sk , 1 κ /dk f + Comk (f ), 2 1 Comk (f ) = − ϑ /d?k+1 f − (ζ + η)e4 f − kηe4 Φf − ξ(e3 f + ke3 (Φ)f ) − kβf. 2 [ /dk , e4 ] =

2. If f ∈ sk−1 , 1 κ /dk f + Com∗k (f ), 2 1 Com∗k (f ) = − ϑ /dk−1 f + (ζ + η)e4 f − (k − 1)ηe4 Φf + ξ(e3 f − (k − 1)e3 (Φ)f ) 2 − (k − 1)βf. [ /d?k , e4 ]f =

We infer [2 , e4 ]ψ   = −e4 (2ωe4 − 2ωe3 + 2ηeθ )ψ + κ4 / 2ψ      1 1 + 2ω − κ 2ωe4 − 2ωe3 + 2ηeθ ψ − 2e4 (ω) + κ2 + ωκ e3 ψ 2 4   1 1 + − κκ + 2ωκ + 2ρ e4 ψ + r−2 d≤1 (Γg )d≤2 ψ 2 2   1 1 = 2e4 (ωe3 ψ) + κ4 / 2 ψ − 2e4 (ω) + κ2 e3 ψ − κκe4 ψ + O(r−4 )d≤1 ψ 4 4 +r−2 Err =

1 1 2ωe4 (e3 ψ) + κ4 / 2 ψ − κ2 e3 ψ − κκe4 ψ + O(r−4 )d≤1 ψ + r−2 Err. 4 4

830

APPENDIX D

This implies [2 , e4 ](rψ) 1 1 2ωe4 (e3 (rψ)) + κ4 / 2 (rψ) − κ2 e3 (rψ) − κκe4 (rψ) + O(r−3 )d≤1 ψ 4 4 +r−1 Err 1 1 = 2ωe3 (e4 (rψ)) + 2ω[e4 , e3 ]rψ + κ4 / 2 (rψ) − κ2 e3 (rψ) − κκe4 (rψ) 4 4 +O(r−3 )d≤1 ψ + r−1 Err 1 1 = 2ωe3 (e4 (rψ)) + κ4 / 2 (rψ) − κ2 e3 (rψ) − κκe4 (rψ) + O(r−3 )d≤1 ψ 4 4 +r−1 Err =

and hence

=

2 (e4 (rψ)) − e4 (r2 ψ)   1 1 2 [2 , e4 ](rψ) − e4 (rκe4 ψ) − e4 (rκe3 ψ) + e4 ψ + O(r−3 )d≤1 ψ 2 2 r +r−1 Err

=

1 1 1 2ωe3 (e4 (rψ)) − e4 (rκe4 ψ) − e4 (rκe3 ψ) + κ4 / 2 (rψ) − κ2 e3 (rψ) 2  2 4 1 2 −3 ≤1 −1 − κκe4 (rψ) + e4 ψ + O(r )d ψ + r Err. 4 r

Next, we compute

=

=

  1 1 1 2 − e4 (rκe4 ψ) − e4 (rκe3 ψ) + κ4 / 2 (rψ) − κ2 e3 (rψ) + e4 ψ 2 2 4 r 1 1 1 1 − e4 (κ(e4 (rψ) − e4 (r)ψ)) − e3 (e4 (rκψ)) − [e4 , e3 ](rκψ) + e4 (e3 (rκ)ψ) 2 2 2 2   1 2 1 2 2 2 +rκ4 / 2 ψ − rκ e3 ψ − κ e3 (r)ψ + 2 e4 (rψ) + e4 rψ 4 4 r r2 +r−1 d≤1 (Γg )d≤2 ψ  1 1 1  r 1 − e4 (κ(e4 (rψ))) + e4 κ (κ + A)ψ − e3 (κe4 (rψ)) − e3 (e4 (κ)rψ) 2 2 2 2 2  1 1 − − 2ωe4 + 2ωe3 − 2(η − η)eθ (rκψ) + e4 (e3 (rκ)ψ) 2 2 4e4 (r) 1 2 1 2r 2 +rκ4 / 2 ψ − rκ e3 ψ − κ (κ + A)ψ + 2 e4 (rψ) − ψ 4 4 2 r r2 +r−1 d≤1 (Γg )d≤2 ψ,

APPENDIX TO CHAPTER 10

i.e.,   1 1 1 2 − e4 (rκe4 ψ) − e4 (rκe3 ψ) + κ4 / 2 (rψ) − κ2 e3 (rψ) + e4 ψ 2 2 4 r 1 1 1 = − e4 (κ(e4 (rψ))) + e4 (rκκψ) − e3 (κe4 (rψ)) 2  4  2 1 1 2 1 − e3 − κ − 2ωκ rψ − ωe3 (rκψ) + e4 (e3 (rκ)ψ) + rκ4 / 2ψ 2 2 2 1 1 2 2κ − rκ2 e3 ψ − rκ2 κψ + 2 e4 (rψ) − ψ + O(r−3 )d≤1 ψ + r−1 Err 4 8 r r 1 1 1 1 = − e4 (κ(e4 (rψ))) + κκe4 (rψ) + e4 (κκ) rψ − e3 (κe4 (rψ)) 2  4  4 2 1 1 2 1 −1 − e3 − κ − 2ωκ rψ − ωe3 (rκψ) + r e3 (rκ)e4 (rψ) 2 2 2 1 1 1 2 / 2 ψ − rκ2 e3 ψ − rκ2 κψ + 2 e4 (rψ) + e4 (r−1 e3 (rκ))rψ + rκ4 2 4 8 r 2κ −3 ≤1 −1 − ψ + O(r )d ψ + r Err. r We infer 2 (e4 (rψ)) − e4 (r2 ψ) 1 1 1 = 2ωe3 (e4 (rψ)) − e4 (κ(e4 (rψ))) + e4 (κκ) rψ − e3 (κe4 (rψ)) 2 4 2    1 1 2 1 −1 − e3 − κ − 2ωκ rψ − ωe3 (rκψ) + r e3 (rκ)e4 (rψ) 2 2 2 1 1 1 2 + e4 (r−1 e3 (rκ))rψ + rκ4 / 2 ψ − rκ2 e3 ψ − rκ2 κψ + 2 e4 (rψ) 2 4 8 r 2κ − ψ + O(r−3 )d≤1 ψ + r−1 Err. r Since e4 (rψ) = rΥˇ e4 ψ, this may be rewritten as 2 (rΥˇ e4 ψ) − e4 (r2 ψ) 1 1 1 = 2ωe3 (rΥˇ e4 ψ) − e4 (rΥκˇ e4 ψ) + e4 (κκ)rψ − e3 (rΥκˇ e4 ψ) 2 4 2    1 1 1 − e3 − κ2 − 2ωκ rψ − ωe3 (rκψ) + e3 (rκ)Υˇ e4 ψ 2 2 2 1 1 1 2 + e4 (r−1 e3 (rκ))rψ + rκ4 / 2 ψ − rκ2 e3 ψ − rκ2 κψ + Υˇ e4 ψ 2 4 8 r 2κ − ψ + O(r−3 )d≤1 ψ + r−1 Err. r Now, since 2 ψ

  1 1 = −e3 e4 ψ + 4 / 2 ψ + 2ω − κ e4 ψ − κe3 ψ + 2ηeθ ψ, 2 2

831

832

APPENDIX D

we have  1 1 rκ2 ψ + rκe3 e4 ψ − rκ 2ω − κ e4 ψ + rκ2 e3 ψ 2 2 

rκ4 / 2ψ

=

+r−1 d≤1 (Γb )d≤1 ψ 1 rκ2 ψ + rκe3 (r−1 e4 (rψ)) − rκe3 (r−1 e4 (r)ψ) + κκe4 (rψ) 2 1 1 2 −1 ≤1 ≤1 − κκe4 (r)ψ + rκ e3 ψ + r d (Γb )d ψ 2 2 1 1 r = rκ2 ψ + rκe3 (Υˇ e4 ψ) − rκe3 (κ)ψ + κκrΥˇ e4 ψ − κ2 κψ 2 2 4 +r−1 d≤1 (Γb )d≤1 ψ =

and hence 2 (rΥˇ e4 ψ) − e4 (r2 ψ) 1 1 1 = 2ωe3 (rΥˇ e4 ψ) − e4 (rΥκˇ e4 ψ) + e4 (κκ)rψ − e3 (rΥκˇ e4 ψ) 2 4 2    1 1 1 − e3 − κ2 − 2ωκ rψ − ωe3 (rκψ) + e3 (rκ)Υˇ e4 ψ 2 2 2 1 1 r + e4 (r−1 e3 (rκ))rψ + rκ2 ψ + rκe3 (Υˇ e4 ψ) − rκe3 (κ)ψ − κ2 κψ 2 2 4 1 2 1 2 2κ −3 ≤1 −1 − rκ e3 ψ − rκ κψ − ψ + O(r )d ψ + r Err. 4 8 r Next, we compute    1 1 1 2 1 e4 (κκ)rψ − e3 − κ − 2ωκ rψ − ωe3 (rκψ) + e4 (r−1 e3 (rκ))rψ 4 2 2 2 1 r 2 1 2 1 2 2κ − rκe3 (κ)ψ − κ κψ − rκ e3 ψ − rκ κψ − ψ 2 4 4 8 r r = κρψ + r−1 d≤1 (Γb )ψ 2 = O(r−3 )ψ + r−1 d≤1 (Γb )ψ so that 2 (rΥˇ e4 ψ) =

1 1 e4 (r2 ψ) + rκ2 ψ + 2ωe3 (rΥˇ e4 ψ) − e4 (rΥκˇ e4 ψ) − e3 (rΥκˇ e4 ψ) 2 2 1 + e3 (rκ)Υˇ e4 ψ + rκe3 (Υˇ e4 ψ) + O(r−3 )d≤1 ψ + r−1 Err. 2

Since 2 (r2 eˇ4 ψ)

= rΥ−1 2 (rΥˇ e4 ψ) − e3 (rΥ−1 )e4 (rΥˇ e4 ψ) − e4 (rΥ−1 )e3 (rΥˇ e4 ψ) +0 (rΥ−1 )rΥˇ e4 ψ + d≤1 (Γg )d≤2 ψ,

833

APPENDIX TO CHAPTER 10

we infer 2 (r2 eˇ4 ψ)

= rΥ−1 e4 (r2 ψ) + r2 Υ−1 κ2 ψ + 2rΥ−1 ωe3 (rΥˇ e4 ψ) 1 −1 1 −1 1 − rΥ e4 (rΥκˇ e4 ψ) − rΥ e3 (rΥκˇ e4 ψ) + re3 (rκ)ˇ e4 ψ 2 2 2 +r2 Υ−1 κe3 (Υˇ e4 ψ) − e3 (rΥ−1 )e4 (rΥˇ e4 ψ) − e4 (rΥ−1 )e3 (rΥˇ e4 ψ) +0 (rΥ−1 )rΥˇ e4 ψ + O(r−2 )d≤1 ψ + Err.

Now, we have 1 2rΥ−1 ωe3 (rΥˇ e4 ψ) − rΥ−1 e4 (rΥκˇ e4 ψ) 2 1 1 − rΥ−1 e3 (rΥκˇ e4 ψ) + re3 (rκ)ˇ e4 ψ + r2 Υ−1 κe3 (Υˇ e4 ψ) 2 2 −e3 (rΥ−1 )e4 (rΥˇ e4 ψ) − e4 (rΥ−1 )e3 (rΥˇ e4 ψ) + 0 (rΥ−1 )rΥˇ e4 ψ ( 3m 1− r 1 1 = 2r e4 (ˇ e4 ψ) + 2rΥ−1 ωe3 (rΥ) − rΥ−1 e4 (rΥκ) − rΥ−1 e3 (rΥκ) Υ 2 2 1 + re3 (rκ) + r2 Υ−1 κe3 (Υ) − e3 (rΥ−1 )e4 (rΥ) − e4 (rΥ−1 )e3 (rΥ) 2 ) +0 (rΥ−1 )rΥ eˇ4 ψ + Err. Also, we have

=

1 1 1 2rΥ−1 ωe3 (rΥ) − rΥ−1 e4 (rΥκ) − rΥ−1 e3 (rΥκ) + re3 (rκ) 2 2 2 +r2 Υ−1 κe3 (Υ) − e3 (rΥ−1 )e4 (rΥ) − e4 (rΥ−1 )e3 (rΥ) + 0 (rΥ−1 )rΥ 4 + O(r−1 ) + rΓb

= −r2 κκ + O(r−1 ) + rΓb . We infer (2 + κκ)(r2 eˇ4 ψ)

= rΥ−1 e4 (r2 ψ) + r2 Υ−1 κ2 ψ + 2r +O(r−2 )d≤1 ψ + Err.

In view of the wave equation satisfied by ψ, i.e., 2 ψ + κκψ

=

N,

1 − 3m r e4 (ˇ e4 ψ) Υ

834

APPENDIX D

we have 1 − 3m r e4 (ˇ e4 ψ) Υ 2 1 − 3m r rΥ−1 e4 (r(N − κκψ)) + r2 Υ−1 κ(N − κκψ) + e4 (r2 eˇ4 ψ) r Υ 1 − 3m r −4 e4 (r)ˇ e4 ψ Υ 2 1 − 3m 4m r rΥ−1 e4 (rN ) + r2 Υ−1 κN + e4 (r2 eˇ4 ψ) + eˇ4 ψ − 2r2 Υ−1 κρψ r Υ r +d≤1 (Γb )d≤1 ψ   3 2 1 − 3m 2 −1 r r Υ e4 (N ) + N + e4 (r2 eˇ4 ψ) + O(r−2 )d≤1 ψ + d≤1 (Γb )d≤1 ψ, r r Υ rΥ−1 e4 (r2 ψ) + r2 Υ−1 κ2 ψ + 2r =

=

=

from which we deduce (2 + κκ)(r2 eˇ4 ψ)

  3 2 1 − 3m r = r2 Υ−1 e4 (N ) + N + e4 (r2 eˇ4 ψ) r r Υ +O(r−2 )d≤1 ψ + Err.

Since f2 ψˇ = f2 eˇ4 ψ = 2 r2 eˇ4 ψ, r we infer (2 + κκ) ψˇ =

    f2 f2 f2 2 2 (2 + κκ)(r eˇ4 ψ) − e3 e4 (r eˇ4 ψ) − e4 e3 (r2 eˇ4 ψ) r2 r2 r2       f2 f2 f2 2 ? 2 +eθ /d2 (r eˇ4 ψ) − eθ /d3 (r eˇ4 ψ) + 0 r2 eˇ4 ψ r2 r2 r2

and hence (2 + κκ) ψˇ ( )   3 f2 2 1 − 3m −1 2 −2 ≤1 r = f2 Υ e4 (N ) + N + 2 e4 (r eˇ4 ψ) + O(r )d ψ + Err r r r Υ     f2 f2 2 −e3 e4 (r eˇ4 ψ) − e4 e3 (r2 eˇ4 ψ) r2 r2       f2 f2 f2 2 ? 2 +eθ /d2 (r eˇ4 ψ) − eθ /d3 (r eˇ4 ψ) + 0 r2 eˇ4 ψ. r2 r2 r2 Now, recall that Err is defined by Err

= r2 Γg e4 e3 ψ + rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ,

835

APPENDIX TO CHAPTER 10

so that (   3 f 2 1 − 3m 2 −1 r (2 + κκ) ψˇ = f2 Υ e4 (N ) + N + 2 e4 (r2 eˇ4 ψ) r r r Υ +O(r−2 )d≤1 ψ + r2 Γg e4 e3 ψ + rΓb e4 dψ + d≤1 (Γb )d≤1 ψ ) +rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ 

   f2 f2 2 −e3 e4 (r eˇ4 ψ) − e4 e3 (r2 eˇ4 ψ) r2 r2       f2 f2 f2 2 ? 2 +eθ /d2 (r eˇ4 ψ) − eθ /d3 (r eˇ4 ψ) + 0 r2 eˇ4 ψ. r2 r2 r2 In view of 

2 ψ

 1 1 = −e4 e3 ψ + 4 / 2 ψ + 2ω − κ e3 ψ − κe4 ψ + 2ηeθ ψ, 2 2

we have 2

r Γg e4 e3 ψ



2

= r Γg



  1 1 −2 ψ + 4 / 2 ψ + 2ω − κ e3 ψ − κe4 ψ + 2ηeθ ψ 2 2

= −r2 Γg N + rΓg e3 ψ + Γg d≤2 ψ and hence (   3 f 2 1 − 3m 2 −1 r (2 + κκ) ψˇ = f2 Υ e4 (N ) + N + 2 e4 (r2 eˇ4 ψ) + O(r−2 )d≤1 ψ r r r Υ ) +rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ 

   f2 f2 2 e (r e ˇ ψ) − e e3 (r2 eˇ4 ψ) 4 4 4 r2 r2       f2 f2 f2 2 ? 2 +eθ / d (r e ˇ ψ) − e / d (r e ˇ ψ) +  r2 eˇ4 ψ. 2 4 θ 3 4 0 r2 r2 r2 −e3

In particular, we have for r ≥ 6m0     3 2 3m 2 −1 ˇ (2 + κκ) ψ = r Υ e4 (N ) + N + 1− e4 ψˇ + O(r−2 )d≤1 ψ r rΥ r +rΓb e4 dψ + d≤1 (Γb )d≤1 ψ + rd≤1 (Γg )e3 ψ + d≤1 (Γg )d2 ψ and for 4m0 ≤ r ≤ 6m0 ,   3 −1 ˇ (2 + κκ) ψ = f2 Υ e4 (N ) + N + O(1)d2 ψ. r This concludes the proof of Proposition 10.47.

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