The Global Nonlinear Stability of the Minkowski Space (PMS-41) [Course Book ed.] 9781400863174

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The Global Nonlinear Stability of the Minkowski

Princeton Mathematical Series EDITORS: LUIS A. CAFFARELLI, JOHN N. MATHER, and HLIAS M. STEIN I. The Classical Groups by Hermann Weyl 3. 4. 8. 9. 10. II.

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Topological Analysis by Gordon T. Whybum (revised 1964) Analytic Functions by Ahlfors, Behnke, Bers, Grauert et al. Continuous Geometry by John von Neumann Riemann Surfaces by L. Ahlfors and L. Sario Differential and Combinatonal Topology edited by S. S. Cairns Convex Analysis by R. T. Rockafellar Global Analysis edited by D. C. Spencer and S. Iyanaga Singular Integrals and Differentiability Properties of Functions by E. M. Stein Problems in Analysis edited by R. C. Gunning Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss

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Copynght ©1993 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data Christodoulou, Demetrios, 1951— The global nonlinear stability of the Minkowski space / Demetrios Christodoulou and Sergiu Klainerman. p. cm. — (Princeton mathematical series :41) Includes bibliographical references. ISBN 0-691-08777-6 (CL) 1. Space and time—Mathematics. 2. Generalized spaces. 3. Nonlinear theories. I. Klainerman, Sergiu, 1950-. II. Title. III. Series. QC173.59.S65C57 1993 530.1 Ί—dc20 92-15664 CIP This book has been composed in Times, Perpetua, and Computer Modern using TgX Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Pnnted in the United States of America 10 9 8 7 6 5 4 3 2 1

n

CONTENTS

II. Bianchi Equations in Space-Time 7

The Comparison Theorem 7.1 Preliminary Results 7.2 The Electric-Magnetic Decomposition 7.3 Null Decomposition of a Weyl Field 7.4 The Null-Structure Equations of a Space-Time 7.5 Ricci Coefficients and the Vectorfields K. S. T. and O 7.6 The Statement of the Comparison Theorem 7.7 Proof of the Comparison Theorem

135 135 143 146 165 170 180 182

8

The Error Estimates 8.1 Preliminaries 8.2 Statement of the Boundedness Theorem 8.3 Proof of the Theorem

205 205 222 229

III. Construction of Global Space-Times. Proof of the Main Theorem 9

Construction of the Optical Function 9.1 Construction of the Exterior Optical Function 9.2 Interior Construction of the Optical Function 9.3 The Initial Cone C 0

261 261 275 282

10

Third Version of the Main Theorem 10.1 Basic Notations. Norms 10.2 Statement and Proof of the Main Theorem

284 284 298

11

Second Fundamental Form 11.1 Preliminaries 11.2 The Exterior Estimates 11.3 The Interior Estimates 11.4 Estimates for the Time Derivatives

311 311 320 332 337

12

The Lapse Function

341

13

Derivatives of the Optical Function 13.1 Higher Derivatives of the Exterior Optical Function 13.2 Derhatives of the Interior Optical Function

351 351 386

14

The Last Slice

411

CONTENTS

15

The Matching

443

16

The Rotation Vectorfields 16.1 Estimates in the Exterior 16.2 Estimates in the Interior

466 468 484

17

Conclusions

491

Bibliography

513

Acknowledgments

We would like to express our gratitude to C. Fefferman for his interest in our work and the precious time he spent in hearing and reading details of our proof. His comments helped us clarify obscure points and make the book easier to read. We would also like to thank S.T. Yau for his early interest and encouragement. Many thanks to W. T. Shu for reading and correcting parts of our manuscript. Finally, we want to express our gratitude to our families for their loving support given to us during the long years in which our present work has matured.

CHAPTER 1

Introduction

The aim of this book is to provide a proof of the nonlinear gravitational stabil­ ity of the Minkowski space-time. More precisely, our work accomplishes the following goals: 1. It provides a constructive proof of global, smooth, nontrivial solutions to the Einstein-Vacuum equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singu­ larities. 2. It provides a detailed description of the sense in which these solutions are close to the Minkowski space-time in all directions and gives a rigorous derivation of the laws of gravitational radiation proposed by Bondi. It also describes our new results concerning the behavior of the gravitational field at null infinity. 3. It obtains these solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the initial data set of the Minkowski space-time, and thus it establishes the global dynamic stability of the latter. 4. Though our results are established only for developments of initial data sets which are uniformly close to the trivial one, they are in fact valid in the complement of the domain of influence of a sufficiently large compact subset of the initial manifold of any "strongly asymptotically flat" initial data set. According to Einstein, the underlying geometry of space-time is that given by a pair (M, g) where M is a 3+1-dimensional manifold and g is an Einstein metric on M, that is, a nondegenerate, 2-covariant tensorfield with the property that at e a c h point o n e c a n c h o o s e 3 + 1 vectors eo, e\ , β2, e3 s u c h that g ( e a , ββ) = η α β\ α, β = 0,1,2,3 where η is the diagonal matrix with entries -1,1,1,1. The Einstein metric divides the nonzero vectors X in the tangent space at each point into timelike, null, or spacelike vectors according to whether the quadratic form (X, X) = ξαβΧαΧ0 is, respectively, negative, zero, or positive. The set of null vectors forms a double cone, called the null cone of the cor­ responding point. The set of timelike vectors forms the interior of this cone. It

2

THE GLOBAL NONLINEAR STABILITY OF THE MINKOWSKI SPACE

has two connected components whose boundaries are the corresponding com­ ponents of the null cone. The set of spacelike vectors is the exterior of the null cone, a connected open set. Any physically meaningful space-time should be time orientable, that is, one can choose in a continuous fashion a future-directed component of the set of timelike vectors. This allows us to specify the causal future and past of any point in space-time. More generally, the causal future of a set S C M, denoted by J + (S), is defined as the set of points q that can be reached by a future-directed causal curve that initiates at S.1 Similarly, J ~ ( S ) consists of the set of all points q that can be reached from 5 by a past-directed causal curve. The boundaries of past and future sets of points in M are null geodesic cones, often called light cones. Their specification defines the causal structure of the space-time, which, up to a conformal factor, uniquely determines the metric. A hypersurface M in M is said to be spacelike if its normal direction is timelike at every point on M. We denote by g the Riemannian metric induced by g on M The covariant differentiation on the space-time M will be denoted by D, while that on M will be written with the symbols D or V. Similarly, we denote by R, respectively R, the Riemann curvature tensors of M, respectively M. Recall that for any given vector fields X, Υ, Z on (M, g), D x D y - D y DχZ = R(X, Y)Z + D[XX]Z, or, in components, relative to an arbitrary frame e a , a - 0,1, 2,3, D^D q Z 7 = D q D^ + Rl 0a Z". The extrinsic curvature, or second fundamental form, of M in M will be denoted by k. Recall that if T denotes the future-directed unit normal to M, we have k t3 = -(De.r.ej) = (T, De, e^} with e x , i - 1,2,3, an arbitrary frame on M. We will use the notation Ε α β Ί s to express the components of the volume element άμη relative to an arbitrary frame. Similarly, if e t , i = 1,2,3 is an arbitrary frame on M, then e ljk = E oljk are the components of άμ Μ , the volume element of M, with respect to the frame eo =T , e\, e j , e^. The Riemann curvature tensor R of the space-time satisfies the following:

1 A differentiable curve λ(ί) whose tangent at every point is a future-directed timelike or null vector.

INTRODUCTION

Biarichi

3

Identities

The traceless part of the curvature tensor is

where the 2-tensor and scalar R are respectively the Ricci tensor and the scalar curvature of the space-time. We call this the conformal curvature tensor of the space-time. We notice that the Riemann curvature tensor has 20 independent components while the conformal curvature and Ricci tensors have 10 components each. The conformal curvature tensor is a particular example of a Weyl tensor. These refer to an arbitrary 4-tensors W that satisfy all the symmetry properties of the curvature tensor and in addition are traceless. We say that such W s satisfy the Bianchi equation if, with respect to the covariant differentiation on M, Bianchi

Equation

For a Weyl tensorfield W the following definitions of left and right Hodge duals are equivalent:

where are the components of the volume element in M. One can easily T check that is also a Weyl tensorfield and . Given an arbitrary vectorfield X, we can define the electric-magnetic decomposition of W to be the pair of 2-tensors formed by contracting W with X according to the formulas and These new tensors are symmetric, traceless, and orthogonal to X. Moreover, they completely determine W, provided that X is not null (see [CH-K]).

4

THE GLOBAL NONLINEAR STABILITY O F THE MINKOWSKI SPACE

Given a vectorfield X and a Weyl field W, Cx W is not, in general, a Weyl field, since it fails to be traceless. To compensate for this, we define its modified Lie derivative

where n is the deformation tensor of X, that is

One can associate [Be-Ro] to the conformal curvature tensor or, more generally, to any Weyl tensorfield W, a 4-tensor that is quadratic in W and plays precisely the same role for solutions of the Bianchi equations as the energymomentum tensor of an electromagnetic field plays for solutions of the Maxwell equations. Bel-Robinson

Tensor

Q is fully symmetric and traceless; moreover, it satisfies the positive energy condition, namely, is positive whenever X, Y, Z, I are futuredirected timelike vectors (see [Ch-K]) for a proof of these properties of Q). Moreover, whenever W satisfies the Bianchi equations. This remarkable property of the Bianchi equations is intimately connected with their conformal properties. Indeed, they are covariant under conformal isometries. That is, if is a conformal isometry of the space-time, that is, for some scalar fi and W is a solution, then so is W. It is well known that the causal structure of an arbitrary Einstein spacetime can have undesirable pathologies. All these can be avoided by postulating the existence of a Cauchy hypersurface in M — a hypersurface £ with the property that any causal curve intersects it at precisely one point. 2 Einstein space-times with this property are called globally hyperbolic. Such space-times are, in particular, stable causal, that is, they allow the existence of a globally defined differentiate function t whose gradient Di is timelike everywhere. We call t a time function, and the foliation given by its level surfaces a t-foliation. We denote by T the future-directed unit normal to the foliation. - In particular, 52 is a s p a c e l i k e h y p e r s u r f a c e

INTRODUCTION

5

Topologically, a space-time foliated by the level surfaces of a time function is d i f f e o m o r p h i c to a product manifold 3? x E where E is a 3-dimensional manifold. Indeed, the space-time can be parametrized by points on the slice t = 0 by following the integral curves of D i . Moreover, relative to this parametrization, the space-time metric takes the f o r m (1.0.1) where

are arbitrary coordinates on the slice The function is called the lapse function of the foliation; gl3 is its first f u n d a m e n t a l form. We refer to 1.0.1 as the canonical f o r m of the space-time metric with respect to the foliation. T h e foliation is said to be normalized at infinity if Normal

Foliation

Condition

on each leaf T h e second f u n d a m e n t a l form of the foliation, the extrinsic curvature of the leaves is given by (1.0.2) W e d e n o t e by V the induced covariant derivative on the leaves E t , and by RlJ the corresponding Ricci curvature tensor. Relative to an orthonormal f r a m e tangent to the leaves of the foliation, we have the formulas

where denotes the projection of to the tangent space of the foliation. It is convenient to calculate relative to a frame for which Since E t is three dimensional, we recall that the Ricci curvature RlJ completely determines the induced Riemann curvature tensor RtJki according to the f o r m u l a

w h e r e R is the scalar curvature . The second fundamental f o r m k, the lapse function 4>> and the Ricci curvature tensor i? t J of the foliation are connected to the space-time curvature tensor R a /? 7 6 according to the following equations: 3

This frame is called Fermi propagated.

6

THE GLOBAL NONLINEAR STABILITY O F T H E MINKOWSKI SPACE

The Structure

Equations

of the

Foliation

(1.0.3a) (1.0.3b) (1.0.3c) where dt denotes the partial derivative with respect to t, and are the components and respectively, of the space-time curvature relative to arbitrary coordinates on Equation 1.0.3a is the second-variation formula, while 1.0.3b and 1.0.3c are, respectively, the classical Gauss-Codazzi and Gauss equations of the foliation. In view of 1.0.3c, the equation 1.0.3a becomes (1.0.3d) Taking the trace of the equations 1.0.3c, 1.0.3b, and 1.0.3a, respectively, we derive (1.0.4a) (1.0.4b) (1.0.4c) where In contrast to Riemannian geometry, where the basic covariant equations one encounters are of elliptic type, in Einstein geometry the basic equations are hyperbolic. The causal structure of the space-time is tied to the evolutions feature of the corresponding equations. This is particularly true for the Einstein field equations, where the space-time itself is the dynamic variable. The Einstein field equations were proposed by Einstein as a unified theory of space-time and gravitation. The space-time (M, g) is the unknown; one has to find an Einstein metric g such that Einstein

Field

Equations

where is the tensor the Ricci curvature of the metric, R its scalar curvature, and the energy momentum tensor of a matter field (e.g., the Maxwell equations). Contracting twice the Bianchi identities we derive

INTRODUCTION

Contracted

Bianchi

7

Identities

which are equivalent to the divergence equations of the matter field

In the simplest situation of the physical vacuum, take the form Einstein-Vacuum

the Einstein equations

Equations

In view of the four contracted Bianchi identities mentioned previously, the Einstein-Vacuum equations, or E-V for short, can be viewed as a system of 1 0 - 4 = 6 equations for the 10 components of the metric tensor g. The remaining 4 degrees of freedom correspond to the general covariance of the equations. Indeed, if is a diffeomorphism, then the pairs ( M , g) and < (M, &*g) represent the same solution of the field equations. Written explicitly in an arbitrary system of coordinates, the E-V equations lead to a degenerate system of equations. Indeed, the principal part of the Ricci curvature is

Thus, for a given metric g, the symbol o^ at a point p of the space-time manifold (M, g) and a covector £ at p is the linear operator on the space covariant symmetric tensors h at p given by

where i^h is the covector at p obtained by contracting h on the left by the vector corresponding to We see that for any given £ and any other covector belongs to the null space N ( a o f the symbol at Nevertheless, the E-V equations are seen to be hyperbolic when account is taken of the geometric equivalence of metrics related by a diffeomorphism. Since any 1-parameter group of diffeomorphisms is generated by a vector field X and the infinitesimal action of the group on the space of metrics is the Lie derivative

the symbol of which is we consider the quotient by the following equivalence relation: h\ ~ hi if and only if

8

THE GLOBAL NONLINEAR STABILITY OFTHE MINKOWSKI SPACE

+ X ξ for some covector X at p. The symbol σ£ when reduced to Qξ is seen to have zero null space whenever ξ is not null. Moreover, when ξ corresponds to a nonzero null vector, choosing a null conjugate ξ to ξ, that is, another null vector in the same component of the null cone at ρ such that (ξ, ξ) = -2, we can identify with the space of all h e SiiM p ) such that i^h = 0. Then Ν(σς) is found to be S^(II), that is, the space of trace-free symmetric 2-covariant tensors on the 2-plane Π, defined to be the orthogonal complement of the plane spanned by ξ and ξ. Therefore the 2-dimensional space ^(Π) is the space of dynamical degrees of freedom of the gravitational field at a point. This discussion indicates that it must be possible to choose coordinates rel­ ative to which the E-V equations can be written as a system of nonlinear wave equations. One well-known such choice is that of "wave coordinates," 4 namely, coordinates χ μ which satisfy the wave equation ξαβΌαΌβχμ =0. In this case the principal part of Κ μι/ becomes — The wellposedness of the local Cauchy problem was proved by Y. Choquet-Bruhat (see [Brl]) in wave coordinates, yet as she pointed out later, these coordinates are unstable (see [Br2]) in the large. The problem of finding globally stable, wellposed coordinate conditions is the first major difficulty one has to overcome in the construction of global solutions to the Einstein equations. To emphasize the dynamic character of the E-V equations, it is helpful to express them in terms of the parameters φ, g, k of an arbitrary ί-foliation. Thus, assuming that the space-time (M, g) can be foliated by the level surfaces of a time function t, and writing g in its canonical form 1.0.1, the E-V equations are equivalent to the following: Constraint Equations for E-V - V,irfc = 0

(1.0.5a)

R - \ k \ + ( t r k f = 0.

(1.0.5b)

2

Evolution Equations for E-V dtgZ] = -2kl3

(1.0.6a)

dtki:i = — V 1 V j ^ + (p(R Z J + t r k k t ] —2 k i a k ^ ) .

(1.0.6b)

Indeed, the equivalence of the equations 1.0.5a, 1.0.5b, 1.0.6a, and 1.0.6b with the E-V is an immediate consequence of 1.0.3d, 1.0.4a, and 1.0.4b. 4

These are inappropriately called harmonic coordinates in the literature.

INTRODUCTION

9

Also, 1.0.4c becomes dttrk

= — Δ

φ + (# + ( t r k ) 2 ) .

(1.0.7)

Given a ί-foliation, we denote by Ε, H the electric-magnetic decomposition of the curvature tensor R of an E-V manifold with respect to T, the future oriented unit normal to the time foliation. Clearly Ε, H are symmetric, traceless 2-tensors tangent to the foliation. In view of these definitions the equations 1.0.3b and 1.0.3c become (1.0.8a) (1.0.8b) Remark that the total number of unknowns in the evolution equations 1.0.6a and 1.0.6b is 13 while the total number of equations is only 12. This discrep­ ancy corresponds to the remaining freedom of choosing the time function t , which defines the foliation. To emphasize the crucial importance of making an appropriate choice of time function, we note that the natural choice φ = 1, corresponding to the temporal distance function from an initial hypersurface, leads to finite time breakdown. This can be seen from equation 1.0.7, which becomes, in view of 1.0.5b, d t t r k = \ k \ 2 > -(trk ) 2 . We also remark that, in view of the twice-contracted Bianchi identities, if g , k satisfy the evolution equations, then the constraint equations 1.0.5a and 1.0.5b are automatically satisfied on any E i provided they are satisfied on a given initial slice Σ ίο . Therefore they can be regarded as constraints on given initial conditions for g and k. According to this an initial data set for E-V is defined to be a triplet (Σ, g, k) consisting of a 3-dimensional manifold Σ together with a Riemannian metric g and covariant symmetric 2-tensor k, which satisfy the constraint equations 1.0.5a and 1.0.5b on Σ. A development of an initial data set consists of an Einstein-Vacuum spacetime (M, g) together with an embedding i : Σ —> M such that g and k are the induced first and second fundamental forms of Σ in M. The central problem in the mathematical theory of E-V equations is the study of the evolution of general initial data sets. The simplest solution of E-V equations is the Minkowski space-time R3+1, that is, the space R 4 together with a given Einstein metric (,) and a canonical coordinate system (x°, x\x 2 ,x 3 ) such that {9α,0β) = ηαβ·,

α,/3 = 0,1,2,3.

The issue we want to address in our work is that of the global nonlinear stability of the Minkowski space-time. More precisely, we want to investigate whether

10

THE GLOBAL NONLINEAR STABILITY OF THE MINKOWSKI SPACE

Cauchy developments of initial data sets that are close, in an appropriate sense, to the trivial data set lead to global, smooth, geodesically complete solutions of the Einstein-Vacuum equations that remain close, in an appropriate, global sense, to the Minkowski space-time. We stress the fact that at the present time it is not even known whether there are, apart from the Minkowski space-time, any smooth, geodesically complete solution, which becomes flat at infinity on any given spacelike direction. Any attempt to simplify the problem significantly by looking for solutions with additional symmetries fails as a consequence of the well-known results of Lichnerowicz for static solutions 5 and Birkhoff for spher­ ically symmetric solutions. According to Lichnerowicz, a static solution that is geodesically complete and flat at infinity on any spacelike hypersurface must be flat. The Birkhoff theorem asserts that all spherically symmetric solutions of the E-V equations are static. Thus, disregarding the Schwarzschild solution, which is not geodesically complete, the only such solution that becomes flat at spacelike infinity is the Minkowski space-time. The problem of stability of the Minkowski space-time is closely related to that of characterizing the space-time solutions of the Einstein-Vacuum equations, which are globally asymptotically flat—as defined in the physics literature, space-times that become flat as we approach infinity in any direction. Despite the central importance that such space-times have in General Relativity as cor­ responding to isolated physical systems, it is not at all settled how to define them correctly, consistent with the field equations. Attempts to develop such a notion, however, have been made in the last 25 years (see [Ne-To] for a survey) beginning with the work of Bondi ([Bo-Bu-Me], [Bo]; see also [Sa]), who introduced the idea of analyzing solutions of the field equations along null hypersurfaces. The present state of understanding was set by Penrose ([Pel], [Pe2]), who formalized the idea of asymptotic flatness by adding a boundary at infinity attached through a smooth conformal compactification. However, it remains questionable whether there exists any nontrivial 6 solution of the field equations that satisfies the Penrose requirements. Indeed, his regularity assump­ tions translate into fall-off conditions of the curvature that may be too stringent and thus may fail to be satisfied by any solution that would allow gravitational waves. Moreover, the picture given by the conformal compactification fails to address the crucial issue of the relationship between conditions in the past and behavior in the future. We believe that a real understanding of asymptotically flat spaces can only be accomplished by constructing them from initial data and studying their asymptotic behavior. In addition, only such a construction can address the 5 A space-time is said to be stationary if there exists a one-parameter group of isometries whose orbits are timelike curves It is said to be static if, in addition, the orbits of the group are orthogonal to a spacelike hypersurface. 6 Namely, a nonstationary solution

INTRODUCTION

11

crucial issue of the relationship between conditions in the past and behavior in the future, an issue that the conformal compactification leaves entirely open. This is precisely the objective we set out to achieve. To bring our discussion more to the point, we have to introduce the notion of an asymptotically flat initial data set. By this we mean an initial data set (E,g,k) with the property that the complement of a finite set in £ is diffeomorphic to the complement of a ball in i? 3 (i.e., £ is diffeomorphic to R3 at infinity), and the notion of energy, linear, and angular momentum are well defined and finite. These can be unambiguously defined for the following class of initial data sets, which we will refer to as strongly asymptotically flat. We say that an initial data set (£,