The Art of Gluing Space-Time Manifolds: Methods and Applications 9783031486111, 9783031486128

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SpringerBriefs in Physics Samad Khakshournia · Reza Mansouri

The Art of Gluing Space-Time Manifolds Methods and Applications

SpringerBriefs in Physics Series Editors Egor Babaev, Department of Physics, Royal Institute of Technology, Stockholm, Sweden Malcolm Bremer, H. H. Wills Physics Laboratory, University of Bristol, Bristol, UK Xavier Calmet, Department of Physics and Astronomy, University of Sussex, Brighton, UK Francesca Di Lodovico, Department of Physics, Queen Mary University of London, London, UK Pablo D. Esquinazi, Institute for Experimental Physics II, University of Leipzig, Leipzig, Germany Maarten Hoogerland, University of Auckland, Auckland, New Zealand Eric Le Ru, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn, Wellington, New Zealand Dario Narducci, University of Milano-Bicocca, Milan, Italy James Overduin, Towson University, Towson, MD, USA Vesselin Petkov, Montreal, QC, Canada Stefan Theisen, Max-Planck-Institut für Gravitationsphysik, Golm, Germany Charles H. T. Wang, Department of Physics, University of Aberdeen, Aberdeen, UK James D. Wells, Department of Physics, University of Michigan, Ann Arbor, MI, USA Andrew Whitaker, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, UK

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Samad Khakshournia · Reza Mansouri

The Art of Gluing Space-Time Manifolds Methods and Applications

Samad Khakshournia Nuclear Science and Technology Research Institute Tehran, Iran

Reza Mansouri Department of Physics Sharif University of Technology Tehran, Iran Institute for Studies in Physics and Mathematics (IPM) Tehran, Iran

ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs in Physics ISBN 978-3-031-48611-1 ISBN 978-3-031-48612-8 (eBook) https://doi.org/10.1007/978-3-031-48612-8 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

In 1996, one of us (RM), while on sabbatical at Potsdam University, Germany, gave a series of lectures with the same title as this monograph. Thereafter came the idea of writing it up as a review and the first draft of the beginning chapters was even written down. We are very glad to see now, exactly 100 years after the first paper on the problem of gluing manifolds published by Lanczos in Berlin (1922), not far from Potsdam, and 27 years after those lectures, we were successful in finishing our longstanding effort of reviewing this very special topic with widespread application in vast areas of gravitation and cosmology. In the centennial of conceiving this “art”, it is a pleasure to dedicate this monograph posthumously to Lanczos and his student Sen, who did an excellent job in a flourishing academic atmosphere in Berlin. We would like to acknowledge the Cosmology Group at the Department of Physics, Sharif University of Technology, especially past students and new colleagues Mohammad Khorrami, Sima Ghassemi, Shahram Khosravi, and Kourosh Nozari, for all the lively discussions and contributions to the topic of this work. RM would like also to thank IPM for providing a motivating atmosphere and financial support for this work. SKh would like to thank Nuclear Science and Technology Research Institute for the support over the years. Tehran, Iran April 2023

Samad Khakshournia Reza Mansouri

v

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Definition of the Problem and Junction Conditions . . . . . . . . . . . . . . .

5

3

Gauss–Codazzi Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Darmois-Israel Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 9

4

Distributional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problems with 1-and 2-Dimensional Concentrated Sources in Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Formulation of the Distributional Method . . . . . . . . . . . . . . . . . . . .

13 13

Null Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definition of the Problem and Related Geometric Quantities . . . . 5.2 Energy-Momentum Tensor of Null Shell and the Gluing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Some Characteristic Features of Null Shells . . . . . . . . . . . . . . . . . . 5.3.1 Affine Parameter on Null Generators . . . . . . . . . . . . . . . . 5.3.2 Reparametrization of Null Generators and the Gluing Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 A Hierarchical Classification of Null Hypersurfaces Being History of Both Wave and Null Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Singular Part of Weyl Tensor and Gravitational Wave Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Influence of Null Shell and Impulsive Gravitational Wave on Matter . . . . . . . . . . . . . . . . . . . . . . .

23 25

Gluing Conditions from the Variational Principle . . . . . . . . . . . . . . . . 6.1 Action in the Presence of a Thin Shell . . . . . . . . . . . . . . . . . . . . . . . 6.2 Variation of the Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 43

5

6

15 17

28 32 32 33

34 35 37

vii

viii

Contents

7

Gluing Conditions in Einstein–Cartan Theory of Gravity . . . . . . . . . 7.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Gluing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Non-null Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48 49 49

8

Gluing Conditions in f(R) Theories of Gravity . . . . . . . . . . . . . . . . . . . 8.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Geometrical Prerequisites and Junction Requirements . . . . . . . . . 8.3 Gluing Conditions for the Generic Case: f R R R (R) = 0 . . . . . . . . 8.4 Gluing Conditions for the Special Case: f R R R (R) = 0 . . . . . . . . .

55 55 56 58 60

9

Gluing Conditions in Quadratic Theories of Gravity . . . . . . . . . . . . . . 9.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Geometrical Prerequisites and Junction Requirements . . . . . . . . . 9.3 Dynamical Gluing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Three Special Types of Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 66 67 70 73

10 Special Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Timelike Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Planar Shell in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Gluing Two Different FRW Spacetimes and Bubble Dynamics in Cosmology . . . . . . . . . . . . . . . . 10.1.3 Embedding a Spherical Inhomogeneous Region into a FRW Background Universe . . . . . . . . . . . . . . . . . . . 10.1.4 Moving Brane in the Static Schwarzschild-AdS Bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5 Gravitational Collapse of a Rotating Thin Shell in 5D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.6 Cylindrically Thin Shell Wormholes . . . . . . . . . . . . . . . . . 10.1.7 Collapsing Stars in f(R) Theories of Gravity . . . . . . . . . . 10.1.8 Gravitational Double Layers in Quadratic f(R) Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Spacelike Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Emerging a de Sitter Universe Inside a Schwarzschild Black Hole Through a Spacelike Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Null Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Smooth Gluing LTB and Vaidya Spacetimes Through a Null Hypersurface . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Coexistence of Matter Shell and Gravitational Wave on a Null Hypersurface . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Null Shells Straddling a Common Horizon . . . . . . . . . . .

79 79 79 81 85 88 91 94 96 98 101

101 104 104 107 111

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1

Introduction

Modeling natural phenomena using mathematical language is the basis for understanding nature. Now, many natural features and phenomena within spacetime are not by any means as smooth as could be defined mathematically in a way suitable to be used in a physical model. Cosmology as a dynamic and flourishing science could not exist without smearing out all non-smooth structures like planets, stars, galaxies and so on. This, however, is the beginning of a long journey called the science of cosmology, its dynamics, and astrophysics, including all different structures, using a wide spectrum of mathematics, details of which are the aim of this study. Manifolds and the simplified geometries on them used in cosmology give us a vast possibility to grasp notions like Branes, walls, domain walls, solitonic objects, D-branes, p-branes, phase transitions in the early universe, and the formation of topological defects, just to name some of our needs in physics. Early attempts to model such phenomena go back to the dawn of relativity [59, 135, 136, 187]. Since then, understanding localized matter distributions in astrophysics and cosmology has always been of interest. Realizing the difficulty of handling thick shells mathematically, it was too natural to consider the idealization of a singular hypersurface as a thin shell and try to formulate its dynamics within general relativity, though Einstein and Straus made the first attempt to implicitly use the concept of a thick shell [75] to embed a spherical star within a FRW universe (see also [119]). It was then too natural to continue studying singular hypersurfaces, started first by Sen [187], Lanczos [135, 136], and Darmois [59], a development which then was summed up by Israel [112]. The next era of intense interest in thin shells began with ideas related to phase transitions in the early universe and the formation of topological defects [126]. Strings and domain walls were assumed to be infinitesimally thin [55] (see [206] for a review), mainly due to technical difficulties. In most of the above examples, the way to make a reasonable model is to glue two separate spacetime manifolds, usually submanifolds of existing solutions to Einstein equations, at a common boundary. We may differentiate two distinct cases of boundaries: hypersurfaces without or with supporting energy-momentum. The first case is simply called a boundary surface, and the second one being a singular hypersurface is coined a thin shell or a surface layer. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_1

1

2

1 Introduction

Thin shell formalism has been used to play an important role in various dynamical contexts ranging from microscopic to astrophysical scales [91]. For instance, by applying a charged shell as an electron model, one may avoid the appearance of negative gravitational mass caused by the concentration of charge at the center [141, 202]. Macroscopically stable quark-gluon matter can also be studied with a toy model in which relativistic thin shells and the MIT bag model are combined [211]. Accordingly, the quantization of systems comprising thin shells is shown to be tractable [29, 53, 69]. On the other side, this formalism is also suitable to describe cosmic bubble dynamics and interior structures of black holes (see Chap. 10 and references therein), gravitationally-induced decoherence [98], exotic objects such as gravastars [150] and wormholes (see Chap. 10 and references therein). Moreover, gluing spacetime domains have also been considered to analyze cosmological phase transitions in the early universe, to describe cosmological voids, to construct semiclassical creation models avoiding the initial singularity of the Big Bang scenario by quantum tunneling (see Chap. 10 and references therein), etcetera. Now, gluing spacetime manifolds in a mathematically consistent way needs geometrical assumptions and leads to dynamical conditions on the gravitational and matter fields related to the gluing hypersurface. These assumptions and conditions are usually summed up under terms such as junction- or matching conditions, originally studied by Darmois [59] and Lichnerowiz [140], and much later for non-null thin shells derived by Israel [112], and later extended to null thin shells by Barrabes and Israel [23]. Although the geometrical part of these so-called conditions is just requirements inherent to the definition of a manifold, the dynamical gluing conditions depend on the gravitational field theory and in each case have to be derived independently. Today, the technology of gluing manifolds in order to model localized phenomena, being space-, time-, or light-like, shells, has grown into a standard research tool to be used widely. Having this in mind, we elaborate on this technology with the aim of having a sound foundation to be used for further research on any area of physics in which N-dimensional manifolds and their geometry, being a configuration- or momentum-space, play a role. This monograph is organized as follows: Chapter 2 is devoted to relevant concepts and definitions needed to set up the necessary ingredients for gluing spacetimes, including a brief historical survey of different junction conditions differentiating geometrical requirements from dynamical conditions. Depending on the way we choose the coordinates on manifolds at each side of the timelike/spacelike hypersurface to be glued, different approaches may be used to obtain basic equations governing the dynamics of thin shells. In the case of arbitrary coordinates, the pill-box integration of Gauss–Codazzi equations over the singular hypersurface leads us to the desired junction conditions [112]. This approach is elaborated on in Chap. 3. It is, however, possible and sometimes more feasible to use a unique coordinate system and metric on the manifolds supposed to be glued. This gives us the opportunity to use distribution-valued tensors on the glued manifold including the boundary surface, leading to a simple and aesthetic form of the gluing conditions similar to the Einstein equations [145]. This approach is introduced in

1 Introduction

3

Chap. 4. Both chapters are limited to timelike/spacelike boundary surfaces due to the intricate behavior of lightlike hypersurfaces. The case of null hypersurfaces is studied in Chap. 5. We will see there that in this case there is no other way than to start with continuous coordinates and use a distributional approach. As the induced metric on the null hypersurface is degenerate, its normal vector is at the same time tangent to it. Therefore, the extrinsic curvature is not defined uniquely. Addressing this subtle point, the necessary prescription similar to Chap. 4 is given, while highlighting some peculiar features of null shells, such as a related impulsive gravitational wave absent in timelike/spacelike shells, and the gluing freedom for a null shell placed at the horizon of static black holes. To have a deeper insight into the dynamics of boundary hypersurfaces, we use the variational principle to derive the junction conditions for gluing spacetimes through a timelike or spacelike hypersurface in Chap. 6. Gravity theories other than general relativity (GR) have always been of interest for different reasons. In Chaps. 7–9, we study the gluing technology in widely discussed alternatives to GR. Chapter 7 is devoted to the gluing conditions in Einstein–Cartan theory of gravity, taking into account the spacetime torsion. Our review covers both non-null and null hypersurfaces. In Chap. 8, we study higher-order gravity theories being motivated due to difficulties of general relativity in providing well-accepted explanations to problems such as singularities, the nature of dark matter, understanding the accelerated expansion of the universe, or quantum gravity. We constrain ourselves to the simplest proposal, the so-called . f (R) theories, where higher powers of the Ricci scalar are added to the standard Einstein–Hilbert action. The study requires special attention due to the unavoidable products of singular distributions to be handled for a consistent mathematical framework needed to derive the gluing conditions. Among several contributions, the work of Senovilla is the first to show the presence of a dipole-type term in the energy-momentum content supported on the shell. Chapter 9 is then confined to the most general quadratic theory, which is defined by the addition of terms quadratic in curvature to the standard general relativity action. Finally, in Chap. 10, we give various examples covering timelike/spacelike and null shells to illustrate the methods presented, highlighting the significance of gluing spacetimes. This monograph is not the beginning of an end! Many topics have been omitted, such as intersecting thin shells [107, 138], signature change on thin shells and its controversy [70, 80, 81, 105, 146], general hypersurfaces whose timelike, spacelike or null character can change from point to point [149, 190], thick shells and their thin shell limit [72, 90, 121], and deriving gluing conditions in more complicated higher-order gravitational theories such as . f (T ) [204] . f (R, T ) [25, 180], Palatini . f (R) theory [161], Palatini . f (R, T ) theory [181], and last but not least, the Penrose junction conditions corresponding to metrics with a delta distribution representing an impulsive gravitational wave [170].

4

1 Introduction

Conventions and Definitions: We use the signature .(− + ++), and adopt the curvature conventions of Misner, σ = Thorn, and Wheeler (MTW) [151], with a Riemann tensor defined by . Rμρν σ σ [μν,ρ + · · · , and a Ricci tensor defined by . Rμν = Rμσ ν . Greek indices run from .0 to .3 and Latin indices from .1 to .3. A semicolon indicates the covariant derivative with respect to either the four-metric of whole spacetime or to the three-metric of shell. There will, however, be no confusion because the kind of indices used makes the difference transparent. Symbol .∇ ± denotes the covariant derivative with respect to either of the metrics of partial manifolds .M± which are to be glued together. The square brackets.[F] are used to indicate the jump of any quantity. F at the layer, and bars . F is the arithmetic mean of it. As we are going to work with distributionvalued tensors, there may be terms in a tensor quantity . F proportional to some ˘ .δ-function. These terms are indicated by . F.

Chapter 2

Definition of the Problem and Junction Conditions

Addressing any question about the technology of gluing may start with any two manifolds with boundaries given they are isometric, or with a given manifold having a singular hypersurface embedded in it, depending on the physical problem we face. In any of these approaches, the isometric feature of the boundary or singular hypersurface is a trivial assumption with some tricky details depending on the timelike/spacelike or lightlike hypersurface which we will discuss here or in Chap. 5, respectively. Once the gluing manifold is assumed to be a solution to our gravity theory, being Einstein or any other generalization or modification to it, we expect other extra gluing conditions coming from dynamics. We will differentiate between isometric requirements, called junction conditions, due to geometry and what we call “gluing conditions” coming from dynamics to remedy misconceptions originating from an interchange of basic geometry and dynamics. Consider the general case of having two .C 3 orientable spacetime manifolds .M+ and .M− each decomposed into two distinct parts along non-null boundaries .∑ + and − .∑ , respectively. The null case will be studied in Chap. 5. Let us denote the distinct + − − parts of .M+ (.M− ) due to .∑ + (.∑ − ) as .M+ 1 and .M2 (.M1 and .M2 ). Coordinates α on two spacetime manifolds are defined independently as .x+ and .x−α , and the corresponding .C 2 metrics are denoted by .g +μν (x+α ) and .g −μν (x−α ). The induced metrics on the boundaries are called .g +i j (ξ+k ) and .g −i j (ξ−k ), where .ξ±k are intrinsic coordinates on .∑ ± , respectively. Now to construct a new glued manifold .M by gluing one of the distinct parts of .M+ to another part of .M− , we trivially demand the boundaries to be isometric having the same coordinates .ξ+k = ξ−k = ξ k . Then, the natural iden+ − .∑ = ∑U =: ∑ leavesUus four differentU possibilities for glued manifolds: tification +U − + + − + .M1 , . M , and . M M1 , .M1 M− M M− 2 2 1 2 2. Historically, within the context of general relativity, to glue spacetimes irrespective of the singular hypersurface being a pure boundary or a shell supporting an energy-momentum, three different sets of junction conditions have been mostly used: Darmois (D), Lichnerowicz (L), and O’Brien–Synge (OS). To have a taste of how these different related concepts have been developed in the last century (see [42]), let us elaborate on them while indicating conceptual development so far, and the updated © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_2

5

6

2 Definition of the Problem and Junction Conditions

versions to be ready building up on the vast application of the gluing technology in any theory of gravitation proposed so far. (i) Darmois junction condition (D) The Darmois junction condition [59] for gluing .M+ to .M− , or equivalently for the existence of a singular hypersurface within the glued manifold .M requires the continuity of intrinsic metric- also called first fundamental form .gi±j of .∑ inherited from both sides computed in the intrinsic coordinate system .ξ i [gi j ] = 0.

.

(2.1)

This general condition originates from the geometric property of boundaries being isometric; therefore, it is a mere requirement to ensure the glued configuration is a manifold. The boundaries may be a pure geometrical boundary or a hypersurface supporting an arbitrary matter or wave called a thin shell. From now on, we will take this continuity of 3-metric as the proper D-junction requirements/conditions. It has been used to add the continuity of the second fundamental form (extrinsic curvature tensor) of .∑ .[K i j ] = 0, (2.2) as an extra junction condition. This requirement, however, goes beyond geometry and needs to constrain dynamics; a point to be studied in the following chapters; we, therefore, do not add it to the D-requirements. An example of using both Darmois requirements for a general spacetime is the misnomer matchable spacetimes used in [84]; a term only valid in case .∑ is a boundary surface and the Einstein dynamics holds, but not in general. It has been conjectured and shown partially for special cases that the Darmois requirement (2.1) holds on any thin shell separating spacetimes containing only positive energy matter [148]. Cases where this condition may not hold are out of the scope of this monograph. (ii) Lichnerowicz junction condition (L) Assuming a unique coordinate system on the glued manifold .M, or the coordinates on both manifolds .M± be continuous at .∑ called admissible by Lichnerowicz, then [gμν ] = 0.

.

(2.3)

This very general requirement is the proper L-junction condition, implying the isometric property of the boundaries to be glued. Note that D- and L-junction requirements/conditions have the same nature and are identical in case there is a continuous coordinate system on the glued manifold. Therefore, we may use the term DL-junction requirements/conditions in general to name the isometric property of the boundary surfaces necessary to have a glued manifold, without specifying the coordinates on it. Historically it was used to add the continuity of first-order partial derivatives of the metric across .∑ as another junction condition [140]

2 Definition of the Problem and Junction Conditions

[

∂gμν ∂x α

.

7

] = 0.

(2.4)

We know, however, that this depends on the dynamics of spacetime and does not hold generally. The equivalence of proper D and L requirements has been discussed and proved by Lake using Gaussian normal- and continuous-coordinates related to it via the .(C∑2 , C 4 ) transformations [133]. (iii) O’Brien–Synge junction conditions (OS) O’Brien and Synge [195] defined hybrid conditions as having some of the D and L characteristics. Moreover, they also mixed the boundary surface and thin shell characteristics, leading to some misunderstanding in the literature [159]. These conditions were formulated for continuous coordinates. Let the boundary be given by 0 0 . x = const, where . x need not be a timelike coordinate. According to OS, the gluing is allowed if .[gμν ] = 0, (2.5) [

∂gi j ∂x 0

.

] = 0,

(2.6)

[Tμ0 ] = 0,

(2.7)

[giμ Tνi − giν Tμi ] = 0.

(2.8)

.

.

Assuming a symmetric energy-momentum tensor, the last condition being derived from (2.5) and (2.7), is therefore unnecessary. These junction conditions were revised later by Synge [196] and Kumar and Singh [130] following a discussion of Lichnerowicz junction conditions and his continuous coordinates and brought to a more covariant form called the generalized OS junction conditions by Israel [112] (see [111] for corrections). Nariai [159] was the first who gave detailed attention to the OS, and showed that conditions (2.5) and (2.6) are necessary and sufficient in order that the curvature tensor is free from delta singularities as expected for boundary surfaces. Robson [177] showed the complete equivalence of OS and L, having in mind always boundary surfaces. In the same spirit, Kumar and Singh [130] showed that conditions (2.6) and (2.7) are not independent: each of them is a necessary and sufficient condition for the other. Bonnor and Vickers [42], discussing these three different junction conditions, concluded that L and D are equivalent, but OS is stronger than the other two and is unsatisfactory in that it may rule out some physically plausible junctions. They had in mind the junction of two different Friedman models across a spherically symmetric layer, which is an example of a thin shell and not a boundary surface. To sum up, we use, in general, the notion of DL-junction requirement to stress the isometric property of the boundary surface or layer within a glued manifold without specifying the coordination on it. D-requirement is the minimum requirement for two manifolds with arbitrary coordinates to be glued. L stands for the case of continuous

8

2 Definition of the Problem and Junction Conditions

ones across the common hypersurface .∑. The OS and generalized OS, being historically interesting, are irrelevant to our further studies. There are mathematical cases of gluing manifolds without any sound dynamics based on the so-called Penrose junction which is out of the scope of this review [170]. DL-junction requirements are general enough to be used for any manifold with any dynamics different from general relativity, such as modified gravity. Extra gluing conditions coming from dynamics and depending on the physical property of the boundary have to be obtained separately. Typically, generalized Einstein gravity theories have additional complexities related to the degrees of freedom, such as Einstein–Cartan theory (Chap. 7), or higher-order equations of motion, such as f(R) theory (Chap. 8) and quadratic gravity (Chap. 9) where extra and different gluing conditions are expected.

Chapter 3

Gauss–Codazzi Approach

3.1 General Remarks The main motivation in this approach is matching two different solutions of Einstein equations along a timelike/spacelike hypersurface of discontinuity, much too similar to the corresponding well-known approach in electrodynamics. Continuity of field equations along the hypersurface is then reached by a limiting process, in contrast to the distributional method (see Chap. 4), where matching is done at the level of metric defined on the entire glued manifold containing terms with distribution-valued tensors. In either case, the result is gluing conditions for the singular hypersurface, being a pure geometrical surface or a layer, to be embedded within a glued manifold. Take a singular hypersurface, being a pure boundary or a layer, embedded in the manifold .M. The Gauss–Codazzi approach is based on a decomposition of the curvature along this hypersurface in terms of geometric quantities defined on it. The four-dimensional coordinates on the manifolds at each side of the hypersurface may be chosen freely and independently, adapted to the symmetry requirements [112].

3.2 Darmois-Israel Formalism The Darmois condition (2.1) is the geometric requirement for gluing two spacetime manifolds with a boundary to build a spacetime manifold with a specific structure reflected in features of the boundary. This is just a purely geometric necessity to fit manifolds together, meaning that distances on the junction surface are the same when measured on each side. As discussed in Chap. 2, the relation (2.2) used to be named the second Darmois junction condition is a result of the underlying dynamics of gravitation being general relativity or any other ones. Our aim in this chapter is, assuming general relativity is valid all over the glued manifold, including the singular hypersurface, to obtain conditions coming from dynamics once we assume how our glued manifold is defined based on the existing solutions of Einstein equations. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_3

9

10

3 Gauss–Codazzi Approach

Consider a non-null hypersurface .∑ along which two different spacetime manifolds are going to be glued. Define a three-bein .ei = ∂ξ∂ i tangent to .∑ having the μ

∂x

μ

components .ei |± = ∂ξ±i . The induced metric on .∑ is then given by the scalar product μ ν .gi j = gμν ei e j |± . According to D-requirement (2.1), this metric is the same on both sides of .∑: .[gi j ] = 0. The unit normal four-vectors .n μ |± are given up to a sign by μ

n μ n μ |± = ∈,

n e | = 0,

. μ i ±

(3.1)

with .∈ = +1 or .−1 depending on the time- or space-like character of .∑, respectively. Conditions (3.1) do not specify the orientation of normal, i.e. its sign. If .∑ glues a − − − part of .M+ to a part of .M− , say .M+ 2 to .M1 , we require .n μ to point from .M1 + + outwards, then .n μ points inwards over .M2 . Hence, an understanding of which side − + .n μ or .n μ points into is crucial. In the spherical symmetric case, the direction of normal is taken from smaller to larger radii; inside-outside. But in more complicated cases due care must be exercised. A slight modification of DI formalism treating both sides of matching on equal footing with the normal vectors having uniquely defined orientation, is presented in [97]. Now, the extrinsic curvature tensor of .∑ can be defined by μ

K ±i j = e i eνj ∇ μ n ν |± ) ( 2 σ μ ν ± ∂ x ± (ξ) ±σ ∂x ± (ξ) ∂x ± (ξ) = −n σ + [μν . ∂ξ i ∂ξ j ∂ξ i ∂ξ j

(3.2)

This symmetric tensor is a measure of how the hypersurface.∑ bends in the surrounding spacetime .M+ or .M− in which it is considered to be embedded. Having all the geometric prerequisites, we may go to dynamics and write 10 Einstein equations for the hypersurface in components normal and tangent to the hypersurface. In a Gaussian normal coordinates .x α ≡ (ξ i , ±n) based on .∑ (n being geodesic distance normal to .∑) with .g nn = gnn = 1 and .g ni = gni = 0, the Einstein tensor .G μν has the following components in terms of the intrinsic and extrinsic curvatures of .∑ [112]: .

G μν n μ n ν |± = .

1 2 ij (K − K i±j K ± − ∈ 3R), 2 ±

μ

G μν e i n ν |± = K i; j − K ,i± ,

μ ν ± ± i e j |± = −∈(K i j − gi j K ),n +

.G μν e

j±

( 3G

(3.3) (3.4)

) 1 l± ± 2 + K ± K lm ) , g + ∈K K + ∈ (K ij ij ± i lj lm ± 2

(3.5) where .3R and .3G i j are the Ricci scalar and Einstein tensor of the three metric .gi j of i .∑, respectively, with . K = K i . Equations (3.3) and (3.4), the so-called contracted Gauss–Codazzi equations [112, 151], act as constraints: the first one (3.3) is the so-called “Hamiltonian”- and the second one (3.4) the “ADM”-constraint. Note that these equations are valid in .M+ and .M− on taking the limits as one approaches the

3.2 Darmois-Israel Formalism

11

shell .∑. Now, to see the effect of energy-momentum tensor . Si j of .∑ on the spacetime geometry, we perform a “pill-box” integration of Einstein equations .G μν = κTμν across .∑: ∫ ∑ ∫ ∑ 1 . Sμν = lim Tμν dn = lim G μν dn, (3.6) ∑→0 −∑ κ ∑→0 −∑ where again n is the proper distance through .∑ in the direction of normal .n μ . . Sμν is the associated energy-momentum 4-tensor of the shell. The same pill-box integration on the components of Einstein tensor (3.3) and (3.4) yields, respectively: .

1 lim κ ∑→0

1 . lim κ ∑→0

∫

∑ −∑

∫

∑

−∑

G μν n μ n ν |± dn = κSμν n μ n ν = 0, μ

μ

G μν e i n ν |± dn = κSμν e i n ν |± = 0.

(3.7)

(3.8)

The physical interpretation of these equations is that no moment associated with the thin shell flows out of .∑. Therefore, . Sμν vanishes off the hypersurface .∑, which is expressed as . Sμν n ν = 0. This can be considered as the definition of a thin shell or surface layer. The energy-momentum 4- and 3-tensors are related as .

μ

S μν = S i j ei eνj .

(3.9)

Similarly, the corresponding 4-dimensional tensor is associated with the 3-tensor . K i j defined on .∑: μ μν .K = K i j ei eνj , (3.10) satisfying .

K μν n ν = 0.

(3.11)

The pill-box integration of the (ij) components of the Einstein tensor (3.5) leads to the following non-vanishing result ∫ ∑ μ G μν ei eνj dn = lim ∑→0 −∑ ( ) −∈ [K i j ] − gi j [K ] = κSi j ,

(3.12)

where in the passage to limit.∑ → 0, the boundedness of the bracket term in the righthand side of (3.5) has been used. This equation relating surface energy-momentum tensor to the jump of extrinsic curvature on .∑ is called the Lanczos equation after the pioneering work of Lanczos and his student Sen early 1920s [135, 145, 187]. Now, substituting the Hamiltonian constraint (3.3) into the Einstein equation and taking the difference of .∑ on both sides using the Lanczos equation, we obtain the evolution identity

12

3 Gauss–Codazzi Approach .

S i j K i j = ∈[Tμν n μ n ν ],

(3.13)

where .[AB] = A+ |∑ [B] + [A]B − |∑ has been used. Similarly, the ADM constraint (3.3) gives the conservation identity .

μ

S i; j = −∈[Tμν ei n ν ]. j

(3.14)

Equations (3.13) and (3.14), being derived directly from the momentum and Hamiltonian constraints imposed on .∑, are identities satisfied throughout the time evolution of the shell and not genuine dynamical equations [156]. Dynamics of the shell is encoded in the Lanczos equation (3.12). The absence of the gravitational constant .G from the identities (3.13) and (3.14) is a sign of their geometric or conservative nature [11]. Noting that the normal component of the energy-momentum tensor is pressure acting on the shell, one may interpret equation (3.13) as the equality of net inward normal pressure on the shell to a combination of geometrical quantities and surface energy-momentum of the shell; it can also be interpreted as a general relativistic analogue to Kelvin’s relation [186]). Specifically, in the case of a spacelike transition layer, Eq. (3.13) gives the equality of change in energy density with the negative work done by impulsive stress. Equation (3.14) indicates the change of momentum density in the continuum resulting from an impulsive stress gradient. In contrast, for a timelike shell, the time component of (3.14) gives the energy conservation equation. To summarize, the following steps are required to solve Einstein equations for glued spacetime manifolds through a non-null hypersurface .∑: i. Describing matter on .M± and .∑, including the equations of state, ii. Solving Einstein equations off .∑, iii. Choosing the normal vector sign in (3.1) and solving Lanczos equations on .∑ (see the applications in Chap. 10 for choosing the normal vector sign), iv. Checking evolution and conservation identities on .∑.

Chapter 4

Distributional Approach

Many physical systems, classical and relativistic, undergo a rapid transition in their state of motion- shock waves in hydrodynamics may be one of the oldest ones. Distributions, although not needed to describe the state of such systems, may be well-suited to understand their important aspects. Usually, such a description is more amenable to treatment than using smooth functions to represent the physical state. It is desirable to use distributions in non-linear dynamical equations we encounter in describing spacetime dynamics in the presence of gravity as well. In addition to hydrodynamics, distributions have also been used in classical electrodynamics being relativistic. There, we encounter rapid changes in physical quantities. One might have, for example, some charge distribution being confined to a one-or two-dimensional region of space with the length of their extra dimensions small compared with characteristic lengths of the problem. Such charge distributions can be replaced by concentrated sources, and the problem is formulated in the sense of distributions, not smooth functions. Recall that linear operations, including differentiations, make sense when applied to distributions. Hence, Maxwell equations, by virtue of their linearity in both fields and sources, make sense as equations based on distributions. This means that the machinery of distribution theory is available and applicable in electromagnetism and guarantees that Maxwell fields with distributional charge-currents make sense physically approximating our real smooth but concentrated charge density distribution. It is very desirable to have this machinery at our disposal in general relativity, where the field equations are non-linear, and the application of distributional calculus is not trivial. We will see in this chapter how distribution-valued tensors can be used in curved spacetimes.

4.1 Historical Remarks Efforts to implement distributional methods in general relativity go back to the works of Papapetrou and Treder [166] and Nariai [159]. Papapetrou and Hamoui [164, 165] then tried to formulate a general method with application to spherical symmetric © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_4

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14

4 Distributional Approach

thin layers. Their method has then been reformulated and corrected by Evans [82]. Taub [198], being mainly interested in relativistic hydrodynamics and shock waves gave a short summary of the literature on concentrated sources. Goldberg [96] tried to introduce sources represented by gravitating point particles, i.e. one-dimensional concentrated sources in spacetime to find equations of motion for these point sources. To this end, there was introduced a class of metrics, specified by their behavior on approaching a singular world line, to describe the near-field of such a particle, leaving aside the existence of a mathematical framework to formulate the matter source concentrated on one dimension in general relativity. Israel [113] and Taub [198] presented a formulation for 2-dimensional concentrated mass distribution. Attempts to define concentrated mass distributions in general relativity have been criticized by Geroch and Traschen [92] who gave an extensive and thorough analysis of the subject in general relativity. Their work is a milestone in all discussions about the validity of distributional Einstein field equations and their applications to concentrated sources. The regularity conditions for metrics, defined by Geroch and Traschen were criticized in the case of line source (see [104]). The (2+1)-dimensional case, in contrast, should not cause any problem, as far as the continuity of metric is assured. Hence, in the case of gluing manifolds along a 2+1 dimensional hypersurface the distributional tensors have been widely used to solve Einstein equations directly, without any use of Gauss–Codazzi decomposition. Although the coordinates have to be “prepared” to make them continuous at the hypersurface of discontinuity, the method may be applied with preference in special cases, and it is the only way to define gluing in case of lightlike hypersurfaces for any field equations. It is noteworthy to pay attention to the first implicit use of distribution-valued tensors to obtain conditions for a spherically symmetric layer as a solution of Einstein equations. Sen [187], a Ph.D. student of Lanczos in Berlin, used already in 1924 continuous 4-dimensional coordinates to solve this problem using implicitly the notion of distribution. Without any reference to a 3+1 decomposition, he could obtain dynamics of the layer in a form which could be written as reduced Einstein equations for a 2+1 dimensional manifold identical to what was later called Darmois–Israel junction conditions [145]). In principle, there should be no difference in the results using either of distributional and Gauss–Codazzi methods. A complete equivalence of the two methods has been demonstrated explicitly [145]. With the results of Geroch and Traschen in mind, it was shown that all dynamical and constraint equations derived by the Darmois–Israel method come out of distributional formalism without any need to define a new covariant derivative or ambiguities in defining the energy-momentum tensor of the shell (see also [120]).

4.2 Problems with 1-and 2-Dimensional Concentrated Sources in Spacetime

15

4.2 Problems with 1-and 2-Dimensional Concentrated Sources in Spacetime There have been different attempts to use the distributional method to formulate 1- and 2-dimensional concentrated sources in general relativity. Here we report on the problems in interpreting such solutions [92]. Consider the metric ds 2 = −dt 2 + dρ2 + β 2 (ρ)dφ2 + dz 2 ,

.

(4.1)

where .φ, .ρ, and .z are cylindrical coordinates with .0 ≤ φ < 2π. This is a static metric with cylindrical symmetry. Let ∫ β(ρ) =

sin( γρ ), l (ρ − l + γl tan γ) cos γ, l γ

if ρ ≤ l, if ρ > l,

(4.2)

where .l > 0 and .γ ∈ (0, π2 ). This metric is .C 1 across .ρ = l, and .C ∞ elsewhere. Substituting the above metric in the Einstein equations leads to .

− T00 = T33 =

γ2 , l2

for ρ ≤ l,

(4.3)

with all the other components of the energy-momentum tensor being zero. For .ρ > l, the spacetime is flat consisting of a portion of Minkowski spacetime with a .φ-angular deficit of.2π(1 − cos γ). Thus, the spacetime consists of a static massive fluid cylinder with a conical flat exterior metric. The mass per unit length of the cylinder, defined as the integral of this mass density over the two-surface of constant .t and .z is given by ∫ l ∫ 2π γ2 (4.4) .μ = dρ dφβ(ρ) 2 = 2π(1 − cos γ), l 0 0 which is the same as angular deficit. Now fix .γ, and consider the limit .l → 0. Then the exterior metric becomes ds 2 = −dt 2 + dρ2 + ρ2 cos γ 2 dφ2 + dz 2 ,

.

(4.5)

i.e. Minkowski spacetime with a .φ-angular deficit of .2π(1 − cos γ). The result (4.4) is independent of .l. Therefore, one can think of the source as approaching a line of mass density equal to .2π(1 − cos γ). One might regard this limit as representing a source in general relativity concentrated on a (1+1)-dimensional surface in spacetime. The equality of angular deficit and the mass per unit length is a relation between the external field and the strength of the source analogous to the relation between the jump in the normal component of the electric field and the surface density in electrostatics. Now, the question is if such isolated examples are reliable. Geroch and Traschen [92] illustrate in an example how one could construct a solution to the

16

4 Distributional Approach

Newtonian gravitation for a line distribution of mass which results in a zero external field: an unusual relationship between field and source! In the case of metric (4.5), one would like to have a distributional energy-momentum tensor and a rule to connect its proportional factor to the mass density .μ. It is, however, possible to construct internal metrics leading to a mass density strictly less than .2π(1 − cos γ) [92]. Therefore, detailed rules for the allowed limiting behavior of source and field are necessary. There is another pathological feature of metrics like (4.5) we would like to mention [125]: it has no Newtonian limit! As one goes far away from the source (.ρ → ∞), or for small .γ when the source is weak, the metric remains conically flat: test particles do not feel any gravitational field. One encounters an analogous problem with point mass in general relativity. People have tried to introduce sources representing point masses to find the equation of motion for such point particles. To this end, there was introduced a class of metrics, specified by their behavior on approaching a singular world line to describe the nearfield of such particles. We know, however, metrics in that class may not be physically realistic: any such concentration of mass would result in collapse through a horizon. The relation between Schwarzschild mass, as a parameter defining the field, and the integral over energy-momentum tensor is not defined for a point mass. In addition, the Riemann tensor is singular at the point mass. That is why, as it seems likely, there is in general relativity no mathematical framework to formulate sources concentrated on one-dimensional surfaces in spacetime. Assuming a solution for concentrated matter source of any dimension, what are the conditions that there is a Newtonian limit to it? The linearized Einstein equations may be written in the form .

L M I G G μν = κ(Tμν + Tμν + Tμν ).

(4.6)

L M stands for the linearized part of the Einstein tensor, .Tμν for the energyHere .G μν I momentum of matter in the absence of gravitation, .Tμν for the interaction of matter G with gravitation, and .Tμν for the energy-momentum tensor of the gravitational field. In the weak field limit, the second and third terms on the right-hand side can be omitted. The result is familiar Fierz–Pauli field equations for a spin-2 particle in flat Minkowski spacetime. Now, consider a linear mass distribution with an infinite extension. In the static case, the Newtonian potential is given by

.

V (ρ) = 2λLn(

ρ ), ρ0

(4.7)

where .ρ is the cylindrical radius, .ρ0 = const., and .λ is the linear mass density. 2 The gravitational energy density is then proportional to . λρ2 , which is not locally G may integrable. Hence, the energy-momentum tensor of a gravitational field, .Tμν not be a distribution, and the next approximation to the Einstein equations including distributions is meaningless. Therefore, the Newtonian limit does not exist for (1+1)dimensional concentrated mass.

4.3 Formulation of the Distributional Method

17

The same is true for a point mass. There the energy density of the gravitational field is proportional to .1/r 4 , which is not locally integrable either. This illuminates the root of ambiguities in using distributions to find solutions of the Einstein equations for concentrated sources of dimension less than three. Geroch and Traschen [92] making a detailed study of the problem, come to the conclusion that for regular metrics in general relativity, it is always possible to define the curvature tensor as a distribution. A metric .gμν is regular provided (i) its inverse .g μν exists everywhere and both .gμν and .g μν are locally bounded, and (ii) the first derivative of .gμν exists and is locally square-integrable. It turns out that metrics for thin shells of matter or radiation are regular. Their main theorem asserts that if the metric corresponding to a source concentrated on some submanifold of spacetime is regular, it has to be of dimension three. Thus, point particles and strings are not permitted as sources. We should also mention very special cases of the so-called semi-regular metrics, in contrast to the regular ones defined by Geroch and Traschen, having the curvature as a distribution [89]. These cases, however, do not change our conclusion of being able to define distributional metrics for (1+2)-dimensional thin shells and solve Einstein equations without using the Gauss–Codazzi decomposition of the manifold. We now turn to the formulation of the distributional method to solve Einstein equations for concentrated matter sources of dimension (1+2). The result should be equivalent to Darmois-Israel formalism based on the Gauss–Codazzi decomposition. For the sake of our analysis, we borrow, to a large extent, the notation introduced in Refs. [145, 172].

4.3 Formulation of the Distributional Method The problem we are going to solve has two sides: geometry and dynamics given on a spacetime manifold assuming Einstein field equations; other theories of gravity will be considered in Chaps. 7–9. We use the notion of “spacetime manifold” to indicate a gravity theory defined on it with Minkowskian signature. Therefore, the dynamics has to be valid all over it, and the manifold will retain all its familiar features even in the presence of a singular hypersurface supporting an energy-momentum tensor. To understand fully the essence of distributional formalism, let us first start with geometrical quantities we need to formulate independent of dynamics. Geometrical Quantities Consider a glued spacetime manifold .M with a singular timelike/spacelike hypersurface .∑ embedded in it; one may look at it as if being consisted of overlapping domains .M− and .M+ cut from spacetime manifolds .M1 and .M2 with correspond− + (x−σ ) and .gμν (x+σ ) in terms of independent disconnected charts .x−σ ing metrics .gμν σ and .x+ , respectively. The common boundary of the domains is called .∑. In other words, the manifolds .M− and .M+ are glued together along the timelike or spacelike hypersurface .∑. Now, assuming a continuous coordinate system .x σ -distinct from .x±σ covering the overlap and reaches into both domains- we can define .n μ to be the unit

18

4 Distributional Approach

normal vector on .∑ pointing in the direction from .M− to .M+ . Let .l(x μ ) denote the proper distance for timelike hypersurface .∑ (or proper time for spacelike hypersurface .∑) along the geodesics perpendicular to .∑, and choose .l to be zero at .∑, negative in the domain .M− , and positive in .M+ . The displacement away from .∑ along the geodesics parametrized by .l is given by .d x μ = n μ dl, and .n μ = ∈∂μl, where .∈ is either .+1 or .−1 depending .n μ is a spacelike or timelike vector, respectively, i.e., μ .n μ n = ∈. Introducing the Heaviside distribution function .θ(l), equal to .0 if .l < 0, and .+1 if .l > 0, we note the following identities θ2 (l) = θ(l),

θ(l)θ(−l) = 0

.

d θ(l) = δ(l), dl

(4.8)

where .δ(l) is the Dirac distribution function. We must also note that the nonlinear products .θ(l)δ(l) and .δ(l)δ(l) are not defined as distributions.1 By applying the coordinate transformations .x±σ = x±σ (x λ ) on the corresponding domains, a pair of − + (x σ ) and.gμν (x σ ) is formed over.M− and.M+ , respectively, each suitably metrics.gμν 3 smooth(say .C ) [149, 190]. Let us assume that a set of coordinates .ξ i are defined on both sides of .∑, where Latin indices run from 1 to 3. The relevant jumps on μ μ .∑ expressed in the continuous coordinates . x must vanish .[x μ ] = [n μ ] = [ei ] = μ ∂x μ [∈] = 0, where .ei = ∂ξi are basis vectors on .∑. Referring to the work of Geroch and Traschen [92], we are now allowed to express the metric .gμν (x σ ) over .M as a distribution-valued tensor: g

. μν

+ − = θ(l)gμν + θ(−l)gμν ,

(4.9)

± where the metrics .gμν are expressed in the continuous coordinates .x μ . It must be verified that geometrical quantities constructed from (4.9), including the Riemann curvature tensor, can be properly defined as distributions. By virtue of the identities d . θ(l) = δ(l) and .n μ = ∈∂μ l, the partial derivatives of .gμν can be written in the form dl

g

. μν,σ

+ − = θ(l)gμν,σ + θ(−l)gμν,σ + ∈δ(l)[gμν ]n σ .

(4.10)

The appearance of a term proportional to .δ(l) in these derivatives is problematic. Computing Christophel symbols associated with the distribution-valued metric tensor .gμν , generates products of the form .δ(l)θ(l), which are undefined [ ]in the distribution formalism. These pathological terms, however, vanish for . gμν = 0, which is the L-requirement discussed in Chap. 2 and holds for any glued manifold with continuous coordinates .x σ . This use of continuous coordinates across the glued manifold does not limit the applicability of distributional formalism and may be treated as merely a gauge condition imposed on the coordinate system. Using μ ν μ ν .[gμν ]ei e j = [gμν ei e j ] = 0, we obtain

Notice that some authors use .θ(l)δ(l) = 21 δ(l) as a distributionally valid property. For this case, see Chaps. 8 and 9.

1

4.3 Formulation of the Distributional Method .

[ ] gi j = 0,

19

(4.11)

implying that the three-dimensional induced metric is the same on both sides of .∑, as clearly required if two spacetimes .M− and .M+ are to be tailored together along the hypersurface .∑. This D-junction condition (4.11) is expressed independently from μ μ .[x ] = 0 to Eq. (4.11) the coordinates .x± and .x μ . Adding four coordinate [ conditions ] we arrive at ten expected L-junction conditions . gμν = 0. Although the four-metric is continuous on .∑, its[derivatives, and so the corre] sponding connections, are discontinuous. Imposing . gμν = 0 into Eq. (4.10), one can now construct the connections without giving rise to undefined terms as −ρ [ ρμν = θ(l)[ +ρ μν + θ(−l)[ μν ,

.

(4.12)

± where .[ ±ρ μν are the Levi–Civita connections on .M , respectively, constructed from ± .gμν . Taking the partial derivative of Eq. (4.12), one obtains −ρ ρ [ ρμν,σ = θ(l)[ +ρ μν,σ + θ(−l)[ μν,σ + ∈δ(l)[[ μν ]n σ .

.

(4.13)

Then, using (4.13), the Riemann tensor associated with the connection (4.12) is calculated to be .

R αμσν = R +α μσν θ(l) + R −α μσν θ(−l) + R˘ αμσν δ(l),

(4.14)

± where . R ±α μσν are the Riemann tensors on .M , respectively, and

.

R˘ αμσν = ∈([[ α μν ]n σ − [[ αμσ ]n ν ).

(4.15)

We see that the Riemann curvature tensor can be defined as a distribution including a .δ-function singular term. We now try to obtain its explicit expression. The metric continuity condition .[gμν ] = 0 in the continuous coordinates .x μ guarantees that the tangential derivatives of the metric are continuous: .[gμν,σ ]eaσ = 0. Therefore, any discontinuity of the metric has to be directed along the normal vector .n α : [gμν,σ ] = γμν n σ ,

.

(4.16)

where the tensor .γμν is given explicitly by .γμν = ∈[gμν,σ ]n σ . Now, the discontinuity of Christoffel symbols across .∑ is given by [[ σμν ] =

.

1 σ (γ n ν + γ σν n μ − γμν n σ ). 2 μ

(4.17)

Substituting Eq. (4.17) into (4.15), the singular part of the Riemann tensor takes the explicit form

20

4 Distributional Approach

∈ R˘ αμσν = − (γ ασ n μ n ν − γνα n μ n σ + γμν n α n σ − γμσ n α n ν ) . 2

(4.18)

Contraction of Eq. (4.18) yields the singular part . R˘ μν of the Ricci tensor ∈ R˘ μν = − (γn μ n ν − γν n μ + ∈γμν − γμ n ν ) , 2

(4.19)

where .γ = γμν g μν and .γν = γμν n μ . Similarly, the singular part . R˘ of the Ricci scalar is obtained to be R˘ = g μν R˘ μν = −∈(∈γ − γ † ),

(4.20)

where.γ † = γμ n μ = γμν n ν n μ . Then, one arrives at the following form for the singular part .G˘ μν of the Einstein tensor: 1 G˘ μν = R˘ μν − gμν R˘ 2 ∈ = − (γn μ n ν + gμν γ † − γν n μ − γμ n ν + ∈γμν − ∈γgμν ). 2

(4.21)

Note that the expression (4.21) is invariant under the gauge transformation ' γμν → γμν = γμν + λμ n ν + λν n μ ,

(4.22)

where .λμ are components of an arbitrary vector field over .∑. Under the gauge transformation (4.22) the following quantities transform as '

γ † = γ † + 2∈λν n ν

.

'

γ μ = γ μ + ∈λμ + λν n ν n μ ,

(4.23)

while .γ † − ∈γ remains invariant. Therefore, in general, it is seen that .γ μ ≡ γ μν n ν (and so .γ † ) can be transformed to zero by the gauge transformation (4.22) for any non-lightlike .∑(∈ /= 0). We emphasize again that all expressions above are purely geometric, independent of any field equations, and thus valid in any theory of gravity based on a Lorentzian manifold. Dynamics Now, for the Einstein equations to be valid across the glued manifold including the surface layer .∑ with a singular part of the Einstein tensor (4.21), the energymomentum .Tμν has to have the following form: T

. μν

+ − = Tμν θ(l) + Tμν θ(−l) + Sμν δ(l),

(4.24)

± where.Tμν are the energy-momentum tensors on.M± , respectively. Therefore. Sμν has to be the four-dimensional energy-momentum tensor describing matter concentrated

4.3 Formulation of the Distributional Method

21

on .∑. The resulting singular part of the Einstein equations may then be written as ˘ μν = 8πSμν , .G (4.25) giving the dynamics of .∑. Any information regarding the possibility of gluing must therefore be simply a result of it. Hence, the use of distribution-valued tensors without any presumptions gives us a very simple and familiar result to understand the dynamics of embedded shells in any application of general relativity. The equation (4.25), having the elegant form of the Einstein equations, is the final form of the so-called junction conditions using the distributional approach and continuous coordinate system [145]. Historically, it was first derived for the timelike case of a spherically symmetric solution of Einstein equations, without using the hitherto unknown distributional calculus, by Sen [187]. Hence, authors in [145] coined it the Sen equation, which may also be called Sen junction conditions. Substituting the expression (4.21) for the singular part of the Einstein tensor, the Sen equation takes the explicit form 1 8π∈Sμν = − (γn μ n ν + gμν γ † − γν n μ − γμ n ν + ∈γμν − ∈γgμν ). 2

(4.26)

It is worth noticing that . Sμν is tangent to the hypersurface .∑: S n μ = 0.

(4.27)

. μν

The Sen junction conditions may also be written in the more familiar form. Based μ on the decomposition . Si j = Sμν ei eνj , the projection of (4.26) onto .∑ leads to the following expression for the three-dimensional intrinsic energy-momentum of the shell: 16πSi j = −∈γ † gi j − γi j + γgi j , = −γi j + γgi j ,

(4.28)

where in the second line .γ † has been transformed to zero by the gauge transformation (4.22). On the other hand, the extrinsic curvature tensor . K i j may now be introduced with the jump across .∑ given by μ

[K i j ] = [∇μ n ν ]ei eνj , μ

= −[[ σμν ]n σ ei eνj , ∈ = γi j , 2

(4.29)

where Eq. (4.17) has been used. Using this equation for the jump of extrinsic curvature and the final form of Eq. (4.28), one ends up with the Lanczos equation:

22

4 Distributional Approach

[K i j ] − gi j [K ] = −8π∈Si j ,

.

(4.30)

where . K = g i j K i j is the trace of . K i j . It is important to note that the junction condiμ tions (4.30) are independent of the coordinates .x± and .x μ . A smooth gluing of two spacetimes at .∑ requires .[K i j ] = 0. It can be shown that the full Riemann tensor is then nonsingular at .∑, meaning that . R˘ αμσν = 0. Therefore, we have seen the explicit method of writing Einstein equations for a regular metric being continuous on .∑ without having continuous derivatives leads to Eq. (4.30), which is equivalent to the DI-method based on the Gauss–Codazzi formalism. It has also been shown that the jump conditions are a consequence of the Bianchi identities and, therefore, implicit in the formalism once the Lichnerowicz conditions are satisfied [145]. The requirement of coordinates to match continuously on the shell, giving a deep insight into the problem of glued manifolds, is not always a disadvantage. In dealing with complicated cases like cylindrical spacetimes with rotation, the DI method turns out to be very cumbersome to apply, while using the Sen equation is more straightforward. We will also see in Chap. 5 for null shells that the distributional approach is the only recommended way of understanding how to deal with such glued manifolds. Therefore, depending on the problem we are looking to solve, it may well pay off the disadvantage of choosing continuous coordinates across the shell [125, 169].

Chapter 5

Null Shells

Null hypersurfaces are peculiar. The normal vector .n μ to a null hypersurface needed to be used in understanding the dynamics of an embedded hypersurface, as we have seen in the last two chapters, is at the same time tangent to it. As a result, the extrinsic curvature lacks information about the extrinsic geometry of a null hypersurface, i.e. about its embedding in a four-dimensional spacetime. Thus, the extrinsic curμ vature tensor defined by . K i j = ei eνj ∇μ n ν turns out to be a tangential derivative; it is necessarily continuous across the null hypersurface and cannot be related to the energy-momentum tensor of the shell, as done for timelike/spacelike shells. Degeneration of the induced metric to a matrix of rank .2, and the non-triviality of defining the normal distance to the null shell are other complications one has to face in the case of null shells. The definition of connection, Riemann tensor, and their jumps across the null hypersurface also needs more attention. In the case of timelike/spacelike shells, according to the Lanczos equation, there is a one-to-one correspondence between the 3- or 4-dimensional expressions of the surface energy-momentum tensor and jump of the extrinsic curvature across the shell; a fact that is not so trivial and does not hold generally is the case of null shells. A null shell is generally accompanied by an impulsive gravitational wave such that the null hypersurface .∑ is at the same time the history of both matter and wave. As an example, one may model an astrophysical event such as a supernova explosion producing a burst of neutrinos together with a burst of outgoing gravitational wave by a null shell accompanied by an impulsive wave. Interest in null hypersurfaces intensified after general relativity with its astrophysical and cosmological applications flourished with the discovery of cosmic microwave radiation in 1965. Dautcourt [60] analyzed null hypersurfaces of discontinuity. Penrose in a classic paper [168] introduced the “cut-and-paste”method to construct null thin shells. In this way, he was able to study certain classes of impulsive planefronted and spherically-fronted waves in a Minkowski background. Baston [26, 27] considered shock waves in both general relativity and Yang–Mills theories. Redmount [175] studied the construction of a wormhole model with two Schwarzschild

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_5

23

24

5 Null Shells

geometries glued at a lightlike hypersurface. Clarke and Dray [51] investigated junction conditions for null hypersurfaces. Berezin et al. [33] studied spherically symmetric null shells. Barrabès [11] presented a unified description of hypersurfaces of discontinuity. However, to get a definite and succinct description of null shells one had to wait until the nineties last century. The work of Barrabès and Israel(BI) in 1991 [23] is the start of a new era of comprehending cases related to gluing manifolds through a null shell. Later, a reformulation of BI formalism was presented by Poisson in [171] to make it more user-friendly for typical applications. Using distribution-valued tensors, as described in the previous chapter was essential to this development. The following cases are examples of using the BI formalism for lightlike shells: Modeling an exploding white hole [12], investigating conditions for nucleation of a de Sitter expanding universe inside a Schwarzschild black hole [16], calculating the impulsive gravitational waves propagating due to (i) abrupt changes of the multiple moments in the axisymmetric Weyl solutions as a model for supernova explosion [15], (ii) jumps in mass and angular momentum in Kerr spacetime supposing to model pulsar glitches [14]; speculating a spherical null shell or “relativistic fireball” radiating outwards due to the sudden deceleration of Schwarzschild black hole to a complete stop [18, 19]; analyzing the detection of an impulsive null signal by a system of neighboring test particles [20, 21]. A description of the event horizon of a perturbed Schwarzschild black hole in terms of the intrinsic and extrinsic geometries of the null hypersurface was given in [203]. Frolov et al. constructed a hybrid black hole by gluing the external Kerr metric and the internal Weyl-distorted metric along their common horizon represented by a null shell [85]. Blau et al. [40] revisited the BI formalism of null shells with a particular emphasis on the gluing freedom for null shells placed at horizons of black holes (see also [35–37]). Gluing the LTB and FRW spacetimes through a null shell studied in [122]. The model of universe as a black hole and the collapse of a rotating null shell in the cosmic string spacetime were investigated using the BI null shell formalism in [116, 118], respectively. Manzano and Mars [147] showed that the original cut-and-paste construction by Penrose, namely the plane-fronted impulsive wave in Minkowski spacetime fits into the standard gluing procedure and, hence, can also be understood in terms of it. Musgrave and Lake [157] developed the computer package GRJunction based on the BI formalism to study null shells in general relativity. As lightlike discontinuities propagate along characteristics of the field equations, it is a nontrivial matter to disentangle the lightlike discontinuities due to the matter shell from the accompanying gravitational waves, a fact we have tried to deal with care in this chapter for Einstein’s gravity. Thus far, the concept of singular null hypersurfaces has also been extended to the case of scalar-tensor theories of gravity [13, 45] as well as the Einstein–Cartan theory [46, 124]. Later, Barrabes and Hogan [22] presented a description of singular hypersurfaces in the Einstein–Gauss–Bonnet theory of gravitation. Their work is aimed to apply both timelike and lightlike hypersurfaces in Brane cosmology (see also [106] and references therein). Very recently, null shells in quadratic gravity have been studied utilizing the variational principle [34].

5.1 Definition of the Problem and Related Geometric Quantities

25

We are now going to review the formalism of glued manifolds across a null hypersurface taking care of subtleties inherent to the null character of the singular hypersurfaces.

5.1 Definition of the Problem and Related Geometric Quantities Consider first the geometric problem of how to formulate quantities related to a glued manifold .M including a null hypersurface. Take the null hypersurface .∑ to be the isometric gluing of two null hypersurfaces .∑ + and .∑ − bounding two distinct spacetimes .M+ and .M− as parts of the glued manifold .M, respectively. Let us adopt the convention that .M− is in the past of .∑, and .M+ in its future. The notion of inside and outside is now better described by indicating whether the region being glued is to the past or future of .∑. Therefore, a null hypersurface is always a “oneway membrane”such that any future-directed causal curve can move from region − + .M to .M , but not in the reverse way. Coordinates on two spacetime manifolds, μ i.e. on both sides of the bulk, may be defined in general independently as .x+ and μ μ μ + − . x − , and the metrics denoted by .gαβ (x + ) and .gαβ (x − ). The induced metric on the null hypersurface is called .gi j (ξ k ), where .ξ k are intrinsic coordinates on .∑. As seen from μ μ i ± .M , .∑ is described by a set of parametric equations . x ± = x ± (ξ ). Now, handling null shells is too tricky to start with any form of a 3+1 decomposition. In fact, we will see in the following that the 4-dimensional Sen equation includes more information than its 3-dimensional counterpart. Therefore, we better assume from now to have continuous coordinates across the whole glued manifold .M, as if we are handling a single spacetime with a null hypersurface .∑ embedded in it. This makes the Ljunction condition (.[gμν ] = 0) trivial. The use of a distribution-valued metric tensor is of no concern, as we have already seen in the last chapter. Now, defining the μ μ three-bein .ei = ∂ξ∂ i tangent to .∑ having components .ei = ∂∂ξx i , the induced metric μ ν .gi j = gμν ei e j on .∑ is the same on both sides of the hypersurface, i.e. the D-junction is also satisfied: [gi j ] = 0.

(5.31)

.

For a null hypersurface .∑, its normal vector .n μ satisfies n n μ = 0,

. μ

μ

n μ ei = 0,

(5.32)

where we have dropped the.± subscripts so as not to overload the notation. Any vector orthogonal to .n μ is tangent to .∑. According to (5.32), .n μ is orthogonal to itself as well as tangent to .∑, hence does not convey information about the way in which the hypersurface is embedded in the surrounding spacetime manifold. Therefore, it cannot be used to define extrinsic geometrical entities as in the case of non-null

26

5 Null Shells

hypersurfaces. To extract information about the geometry of embedding the null hypersurface in .M± , it is appropriate to introduce a transverse vector . N μ defined on .∑ with equal projections on both sides of .∑: μ

[Nμ ei ] = 0,

.

[Nμ N μ ] = 0.

(5.33)

Equation (5.33) are needed to have a geometrical quantity defining the embedding. To ensure that . N is transverse to .∑, it must have a nonzero scalar product with the normal .n: μ −1 . N nμ = η = −1. (5.34) where .η is some nonzero constant practically taken to be -1. Notice that the minus sign amounts to requiring . N and .n to be directed toward the same side of the null hypersurface .∑, e.g., both future-directed in our case. From (5.34), we see that . N μ is determined up to a tangential displacement . N |→ N + λi (ξ j )ei with arbitrary functions .λi . Notice that any displacement along a generator necessarily gives .ds 2 = 0. Hence, the induced metric.gi j on the null hypersurface is effectively two-dimensional and degenerate. Therefore, its inverse as a raising indices operator cannot be defined. ij Let us introduce the pseudo-inverse symmetric tensor .g∗ by μ

g ik g jk = δ ij + n i Nμ e j ,

. ∗

(5.35)

ij

where one can easily see that .g∗ gi j = 2. The 3-vector .n i is given by decomposing μ the normal .n μ in terms of the basis .{N μ , ei }: n μ = n i eiμ .

.

(5.36)

μ

Since .n μ ei = 0, this decomposition leads to g n j = 0.

. ij

(5.37)

ij

As a result, we see that .g∗ is not uniquely defined by (5.35). Applying a transforij ij mation like .g∗ → g∗ + 2χn i n j , where .χ is an arbitrary function, doesn’t change μ (5.35). The basis .{N μ , ei } gives the following completeness relation: g μν = g∗i j eiμ eνj − 2n i ei(μ N ν) .

.

(5.38)

We are now ready to define expressions related to embedding and other geometric quantities we need to define the dynamics, being Einstein equations or any other generalization or modification of it. Let us introduce first a continuous coordinate system .x μ in .M including the null hypersurface .∑. The parametric equation of .∑ is then written as .Φ(x μ ) = 0, where .Φ is a smooth function and the domains in which + .Φ is positive or negative are contained in .M or .M− , respectively. The normal μ −1 vector .n is then defined by .n μ = α ∂μ φ, with .α being a function on .∑, non-zero

5.1 Definition of the Problem and Related Geometric Quantities

27

but otherwise arbitrary with its sign negative in order to have .n μ future-directed. We will see that the function .α will play a role in extracting physical quantities on the matter shell observed by a congruence of timelike observers; again a distinction to the timelike/spacelike case where the normal vector can be normalized. We expect the relevant jumps on .∑ expressed in the continuous coordinates .x μ to be vanishing: μ μ μ .[gμν ] = [n ] = [ei ] = [N ] = [α] = 0. Note that the condition .[gμν ] = 0 implies that the tangential derivatives of the metric are continuous: .[gμν,σ ]eiσ = 0. However, the transverse derivative of the metric, namely .gμν,σ N σ , may be discontinuous. We may use this discontinuity to define the tensor field .γμν by γ

. μν

= [gμν,σ ]N σ .

(5.39)

It then follows that 1 σ [[μν ] = − (γμσ n ν + γνσ n μ − γμν n σ ). 2

.

(5.40)

Note that in the very trivial case of vanishing transverse derivative of the metric, i.e. in case of smooth gluing with a pure hypersurface without any extra support of energy-momentum on it, this .γμν is not defined. Following steps in the distributional method described in the previous chapter, we derive the Riemann tensor containing a singular part given by α −1 R˘ αμσ ν = [[ α μν ]n σ − [[ αμσ ]n ν .

.

(5.41)

Substituting (5.40) into (5.41), one gets 1 α (γ n μ n ν − γν α n μ n σ + γμν n α n σ − γμσ n α n ν ) . 2 σ

α −1 R˘ αμσ ν =

.

(5.42) Similarly α −1 R˘ μν =

.

1 (γ n μ n ν − γνσ n σ n μ − γμσ n σ n ν ) , 2

(5.43)

and also the Ricci scalar α −1 R˘ = α −1 g μν R˘ μν = −γσ ν n σ n ν .

.

(5.44)

The singular part .G˘ μν of the Einstein tensor then takes the form 1 (γ n μ n ν − γνσ n σ n μ − γμσ n σ n ν + gμν γρσ n ρ n σ ) 2 1 = (γ n μ n ν − γμ n ν − γν n μ + γ † gμν ), 2

α −1 G˘ μν =

.

(5.45)

28

5 Null Shells

where γ μ = γ μν n ν ,

γ † = γ μnμ,

.

γ = γμν g μν .

(5.46)

5.2 Energy-Momentum Tensor of Null Shell and the Gluing Conditions We now assume the Einstein field equations to be valid all across the manifold .M, including the null hypersurface. Given that the left-hand side includes a singular term proportional to the Dirac distribution, we conclude that the energy-momentum tensor must also have a singular term written as T˘

. μν

= ηαSμν ,

(5.47)

where .η = −1, and . Sμν is the surface energy-momentum tensor of the null shell expressed in the continuous coordinates .x μ . Now, the Sen equation, .G˘ μν = −ακ Sμν , trivially valid due to the distributional formalism reflects the gluing conditions on the null hypersurface. Substituting Eq. (5.45) into the Sen equation for null shell .∑, we arrive at the following expression for the 4-dimensional energy-momentum tensor of the null shell: 16π S μν = −γ n μ n ν − γ † g μν + γ μ n ν + γ ν n μ ,

.

(5.48)

Note that . S μν given by Eq. (5.48) is invariant under the gauge transformation (4.22). However, in contrast to timelike/spacelike shells, we can no longer gauge .γ μ to zero for null shells .(∈ = 0), indicating physically significant information. It is also seen from Eq. (5.48) that the surface energy-momentum tensor . S μν is symmetrical and tangent to the null hypersurface .∑ as expected1 : .

S μν n ν = 0.

(5.49)

Using the completeness relation (5.38), one can decompose the vector .γμα n μ : γ α nμ =

. μ

1 σ μ μ (γ − g∗i j γμν ei eνj )n α − γ † N α + (g∗i j γμν e j n ν )eiα . 2 σ

(5.50)

Substituting this into Eq. (5.48) while using once more the completeness relation (5.38), we finally end up with the following expression for the energy-momentum tensor of the shell in terms of geometric quantities:

Notice that in the presence of torsion, within the Einstein–Cartan theory, . S μν is no longer symmetric, as mentioned in Chap. 7.

1

5.2 Energy-Momentum Tensor of Null Shell and the Gluing Conditions .

μ

S μν = μn μ n ν + pg∗i j ei eνj + ∏μν ,

29

(5.51)

where the 6 physical quantities related to the null shell in 4 dimensions are 16π μ = −γ − 2γμν n (μ N ν) , † .16π p = −γ , μν .16π ∏ = g∗i j eαj n β (n μ eiν + n ν eiμ )γαβ . .

(5.52) (5.53) (5.54)

Quantities .μ, . p, and .∏μν are used to be identified as the mass density, isotropic pressure, and anisotropic stress related to the existence of null shell, respectively: any physical information extra to the mass density and pressure associated with .∑ is included in.∏μν , representing the 4-dimensional anisotropic stress of the null shell. To have a good understanding of this fact, a detailed study of 3-dimensional geometrical entities and their tricky behavior is required. Note that, although the surface energymomentum tensor (5.51) is constructed in the continuous coordinate system .x μ , being a tensorial equation, it can be expressed in any coordinate system. Particularly, viewed from .M± , the surface energy-momentum tensor can be expressed in the original coordinates .x±μ . However, surface quantities within Eqs. (5.52)–(5.54) based on the tensor .γμν given by Eq. (5.39) are defined in the continuous coordinates. Let us try to obtain the surface quantities of the null shell as functions of the shell’s intrinsic 3-dimensional coordinates. In order to do so, the following transverse extrinsic curvature .Ki j is introduced: Ki j = eiμ eνj ∇μ Nν .

(5.55)

.

Recalling that . N μ is determined up to a tangential displacement with arbitrary functions .λi , we note that .Ki j is not uniquely defined. However, its jump across the null hypersurface is independent of functions .λi in the choice of transverse vector . N μ : μ

μ

σ [Ki j ] = [Nν;μ ]ei eνj = −[[μν ]Nσ ei eνj 1 = γμν eiμ eνj 2 1 = γi j , 2

.

(5.56)

where we have used Eq. (5.40) and introduced the 3-tensor .γi j as the projection of γ on .∑ (.γi j = γμν eiμ eνj ). The 3-tensor .γi j is independent of the choice of transverse vector. N μ reflecting properly the embedding properties of null hypersurface.∑ having 6 independent components as.γμν . Now, we decompose the 4-dimensional tensor. S μν in the following way: μ μν .S = S i j ei eνj , (5.57)

. μν

μ

with . S i j being components of the decomposition relative to three basis vectors .ei on the hypersurface. Notice that . S i j is not simply obtained from the projected 3-

30

5 Null Shells μ

dimensional . Si j = ei eνj Sμν by raising the three indices; a fact due to the metric being degenerate. Starting from the 4-dimensional expression (5.48), using relations (5.46) and (5.57 ) together with completeness relation (5.38), one can uniquely express this so-called 3-dimensional energy-momentum tensor in terms of geometric quantities: 16π S i j = (g∗ik n j n l + n i n k g∗jl − g∗i j n k n l − n i n j g∗kl )γkl ,

.

(5.58)

Although .γkl , in general, has 6 components, the resulting . S i j has just 4. To see this, let us define [15] 1 1 γˆ = γi j − g∗kl γkl gi j + 2n k γk(i N j) + γ † Ni N j − γ † (Nμ N μ )gi j , 2 2

. ij

(5.59)

satisfying four constraints: γˆ n j = 0,

g∗i j γˆi j = 0,

. ij

(5.60)

leading to just two independent components for .γˆi j . Notice that the conditions (5.60) ij on .γˆi j are independent of the freedom in .g∗ mentioned following (5.37) above. Therefore, .γˆi j does not contribute to the surface energy-momentum tensor (5.58). That is why . S i j has just 4 components instead of 6 which one would expect in the case of a timelike/spacelike shell. While .γˆi j does not contribute to . S i j , the remaining four components of .γi j determine the four independent components of . S i j . This is a crucial difference to the timelike/spacelike cases. It will be shown in the following subsection that the two independent components of.γˆi j reflect two degrees of freedom of polarization of gravitational waves generally present in such a gluing. The expression (5.58) does not depend on the choice of transversal vector and pseudo-inverse metric. Therefore, it may be written in the following general form [106]: ij .S = μn i n j + pg∗i j + ∏i j , (5.61) with 16π μ = −g∗kl γkl ,

.

.16π p = −γ , 16π ∏i j = (g∗ik n j n l + g∗jl n i n k )γkl .

†

.

(5.62) (5.63) (5.64)

It is easily seen that the energy density.μ and isotropic surface pressure. p are the same as defined in (5.52) and (5.53), respectively. However, the 3-dimensional anisotropic stress .∏i j , with just 2 independent components does differ from his 4-dimensional counterpart .∏μν given in (5.54) having 4 independent components. The 3-dimensional energy-momentum tensor . S i j may alternatively be expressed as ij .S = μn i n j + pg∗i j + J i n j + J j n i , (5.65)

5.2 Energy-Momentum Tensor of Null Shell and the Gluing Conditions

31

with 16π J i = g∗ik γk .

.

(5.66)

Due to the absence of a rest-frame on null shell .∑, there is an ambiguity in the operational interpretation of components in Eq. (5.65) or (5.61) (as well as the 4dimensional expression (5.51)). This ambiguity, being an unavoidable feature of null shells, is associated with the congruence of timelike geodesics intersecting .∑ performing measurement on it: any such congruence is defined up to a multiplicative factor (see the parameter .α in (5.47)). As a consequence, different observers upon crossing the shell will measure re-scaled .μ, . p, and . J i , as described in [171]. A preferred and convenient choice of the 3-dimensional coordinates is.ξ i = (ξ 1 = λ, ξ A ) adapted to the null generators of .∑ with .λ indicating on them, and .ξ A (with μ . A = 2, 3) labeling the individual generators [171]. The normal vector .n may then ∂ μ be defined by .n = eλ = ∂λ . The transverse vector . N is also uniquely determined by μ four conditions . Nμ n μ = −1, . Nμ e A = 0, and . Nμ N μ = 0. Hence, both null vectors μ μ .n and . N become future-directed. Let us now make a correspondence between ij the degenerate .g∗ and .g AB being inverse of two-dimensional non-degenerate .g AB ij AB i j through .g∗ = g e A e B . The completeness relation (5.38) is then written as .g μν = μ g AB e A eνB − 2n (μ N ν) . Now, the components of . S i j (5.65) interpreted as the surface energy density.μ, pressure. p and surface energy current. J A take the following forms: 16π μ = −g AB γ AB , † .16π p = −γ , A AB .16π J = g γB . .

(5.67) (5.68) (5.69)

μ

From (5.56) we have .γ AB = γμν e A eνB = 2[KAB ], .γ † = γμν n μ n ν = 2[Kλλ ], .γ B = γμν eμB n ν = 2[KλB ]. Hence, the physical quantities of matter shell can be expressed in terms of the jumps in transverse 2-dimensional extrinsic curvature: 1 AB g [KAB ], 8π 1 [Kλλ ], .p = − 8π 1 AB A g [KλB ]. .J = 8π μ=−

.

(5.70) (5.71) (5.72)

Note that in contrast to the timelike/spacelike case for null embedded hypersurfaces, this set of Eqs. (5.70)–(5.72) does not include all gluing information and conditions. The anisotropic stresses present in .∏μν are well-distinguished in components of the Weyl tensor on .∑. Hence, the vanishing of . S i j does not lead to . S μν = 0. Vanishing of . S μν , however, leads to the vanishing of the distributional part of Einstein tensor which means there is no shell and .∑ is the history of an impulsive gravitational wave provided .γˆi j is nonzero, otherwise, .∑ is just a hypothetical surface within the manifold.

32

5 Null Shells

5.3 Some Characteristic Features of Null Shells In the following, we review some more interesting features peculiar to null shells.

5.3.1 Affine Parameter on Null Generators Let us start with the fact that normal vector .n is tangent to the null generators of the hypersurface .∑. Null generators are both null and hypersurface orthogonal [175], therefore geodesics. Given that .λ is an arbitrary parameter on a null generator, the geodesic equation on either side of the hypersurface may be written as n ν ∇ν n μ = κn μ ,

.

(5.73)

where .κ ± vanishes whenever .λ is an affine parameter on the .M± side of .∑. Note that .λ may be chosen to be affine on one side of the hypersurface; it is, however, in general not possible to do it on both sides. This can be seen from Raychaudhuri’s equation describing the evolution of congruence of null generators (see [208]): .

1 dθ + θ 2 + σ μν σμν = κθ − 8π Tμν n μ n ν , dλ 2

(5.74)

where .θ and .σμν are expansion and shear of the congruence, respectively, and .Tμν is the energy-momentum tensor of the surrounding medium. Since the congruence is hypersurface orthogonal, the term related to vorticity is set to zero. Now it is seen that the left-hand side of Eq. (5.74), depending only on the intrinsic geometry of .∑, is necessarily continuous having no jump across .∑. Then, the continuity of the right-hand side yields μ ν .[κ]θ = 8π [Tμν n n ], (5.75) expressing the fact that all the energy absorbed by the shell from its surroundings goes into work done on the expanding or contracting shell by the surface pressure because the rest mass of the null shell material must be kept at zero. Using Eqs. (5.73), (5.36), (5.34), (5.56)–(5.57), (5.46), and by virtue of.[∇n η] = 0, the jump .[κ] is calculated to be μ

[κ] = −[Nμ n ν n μ ;ν ] = −n j [Nμ eνj (n i ei );ν ] = n i n j [Ki j ] =

.

1 μ ν n n γμν 2 1 = γ † . (5.76) 2

Note that according to Eq. (5.63), .γ † is associated with the isotropic pressure of the null shell. We see then from Eq. (5.76) that a null shell is affinely conciliable, i.e. .λ can be an affine parameter on both sides of .∑ (.κ + = κ − ) if and only if the

5.3 Some Characteristic Features of Null Shells

33

shell is pressureless, i.e., .γ † = 0. According to Eq. (5.75), this is possible when μ ν .[Tμν n n ] = 0, i.e., only if there is no net exchange of energy-momentum between the shell and its surroundings. In the case of stationary null shell .(θ = 0), this result is not applicable.

5.3.2 Reparametrization of Null Generators and the Gluing Freedom We now ask whether there is any freedom to “glue” two manifolds along a given null hypersurface or, given the junction conditions between two manifolds, the gluing is completely specified. It was first pointed out in [23]- by way of an example- that for specific null shells (horizon shells) situated at the horizon of static black holes, there is considerable freedom to glue manifolds. Isolated examples were already known in literature such as the Dray ’t’Hooft gluing of two equal-mass Schwarzschild black holes along their horizon [71]. Poisson [171] investigated the effect of reparametrization of null generators on the physical quantities of a null shell, without referring to the horizon shells for which this gluing freedom occurs. A detailed analysis of the phenomenon was then presented in [40]. To address the question of freedom in gluing null shells, following [40], we seek possible allowed coordinate transformations on either side of a shell, preserving the geometrical junction condition (5.31). This amounts to solving for Killing vectors i . Z of the induced metric .gi j on .∑: .

L Z gi j = Z k ∂k gi j + (∂i Z k )gk j + (∂ j Z k )gki = 0.

(5.77)

For a timelike or spacelike shell, gluing is unique up to isometries of the induced metric determined by the solution of this Killing equation leaving all physical quantities invariant. However, for a null shell, this is not so trivial. Taking the degenerate metric as .gλλ = giλ = 0 and the non-degenerate .g AB , Eq. (5.77) becomes .

L Z gi j = Z k ∂k gi j + (∂i Z C )gC j + (∂ j Z C )gCi = 0,

(5.78)

with its .i = λ− or . j = λ− component .

L Z gλj = L Z giλ = 0

⇒

∂λ Z A = 0.

(5.79)

This rules out any .λ-dependent transformations of the spatial coordinates .ξ A , meaning that the isometry transformations along the spatial directions of the null hypersurface are independent of .λ. Next, considering spatial components . AB of (5.78), we get .

L Z g AB = 0

⇒

Z λ ∂λ g AB + Z C ∂C g AB + (∂ A Z C )gC B + (∂ B Z C )g AC = 0. (5.80)

34

5 Null Shells

Assuming .λ-dependence of .g AB , Eq. (5.80) implies . Z λ = 0. Therefore, only .λindependent isometries of the metric .g AB are allowed. Similar to the case of non-null shells, such gluing, when it exists, is unique (see [35] for some special cases in which a .λ-dependent metric can still possess non-trivial gluing freedom). In the special case metric .g AB being independent of .λ, such as horizon-straddling null shells according to Eq. (5.80), the component . Z λ turns out to be completely unconstrained. Therefore, it is possible to perform arbitrary reparametrization of .λ on .∑: .∂λ g AB = 0 ⇒ λ −→ λ¯ (λ, ξ A ) allowed, (5.81) representing an independent change of parameter .λ on each null generator of .∑. In this case, the gluing is not unique and one can construct infinitely many null shells related to each other via arbitrary transformations (5.81). This happens in the case of Killing horizons of stationary black holes, Rindler horizons, and other quasi-local notions of horizons such as non-expanding horizons and isolated horizons (see [7] for a review). As a consequence, we see that in this special case the gluing group (group of the coordinate transformations of .λ on .∑ given by (5.81)) is infinite-dimensional. The case of two black hole spacetimes glued through a stationary null hypersurface each given by gluing transformations (5.81), leading to a null horizon shell is such an example with an infinite number of distinct ways to glue, as already pointed out in [23].

5.3.3 A Hierarchical Classification of Null Hypersurfaces Being History of Both Wave and Null Matter The geometry of null hypersurfaces shows an interesting hierarchical structure we would like to review. Due to this hierarchy classification of the null hypersurface .∑ inherited by .M+ and .M− , introduced by Penrose [168], one can distinguish three different intrinsic geometries of the null shells in order of increasing structure: Type I: the general case of a null shell embedded in an arbitrary spacetime with the induced metric being the same on both sides of .∑ according to the basic junction requirement (5.31). Type II: type I and the normal vector .n being parallel transported along the null generators to be matched across the null hypersurface [n μ ∇μ n ν ] = 0.

.

(5.82)

Type III: type II and any tangent vector .v α with .n α v α = 0 being parallel transported along the null generators to be matched across the null hypersurface [n μ ∇μ v ν ] = 0.

.

(5.83)

5.3 Some Characteristic Features of Null Shells

35

It is clear that a type I geometry on .∑ is the most general one always assumed to be held in the present review. A type II geometry implies.[κ] = 0 as seen from Eqs. (5.73) and (5.82). This means that the null generators may be affinely parametrized on both sides of .∑, being denoted as affinely conciliable. In this case, as seen from (5.76), † .γ = 0, and consequently, from (5.63), this type II induced geometry corresponds to a pressure-free null shell, i.e. a shell having only a surface energy density and anisotropic surface stresses. In the case of type III geometry on.∑, using Eq. (5.40) and the condition (5.83), one gets .γμ = An μ , where . A is an arbitrary function on .∑. We see then from Eq. (5.48) that the surface energy-momentum tensor reduces to .−16π −1 Sμν = (2 A − γ )n μ n ν . Therefore, there are no surface stresses and the shell merely admits a surface energy density .−(2 A − γ )/16π vanishing for . A = γ /2.

5.3.4 Singular Part of Weyl Tensor and Gravitational Wave Component Let us concentrate now on identifying gravitational waves in the glued spacetime, including a null hypersurface. We have already seen from Eq. (5.58) that the symmetric tensor . Si j has just four independent components, in contrast to the case of timelike/spacelike shells where there is a one-to-one correspondence between the surface energy-momentum tensor and .γi j as seen in Eq. (4.28). This means that a part of .γi j denoted by.γˆi j (see Eq. 5.59) with two independent components decoupled from the matter may contribute to two degrees of freedom needed for an impulsive gravitational wave on .∑. To see this, we must study the Weyl tensor related to our glued spacetime given by .C κλμν

= Rκλμν +

1 1 (gκν Rμλ + gλμ Rνκ − gκμ Rνλ − gλν Rμκ ) + R(gκμ gνλ − gκν gμλ ), 2 6

(5.84) can be written as distribution-valued tensor for the glued spacetime .M+ ∪ M− : + − Cκλμν = Cκλμν θ (φ) + Cκλμν θ (−φ) + ηC˘ κλμν δ(φ),

.

(5.85)

where .η = −1, and using (5.42)–(5.44) and (5.48), the singular part .C˘ κλμν concentrated on .∑ is found to be [23, 106] ) 16π Sσσ ( gκ[μ gν]λ . (5.86) α −1 C˘ κλμν = 2n [μ γν][κ n λ] − 8π Sκ[μ gν]λ − Sλ[μ gν]κ + 3

.

The singular part of the Weyl tensor (5.86) can be conveniently written as a sum: Wκλμν + Mκλμν . Here .Wκλμν is related to the gravitational wave part2 and . Mκλμν to the surface energy-momentum tensor . Sμν representing the matter part. Different

.

2

According to [15] .Wκλμν is a part of the first term in (5.86), without more clarification.

36

5 Null Shells μ

components of these contributions on the oblique basis (.ei , N μ ) were calculated in [15]. It is shown that .

1 μ Wκλμν eiκ N λ e j N ν = − γˆi j . 2

(5.87)

Therefore, the wave part of .C˘ κλμν is constructed from .γˆi j . In addition .

Wκλμν n κ = 0,

(5.88)

indicating that the wave part .Wκλμν is of type . N in the Petrov classification with .n μ as a quadruply repeated principal null direction. Hence, this part of the Weyl tensor in (5.86) is related to an impulsive gravitational wave with propagation direction .n μ in spacetime and with .∑ as its history. Alternatively, it is convenient to use the Newman–Penrose components of the Weyl tensor on a null tetrad. Taking advantage of continuous coordinates.x μ covering both sides of .∑, we first introduce a transverse null vector field .l μ on .∑ satisfying the condition .l α n α = −1. Let .m μ and .m¯μ be two covariant null complex conjugate vector fields (.m α m α = m¯ α m¯ α = 0), tangent to .∑, orthogonal to .n μ and .l μ , satisfying .m α m ¯ α = 1. Now, the null tetrad {.n α , l μ , m μ , m¯ μ } on .∑ may be used to calculate the Newman–Penrose components of the Weyl tensor in .M± denoted as .Ψ ± A .(A = 0, 1, 2, 3, 4) [194]: .Ψ0

= Cμνρσ n μ m ν n ρ m σ ,

Ψ1 = Cμνρσ n μ m ν n ρ l σ , μ

Ψ2 = −Cμνρσ n μ m ν l ρ m¯ σ ,

ν ρ σ

Ψ4 = Cμνρσ l μ m¯ ν l ρ m¯ σ .

Ψ3 = Cμνρσ l m¯ l n ,

(5.89)

From (5.86), the corresponding Newman–Penrose components of the singular part ˘ A are then given by [17]: of the Weyl tensor .C˘ μνρσ denoted by .Ψ .

˘ 0 = 0, Ψ

˘3 = Ψ

1 γμ m¯ μ , 2

˘ 1 = 0, Ψ ˘4 = Ψ

˘2 = Ψ

1 † γ , 6

1 γμν m¯ μ m¯ ν , 2

(5.90)

where the first four of these components describe the matter part of the Weyl tensor ˘ 4 describes the wave part. We may now look at the classification of induced and .Ψ geometry on .∑ and its relation to the Petrov type of the singular part of the Weyl tensor: (a) Type I on .∑: the singular part of the Weyl tensor .C˘ μνρσ is of Petrov type II with ˘ 0 , and.Ψ ˘ 1 , while satisfying the criterion.n [λ C˘ μ]νρσ n ν n ρ = two vanishing coefficients.Ψ 0. Then, a matter null shell and an impulsive gravitational wave coexist on the null hypersurface .∑. ˘ 2 , then .Ψ ˘ A is Petrov type III (b) Type II on .∑: .γ † = 0, leading to vanishing .Ψ ˘ 0 , .Ψ ˘ 1 , and .Ψ ˘ 2 , while satisfying the criterion with the three vanishing coefficients .Ψ

5.3 Some Characteristic Features of Null Shells

37

n C˘ μ]νρσ n ν = 0. Then the null hypersurface .∑ is the history of both a null shell admitting a surface energy density and anisotropic surface stresses, and an impulsive gravitational wave. (c) Type III on .∑: .γ † = 0, and .γμ m¯ μ = 0 (recall that for type III, .γμ = An μ ). ˘ A is Petrov type . N with the four vanishing coefficients .Ψ ˘ 0 , .Ψ ˘ 1 , .Ψ ˘ 2, Therefore, .Ψ ˘ 3 , meaning that the matter part of the Weyl tensor vanishes, while satisfying and .Ψ the criterion .C˘ μνρσ n μ = 0. This case happens when the surface energy-momentum tensor is isotropic (with no surface stresses, isotropic or anisotropic). Then, the null hypersurface .∑ is the history of both a null shell admitting only a surface energy density and an impulsive gravitational wave. In the special case . A = γ /2, there is ˘ 4 /= 0. no shell and .∑ is the history of an impulsive gravitational wave provided .Ψ

. [λ

5.3.5 Influence of Null Shell and Impulsive Gravitational Wave on Matter We now study the interaction of a null shell and/or impulsive gravitational wave whose history is the null hypersurface .∑ with the matter by examining the status of optical parameters of the congruence of timelike geodesics such as histories of galaxies in a cosmological model, crossing the null hypersurface .∑. Following [15], we consider an arbitrary congruence of timelike geodesics with the unit tangent vector.u μ , intersecting.∑. To ensure smoothness of the congruence at null hypersurface .∑, we require equality of the tangential projection of this vector field μ onto .∑: .[u μ ei ] = 0. The transverse projection of the vector field, .u μ N μ , however, and its transverse derivative . N μ ∂μ u ν , may be discontinuous across .∑. Describing the jump of the latter one by the vector .V ν on .∑, we have

or

.

[N μ ∂μ u ν ] = V ν ,

(5.91)

[∂μ u ν ] = −n μ V ν .

(5.92)

.

Using this result as well as Eqs. (5.34) and (5.40), the jump in 4-acceleration .a μ = u ν ∇ν u μ of the timelike congruence across .∑ is obtained to be 1 [a μ ] = sV μ + sU μ + (u ν Uν )n μ , 2

.

(5.93)

where we have set .s = −u σ n σ > 0 and .Uμ = γμν u ν . Using .u σ a σ = 0 (by virtue of u u μ = −1 show that .u σ u ν ∇ν u σ = 0) and Eq. (5.93), one obtains:

. μ

u U μ = −2u μ V μ .

. μ

(5.94)

38

5 Null Shells

Introducing now the projection tensor .h μν = gμν + u μ u ν , the jump in expansion Θ = ∇μ u μ of the timelike congruence (null geodesics of .∑ being intrinsic have no jump in their expansion: .[θ ] = 0) across .∑ is found to be [17]

.

[Θ] =

.

1 sγ − V σ n σ . 2

(5.95)

Similarly, the jump of shear tensor .σμν = ∇(μ u ν) + a(μ u ν) − 13 Θh μν of the timelike congruence across .∑ is given by s γμν − V(μ n ν) + sU(μ u ν) 2 1 1 + (u σ Uσ )n (μ u ν) + sV(μ u ν) − [Θ]h μν , 2 3

[σμν ] =

.

(5.96)

and the vorticity tensor .ωμν = ∇[μ u ν] + a[μ u ν] of the timelike congruence across .∑: [ωμν ] = −U[μ n ν] − V[μ n ν] + sU[μ u ν] 1 + (u σ Uσ )n [μ u ν] + sV[μ u ν] . 2

.

(5.97)

Defining the vorticity vector as ωα =

.

1 αβμν η u β ωμν , 2

(5.98)

with .ηαβμν = (−g)−1/2 ∈αβμν , .g = det (gμν ), and .∈αβμν as the four-dimensional LeviCivita permutation symbol, Eq. (5.97) may be expressed as the jump in vorticity vector 1 αβμν α .[ω ] = − η u β (Uμ + Vμ )n ν . (5.99) 2 We are now in the position to relate jumps of kinematical quantities of the timelike congruence intersecting the null hypersurface .∑ given by (5.93), (5.95)–(5.97), and ˘ A of the singular part of the Weyl (5.99), with the Newman–Penrose components .Ψ tensor given by (5.90). The following is inferred easily [17]: ˘ 4 /= 0 gravitational wave is present; (1) .[σμν m μ m ν ] /= 0 ⇐⇒ Ψ ˘ 4 = 0 no gravitational wave, (2) if .[σμν ] = 0, then .Ψ ˘ 3 /= 0, null shell is present; (a) .[ωμ ] /= 0 ⇐⇒ Ψ μ ˘ 3 /= 0, null shell is present (b) .[a m μ ] /= 0 ⇐⇒ Ψ μ ˘3 = Ψ ˘ 4 = 0 and (3) if .[σμν ] = 0 and .[a ] = 0, .[ωμ ] = 0 then .Ψ ˘ 2 /= 0. .[Θ] / = 0 ⇐⇒ Ψ Therefore, according to the case (1) above, there is a close relationship between the presence of a wave and shear of the timelike congruence such that if .∑ is the

5.3 Some Characteristic Features of Null Shells

39

history of a gravitational wave, then its impact on any timelike congruence, such as world lines of the cosmic fluid in a cosmological model is to cause a jump in shear of congruence across .∑. This is a generalization of the result by Penrose [168] expressing that for a null geodesic congruence crossing .∑ with a continuous tangent, the existence of a jump in the complex shear is necessary for .∑ to be the history of an impulsive gravitational wave. In case there is no jump in the fluid shear, it follows from the case (2) above that the null hypersurface is just the history of a lightlike shell of matter with a Petrov type III delta function in the Weyl tensor if the vorticity vector of the fluid ((a) of the case (2) above) or if a complex component of the fluid 4-acceleration ((b) of the case (2) above) jumps across .∑. In the case that we are facing just a jump in the expansion of cosmic fluid across .∑, then the case (3) above shows that the delta function in the Weyl tensor is Petrov type II.

Chapter 6

Gluing Conditions from the Variational Principle

The principle of least action provides us with a new way to make mathematical models describing dynamics which is fundamental to understanding natural phenomena. Having so far derived the basic gluing conditions for dynamics of thin shells within a manifold using pill-box integration of the Gauss–Codazzi equations and also from the distributional method, it is now interesting to see how the variational principle applied to an action can be employed for this purpose. Gluing conditions for thin shells have been obtained via variational methods by several authors [3, 50, 94, 95]. It is also widely used to construct quantum models of shells [4, 29, 54, 100, 101, 127]. To develop a quantum description it is not enough to know the equations of motion, but the knowledge of an action for the shell degrees of freedom is needed. The least action principle is also used to obtain the equations of gravitational double layers arising in the quadratic theories of gravity [30, 31]. By applying the variational principle the dynamical equations of a physical system composed of a singular hypersurface and the surrounding spacetime appear with the shell equation being the natural boundary condition. This is particularly useful for braneworld scenarios, where the Lanczos equation on the brane is a part of the equations of motion and, thus, the brane and bulk dynamics arise from a unified variational principle [24]. Up to now, there has not much attention to the variational method for null shells. Jezierski et al. [115] considered the variational treatment of null shells; however, they did not demonstrate that their results reproduce those obtained by Barrabes and Israel [23] in the null limit. In this case, the trouble with standard prescription is that a variation in the metric leads to a variation in the causality and such variations do not preserve the nullity of the hypersurface (see [167] for more detail). Racsko [173] investigated the variational principle for the gravitational field in the presence of generic shells with completely unconstrained signature. Here, we review the variational principle only for timelike/spacelike shells.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_6

41

42

6 Gluing Conditions from the Variational Principle

6.1 Action in the Presence of a Thin Shell The Einstein–Hilbert action contains second derivatives of the metric tensor which is a non-typical feature of field theories. Since a second-order Lagrangian normally produces equations of motion of order four in the metric, boundary conditions pertinent to the variational problem are that of a fourth-order differential equation, requiring fixing of both metric and its normal derivative at the boundary. However, Gibbons and Hawking [93] and York [210] realized that a well-posed variational procedure for the Einstein–Hilbert action on manifold .M with boundary .∂M (see [73] for discussion on criteria for a well-posed action) requires fixing the metric but not its normal derivative on the boundary. Therefore, they added to the Einstein–Hilbert action a boundary term the role of which is to compensate for variations of the normal derivatives of the metric on.∂M stemming from the variation of Einstein–Hilbert action after integrating by parts. Now, we consider a physical system consisting of two given spacetimes .M− and .M+ glued together along a non-lightlike hypersurface .∑, representing the world history of the shell with a singular distribution of matter on it. Therefore, boundary of the glued spacetime manifold hypersurface U.∑ is+ considered an interior − M such that .∂M− = ∂M+ = ∑. Following Berezin et al. [30], let .M ≡ M us introduce the Gaussian normal coordinate system concentrated on .∑: ds 2 = ∈dn 2 + gi j (n, x)d x i d x j ,

(6.1)

.

where .x i ∈ ∑, and .∈ = +1 or -1, depending on whether .∑ is timelike or spacelike, respectively, and .n the proper distance through .∑ in the direction of the outer normal vector. The gravitational action of the physical system is the Einstein–Hilbert action for the regions .M± together with a Gibbons–Hawking–York boundary term: S =

. H

1 2κ

⎧∫ M±

√ R −gd 4 x − ∈

∫ ∑

) √ 2[K ] |3 g|d 3 x ,

(6.2)

where .g is the determinant of four-dimensional metric, .3 g that of three-dimensional metric on .∑, . R the scalar curvature in .M± , and . K = g i j K i j the trace of extrinsic curvature of .∑. The total action is the sum of gravitational action (6.2) and the action for all matter fields, . Stot = S H + Sm . This matter part of the action may be written as ∫ √ (6.3) . Sm = Lm −gd 4 x, M

where .Lm is the matter field Lagrangian density, and .M ≡ .M−

U

∑

U

M+ .

6.2 Variation of the Action

43

6.2 Variation of the Action Let us first focus on the gravitational part of the action. Due to the presence of Gibbons–Hawking–York surface term, one can obtain the field equations from the variational principle in such a way that only the metric on the boundary is held fixed: .δgμν |∑ = 0. Varying the gravitational action (6.2) with respect to the metric gives ∫ 2κδS H =

.

=

±

∫M

M±

√ δ(g μν Rμν −g)d 4 x − ∈ (

∫ (

∫ ∑

( √ ) δ 2[K ] |3 g| d 3 x

√ ) √ √ Rμν −gδg μν + g μν −gδ Rμν + Rδ −g d 4 x

√ ) √ (2δ[K ]) |3 g| + 2[K ]δ |3 g| d 3 x −∈ ∫ ∑ √ 1 (g μν δ Rμν + (Rμν − gμν R)δg μν ) −gd 4 x = ± 2 ∫M √ ∈(2[δ K ] − [K ]gi j δg i j ) |3 g|d 3 x, −

(6.4)

∑

√ √ √ where in the last step relations .δ( −g) = − 21 −ggμν δg μν and .δ( |3 g|) = √ 1 |3 g|gi j δg i j have been used (see Chap. 4 in [172]). The first term in the right2 hand side (6.4) can be written as ∫ ∫ ( )√ √ g μν ∇ρ (δ[ ρμν ) − ∇ν (δ[ ρμρ ) −gd 4 x (6.5) . g μν δ Rμν −gd 4 x = M± M± ∫ ( )√ ∇ρ (g μν δ[ ρμν ) − ∇ν (g μν δ[ ρμρ ) −gd 4 x = ± ∫M ) ( √ √ ∂ρ (g μν δ[ ρμν −g) − ∂ν (g μν δ[ ρμρ −g) d 4 x = ± ∫ ∫M ρ ρ g μν [δ[μν ]d Sρ − g μν [δ[μρ ]d Sν = ∑ ∑ ∫ ∫ √ √ ρ ρ ∈g μν [δ[μρ ]n ν |3 g|d 3 x = ∈g μν [δ[μν ]n ρ |3 g|d 3 x − ∑ ∑ ∫ √ ij n i = − (g [δ[i j ] − ∈[δ[ni ]) |3 g|d 3 x ∫∑ √ = ∈ (g i j [δ K i j ] + [δ K ]) |3 g|d 3 x ∫∑ √ = ∈ (2[δ K ] − δg i j [K i j ]) |3 g|d 3 x, ∑

where in the first line we have inserted the Palatini identity as .δ Rμν = ∇ρ (δ[ ρμν ) − ∇ν (δ[ ρμρ ) [172]. The second line has been written by virtue of the metricity condition √ ν μν ν√ −g),ν satisfied by any .g ;σ = 0. In the third line, we have used . −g A ;ν = (A

44

6 Gluing Conditions from the Variational Principle

ν vector In the fourth line, using the Stocks’ theorem ∫ . A√ (noteμthat .4δ[ is ∫a tensor). as . M± ( −g A ),μ d x = ∑ Aμ d Sμ , the full derivative terms have been converted √ into surface integrals. In the fifth line, .d Sρ = ∈n ρ 3 gd 3 x has been substituted. The sixth line is written in terms of the Gaussian normal coordinate system (6.13). In the seventh line, relations .[ ni j = −∈ K i j and .[ i n j = K ij , have been substituted. Finally, in the last line, we have used the fact that the induced metric on .∑ is continuous. Thus (6.4) takes the form

.δS H

=

−∈ 2κ

) ( ∫ √ √ 1 1 ([K i j ] − [K ]gi j )δg i j |3 g|d 3 x + Rμν − gμν R δg μν −gd 4 x. 2κ 2 ∑ M±

∫

(6.6) Note that arbitrary variation .δ K containing the normal derivatives of the metric has completely disappeared in the expression (6.6). Now, turning to the matter part of the action, we obtain the variation ∫

√ δ(Lm −g)d 4 x ) ∫M ( √ δLm μν √ = δg −g + Lm δ −g) d 4 x μν M δg ) ∫ ( √ 1 δLm − Lm gμν δg μν −gd 4 x. = μν 2 M δg

δSm =

.

(6.7)

Defining the matter energy-momentum tensor .Tμν by T

. μν

≡ −2

δLm + Lm gμν , δg μν

(6.8)

Equation (6.7) takes the form 1 .δSm = − 2

∫ M

√ Tμν δg μν −gd 4 x.

(6.9)

On the other hand, the overall energy-momentum tensor can be written as T

. μν

+ − = Sμν δ(n) + Tμν θ (n) + Tμν (1 − θ (n)),

(6.10)

± where . Sμν is the surface energy-momentum tensor of matter on .∑, and .Tμν is the ± energy-momentum tensor in .M . Then one gets .δSm

∫ ∫ √ √ 1 1 Sμν δg μν |3 g|d 3 x − T ± δg μν −gd 4 x (6.11) 2 ∑ 2 M± μν ∫ ∫ √ 1 1 ± δg μν √−gd 4 x. =− (Si j δgi j + Snn δg nn + 2Sni δg ni ) |3 g|d 3 x − Tμν ± 2 ∑ 2 M

=−

6.2 Variation of the Action

45

Variation of the full action .δS H + δSm becomes .

∫ ∫ √ √ 1 −∈ ([K i j ] − [K ]gi j )δg i j ) |3 g|d 3 x − (Si j δg i j + Snn δg nn + 2Sni δg ni ) |3 g|d 3 x 2κ ∑ 2 ∑ ) ( ∫ ∫ √ √ 1 1 1 + (6.12) T ± δg μν −gd 4 x = 0. Rμν − gμν R δg μν −gd 4 x − 2κ M± 2 2 M± μν

It is seen that the Einstein equations hold in .M± . The absence of terms with .δg nn and ni .δg in the first term .δS H yields the results . Snn = Sni = 0, meaning that the normal components of the surface energy-momentum tensor are zero, as expected. Finally, we end up with the Israel junction conditions expressed as Lanczos equation: ∈([K i j ] − gi j [K ]) = −κ Si j .

.

(6.13)

Therefore, we have seen that in the most general case using Einstein–Hilbert action, the Lanczos equation and Israel junction conditions are derived from the variational principle. Note that the correct set of equations in the coordinate system chosen is derived by action, provided the position of hypersurface and coordinates in a neighborhood of the hypersurface are fixed by the Gaussian normal coordinate condition. In a different treatment [153], an action principle of singular hypersurfaces in general relativity was presented which is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. Then, the position of hypersurface measured from one side of the hypersurface and that measured from another side can be independently variated as required by the double covariance.

Chapter 7

Gluing Conditions in Einstein–Cartan Theory of Gravity

The Einstein–Cartan (EC) theory is a natural generalization of general relativity that accounts for the presence of spacetime torsion. Hence, in the EC theory geometry of spacetime is determined by both metric and torsion. While curvature is assumed to originate from matter, spacetime torsion is usually assumed to be triggered by the intrinsic angular momentum of the matter [191] (see also [39] for a review). The study of gluing conditions in spacetime manifolds with torsion, based on geometric DL junction requirements, is the purpose of this chapter. A first generalization of the gluing technology to EC gravity was made in [6] by applying the language of differential forms. The paper is, however, restricted to the case of boundary surfaces in general relativity where the extrinsic curvature has to be continuous across the hypersurface. Later, Bressange [46] extended the unified description for thin shells of any type, including the null case, provided by Barrabes–Israel [23] to the presence of torsion. In his approach, the contorsion discontinuity, naturally appearing in the expression for the shell’s energy-momentum tensor is taken to be orthogonal to the hypersurface, yielding a modification to the surface energy-momentum tensor and the resulting gluing conditions. According to this modification, the Israel junction conditions turn out to be formally the same as in general relativity [112], but the tensor representing the jump of the normal derivatives of the metric across a non-lightlike shell is replaced with a non-symmetric one splitting into a Riemann part and a Cartan part (see also [205]). On the other hand, Hoff da Silva et al. presented an extension of junction conditions for a timelike hypersurface to cover the EC gravity in the context of braneworld scenarios [57] (see also [56]). They assumed an orthogonal form for the jump of the torsion tensor across the brane not reflecting the required antisymmetric properties, resulting in no torsion contribution to the jump of the brane extrinsic curvature tensor. Maier et al. [144] (see also [143]) again in the context of braneworld models with . Z 2 symmetry assumed the torsion in bulk to be continuous across the brane just as the metric tensor allowed, however, its first derivatives appearing in the effective field equations on the 3-brane to be discontinuous. Here, the Israel junction conditions are extended such that the extrinsic curvature tensor splits into a symmetric part connecting to the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_7

47

48

7 Gluing Conditions in Einstein–Cartan Theory of Gravity

matter distribution on the brane and an antisymmetric part including the 5-dimension torsion tensor projected onto the brane. However, they presupposed the brane energymomentum tensor in the presence of torsion to be symmetric and tangent to the brane. With a choice of torsion discontinuity taken to be orthogonal to the hypersurface and consistent with the antisymmetric properties of torsion and contorsion tensor, Khakshournia et al. [124] showed that the generalized asymmetric surface energymomentum tensor turns out to be automatically tangent to the hypersurface while resulting generalized gluing conditions are modified. In this chapter, we review the gluing conditions in EC theory following the approach of [123, 124].

7.1 Field Equations In a Riemann–Cartan spacetime manifold, the torsion tensor is defined by the antisymmetric component of the affine connection: .

T μ νσ = [ μ νσ − [ μ σν .

(7.1)

Demanding the metricity condition .∇σ gμν = 0, following the decomposition of the asymmetric connection is obtained: [ μ νσ = [˚μ νσ + K μ νσ ,

.

(7.2)

where .[˚σ νμ denotes the Christoffel symbols and . K σ μν are the components of contorsion or defect tensor of the connection given in terms of the torsion components: .

K μ νσ =

1 μ (Tν σ + Tσ μ ν − T μ νσ ), 2

(7.3)

with . K μνσ = −K νμσ . The Einstein–Cartan field equations can be derived from the variation of the Einstein–Hilbert action. Variation with respect to the metric tensor gives the Einstein equations: . G μν = κTμν , (7.4) and variation with respect to the torsion tensor leads to the Cartan equations: .

T μ νσ + δνμ T ρ σρ − δσμ T ρ νρ = κS μ νσ ,

(7.5)

where . S μ νσ is the spin tensor representing the density of intrinsic angular momentum in the matter distribution related to the torsion tensor in a purely algebraic way. Since the Cartan field equations (7.5) are linear and algebraic, the torsion tensor vanishes

7.2 Gluing Conditions

49

outside material bodies where the spin density is zero. It must be noted that although Eq. (7.4) is apparently identical to its general relativistic counterpart, here both .G μν and .Tμν are generally asymmetric due to the presence of torsion.

7.2 Gluing Conditions We start with the familiar terminology of gluing manifolds. A hypersurface .∑, being either timelike/spacelike or lightlike, separates two Reimann–Cartan spacetimes.M− − and.M+ ..M− is endowed with a metric tensor.gμν , a non-symmetric connection.[ −σ μν , μ −σ and a torsion tensor .T μν with respect to a local coordinate system .x− . Similarly, + + +σ .M is endowed with the metric tensor .gμν , a non-symmetric connection .[ μν , and μ a torsion tensor .T +σ μν with respect to a local and independent coordinate system . x + . As we are going to use the distributional approach (see Sect. 4.3), the four-metric μ μ .gμν in the coordinate system . x , distinct from . x ± covering the overlap and reaches into both domains, is assumed to be continuous across .∑. In the presence of torsion in addition to this geometric L-requirement, it is required that the purely tangential part of the torsion tensor be continuous across .∑: μ

[ei eνj ekσ Tμνσ ] = [Ti jk ] = 0.

.

(7.6)

This may be called a generalization of L-conditions to EC geometry, as it may look different for other geometries we are interested in physics. Let us first review the case of timelike/spacelike shell before going to the null shell case.

7.2.1 The Non-null Shell Taking the continuity of the tangential part of the torsion tensor as expressed in (7.6), any possible discontinuity must be normal to the timelike/spacelike hypersurface .∑. Hence, the discontinuity of the torsion tensor as a primary geometric object across .∑ can be decomposed as σ σ .[T μν ] = ζμν n , (7.7) for some tensor .ζμν given explicitly by .ζμν = ∈n σ [T σ μν ], and .∈ = +1(−1) for a timelike (spacelike) hypersurface. Consistency with the antisymmetric property of σ . T μν on the last two indices requires the tensor .ζμν to be antisymmetric. Now using Eq. (7.3), the contorsion discontinuity across .∑ is written as [K σμν ] =

.

1 σ (ζ n μ + ζ σμ n ν − ζμν n σ ). 2 ν

(7.8)

50

7 Gluing Conditions in Einstein–Cartan Theory of Gravity

This contorsion discontinuity induced by the choice of torsion discontinuity (7.7) is automatically consistent with the antisymmetric property of the contorsion tensor on its first two indices. Substitution of (4.17) expressing the discontinuity of Christoffel symbols and (7.8) into Eq. (7.2) leads to the following expression for the discontinuity of non-symmetric connection across .∑: [[ σμν ] =

.

1 σ 1 (γ n ν + γ σν n μ − γμν n σ ) + (ζ σν n μ + ζ σμ n ν − ζμν n σ ). 2 μ 2

(7.9)

Recalling the distributional form of the Riemann tensor, the singular part of it as given in (4.15), can be written using Eq. (7.9) as .

∈ R˘ αμσν = − (γ ασ n μ n ν − γνα n μ n σ + γμν n α n σ − γμσ n α n ν ) 2 ∈ + (ζ αν n μ n σ − ζ ασ n μ n ν − ζμν n α n σ + ζμσ n α n ν ). 2

(7.10)

Contraction of Eq. (7.10) yields the singular part . R˘ μν of the Ricci tensor: ˘ μν .R

=−

∈ ∈ (γn μ n ν − γνσ n σ n μ − γμσ n σ n ν + ∈γμν ) + (ζμσ n ν n σ − ζνσ n μ n σ − ∈ζμν ), 2 2

(7.11) while the singular part . R˘ of the Ricci scalar is given by .

R˘ = g μν R˘ μν = −∈(∈γ − γρσ n ρ n σ ).

(7.12)

We finally arrives at the following form for the singular part .G˘ μν of the Einstein tensor: ˘ μν .G

∈ = − (γn μ n ν + gμν γρσ n ρ n σ − γνσ n σ n μ − γμσ n σ n ν + ∈γμν − ∈γgμν ) 2 ∈ + (ζμσ n ν n σ − ζνσ n μ n σ − ∈ζμν ). 2

(7.13)

Recalling the distribution-valued energy-momentum tensor as given in (4.24) and noting Sen equation .G˘ μν = 8πSμν , the surface energy-momentum tensor of the nonnull shell . Sμν is found to be [124] .8π∈Sμν

1 = − (γn μ n ν + gμν γρσ n ρ n σ − γνσ n σ n μ − γμσ n σ n ν + ∈γμν − ∈γgμν ) (7.14) 2 1 + (ζμσ n ν n σ − ζνσ n μ n σ − ∈ζμν ). 2

Note that in both the Einstein- and energy-momentum-tensor two distinct terms appear, one from metric with the same form as in general relativity, and the other from torsion in EC theory, respectively. By virtue of the antisymmetric property of .ζμν , following two conditions hold simultaneously:

7.2 Gluing Conditions

51

(i)Sμν n μ = 0,

.

(ii)Sμν n ν = 0.

(7.15)

These are the necessary and sufficient conditions for the asymmetric tensor . S μν to be purely tangent to .∑. Therefore, there is no need to impose any additional constraints to ensure that the surface tensor . Sμν is indeed tangent to .∑. Based on the conditions (7.15), the following decomposition can be made: .

μ

S μν = S i j ei eνj .

(7.16)

μ

Now, . Si j = Sμν ei eνj is an asymmetric three-tensor to be evaluated as 16πSi j = −∈gi j γ † − γi j − ζi j + γgi j , = −(γi j + ζi j ) + γgi j ,

.

(7.17)

where in the second line, by virtue of the gauge transformation (4.22) we have gauged γ † = γμν n μ n ν to zero. On the other hand, an asymmetric extrinsic curvature tensor .Ki j may now be introduced with the jump across .∑ given by .

μ

[Ki j ] = [∇μ n ν ]ei eνj ,

.

μ

= −[[ σμν ]n σ ei eνj , ∈ = (γi j + ζi j ), 2

(7.18)

where Eq. (7.9) has been used. Using this equation for the jump of extrinsic curvature and the expression (7.17) for the surface energy-momentum tensor, one ends up with the following form for the generalized gluing conditions in the presence of torsion: [Ki j ] − gi j [K] = −8π∈Si j .

.

(7.19)

Note that .[K] = 2∈ γ due to the antisymmetric property of .ζi j leading to .ζ = 0. Therefore, it is seen that in the context of Cartan gravity, the gluing conditions take the same form of Lanczos equation as in general relativity (see (4.30)) with .[Ki j ] being now a non-symmetric tensor splitting into a Riemann part .γi j and a Cartan part .ζi j [124]. We see once again the power and simplicity of the distributional approach as described in Chap. 4.

7.2.1.1

The Case of Null Shells

When .∑ is a null hypersurface, the discontinuity of torsion tensor across it is taken to be σ σ .[T μν ] = −ζμν n , (7.20)

52

7 Gluing Conditions in Einstein–Cartan Theory of Gravity

for some antisymmetric tensor.ζμν given by.ζμν = Nσ [T σ μν ] with. Nμ n μ = −1. Then, from Eq. (7.3) the contorsion discontinuity takes the form 1 [K σμν ] = − (ζ σν n μ + ζ σμ n ν − ζμν n σ ), 2

.

(7.21)

consistent with the antisymmetric property of the contorsion tensor on the first two indices. Substituting (5.40) for the discontinuity of Christoffel symbols and (7.21) into Eq. (7.2) yields the following expression for the discontinuity of non-symmetric connection across .∑: .[[

σ ] = − 1 (γ σ n + γ σ n − γ n σ ) − 1 (ζ σ n + ζ σ n − ζ n σ ). μν μν μν μ ν ν μ ν μ μ ν

2

2

(7.22)

The singular part of the Riemann tensor for a null shell given by .α−1 R˘ αμσν = [[ α μν ]n σ − [[ αμσ ]n ν (see Sect. 5.3) takes the form 1 α−1 R˘ αμσν = (γ ασ n μ n ν − γνα n μ n σ + γμν n α n σ − γμσ n α n ν ) 2 1 α − (ζ ν n μ n σ − ζ ασ n μ n ν − ζμν n α n σ + ζμσ n α n ν ). 2

.

(7.23)

Contraction of Eq. (7.23) yields the singular part . R˘ μν of the Ricci tensor: .α

−1

1 1 R˘ μν = (γn μ n ν − γνσ n σ n μ − γμσ n σ n ν ) − (ζμσ n ν n σ − ζνσ n μ n σ ), 2 2

(7.24)

The singular part of the Ricci scalar is then given by α−1 R˘ = α−1 g μν R˘ μν = −γσν n σ n ν .

.

(7.25)

The singular part .G˘ μν of the Einstein tensor then takes the form 1 (γn μ n ν + gμν γρσ n ρ n σ − γνσ n σ n μ − γμσ n σ n ν ) 2 1 − (ζμσ n ν n σ − ζνσ n μ n σ ). 2

α−1 G˘ μν =

.

(7.26)

From Sen equation .G˘ μν = −8παSμν for null shells, the energy-momentum tensor of the shell is written as [123]: 1 8πS μν = − (γn μ n ν + g μν γρσ n ρ n σ − γ νσ n σ n μ − γ μσ n σ n ν ) 2 1 + (ζ μσ n σ n ν − ζ νσ n μ n σ ). 2

.

(7.27)

7.2 Gluing Conditions

53

Here again, we have the distinction of terms in the energy-momentum tensor: the first term originated from the metric and is similar to GR and the second one from torsion. It is easy to see that two conditions (7.15) are automatically satisfied for the null shell. Energy-momentum tensor of the null shell can be simplified after μ decomposing it into the basis of .(N μ , ea ). Using the completeness relations (5.38), the following decomposition [171] is found: 1 ν 1 γn − γi j g∗i j n ν + ζλσ N λ n σ n ν − γλσ n λ n σ N ν 2 2 + g∗i j (γλσ − ζλσ )eλj n σ eνi .

(γ νσ − ζ νσ )n σ =

.

(7.28)

Inserting (7.28) into Eq. (7.27) and using once more the completeness relations together with a rearrangement of terms, one finally ends up with the three-dimensional intrinsic form for the surface energy-momentum tensor of the null shell in the presence of torsion [123]: 16πS i j = −g∗kl γkl n i n j − γkl n k n l g∗i j + n i g∗jk (γkl − ζkl )n l + n j g∗il (γkl − ζkl )n k .

.

(7.29) From the expression (7.29), the matter on null shell can be characterized by 1 kl g γkl , 16π ∗

(7.30)

1 γkl n k n l , 16π

(7.31)

1 il g (γkl − ζkl )n k , 16π ∗

(7.32)

1 jk g (γkl − ζkl )n l , 16π ∗

(7.33)

σ=−

.

as the surface energy density, and .

p=−

as the isotropic surface pressure, and Ji =

. 1

j

J =

. 2

as the asymmetric surface energy currents. Note that due to the antisymmetric property of .ζi j , one gets .g∗kl ζkl = 0 and .ζkl n k n l = 0, indicating the vanishing of torsion contribution to the energy density .σ in (7.30) and isotropic surface pressure . p in (7.31). Therefore, in the presence of torsion, the surface energy-momentum tensor of the null shell is modified as given in (7.29). It is found from (7.29) that there is a part .γˆ kl − ζˆkl of .γkl − ζkl not contributing to the matter content on the shell encoded in . S i j . This part satisfies the following seven independent equations:

54

7 Gluing Conditions in Einstein–Cartan Theory of Gravity

g kl γkl = 0,

(7.34)

(γkl − ζkl )n = 0, l .(γkl − ζkl )n = 0.

(7.35) (7.36)

. ∗ .

k

Since .γkl − ζkl has 9 independent components, it follows that .γˆ kl − ζˆkl has two independent components contributing to the Weyl tensor associated with the shell and can be interpreted as representing the two polarization degrees of freedom related to an impulsive gravitational wave traveling along the shell [46]. In summary, based on the assumption of torsion discontinuity orthogonal to the hypersurface consistent with the antisymmetric properties of the torsion tensor, the asymmetric energy-momentum of the non-null/null shell turns out to be unconditionally tangent to the hypersurface .∑. When the hypersurface is non-null, the gluing conditions in the presence of torsion look the same as Lanczos equations in GR; the jump of extrinsic curvature across .∑ is, however, non-symmetric. The energy momentum tensor expressed in geometric quantities, which may be called the generalized Sen equation, splits nicely in a term having the form of Sen conditions in GR and an extra term originating from torsion. In the case of the surface energymomentum tensor of a null shell, the torsion contribution to the surface energy current turns out to be non-zero.

Chapter 8

Gluing Conditions in f(R) Theories of Gravity

Despite the success of general relativity in describing a wide variety of gravitational phenomena in astrophysical and cosmological scales, in the last few decades, we have been witnessing a growing interest in studying alternative theories of gravity in order to address supposed shortcomings of general relativity in certain regimes. The most straightforward modification of Einstein’s equations are . f (R) theories of gravity, in which the Ricci scalar curvature . R in the Einstein–Hilbert action is replaced by an analytic function . f (R). There have been some claims in . f (R) cosmology to explain the evolution of the observed universe without the need for dark matter or dark energy [114, 193]. What are the junction requirements and gluing conditions if dynamics is based on such generalized gravitational fields? Typically, . f (R) theories introduce difficulties in the dynamics having higherorder equations of motion and may lead to extra junction requirements and gluing conditions [67, 188]. It has been demonstrated that additional requirements and conditions are derived, as opposed to standard general relativity. Using a Gaussian coordinate system relative to the matching hypersurface, Deruella et al. [67] derived these gluing conditions. Later, a rigorous account of gluing in . f (R) gravity across a timelike hypersurface using the calculus of tensor distributions was presented by Senovilla [188]. In this section, we aim to review the gluing conditions and requirements for . f (R) theories as deduced by Senovilla.

8.1 Field Equations The action for . f (R) gravity can be written as S =

. H

1 2κ

⎧∫ M

√ f (R) −gd 4 x −

∫ M

) √ Lm −gd 4 x ,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_8

(8.1)

55

56

8 Gluing Conditions in f(R) Theories of Gravity

where .κ = 8π, .g is the determinant of spacetime metric, . f (R) is some function of the Ricci scalar and .Lm denotes the matter Lagrangian density. A variation of the action in Eq. (8.1) with respect to the metric .gμν yields f (R)Rμν −

. R

1 f (R)gμν − ∇μ ∇ν f R (R) + gμν ∎ f R (R) = κTμν , 2

(8.2)

where.Tμν represents the energy-momentum tensor,.∎ ≡ ∇μ ∇ μ denotes D’ Alembert operator, while subscript . R stands for partial derivative of the function . f (R) with respect to the Ricci scalar. R, i.e.. f R ≡ ∂∂ Rf . These field equations are, in general, of the fourth order in derivatives of the metric tensor1 (see [192] and references therein for some mathematical subtleties in deriving Eq. (8.2) from the action (8.1)). Equation (8.2) can be expanded in ( ) 1 f (R)gμν − f R R (R) ∇μ ∇ν R − gμν ∎R 2 ( ) − f R R R (R) ∇μ R∇ν R − gμν ∇ρ R∇ ρ R = κTμν .

f (R)Rμν −

. R

(8.3)

Henceforth, we use Eq. (8.3) to derive the gluing conditions in f(R) theories of gravity.

8.2 Geometrical Prerequisites and Junction Requirements The glued spacetime manifold .M, endowed with a metric .gμν on both sides of the timelike/spacelike hypersurface .∑, has to be properly defined throughout the entire spacetime. Recall that according to the distributional approach (see Sect. 4.3), the four-metric .gμν in the continuous coordinate system .x μ written as a distributionvalued tensor (4.9) is continuous across .∑. This L-junction condition was identified as a geometric requirement. The derivatives, need not be continuous. Hence, the extrinsic curvature of .∑ on both parts of .M± may be different expecting a jump across it. To see this, let’s write the extrinsic curvature tensor living in .M as

.

K μν = h σμ h νρ ∇σ n ρ =

(8.4)

h μσ ∇σ n ν ,

where the second line stems from .h νρ = δνρ − ∈n ν n ρ , using the identity .n ν ∇σ n ν = 0. The jump of extrinsic curvature tensor across .∑ can then be written as

Note that in the case of . f (R) Lagrangian, the Palatini version of the variational principle- treating the metric and affine connection as independent geometrical quantities, leads to the second-order differential equations instead of the fourth-order ones that one gets with the metric variation. To derive the junction conditions corresponding to Palatini’s formulation of f(R) theories see [161].

1

8.2 Geometrical Prerequisites and Junction Requirements

[K μν ] = h μσ [∇σ n ν ]

.

=

57

(8.5)

ρ −h μσ [[σν nρ]

1 = − h μσ (γ ρσ n ν + γ ρν n σ − γσν n ρ )n ρ 2 ∈ = h μσ γσν 2 ∈ = γμν , 2 where in the third line, Eq. (4.17) has been used. In the fourth line, by virtue of the gauge in which .γ μ = 0 (see Sect. 4.3), the first two terms of the third line have been set to zero. In the fifth line, we have again used .h μσ = δμσ − ∈n μ n σ . Now, the distribution-valued Ricci tensor . Rμν may be written as .

+ − Rμν = Rμν θ(l) + Rμν θ(−l) + R˘ μν δ(l).

(8.6)

Its singular part, given by (4.19), in the gauge .γμ = 0 and using Eq. (8.5), takes the form .

∈ R˘ μν = − (γn μ n ν + ∈γμν ) 2 = −[K ]n μ n ν − ∈[K μν ].

(8.7)

Similarly, the Ricci scalar as .

˘ R = R + θ(l) + R − θ(−l) + Rδ(l),

(8.8)

with its singular part given by (4.20), takes the following form in the gauge .γ † = 0 and using again Eq. (8.5): .

R˘ = −γ

(8.9)

= −2∈[K ]. We see that the singular part of the Ricci scalar vanishes if and only if there is no jump in the trace of the second fundamental form. The presence of differential terms .∇μ R, .∇μ ∇ν R and .∎R in the field equations (8.3) indicates the use of first and second covariant derivatives of . R in the distributional formalism. Starting from (8.8), these terms can be computed as .∇ν R .∇μ ∇ν R

˘ = ∇ν R + θ(l) + ∇ν R − θ(−l) + ∈[R]n ν δ(l) + ∇ν ( Rδ(l)), +

−

= ∇μ ∇ν R θ(l) + ∇μ ∇ν R θ(−l) + ∈[∇ν R]n μ δ(l) ˘ + ∈∇μ ([R]n ν δ(l)) + ∇μ ∇ν ( Rδ(l)), ( [ ] ) − + ˘ ∎R = ∎R θ(−l) + ∎R θ(l) + ∈ ∇ρ R n ρ δ(l) + ∈∇ν [R]n ν δ(l) + ∎( Rδ(l)),

(8.10) (8.11)

.

(8.12)

58

8 Gluing Conditions in f(R) Theories of Gravity

where .∇μ θ(l) = ∈n μ δ(l) has been used. Looking at Eq. (8.3), we notice that all terms on the left-hand side are factors of . f (R) or its derivatives, involving the Ricci scalar . R having a distributional part given by Eq. (8.9). Therefore, products of the form 2 .δ(l)δ(l) arise which are not allowed in the distribution formalism, without canceling each other in Eq. (8.3). Hence, in order to make Eq. (8.3) distributionally correct, we have to demand vanishing of the singular part of the Ricci scalar proportional to ˘ = 0. We then have from (8.9) .δ(l): . R [K ] = 0.

(8.13)

.

Therefore, in the case of nonlinear function . f (R), in addition to the continuity of metric on .∑, the trace of extrinsic curvature of .∑ is required to be continuous: an extra algebraic requirement absent in the general relativistic case. It is convenient to know the following identities satisfied in the general . f (R) theories, as shown in [188] + − (K μν − K μν )S μν = 2n ρ n σ [Tρσ ],

(8.14)

h μα ∇α Sμν = −n ρ h σν [Tρσ ].

(8.15)

.

.

These are identical with the evolution (3.13) and conservation (3.14) identities in general relativity. To proceed further looking for gluing equations, we distinguish two cases, depending on whether . f R R R (R) vanishes or not [188].

8.3 Gluing Conditions for the Generic Case: . f R R R (R) / = 0 Products of type .∇μ R∇ν R appear in the field equations (8.3). We then see from Eq. (8.10) that squares of type .([R]δ(l))([R]δ(l)) appear which are not allowed. To avoid it, continuity of the Ricci scalar . R across .∑ has to be required .

[R] = 0.

(8.16)

This is another algebraic requirement not present in general relativistic case. Taking (8.13) and (8.16) into account, Eqs. (8.10), (8.11), and (8.12) are simplified to ∇ν R = ∇ν R + θ(l) + ∇ν R − θ(−l),

.

−

+

∇μ ∇ν R = ∇μ ∇ν R θ(−l) + ∇μ ∇ν R θ(l) + ∈ [∇ν R] n μ δ(l), [ ] ρ − + .∎R = ∎R θ(−l) + ∎R θ(l) + ∈ ∇ρ R n δ(l).

.

2

We will use the relation .θ(l)δ(l) = 21 δ(l) in Chaps. 8 and 9.

(8.17) (8.18) (8.19)

8.3 Gluing Conditions for the Generic Case: f R R R (R) /= 0

59

We now have all the necessary tools to write down the distributional version of the field equations for . f (R) gravity and look for dynamical gluing conditions. Our starting point is Eq. (8.3). The only remaining singular parts on the left-hand side are those coming from the distributional form of the Ricci tensor given by Eqs. (8.6), (8.18), and (8.19). The counter term in the right-hand side from the distributional form (4.24) of the energy-momentum tensor, is . Sμν of the thin shell. Therefore, .

] [ σ ¯ K μν − ∈ f R R ( R)([∇ ¯ − ∈ f R ( R) ν R] n μ − gμν [∇σ R] n ) = κSμν ,

(8.20)

¯ ¯ where we have used .[AB][ = A[B] ] + [A] B (for the proof see [176]). The explicit form for the discontinuity . ∇μ R was obtained by using the orthogonal decomposition in [149, 188] ] [ σ . ∇μ R = bn μ + h μ ∇σ [R], (8.21) where .b is a[function ] on .∑ measuring the jump in the normal derivative of . R across ∑: .b ≡ ∈n ρ ∇ρ R . With (8.16), the second term in (8.21) must be set to zero. Thus, Eq. (8.20) takes the form

.

.

] [ [ ] ¯ K μν + ∈ f R R ( R)n ¯ ρ ∇ρ R h μν = κSμν , − ∈ f R ( R)

n μ Sμν = 0,

(8.22)

with .h μν = gμν − ∈n μ n ν . Using projection vectors, Eq. (8.22) is projected onto the hypersurface .∑, leading to .

[ [ ] ] ¯ K i j + f R R ( R)n ¯ ρ ∇ρ R gi j = ∈κSi j . − f R ( R)

(8.23)

This is the final form of dynamical gluing equations in the . f (R) theory of gravity. We would like to coin it generalized Lanczos condition as it reduces to the familiar Lanczos equations (4.30) for . f (R) → R. We have intentionally distinguished between the algebraic . f (R) junction requirements (8.13) and (8.16), in addition to the geometric L-condition, and the dynamical generalized Lanczos condition. This is another advantage of the distributional formalism! Smooth Gluing at .∑ In the case of timelike singular hypersurfaces we are considering, the smooth gluing is identified with vanishing . Sμν (or . Si j ), i.e. .∑ is just a boundary surface. From (8.22), we need .[K μν ] = 0, (8.24) [ ] and .n ρ ∇ρ R = 0, which by using (8.21) takes the form .

] [ ∇μ R = 0.

(8.25)

Therefore, conditions for smooth gluing in . f (R) theories with . f R R R (R) /= 0 are more demanding compared to the general relativity case: in addition to (8.24), we have the gluing conditions (8.25).

60

8 Gluing Conditions in f(R) Theories of Gravity

8.4 Gluing Conditions for the Special Case: . f R R R (R) = 0 In this case, we may write . f (R) in the quadratic form .

f (R) = R − 2Δ + αR 2 ,

(8.26)

where .Δ and .α are arbitrary constants. Now, while the requirement (8.13) is preserved, the Ricci scalar . R can be discontinuous at .∑ (.[R] /= 0), in contrast to the generic case of . f R R R (R) /= 0. Therefore, the singular part of the second derivative of the Ricci tensor in Eq. (8.11) consists of terms .[∇ν R]n μ δ(l) and .∇μ ([R]n ν δ(l)). Using the discontinuity .[∇μ R] given in (8.21), and taking computation of the latter term given in [188], we obtain ∇μ ([R]n ν δ(l)) = ∈Δμν + δ(l)( K¯ μν − ∈K n μ n ν + n ν h σμ ∇σ )[R],

.

(8.27)

where the first term on the right-hand side (8.27) is a 2-covariant symmetric tensor distribution function with support on .∑ of “delta prime” type, acting on any test function .Y μν ∫ .

M

Δμν Y μν d 4 x = −

∫ ∑

[R]n μ n ν n ρ ∇ρ Y μν dσ,

( ) Δμν = ∇ρ [R]n μ n ν n ρ δ(l) .

(8.28) Putting (8.21) and (8.27) into (8.11), the second derivative of the Ricci scalar . R takes the form ∇μ ∇ν R = ∇μ ∇ν R + θ(l) + ∇μ ∇ν R − θ(−l) { } + ∈ bn μ n ν + n μ h σν ∇σ [R] + n ν h σμ ∇σ [R] + [R]( K¯ μν − ∈K n μ n ν ) δ(l)

.

+ Δμν .

(8.29)

Using the condition (8.13) leading to . K¯ = K , and by virtue of .n ν h σν = 0, the trace of (8.29) gives us ∎R = ∎R + θ(l) + ∎R − θ(−l) + bδ(l) + Δ,

.

(8.30)

where .Δ = g μν Δμν is defined as ∫

∫ ΔY d x = −∈ 4

.

M

∑

[R]n ρ ∇ρ Y dσ

Δ = ∈∇ρ ([R]n ρ δ(l)) .

(8.31)

Collecting all the above, the singular part of the left-hand side of the field equation (8.3) becomes { .

} ¯ μν ] + ∈ K¯ μν [R]) − 2∈α(n μ h σν ∇σ [R] + n ν h σμ ∇σ [R]) + 2α[R]K n μ n ν δ(l) −∈[K μν ] − 2α(−bh μν + ∈ R[K +2αΩμν ,

(8.32)

8.4 Gluing Conditions for the Special Case: f R R R (R) = 0

61

where the quadratic form of . f (R) given in (8.26) and the distributionally valid product .θ(l)δ(l) = 21 δ(l) have been used. .Ωμν is a 2-covariant symmetric tensor distribution defined as Ωμν = gμν Δ − Δμν ( ) = ∈∇ρ [R]h μν n ρ δ(l) ,

.

(8.33)

such that for a test function .Y μν , the action .Ωμν on .Y μν yields ∫ .

M

Ωμν Y μν d 4 x = −∈

∫ ∑

[R]h μν n ρ ∇ρ Y μν dσ.

(8.34)

Now, looking at the singular terms in (8.32), reflecting the left-hand side of the field equations (8.3), we expect a similar structure for the energy-momentum tensor T

. μν

+ − = Tμν θ(l) + Tμν θ(−l) + T˘μν δ(l) + Tμν .

(8.35)

Here, .T˘μν is a symmetric tensor field with support on .∑ proportional to .δ(l), and .Tμν is, by definition, the singular part of .Tμν with support on .∑ not proportional to .δ(l) but of delta derivative type. Motivated by the structure of singular term proportional to .δ(l) on .∑ in (8.32), one is tempted to perform an orthogonal decomposition of the corresponding singular term .T˘μν in (8.35) into the tangent, normal-tangent and normal parts with respect to .∑ as follows T˘

. μν

= Sμν + Tμ n ν + Tν n μ + T n μ n ν ,

(8.36)

with . Sμν

ρ

= h μ h σν T˘ρσ ,

n μ Sμν = 0;

ρ

Tμ = h μ T˘ρν n ν ,

n μ Tμ = 0;

T = n μ n ν T˘μν ,

(8.37) so that (8.35) takes the form T

. μν

+ − = Tμν θ(l) + Tμν θ(−l) + (Sμν + Tμ n ν + Tν n μ + T n μ n ν )δ(l) + Tμν . (8.38)

Therefore, the singular part of the energy-momentum tensor .Tμν consists of two different parts, one of thin-shell type proportional to .δ(l) denoted by .T˘μν , and .Tμν which is of ‘delta-prime’ type. The former splits into three different terms: tangent, tangent-normal and normal to .∑, unlike in GR where only the first term . Sμν arises. Identifying the singular terms on each side of the field equation (8.3) as given in the expressions (8.32) and (8.38) gives us different dynamical gluing conditions. First, we notice similar to GR a relation for the surface energy-momentum tensor which may be called the generalized Sen equation

62

8 Gluing Conditions in f(R) Theories of Gravity .

( ) − [K μν ] + 2α [n σ ∇σ R]h μν − [R K μν ] = ∈κSμν ,

n μ Sμν = 0,

(8.39)

¯ μν ] + K¯ μν [R] has been used. In addition to . Sμν where the expression .[R K μν ] = R[K satisfying the gluing equation (8.39), there are three new dynamical ones [188]: (i) An external energy flux vector .Tμ satisfying ∈κTμ = −2αh νμ ∇ν [R],

.

n μ Tμ = 0.

(8.40)

(ii) An external scalar pressure or tension .T satisfying κT = 2α[R]K .

(8.41)

.

(iii) A two-covariant symmetric tensor distribution .Tμν of delta derivative type satisfying . κTμν = 2αΩμν . (8.42) Using (8.34) one can write the junction equation (8.42) equivalently as ∫ κ

.

M

Tμν Y μν d 4 x = −∈

∫ ∑

2α[R]h μν n σ ∇σ Y μν dσ,

(8.43)

for any test tensor field .Y μν . The contribution .Tμν resembles the energy-momentum content of double layer surface charge or dipole distributions in classical electrodynamics and hence was interpreted by Senovilla as a gravitational double layer [188]. The presence of .Tμν is essential to keep energy-momentum conservation: without double-layer contribution, the total energy-momentum tensor with its distributional parts would not be covariantly conserved as shown in [189]. Similar to dipole distributions in classical electrodynamics, the strength of the gravitational double layer is defined as κPμν = 2α[R]h μν ,

.

Pμν = Pνμ ,

n μ Pμν = 0.

(8.44)

Therefore, double layers arise when an abrupt change, or jump, in the scalar curvature R occurs. Early works on junction conditions in quadratic . f (R) gravity were based on using Gaussian coordinates centered on.∑ [67]. It turned out that the derivatives of Dirac delta supported on the shell are ill-defined in this coordinate system [176]. This is why double layers were not found in quadratic . f (R) or other quadratic theories until they were discovered in [188]. Therefore, in general, while studying singular surfaces in . f (R) theories with . f R R R (R) = 0, we do expect not only a thin shell but also a double layer at the gluing hypersurface. Depending on whether the thin shell and/or double layer are present, three different types of gluing may be considered: .

8.4 Gluing Conditions for the Special Case: f R R R (R) = 0

63

(i) Smooth gluing at .∑ In this case, .∑ is a boundary surface characterized by finite jumps in the energymomentum tensor but neither a thin shell nor a thin double layer is present. This requires . Sμν = 0 for no thin shell, and .Pμν = 0 for no double layer. The following set of gluing conditions is obtained [K μν ] = 0,

.

(8.45)

[R] = 0, [ ] ∇μ R = 0, where Eq. (8.21) has been used in the third line. Therefore, the required conditions for the smooth gluing in . f (R) theories with . f R R R (R) = 0 are the same as we obtained for the case . f ''' (R) /= 0. (ii) Pure thin shell at .∑ In this case . Sμν /= 0. The non-existence of a double layer means its strength .Pμν must be set to zero. From (8.44) we conclude [R] = 0.

.

(8.46)

Note that as there is no double layer, then due to (8.46), there can be neither external energy flux .Tμ nor external pressure/tension .T (see Eqs. (8.40) and (8.41), respectively). (iii) Pure double layer at .∑ Having no thin shell means we have simultaneously . Sμν = 0, .Tμ = 0, .T = 0, while .[R] / = 0. From (8.39), (8.40), and (8.41) following set of gluing conditions is obtained [ ] n ρ ∇ρ R = 0,

.

(8.47)

h νμ ∇ν [R] = 0, K = 0, [K μν ] + 2α[R K μν ] = 0. The first and second lines imply . R ± to be constant on .∑, and the third one implies that .∑ must have zero mean curvature, while the fourth line is the generalized Sen equation (8.39) for vanishing . Sμν . In summary, the non-dynamical junction requirements in generic f(R), in addition to the continuity of metric on .∑, include continuity of trace of the extrinsic curvature and that of scalar curvature . R. In the case of quadratic f(R) the last requirement does not hold and the scalar curvature can have a jump across .∑, leading to a much richer structure of matter content on .∑ and more diversified dynamical gluing conditions. In addition to the standard surface energy-momentum tensor . Sμν , there is an external energy flux vector .Tμ whose timelike component .(∈ = +1) measures the energy flux while its spatial components measure tangential stresses across .∑, an external

64

8 Gluing Conditions in f(R) Theories of Gravity

(normal to the matching hypersurface) scalar tension (or pressure) .T , and another energy-momentum contribution .Tμν signaling the presence of a double layer on .∑. All these requirements are necessary to make the whole energy-momentum tensor divergence-free.

Chapter 9

Gluing Conditions in Quadratic Theories of Gravity

Theories of gravity with terms quadratic in the curvature such as . R 2 , . Rμν R μν and . Rμνρσ R μνρσ being linearly added to the Einstein–Hilbert action are known as quadratic gravity. The motivation behind this natural generalization of general relativity is the low-energy effective action of string theory containing terms quadratic in the curvature [184]. The gluing is then special and should be looked at differently. In general, derivation of gluing conditions for these theories, in which the field equations are fourth-order in the derivatives of metric, leads to the powers of delta function ambiguities and hence is a non-trivial task. Models based on the Lovelock densities, such as the Gauss–Bonnet gravity, are unique in the sense that all these ambiguous terms cancel each other such that gluing conditions are simple again [62, 66]. In this way, Barrabes et al. obtained the junction conditions through timelike/spacelike and null hypersurfaces in quadratic gravity with the Gauss–bonnet term [22]. In the general case of quadratic theories, two following approaches to formulate the junction conditions have been adopted: A. Imposing regularity condition on the metric tensor at the hypersurface such that .∑ is taken to be a singular hypersurface of different orders according to the terminology forwarded by Israel [112]. A singular hypersurface in general relativity with .[K μν ] /= 0 is then of order one. Deruelle et al. [67] and Balcerzak et al. [9] used singular hypersurfaces of order three by requiring the metric on brane to be of class 2 .C to obtain junction condition on it. This means explicitly that the metric and its first derivatives are regular at the hypersurface, while the second one has a kink, the third has a step function discontinuity, and the fourth derivative of it has a singular part proportional to the delta function. B. In an alternative approach, leading to less restrictive regularity conditions, the reduction of the fourth-order theory to a second-order one at the expense of introducing an extra rank-four tensor field called “the tensoron” has been considered [9]. Junction conditions within such a theory are then formulated by either assuming the continuity of the tensoron across the hypersurface or adding a generalized GibbonsHawking boundary term constructed out of the extrinsic curvature and the tensoron without any assumption about the continuity of it on the hypersurface. In the latter © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_9

65

66

9 Gluing Conditions in Quadratic Theories of Gravity

case, assuming the continuity of tensoron on the brane, these most general junction conditions reduce to the ones obtained in the former case [10]. The equivalence of junction conditions for the second-order theory to those of the fourth-order one has been demonstrated in [9]. Works mentioned so far on junction conditions in quadratic gravity are based on Gaussian coordinates centered on .∑. It turns out that the derivatives of Dirac delta supported on .∑ are ill-defined in this coordinate system [176]. This is why no double layer was found in quadratic theories. A rigorous account of junction conditions in quadratic gravity across a timelike hypersurface based on approach A, using the distributions formalism is given in [176], which we will review now.

9.1 Field Equations Consider a quadratic theory of gravity with the linear combination of the curvature terms described by the Lagrangian L=

.

) 1 ( R − 2Δ + a1 R 2 + a2 Rμν R μν + a3 Rμνρσ R μνρσ + Lmatter , 2κ

(9.1)

where .Δ is the cosmological constant, .a1 , a2 , a3 are three constants specific to the theory, and .Lmatter is the Lagrangian density describing the matter fields. .Δ−1 and 2 .a1 , a2 , a3 have physical units of . L . The corresponding field equations are .

G μν + Δgμν + Hμν = κTμν ,

(9.2)

with . Hμν

{ = 2 a1 R Rμν − 2a3 Rμσ Rνσ + a3 Rμρσλ Rν ρσλ + (a2 + 2a3 )Rμρνσ R ρσ ) ) ) ( ( 1 1 a2 + 2a3 ∎Rμν − a1 + a2 + a3 ∇μ ∇ν R + 2 2 { } 1 2 ρσ − gμν (a1 R + a2 Rρσ R + a3 Rρσκλ R ρσκλ ) − (4a1 + a2 )∎R , (9.3) 2

and .∎ ≡ g μν ∇μ ∇ν being D’Alembertian and . Hμν incorporates the deviation from general relativity introduced by the quadratic terms in the action. These equations are fourth order in the derivatives of metric, except for the Gauss–Bonnet combination .a1 = −a2 /4 = a3 yielding second-order field equations.

9.2 Geometrical Prerequisites and Junction Requirements

67

9.2 Geometrical Prerequisites and Junction Requirements Let us look for geometrical requisites to obtain junction requirements for a glued manifold with a timelike/spacelike singular hypersurface .∑ embedded in it by using the distributional formalism. In this way, we need Eq. (9.2) to make sense distributionally. Hence, the tensor . Hμν must be well-defined in a distributional setting, meaning there should be no multiplication of singular distributions such as .δ(l)δ(l). Starting with the distributional form of the Riemann tensor given in (4.14), its singular part . R˘ μνρσ , given in (4.15), can be conveniently rewritten as .

R˘ μνρσ = −n μ [K νσ ]n ρ + n μ [K νρ ]n σ − n ν [K μρ ]n σ + n ν [K μσ ]n ρ ,

(9.4)

where Eq. (8.5) has been substituted. Using the singular part . R˘ μν of the Ricci tensor (8.7) and that of the Ricci scalar . R˘ as computed in (8.9), the singular part .G˘ μν of the Einstein tensor can be written as .

G˘ μν = −∈[K μν ] + ∈h μν [K ].

(9.5)

Note that the expression (4.21) for .G˘ μν takes the form (9.5) after imposing the gauge γ = 0, using Eq. (8.5), and .h μν = gμν − ∈n μ n ν . Now, if either.a2 or.a3 in (9.1) is non-zero,1 then products of the Ricci and Riemann tensor by itself, or by each other, given the singular distributional contribution from (9.4) and (8.7), lead to the appearance of .δ(l)δ(l) terms in (9.3). To get rid of these illdefined terms, one has to demand the vanishing of the corresponding singular terms. This happens if and only if the jump of the second fundamental form vanishes. Hence, it is indispensable to require [ ] . K μν = 0. (9.6)

. μ

Obviously, (9.6) coincides with the smooth gluing condition in general relativity, resulting in a boundary surface with no matter content as follows from the Lancsoz equations (4.30). However, the nature of these requirements/conditions is quite different. While here, we face an algebraic problem solved by this requirement, in the case of GR it is a dynamical condition on the Einstein equations to be valid: requirements for a glued manifold stem from geometry and algebra in contrast to conditions on glued spacetimes depending on dynamics. This distinction paves the way to comprehend details of a glued spacetime manifold with any dynamics. Given the requirement (9.6), the only terms in (9.3) that are relevant to calculate the singular part of . Hμν are .∇μ ∇ν R, .∎Rμν and .∎R. More precisely, one needs to compute the singular part of the expression

1

The special case.a2 = a3 = 0, yielding quadratic f(R) gravity, was treated in the previous chapter.

68

9 Gluing Conditions in Quadratic Theories of Gravity

.

) ( 1 − (2a1 + a2 + 2a3 ) ∇μ ∇ν R + (a2 + 4a3 ) ∎Rμν + 2a1 + a2 gμν ∎R 2 = − (κ1 + κ2 ) ∇μ ∇ν R + 2κ2 ∎Rμν + κ1 gμν ∎R, (9.7)

where the following abbreviations have been introduced κ ≡ 2a1 + a2 /2,

. 1

κ2 ≡ 2a3 + a2 /2.

(9.8)

The quadratic . f (R) gravity corresponds to .κ2 = 0, and the Gauss–Bonnet gravity to κ = κ2 = 0. Assuming the requirement (9.6), the distributional form of the Ricci + − θ(l) + Rμν θ(−l) can be used to compute the relevant tensor written as . Rμν = Rμν covariant derivatives as follows

. 1

.∇σ

+ − Rμν = ∇σ Rμν θ(l) + ∇σ Rμν θ(−l) + ∈[Rμν ]n σ δ(l),

(9.9)

.∇ρ ∇σ Rμν

) ( + − = ∇ρ ∇σ Rμν θ(l) + ∇ρ ∇σ Rμν θ(−l) + ∈n ρ [∇σ Rμν ]δ(l) + ∈∇ρ [Rμν ]n σ δ(l) ,

(9.10)

.∎Rμν

) ( + − = ∎Rμν θ(l) + ∎Rμν θ(−l) + ∈n σ [∇σ Rμν ]δ(l) + ∈g ρσ ∇ρ [Rμν ]n σ δ(l) . (9.11)

Similarly, relevant terms in the distributional form of Ricci scalar . R = R + θ(l) + R − θ(−l) become .∇ν R .∇μ ∇ν R .∎R

= ∇ν R + θ(l) + ∇ν R − θ(−l) + ∈[R]n ν δ(l), = ∇μ ∇ν R + θ(l) + ∇μ ∇ν R − θ(−l) + ∈n μ [∇ν R]δ(l) + ∈∇μ ([R]n ν δ(l)) , = ∎R + θ(l) + ∎R − θ(−l)) + ∈n σ [∇σ R]δ(l) + ∈g μν ∇μ ([R]n ν δ(l)) .

(9.12) (9.13) (9.14)

To proceed further, one needs to write explicit expressions for the singular terms above. Recall that the discontinuity.[∇ν R] and the singular term.∇μ ([R]n ν δ(l)) were calculated in (8.21) and (8.27), respectively. To calculate the discontinuity .[∇σ Rμν ], we may write similar to (8.21) [176] [∇σ Rμν ] = rμν n σ + h λσ ∇λ [Rμν ],

.

where r

. μν

≡ ∈n σ [∇σ Rμν ],

rμν = rνμ ,

(9.15)

(9.16)

representing discontinuities in normal derivatives of the Ricci tensor. Similarly, the second singular term in (9.10) may be written as [176] ( ) ∇ρ [Rμν ]n σ δ(l) = ∈Δρσμν + δ(l)( K¯ ρσ − ∈K n ρ n σ + n σ h λρ ∇λ )[Rμν ], (9.17)

.

9.2 Geometrical Prerequisites and Junction Requirements

69

where we have . K¯ ρσ = K ρσ due to (9.6). The first term on the right-hand side is a 4-covariant symmetric tensor distribution of “delta prime” type with support on .∑, acting on any test function .Y ρσμν ∫ .

M

Δρσμν Y ρσμν d 4 x ≡ −

∫ ∑

[Rμν ]n ρ n σ n λ ∇λ Y ρσμν dσ,

( ) Δρσμν ≡ ∇λ [Rμν ]n ρ n σ n λ δ(l) .

(9.18) It is seen that .Δρσμν = Δσρμν = Δρσνμ . Now, putting (9.15) and (9.17) into (9.10), and using .h μν = gμν − ∈n μ n ν , expression (9.10) takes the form + − .∇ρ ∇σ Rμν = ∇ρ ∇σ Rμν θ(l) + ∇ρ ∇σ Rμν θ(−l) + Δρσμν { } +∈ rμν n ρ n σ + n ρ h λσ ∇λ [Rμν ] + n σ h λρ ∇λ [Rμν ] + [Rμν ](K ρσ − ∈K n ρ n σ ) δ(l). (9.19)

Contraction of (9.19) with .g ρσ leads to + − ∎Rμν = ∎Rμν θ(l) + ∎Rμν (θ(−l)) + rμν δ(l) + g ρσ Δρσμν ,

.

(9.20)

where the .n μ n μ = ∈ and .n μ h μν = 0 have been used, while .g ρσ Δρσμν , a delta prime type distribution with support on .∑, acts on any test function .Y μν ∫ .

g ρσ Δρσμν Y μν d 4 x ≡ −

M

∫ ∑

[Rμν ]n ρ n σ n λ ∇λ (g ρσ Y μν )dσ,

∫ = −∈

∑

) ( g ρσ Δρσμν = ∈∇λ [Rμν ]n λ δ(l) ;

[Rμν ]n λ ∇λ Y μν dσ.

(9.21)

Finally, by taking the trace of (9.20), one obtains ∎R = ∎R + θ(l) + ∎R − θ(−l) + bδ(l) + Δ,

.

(9.22)

where .b = rαα ≡ ∈n ρ [∇ρ R], and the scalar distribution .Δ ≡ g μν Δμν is of delta prime type with support on .∑, acting on any test function .Y ∫

∫

.

M

ΔY d 4 x = −

∑

[R]n σ ∇σ Y dσ,

Δ = ∈∇σ ([R]n σ δ(l)) .

(9.23)

Having computed the relevant singular terms in (9.7), we write the tensor. Hμν defined by (9.3) in the following distributional form .

+ − Hμν = Hμν θ(l) + Hμν θ(−l)) + H˘ μν δ(l) + Hμν ,

(9.24)

where ˘ μν .H

{ } = 2κ2 rμν + κ1 bgμν − ∈(κ1 + κ2 ) bn μ n ν + n μ h λν ∇λ [R] + n ν h λμ ∇λ [R] + [R](K μν − ∈K n μ n ν ) ,

(9.25) is the singular part proportional to .δ(l), and

70

9 Gluing Conditions in Quadratic Theories of Gravity

( ) ( ) Hμν = κ1 gμν Δ − Δμν + κ2 2g ρσ Δρσμν − Δμν ,

.

(9.26)

is the singular term of delta derivative type. Now, by virtue of .gμν = h μν + ∈n μ n ν , it is convenient to perform a natural orthogonal decomposition of the tensor .rμν into tangent, normal-tangent, and normal parts with respect to .∑ r

. μν

= rρσ h ρ μ h σ ν + rρσ h ρ μ n σ n ν + rρσ h ρ ν n σ n μ + n ρ n σ rρσ n μ n ν .

(9.27)

Substituting (9.27) into (9.25) together with a rearrangement of terms, the expression (9.25) for . H˘ μν can also be decomposed in the same way .

H˘ μν = 2κ2 rρσ h ρ μ h σ ν + κ1 bh μν − ∈(κ1 + κ2 )[R]K μν + 2κ2 rρσ (n σ h ρ ν n μ + n σ h ρ μ n ν ) − ∈(κ1 + κ2 )(n μ h λν ∇λ [R] + n ν h λμ ∇λ [R]) ) ( (9.28) + κ2 (2rρσ n ρ n σ − b) + (κ1 + κ2 )[R]K n μ n ν .

Let us now define two new 2-covariant tensor distributions with support on .∑, acting on any test function .Y μν [176] .Ωμν

( ) ≡ gμν Δ − Δμν = ∈∇σ [R]h μν n σ δ ;

∫ M

Ωμν Y μν d 4 x = −∈

∫ ∑

[R]h μν n σ ∇σ Y μν dσ,

(9.29)

and ( ) 1 1 .Φμν ≡ g ρσ Δρσμν − Δμν − Ωμν = ∈∇σ [G μν ]n σ δ(l) ; 2

2

∫ M

Φμν Y μν d 4 x = −∈

∫ ∑

[G αβ ] n σ ∇σ Y αβ dσ,

(9.30) where the metricity condition .g μν ;σ = 0 and .gμν = h μν + ∈n μ n ν have been used. Now, the expression (9.26) for .Hμν can be rewritten as Hμν = (κ1 + κ2 )Ωμν + 2κ2 Φμν ({ } ) = ∈∇σ (κ1 + κ2 )[R]h μν + 2κ2 [G μν ] n σ δ(l) .

.

(9.31)

9.3 Dynamical Gluing Conditions In order to satisfy the dynamical field equations (9.2) on the glued spacetime manifold, including .∑, the structure of singular terms in (9.24) dictates that the energymomentum tensor .Tμν on the right-hand side of (9.2) must have the following form T

. μν

+ − = Tμν θ(l) + Tμν θ(−l) + T˘μν δ(l) + Tμν ,

(9.32)

where .T˘μν is a symmetric tensor field with support on .∑ proportional to .δ(l), and .Tμν is, by definition, a singular part of .Tμν with support on .∑ but of delta derivative type. Motivated by the structure of singular term . H˘ μν proportional to .δ(l) as seen in (9.28),

9.3 Dynamical Gluing Conditions

71

one is tempted to perform an orthogonal decomposition of the corresponding singular term .T˘μν in (9.32) into tangent, normal-tangent, and normal parts with respect to .∑ T˘

. μν

= Sμν + Tμ n ν + Tν n μ + T n μ n ν ,

(9.33)

with . Sμν

ρ

= h μ h σν T˘ρσ ,

n μ Sμν = 0;

ρ

Tμ = h μ T˘ρν n ν ,

n μ Tμ = 0;

T = n μ n ν T˘μν ,(9.34)

so that (9.32) takes the form T

. μν

+ − = Tμν θ(l) + Tμν θ(−l) + (Sμν + Tμ n ν + Tν n μ + T n μ n ν )δ(l) + Tμν . (9.35)

Identifying singular terms on each side of the field equation2 (9.2) as given in (9.24), taking into account . H˘ μν (9.28), .Hμν (9.31), and (9.35), one arrives at the following dynamical gluing conditions [176]: 1. The energy-momentum tensor . Sμν on .∑, describing the tangential part of the distributional energy-momentum tensor .T˘μν supported on .∑ satisfies ∈κSμν = −(κ1 + κ2 )[R]K μν + κ1 n λ ∇λ [R]h μν + 2κ2 n λ ∇λ [Rρσ ]h ρμ h σν . (9.36)

.

S is the familiar quantity defined in the case of thin shells within general relativity; therefore, we may call it the generalized Sen equation for quadratic gravity. Note that for quadratic . f (R) gravity, as a special case of quadratic gravity, .κ1 = 2α and .κ2 = 0, the generalized Sen equation of quadratic gravity (9.36) reduces to (8.39) subject to (9.6), that is a necessary algebraic requirement in quadratic gravity. 2. The external energy flux vector .Tμ , describing the normal-tangent components of ˘μν supported on .∑, satisfies .T . μν

∈κTμ = −(κ1 + κ2 )h λμ ∇λ [R] + 2κ2 n λ ∇λ [Rρσ ]h ρμ n σ .

.

(9.37)

The timelike component of .Tμ describes the normal flux of energy across a timelike hypersurface .∑, while its spatial components measure the normal-tangential stresses. Such a vector does not exist on thin shells within general relativity. In the case of quadratic . f (R) gravity, .κ1 = 2α and .κ2 = 0, the gluing equation (9.37) simply reduces to (8.40). 3. The external pressure or tension .T , describing the normal part of .T˘μν supported on .∑, satisfies κT = (κ1 + κ2 )[R]K + ∈κ2 (2n λ ∇λ [Rμν ]n μ n ν − n λ ∇λ [R]).

.

2

(9.38)

Note that due to the requirement (9.6), the singular part of the Einstein tensor given in (9.5) vanishes.

72

9 Gluing Conditions in Quadratic Theories of Gravity

The scalar .T describes the total normal pressure/tension supported on .∑. Again, such a scalar does not exist on thin shells within general relativity. In the case of quadratic . f (R) gravity, .κ1 = 2α and .κ2 = 0, the gluing equation (9.38) simply reduces to (8.41). 4. The double-layer energy-momentum tensor distribution .Tμν , satisfies κTμν = Hμν .

(9.39)

.

Using (9.31), one can write the gluing equation (9.39) equivalently as ∫ κ

.

M

Tμν Y μν d 4 x = −∈

∫ ∑

{ } (κ1 + κ2 )[R]h μν + 2κ2 [G μν ] n ρ ∇ρ Y μν dσ ,

(9.40) for any test function .Y . Here, .Tμν is a symmetric tensor distribution of “deltaprime” type with support on .∑, resembling the energy-momentum content of a double-layer surface charge distributions, or “dipole distributions”, with strength μν

n μ Pμν = 0, (9.41) depending on the jump of Einstein (or equivalently, the Ricci) tensor at the layer (for quadratic . f (R) gravity .κ2 = 0, then (9.41) reduces to (8.44)). Note that .Pμν is the analogue of strength . D in an electrostatic dipole layer; it also satisfies the following identities (see [176] for more detail) κPμν ≡ (κ1 + κ2 )[R]h μν + 2κ2 [G μν ],

.

Pμν = Pνμ ,

T = −h μσ ∇ σ Pμν ,

(9.42)

T = K μν Pμν .

(9.43)

. ν

.

The following identities are also satisfied by the surface energy-momentum tensor S and the double-layer strength .Pαβ within .∑ in quadratic gravity [176]

. μν

.∈n

μ ρ h ν [Tμρ ] + h μσ ∇σ Sμν

.∈n

μ ν

n [Tμν ] − Sμν K

μν

= −Pμρ h σν ∇σ K μρ + h σρ ∇σ (P μρ K μν ) − h σρ ∇σ (Pμν K μρ ), ( ) = h μρ ∇ρ h νσ ∇σ Pμν + P μν n ρ n σ Rρμσν + K μρ K νρ .

(9.44) (9.45)

Equations (9.44) and (9.45) generalize the conservation identity (3.14) and the evolution identity (3.13) in general relativity, respectively. These identities possess an obvious structure related to energy-momentum conservation relations. It is seen that . Sμν is not divergence-free even in cases where there is no flux of energy from the bulk across .∑ (.n μ h ρν [Tμρ ] = 0) due to new terms on the right-hand side of Eq. (9.44) including the double layer strength .Pμν acting as a source of the energy-momentum contents on .∑, extrinsic geometric properties of .∑, and curvature components of spacetime.

9.4 Three Special Types of Gluing

73

9.4 Three Special Types of Gluing In general, by gluing spacetimes in quadratic gravity, we expect to have a thin shell plus a double layer at the gluing hypersurface. Depending on whether or not thin shells and/or double layers are present, three types of gluing may be differentiated: (i) Pure thin shell at .∑ In this case, . Sμν /= 0, and no double layer is present, meaning its strength .Pμν has to be zero. Then, from (9.42) and (9.43), it is seen that both .Tν and .T must vanish. We then notice if there is no double layer (that is .Pμν = 0), then there can be neither external energy flux .Tμ nor pressure/tension .T . In other words, it is the double layer energy-momentum contribution which actually induces an external energy flux vector and an external scalar pressure/tension on the shell. Now, Eq. (9.41) takes the form .(κ1 + κ2 )[R]h μν + 2κ2 [G μν ] = 0. (9.46) Taking the trace of (9.46) in .(n + 1)-dimensional spacetime, one obtains (nκ1 + κ2 )[R] = 0.

.

(9.47)

The trace of surface energy-momentum tensor given in (9.36) is also computed to be [176] ∈κS = (nκ1 + κ2 )b − K μν Pμν . = (nκ1 + κ2 )b,

.

(9.48) (9.49)

where .b ≡ ∈n ρ [∇ρ R], and the last line is valid for the pure thin shell case. Moreover, Eqs. (9.44) and (9.45) are reduced to ∈n μ h ρν [Tμρ ] = −h μσ ∇σ Sμν ,

(9.50)

∈n μ n ν [Tμν ] = Sμν K μν ,

(9.51)

.

.

which is the same as in general relativity. Depending on whether .κ1 or .κ2 is nonvanishing different cases may be distinguished: • .κ1 /= 0 and .κ2 /= 0 In case .(nκ1 + κ2 ) /= 0, the condition (9.47) leads to .[R] = 0, and then (9.46) yields .[G μν ] = 0. Consequently .[Rμν ] = 0, but .[∇σ Rμν ] /= 0. Therefore, the energy-momentum tensor . Sμν on .∑ given by (9.36) takes the form κSμν = κ1 bh μν + 2κ2 rρσ h ρμ h σν .

.

(9.52)

In the exceptional case.nκ1 + κ2 = 0, from (9.47) one gets.[R] /= 0, and then from (9.46) it is easily found that

74

9 Gluing Conditions in Quadratic Theories of Gravity

1−n [R]h μν , 2n

(9.53)

[R] 1 ( h μν + n μ n ν ). 2 n

(9.54)

[G μν ] =

.

and consequently [Rμν ] =

.

Hence, the discontinuities of the curvature tensor can be written in terms of .[R] as the only degree of freedom. It is seen from (9.48) that the non-zero tangential energy-momentum . Sμν is also traceless. Note that this case is probably irrelevant as the coupling constants .κ1 and .κ2 satisfy the dimensionally dependent condition .nκ1 + κ2 = 0. • .κ2 /= 0 but .κ1 = 0 This case is similar to the previous one. However, the energy-momentum tensor takes the form ρ σ .κSμν = 2κ2 r ρσ h μ h ν . (9.55) • .κ2 = 0 but .κ1 /= 0 In this case, .[R] = 0, but .[G μν ] /= 0, so that jumps in some components of the curvature tensor and its derivatives across .∑ are allowed. The energy-momentum tensor on .∑ then takes the following form κSμν = κ1 bh μν ,

.

(9.56)

indicating the existence of a vacuum energy type of matter content on .∑. (ii) Smooth gluing at .∑ In this case, .∑ is a boundary surface characterized by finite jumps in the energymomentum tensor, but neither a thin shell nor a thin double layer is present. Therefore, in addition to requiring the strength of double layer .Pμν to vanish (i.e. vanishing of both .Tα and .T ), we need to impose the requirement . Sμν = 0 to get rid of a thin shell. We see then from identities (9.44) and (9.45) that n μ [Tμν ] = 0,

.

(9.57)

indicating the normal components of the energy-momentum tensor have to be continuous across .∑, as in general relativity. Different cases may now be distinguished depending on the .κ2 and .κ1 values • .κ2 /= 0 and .κ1 /= 0 If .(nκ1 + κ2 ) /= 0, in addition to .[R] = 0, and consequently .[G μν ] = [Rμν ] = 0, as in the previous case, we see from (9.49) that .b ≡ n ρ [∇ρ R] = 0. Furthermore, with . Sμν = 0, from (9.36) one gets .rμν h μα h νβ = 0, leading to [176] [Rμνρσ ] = 0,

.

[∇α Rμνρσ ] = 0,

(9.58)

9.4 Three Special Types of Gluing

75

indicating the curvature tensor and its covariant derivatives to be continuous across ∑ as a boundary surface. In the exceptional case .(nκ1 + κ2 ) = 0, we see from (9.49) that .b ≡ ∈n ρ [∇ρ R] can be non-zero, leading to discontinuities of curvature tensors given by (9.53) and (9.54). Therefore, while .∑ is a boundary surface, jumps in the curvature tensor and its derivatives across .∑ are expected. However, as previously mentioned, this case turns out to be irrelevant due to its defining condition depending on the dimension .n. • .κ2 /= 0 and .κ1 = 0 Similar to the previous one. • .κ2 = 0 but .κ1 /= 0 As in the previous case, from (9.46), one gets [R] = 0, but .[G μν ] /= 0, therefore .[Rμν ] /= 0. Then, for the smooth matching we are considering and given (9.49), we have .b ≡ ∈n ρ [∇ρ R] = 0. Taking now Eq. (9.37) with .Tμ = 0, we obtain σ .h μ ∇σ [R] = 0, which by using (8.21) leads to.[∇μ R] = 0. It is, however, easily seen that .[∇μ Rμν ] /= 0 [176]. • .κ2 = 0 and .κ1 = 0 In this case, the quadratic part of the Lagrangian (9.1) is the Gauss–Bonnet term .(a1 = a3 = −a2 /4). All junction equations (9.36), (9.37), (9.38), and (9.39) are satisfied identically subject to the requirement (9.6) which is indispensable for quadratic gravity. .

(iii) Pure double layer at .∑ In this extreme case, the double layer term.Tμν in (9.32) is non-zero while the remaining singular part in the energy-momentum tensor, that is .T˘μν , including . Sμν ,.Tμ , and .T simultaneously vanish, meaning a double layer without a thin shell. By vanishing .Tν and .T , Eqs. (9.42) and (9.43) become, respectively, h μ ∇ σ Pμν = 0,

(9.59)

K μν Pμν = 0,

(9.60)

. σ

.

showing the conservation of double layer strength .Pμν . Inserting the expression (9.41) for .Pμν into (9.59) and (9.60) yields (κ1 + κ2 )h νσ ∇ σ [R] + 2κ2 h μσ ∇ σ [G μν ] = 0,

(9.61)

(κ1 + κ2 )[R]K + 2κ2 K μν [G μν ] = 0.

(9.62)

.

.

From (9.60) and vanishing the trace . Sμν given by (9.48), one gets (nκ1 + κ2 )b = 0.

.

Now, we distinguish two different cases:

(9.63)

76

9 Gluing Conditions in Quadratic Theories of Gravity

• .nκ1 + κ2 /= 0 Then .b ≡ ∈n ρ [∇ρ R] = 0, meaning that the jump in the derivative of Ricci scalar is now tangent to .∑, written as (see Eq. (8.21)) [∇μ R] = h σμ ∇σ [R].

.

(9.64)

Vanishing of . Sμν given by (9.36) yields an expression for the tangent part of discontinuity in the normal derivative of Ricci tensor .rμν ≡ n σ [∇σ Rμν ] as h ρ h σν rρσ =

. μ

) ( 1 κ1 [R]K μν . 1+ 2 κ2

(9.65)

In case .κ2 /= 0, the vanishing of .Tμ given by (9.37) yields an expression for the tangent-normal part of discontinuity in the normal derivative of Ricci tensor as h ρ n σ rρσ =

. μ

κ1 1 (1 + )h σμ ∇σ [R]. 2 κ2

(9.66)

Similarly, the vanishing of .T given by (9.38) yields an expression for the total normal part of discontinuity in the normal derivative of Ricci tensor as n μ n ν rμν = −

.

) ( κ1 1 [R]K . 1+ 2 κ2

(9.67)

Taking into account these expressions, without loss of generality, one can write the discontinuity in the normal derivative of the Ricci tensor as ) ( κ1 1 ([R]K μν + n μ h σν ∇σ [R] + n ν h σμ ∇σ [R] − n μ n ν [R]K ). 1+ .r μν = 2 κ2 (9.68) In the case of .κ2 = 0 but .κ1 /= 0, (9.61) and (9.62) may be written as h σ ∇σ [R] = 0.

(9.69)

[R]K μν = 0.

(9.70)

. ν

.

As a consequence of (9.69) and (9.64), we obtain .[∇μ R] = 0. Now, from (9.41), we see that the double layer strength (for .κ2 = 0) is proportional to .[R], meaning that a non-zero .Pμν requires .[R] /= 0, and hence from (9.70) one necessarily gets . K μν = 0. • .nκ1 + κ2 = 0 In this exceptional case, with vanishing. Sμν ,.Tμ , and.T , one easily gets from (9.36), (9.61), and (9.62), respectively,

9.4 Three Special Types of Gluing

77

(1 − n)[R]K μν − bh μν + 2nh ρμ h σν rρσ = 0,

.

(1 −

.

n)h σμ ∇σ [R]

ρσ

− 2nh ∇σ [G ρμ ] = 0, μν .(1 − n)[R]K − 2n K [G μν ] = 0.

(9.71) (9.72) (9.73)

The first equation gives an expression for the tangent part of the discontinuity in the normal derivative of the Ricci tensor, while the two other ones provide constraints on the allowed jumps of the Ricci scalar. In addition, from (9.41), we can see that in (n+1)-dimensional spacetimes, the strength of the double layer is traceless .(P = (nκ1 + κ2 )[R] = 0). In summary, within a quadratic gravity, the continuity of metric on .∑ and the extrinsic curvature, as given in (9.6), is a geometric/algebraic necessity to define a solution of the given dynamics. This is in contrast to GR, where these requirements may happen just for smooth gluing which is a dynamics assumption. Hence, in case the quadratic terms are switched on in the Hilbert action, any spacetime manifold containing a boundary surface will develop a double layer and a thin shell at the gluing hypersurface. Moreover, for the energy-momentum tensor to be divergencefree as a necessary local conservation condition, all terms in (9.35) and the double layer .Tμν have to be included [176].

Chapter 10

Special Applications

We are now in a position to apply the art of manipulating manifolds to embed singular hypersurfaces within them. Such glued spacetime manifolds have a vast spectrum of applications. We will concentrate on some time-, space-, or lightlike examples just to elaborate on the art and show how useful this technology is.

10.1 Timelike Shells Let us start with some typical timelike thin shells embedded in 4- and 5-dimensional spacetimes.

10.1.1 Planar Shell in Vacuum Planar shell or wall with a homogeneous and isotropic gravitational field in just one direction orthogonal to it has always been of interest. Now, the only well-known plane-symmetric vacuum solution to Einstein equations without cosmological constant is the Taub solution [199]. In the plane-symmetric coordinates.x μ = (t, z, x, y), it is given by ds 2 = √

.

1 1 + kz

(−dt 2 + dz 2 ) + (1 + kz)(d x 2 + dy 2 ),

(10.1)

where .k > 0 is an integration constant. This solution (10.1) represents a homogeneous and isotropic gravitational field in the .x − y plane. By a suitable coordinate transformation, the above metric may be recast into the following form [2]

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Khakshournia and R. Mansouri, The Art of Gluing Space-Time Manifolds, SpringerBriefs in Physics, https://doi.org/10.1007/978-3-031-48612-8_10

79

80

10 Special Applications

ds 2 = −

.

1 z 2/3

dt 2 + dz 2 + z 4/3 (d x 2 + dy 2 ),

z /= 0.

(10.2)

As shown by Taub himself, this spacetime is asymptotically flat as .|z| → +∞ and singular at .z = 0. The Kretschmann scalar for (10.1) is found to be .

K ≡ Rμνσγ R μνσγ =

64 . 27z 4

(10.3)

It shows that .z = 0 is a physical singularity. The main properties of Taub solution are: (i) it is curved; (ii) it is plane-symmetric and static; (iii) it has a timelike singularity at .z = 0; (iv) its Kretschmann invariant vanishes for .|z| −→ ∞, signaling it is asymptotically flat at spatial infinity in the .z-direction. Is it possible to replace the singularity with a regular source as the central shell? This assumption has drawn much attention to finding sources for the Taub solution [28, 58]. It has been found that the source of the Taub solution will have a negative energy density if it is required to be static, reflection symmetric, and free of spacetime singularity. To see it, one will use the concept of glued manifold and thin shell formalism to model the source as an infinitely thin plane, so-called the planar shell or planar wall [109]. Using the substitution .z → |z| into the metric (10.1), we cut the spacetime into two regions, .M+ with .z > 0, and .M− with .z < 0, and then glue the Taub domain .z > 0 with a copy of it, so that the resulted glued spacetime .M− ∪ M+ has a reflection symmetry with respect to the surface .z = 0 with .M− on its left side, and .M+ on its right side. In this way, with .ξ A = (τ , x, y) chosen to be the intrinsic coordinates on the timelike hypersurface of junction .∑ situated at .z = 0, the induced 3-metric on .∑ is written as ds 2 |∑ = −dτ 2 + d x 2 + dy 2 ,

(10.4)

.

where .τ denotes the proper time of observers whose world lines lie within .∑. Thus, the first junction condition expressing continuity of the induced metric across .∑ is automatically satisfied. The tangent vectors on .∑ are eμ = u μ |± = (1, 0, 0, 0),

exμ = (0, 0, 1, 0),

eμy = (0, 0, 0, 1), (10.5) where .u μ is the four-velocity of observers on .∑. The unit normal vector to .∑ deterμ mined from .n μ ei = 0 and .n μ n μ = 1, directed from .M− to .M+ , has components . τ

μ

n = (0, 1, 0, 0).

(10.6)

. ±

μ

The extrinsic curvature tensor of the shell is calculated from . K i j = e i eνj ∇ μ n ν (see Eq. (3.2)). Let us first calculate the components of the connection coefficients (Remember that the substitution .z → |z| into the metric (10.1) has already been made)

10.1 Timelike Shells

81

1 [τz τ |+ = −[τz τ |− = − k, 4 1 z z [x x |+ = −[x x |− = − k, 2 1 z [ yy |+ = −[xz x |− = − k. 2

.

(10.7)

Then, the non-vanishing components of the extrinsic curvature are found to be .

1 ± = ± k. K x±x = K yy 2

1 K τ±τ = ± k, 4

(10.8)

Notice that due to the reflection symmetry, we have . K i−j = −K i+j , as expected. Assume that the surface energy-momentum tensor of the planar shell is given by S = (σ + p)u i u j + pgi j ,

. ij

(10.9)

where .gi j is the shell intrinsic metric given in (10.4), .σ is the surface energy density of the shell, and. p is the pressure in.x and. y directions measured by observers who are at rest with respect to the shell. The Lanczos equation (3.12) determines the surface energy density .σ and surface pressure . p of the planar shell σ=

.

−k , 4π

p=

k , 16π

(10.10)

where we have substituted .κ = 8π in Lanczos equation (3.12). This energy density being negative violates the weak-, dominant-, and strong-energy conditions. Therefore, we have to conclude that the planar shell as glued manifold embedded in Taub solution is not a physical solution of the Einstein equations. This result (10.10) is in agreement with the necessary condition . p = − 41 σ, for the static solution of planarsymmetric vacuum solution of Einstein equations as given by Ipser et al. [109, 110].

10.1.2 Gluing Two Different FRW Spacetimes and Bubble Dynamics in Cosmology Spherical bubbles embedded in a spherically symmetric background universe arise in many applications of general relativity, including cosmic inflation, cosmological phase transitions, and gravitational collapse. In these scenarios, the idealized structured spacetime is glued from two different manifolds with distinct metrics across a spherically symmetric thin shell, the so-called bubble wall, whose history coincides with a timelike hypersurface and its evolution is described by the thin shell formalism [32, 41, 48, 68, 83, 128, 132, 134, 142, 160, 163, 197].

82

10 Special Applications

To address this issue, let us look at the dynamics of a general spherical bubble in a cosmological setting. For this purpose, we consider the (2+1)-dimensional timelike hypersurface .∑ as the trajectory of a spherical bubble wall embedded in two glued Friedmann–Robertson–Walker (FRW) spacetimes .M± given by | ( ) | ds 2 | = −dt±2 + a2± (t± ) dχ2± + r±2 (χ± )(dθ2 + sin2 θdϕ2 ) ,

.

±

(10.11)

where .χ is a comoving radial coordinate, and ⎧ closed universe), ⎨ sin χ (k = +1, χ (k = 0, flat universe), .r (χ) = ⎩ sinh χ (k = −1, open universe)·

(10.12)

Both sides of the bubble wall evolve according to the Friedmann equations .

.

H±2 +

8πGρ± k± , = 2 3 a±

1 d 2 a± 4πG =− (ρ± + 3 p± ) , 2 a± dt± 3

(10.13)

(10.14)

± is the Hubble parameter, .ρ± and . p± are the energy density and where . H± = a1± da dt± pressure of matter in .M± , respectively. The induced metric on .∑ is given by

| | ds 2 | = −dτ 2 + R 2 (τ )(dθ2 + sin2 θdϕ2 ),

.

∑

(10.15)

where .τ is the proper time on comoving world lines along which .χ, .θ, and .ϕ remain constant, and . R(τ ) = a(t (τ )r (χ(τ ))|∑ . Continuity of induced metric on the wall requires 2 2 2 . − t˙± + a(t)± χ ˙ ± = −1, (10.16) and .

∑

R(τ ) ≡ a− (t− )r− (χ− ) = a+ (t+ )r+ (χ+ ),

(10.17)

∑

where .= means that both sides of the equality are evaluated on .∑, and overdot denotes derivative with respect to .τ . Solving (10.16) for .t˙, one gets the expression for Lorentz factor .γ γ ≡ t˙± = √

. ±

1 2 1 − v±

,

(10.18)

10.1 Timelike Shells

83

± where.t˙ is assumed to be positive as both.t and.τ are future-directed, and.υ± ≡ a± dχ dt± is the peculiar velocity of the wall observed in .M± . Differentiation of (10.17) with respect to the proper time .τ gives

dR . = γ− dτ

(

dr− v− + H− R dχ−

)

( = γ+

) dr+ v+ + H+ R . dχ+

(10.19)

The tangent vector .u μ of .∑ is defined as μ

u =

. ±

μ ( ) || d x± || | = t˙± , χ˙ ± , 0, 0 | . ∑ dτ ∑

(10.20)

Now, from .n μ n μ = 1 and .n μ u μ = 0, the normal vector .n μ must be ( )| 1 | μ n = ζ± a± χ˙ ± , t˙± , 0, 0 | , ∑ a±

. ±

(10.21)

where .ζ is the undetermined sign of the normal vector .n μ to .∑. Let us focus now on the angular component of extrinsic curvature tensor of .∑, having a crucial role in understanding significant features of geometry and topology of gluing. Defined as θ θ t θ r θ . K = ∇θ n = −[ n − [ n , we use (10.21) to obtain θ tθ rθ |

θ| .Kθ | = ±

γ± ζ± R

(

) dr± + υ± H± R . dχ±

(10.22)

This angular component is expressed in geometrical quantities. We now refer to the first Friedman equation (10.13) to substitute the Hubble function in terms of physical quantities in (10.22) using (10.12) and (10.17). In doing so, one needs to use the square of (10.22), leading to a change in the role of sign function .ζ. The final result is √ | ε± 8πρ± 2 θ| R , (10.23) .Kθ | = 1 + R˙ 2 − ± R 3 with .ε ≡ sgn(K θθ ) being a new sign factor. This form of the angular component of the extrinsic curvature was first obtained in [33], based on the sign .ζ. We will soon analyze the meaning and difference of both sign factors, but let’s first derive explicitly the Lanczos equations. The surface energy-momentum tensor of the wall in perfect fluid form is given by μν .S = (σ + ω)u μ u ν + ωh μν , where .u μ is the four-velocity of any observer whose world line lies within.∑ without energy flux in his local frame. The physical quantities .σ and .ω are surface energy density and pressure measured by that observer. The angular component of the Lanczos equation (3.12) takes then the form √ ε

. −

√ 8πρ− 2 8πρ+ 2 2 ˙ R − ε+ 1 + R˙ 2 − R = 4πσ R, 1+ R − 3 3

(10.24)

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10 Special Applications

where the expression (10.23) for . K θθ has been used. In the spherically symmetric case we are studying, the Lanczos equation has just two independent angular and time components, while the time component of the conservation identity (3.14) is given by ] [ 2 R˙ υ dσ + (σ + ω) = (ρ + p) . . (10.25) dτ R 1 − υ2 ± This can be viewed as an integrability condition for the angular and time components of the Lanczos equation (it was shown in [128] that, for the case of spherical symmetry, the time component of the Lanczos equation always reduces to an identity). Hence, (10.24) and (10.25) can be considered as basic equations for the evolution of spherical bubble walls. We are now ready to explore the significance of sign functions .ε and .ζ. Following the approach of [182], we introduce Gaussian normal coordinates .(n, τ , θ, ϕ), with .n = 0 corresponding to .∑ ds 2 = dn 2 − a2 (n, τ )dτ 2 + R¯ 2 (n, τ )(dθ2 + sin2 θdϕ2 ),

.

(10.26)

¯ where. R(n, τ ) is the radius of 2D spheres with.n = const and.τ = const, in the Gaussian normal coordinates. Given the induced metric on .∑ in the form .ds 2 = −dτ 2 + ¯ τ ) = R(τ ) = a(t (τ ))r (χ(τ ))|∑ , with R 2 (τ )(dθ2 + sin2 θdϕ2 ), one requires . R(0, .a(0, τ ) = 1. The angular component of the extrinsic curvature tensor of the shell in this coordinate system is then written as θ .Kθ

( ) 1 ∂ R¯ || 1 ∂ R¯ ∂χ ∂ R¯ ∂t || + = ∇θ n = | = | . R ∂n ∑ R ∂χ ∂n ∂t ∂n ∑ θ

(10.27)

From (10.27), it is evident that .ε, the sign of . K θθ depends on whether or not the radii of 2D spheres increase in the normal direction. Transforming now the normal vector .n μ written in the Gaussian coordinates .(n, τ , θ, ϕ) to FRW ones .(t, χ, θ, ϕ), we obtain )| ( ∂t ∂χ ∂x μ || | μ , , 0, 0. = (10.28) .n = | | . ∑ ∂n ∑ ∂n ∂n Comparing (10.28) with (10.21), we arrive at .

| ∂t | = ζa χ˙ | , ∑ ∂n

1 || ∂χ = ζ t˙| . ∂n a ∑

(10.29) μ

Taking into account .t˙ > 0, from (10.29) we see that .ζ± , signs of normal vector .n ± , ± which determine the global topology of spacetime: in case of are also the signs of. ∂χ ∂n flat or open universe .ζ± determine the interior-exterior characters of .M± , in contrast to a closed universe where .ζ± can be changed by the coordinate transformation .χ± → π − χ± [182]).

10.1 Timelike Shells

85

Let us now look at different cases. The case .ζ+ > 0 and .ζ− < 0 corresponds to a shell embedded in a spacetime with no center (.χ = 0), so-called wormhole matching or wormhole manifold. If .ζ+ < 0 and .ζ− > 0, then the glued manifold has two centers, one in each region (anti-wormhole matching). In the case that both .ζ+ and .ζ− are positive, the spacetime .M− plays the role of interior containing a point singularity, and .M+ is the exterior part of the glued manifold; this is the case where realistic bubbles nucleate in the actual universe. For both .ζ+ and .ζ− negative, the global structure of spacetime is similar to the last one, with the roles of .M+ and − .M being swapped. Now, let us turn to .ε, the sign of . K θθ . In static spacetimes, it is easily seen that .ε is reduced to .ζ (10.22); therefore, it specifies the spatial topology playing a key role in classifying the global spacetime structures [185] (for a recent classification of spherically symmetric electrically charged thin shells with a Minkowski interior and a Reissner–Nordström exterior see [139]). In the general non-static case, however, it is seen from (10.22) that .ε, the sign of . K θθ , has to be taken different from .ζ: due to the independence of .υ and . R, one can set up any sign of . K θθ . As an example, assume .k + ≤ 0 and take the comoving radius of the shell to decrease in time, meaning the peculiar velocity of the shell observed in .M+ is negative (.υ+ < 0). Then, as the second term in (10.22) being negative becomes large enough, the sign of . K θθ , not taking .ζ into account, can get negative. It requires the physical radius of the shell, . R, to be larger than the horizon length (. R > H+−1 ) due to . H+ R > (dr+ /dχ+ )/|υ+ | > 1 for .k+ ≤ 0, as mentioned in [183]. Therefore, it is concluded that, in general, for the time-dependent spacetimes, like FRW, the signs of . K θθ± do not determine the spatial topology. The Raychaudhuri equation being a follow-up of our dynamic expressing the expansion of null geodesics may be used to give us more direct information on some results of the gluing. According to [102], if the background universe is flat or open, and the cosmic bubble is flat or positively curved, we can see from the Raychaudhuri equation that this gluing is possible only if the bubble expansion rate is smaller than the background . H− < H+ for any bubble size. On the other hand, if the bubble is negatively curved, the obtained constraint appears to put an upper bound on the size of the allowed bubble, rather than the lower bound of getting a superhorizon size, as mentioned above.

10.1.3 Embedding a Spherical Inhomogeneous Region into a FRW Background Universe Consider a glued manifold model of a spherical inhomogeneous structure represented by the Lemaitre–Tolman–Bondi (LTB) metric as .M− glued to a pressure-free FRW background universe as .M+ [119]. The LTB metric in the synchronized comoving coordinates is written as

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10 Special Applications

R '2 dr 2 + R 2 (r, t)(dθ2 + sin2 θdφ2 ), 1 + E(r )

ds−2 = −dt 2 +

.

(10.30)

where . E(r ) > −1 is an arbitrary real function representing a local curvature. The corresponding Einstein field equations are given by ( .

.

∂R ∂t

)2

2M(r ) , R M ' (r ) 4πρ(r, t) = 2 ' , R R = E(r ) +

(10.31) (10.32)

where overdot and prime represent derivatives with respect to .t and .r , respectively, and . M(r ) is defined by ∫ .

R(r,t)

M(r ) = 4π

ρ(r, t)R 2 d R.

(10.33)

0

An average energy density in the interior .M− for the LTB inhomogeneous bubble can be defined as ρ =

. −

M(r ) || . 4π 3 |∑ R 3

(10.34)

The timelike hypersurface upon which these spacetime regions are glued is the bubble wall .∑. The Brezin formula given in [33], for the angular component of the extrinsic curvature tensor . K θθ of .∑ in the LTB interior region yields |

θ| .Kθ |

√ 2M(r ) || ε− 1 + R˙ 2 − = | . − R R ∑

(10.35)

Using the expression(10.34) for the average density, (10.35) takes the form |

θ| .Kθ |

√ 8πρ− 2 ε− R . = 1 + R˙ 2 − − R 3

(10.36)

Having obtained the expressions (10.35) and (10.23) for . K θθ in .M− and .M+ , respectively, we write the angular component of the Lanczos equation as √ ε

. −

√ 8πρ− 2 8πρ+ 2 2 ˙ R − ε+ 1 + R˙ 2 − R = 4πσ R. 1+ R − 3 3

(10.37)

Solving Eq. (10.37) for . R˙ 2 , we obtain .

) ( 8π R˙ 2 = 4π 2 σ 2 (ξ − 1)2 + ρ+ R 2 − 1, 3

(10.38)

10.1 Timelike Shells

87

where .ξ is defined by ξ≡

.

ρ+ − ρ− · 6πσ 2

(10.39)

The surface energy density.σ on .∑ is assumed to be positive. Substituting Eq. (10.38) back into Eq. (10.24), we get ε |ξ + 1| − ε+ |ξ − 1| = 2 ·

. −

(10.40)

From (10.40), it is easily seen that the sign functions .ε± are determined by the value of .ξ as ⎧ ⎨ (+1, +1) for ξ > 1, (+1, −1) for |ξ| < 1, .(ε− , ε+ ) = (10.41) ⎩ (−1, −1) for ξ < −1· Notice that due to the interior and exterior characters of .M− and .M+ , respectively, we have .ζ± = +1. Hence, from (10.22), .ε+ can be determined as ( ε = sgn

. +

) dr+ + υ+ H+ R , dχ+

(10.42)

where . H+ is the Hubble parameter of the background, and .υ+ is the peculiar velocity of .∑ relative to the FRW background. Depending on the average energy density within the bubble wall compared to the background density, the following cases may now be distinguished [119]: (I) .ρ− > ρ+ . An overdense inhomogeneous bubble embedded in a background FRW universe. Obviously .ξ < 0. Thus, from (10.41), we must have .ε+ = −1. This leads to the following cases: (a) .k+ = 0, −1; R < H+−1 . Then it is easily seen from (10.42) that .ε+ = +1. Therefore, there is no matching of an overdense bubble sized smaller than the background Hubble radius to a flat or negatively curved background FRW universe. (b) .k+ = 0, −1; R > H+−1 , and the peculiar velocity .υ+ < 0. It is seen from Eq. (10.42) that.ε+ = −1. Therefore, overdense spherical structures with a radius greater than the background Hubble radius and .υ+ < 0, can be glued to a background FRW universe with .k+ = 0, −1. (c) .k+ = +1. Taking the definition (10.12), we see from (10.42) that one may have .ε+ = −1. Therefore matching is possible regardless of .ξ value. (II) .ρ− < ρ+ ; k+ = 0, −1. An overdense inhomogeneous region surrounded continuously by an underdense region .ρ− (r, t) < ρ+ , i.e., a void described by the same LTB metric. The underdense void as part of the LTB region is continuously glued to the background FRW satisfying Eq. (10.24), .ρ− is the overall average density related

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Table 10.1 Classification of junctions of LTB spherical inhomogeneity to the flat and open FRW background R < H+−1 or R > H+−1 , υ+ < 0 −1 . . R > H+ , υ+ > 0 (ε+ = −1) (ε+ = +1) .ρ−

> ρ+

.

.ρ−

= ρ+

.

.ρ−

< ρ+

Thin shell ξ H+−1 . In this case, with the negative peculiar velocity, we have .ε+ = −1. The matching is possible with .0 < ξ < 1, i.e., .σ /= 0. (III) .ρ− = ρ+ ; k+ = 0, −1. The void completely compensates for the overdense region, so that the overall mean density of the inhomogeneous bubble is equal to that of the background FRW universe. (a) . R < H+−1 . It turns out that .σ = 0, the matching is possible without the formation of a thin shell. This situation reminds us of the Einstein–Straus type model [10], where the overdense region is surrounded by a vacuum shell compensating for the mass excess. (b) . R > H+−1 . For the negative peculiar velocity, we have again .ε+ = −1. Both junctions with .σ = 0 and ./= 0 are possible, as it is easily seen from Eq. (10.41). The case .ε− = −1 corresponds to .σ = 0, meaning no thin shell. The case .ε− = +1 is only possible through a thin shell. Table 10.1 summarizes the above results.

10.1.4 Moving Brane in the Static Schwarzschild-AdS Bulk A currently relevant scenario where thin shells play a role is the brane-world cosmology (for a review see [44]). In this scenario, there is a five-dimensional bulk spacetime with an embedded three-brane where matter is confined, and Newtonian gravity is effectively reproduced at large distances. Thus, in the brane-world scenario dynamics of our universe is reflected by that of the brane.

10.1 Timelike Shells

89

To see the impact of thin shell formalism on deriving the brane cosmological equations, we consider the general spherically symmetric solution of five-dimensional Einstein equations .G AB + Δg AB = 0, with the cosmological constant .Δ < 0 known to be static, corresponding to the Schwarzschild-anti-de Sitter spacetime [154]. It is in contrast to the original Randall–Sundrum solution [174], which is a 5D anti-de Sitter spacetime. The Schwarzschild-AdS bulk metric in five dimensions is given by [108, 129] .ds

( ) 2 2 = − f (r )dt 2 + dr + r 2 dχ2 + f (χ)2 (dθ 2 + sin2 θdϕ2 ) , k f (r )

k = −1, 0, 1

(10.43)

with .

f (r ) = k −

Δ 2 M r − 2, 6 r

(10.44)

where . f 1 (χ) = sin χ, . f 0 (χ) = χ, . f −1 (χ) = sinh χ, and . M the Schwarzschild mass parameter. The three-brane is made of gluing two copies of five-dimensional Schwarzschild-AdS spacetime portions along the brane world-sheet while imposing at the same time a . Z 2 symmetry (see [61] for cases without . Z 2 symmetry). In this case, the . Z 2 symmetry means that by approaching the brane from one side and going through it, one emerges into a bulk that looks the same, with the normal vector reversed. The radial coordinate .r is taken to decrease as one moves away from either side of the brane (spacetime with two centers). In doing so, the normal vector .n A points to increasing .r on one side of the brane (the “-” side with the sign of relevant normal vector .ζ− > 0) and to decreasing .r on the other side (the “+” side with the sign of relevant normal vector .ζ+ < 0). Hence, the normal vector must flip across the brane.1 The history of the brane is then a timelike hypersurface .∑; the presence of matter on it, is due to a jump in the extrinsic curvature across the brane, stemming from the flip of the normal vector across .∑. The four-dimensional metric induced on the three-brane is then given by the FRW form ) ( ds 2 = −dτ 2 + R 2 (τ ) dχ2 + f (χ)2 (dθ2 + sin2 θdϕ2 ,

.

(10.45)

meaning that the cosmic time and the scale factor of the corresponding FRW universe are identified with the proper time .τ and radial coordinate of the brane . R(τ ), respectively. The trajectory of the brane is given by .t = T (τ ) and .r = R(τ ), indicating that the three-brane .∑ is moving in the static reference frame (10.43). The tangent vector on the brane from each side is given by ˙ 0, 0, 0), u A = (T˙ , R,

. ±

1

(10.46)

Recall that in the thin shell formalism, the positive orientation of normal vector is chosen such − + + that if .n − μ points outwards from .M , then .n μ points inwards over .M , see Sect. 3.2.

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10 Special Applications

with the dot denoting derivative with respect to .τ . Here . R˙ represents the velocity of the brane measured with the proper time on it. The unit normal vector, defined such that .n A u A = 0 and .n A n A = 1, is then obtained to be A .n −

)| ( ˙ √ R | 2 ˙ = ζ− , f + R , 0, 0, 0 | , ∑ f

(10.47)

( ˙ √ )| R | , f + R˙ 2 , 0, 0, 0 | , ∑ f

(10.48)

n A = ζ+

. +

where .ζ± indicating the interior-exterior characters of .M± is given by .ζ− = +1 and .ζ+ = −1. The spatial components of the extrinsic curvature are computed as √ − .Ki j

=

√

f + R˙ 2 || gi j | , ∑ R

K i+j

=−

f + R˙ 2 || gi j | . ∑ R

(10.49)

The generalization of Lanczos equation (3.12) to the five dimensions reads ( [K μν ] =

.

−κ25

Sμν

) 1 − Sh μν , 3

(10.50)

where .κ25 is the gravitational coupling constant in 5-dimensions. Let us assume the energy-momentum tensor of brane in the form of a prefect fluid . Sμν = (ρb + pb )u μ u ν + pb gμν , with .ρb and . pb being the brane energy density and pressure, respectively. Then, the spatial components of Lanczos equation (10.50) for the brane take the form √ κ2 f (R) + R˙ 2 . (10.51) = 5 ρb . R 6 Substituting (10.44) into (10.51) and rearranging the result, one finds .

κ45 2 Δ C R˙ 2 k ρ + + 4 − 2. = R2 36 b 6 R R

(10.52)

It is seen that the junction equation written for the brane directly leads to the modified Friedmann equation on the brane relating the Hubble parameter to the brane energy density. This is essentially different from the standard four-dimensional Friedmann equation in the sense that the energy density of the brane enters quadratically on the right-hand side, in contrast to the usual linear dependence, as addressed in [38, 137]. The time component of the extrinsic curvature tensor, from each side, is given by .

K τ−τ =

1 d R˙ dτ

√

| | f + R˙ 2 | , ∑

K τ+τ = −

1 d R˙ dτ

√

| | f + R˙ 2 | . ∑

(10.53)

10.1 Timelike Shells

91

Using (10.53), the time component of the Lanczos equation takes the form 1 d . R˙ dτ

(√

f (R) + R˙

) =

−κ25 (2ρb + 3 pb ) . 6

(10.54)

Combining the junction equations (10.51) and (10.54) yields the time component of the conservation identity (3.14) on the brane ρ˙ + 3

. b

R˙ (ρb + pb ) = 0. R

(10.55)

Therefore, using the technology of gluing manifolds for a brane in the static fivedimensional Sch-AdS spacetime (10.43), we see the equivalence of its dynamics to the cosmological evolution in four dimensions.

10.1.5 Gravitational Collapse of a Rotating Thin Shell in 5D There are a few analytic solutions to Einstein’s equations describing the gravitational collapse of rotating matter. Myers and Perry [158] obtained a higher-dimensional generalization of the Kerr solution describing stationary rotating black holes parameterized by a mass parameter .m and .(D − 1)/2 angular momentum parameters .ai . The solution has been subsequently generalized to include a cosmological constant in five dimensions [103]. Five-dimensional spacetimes admit two orthogonal rotation planes and, thus, two independent angular momenta. When these two momenta are equal, the isometry group is enhanced, and the metric functions can be expressed in terms of a single coordinate .r . Spacetimes with this property, commonly referred to as having cohomogeneity 1, are of special interest. Consider a gluing manifold based on Myers–Perry solutions with cohomogeneity 1 in the presence of a cosmological constant in 5D. Metrics describing the two geometries to be glued with two parameters .(m ± and .a± ) in coordinates .x μ = (t, r, ψ, θ, φ) are given by ( )2 2 2 = − f 2 (r )dt 2 + g 2 (r )dr 2 + r (dθ 2 + sin2 θdφ2 ) + h 2 (r ) dψ + 1 cos θdφ − Ω (r )dt , ± ± ± ±

.ds±

4

2

(10.56) where 2 .g± (r )

( 2 )−1 2m ± a± 2m ± Ξ± r2 + , = 1+ 2 − l r2 r4

( 2 ) 2m ± a± , h 2 (r ) = r 2 1 + r4

. ±

Ω± (r ) =

2m ± a± , r 2 h 2± (r )

(10.57)

(10.58)

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10 Special Applications

f (r ) =

. ±

r , g± (r )h ± (r )

Ξ± = 1 −

2 a± . l2

(10.59)

The metric (10.56) is a solution of the vacuum Einstein equations with a cosmological constant equal to .Δ = −6l −2 (for more detail see [131]). Asymptotically, the solution (10.56) approaches anti-de Sitter space with the radius of curvature .l. An asymptotically flat Myers–Perry black hole is recovered by taking .l −→ ∞. Now, two spacetimes with metrics given above are glued across the timelike hypersurface .∑ defined parametrically by the equations .t = T (τ ) and .r = R(τ ), where .τ is the proper time of an observer comoving with the hypersurface. For convenience, one may go to a rotating frame to remove the cross terms .dtdψ and .dtdφ in the induced metric on .∑ by making the coordinate transformation .dψ −→ dψ + Ω± (R(t))dt. The exterior and interior metrics then take the forms r2 2 ds±2 = − f ±2 (r )dt 2 + g± (r )dr 2 + (dθ2 + sin2 θdφ2 ) 4 ]2 [ 1 + h 2± (r ) dψ + cos θdφ − (Ω± (R(t)) − Ω± (r ))dt . 2

.

(10.60)

The induced metric on .∑ in the intrinsic coordinates . y i = (τ , ψ, θ, φ) then becomes )2 ( | 1 R2 2 2 2 2 2 (dθ + sin θdφ ) + h (R) dψ + cos θdφ , .ds | = −dτ + ∑ 4 2 2|

(10.61)

where .h − (R) = h + (R) ≡ h(R) on .∑. The continuity of induced metric then implies .

2 − f ±2 (R)T˙ 2 + g± (R) R˙ 2 = −1,

2 2 m − a− = m + a+ ,

(10.62)

where dot stands for .d/dτ . Therefore, the Darmois junction condition dictates that a rotating exterior (with .m + a+ /= 0) cannot be continuously glued to a non-rotating .(a− = 0) or flat .(m − = 0) interior. In other words, in the presence of rotation, the matched solution requires the interior geometry to be that of a rotating mass distribution. μ ˙ 0, 0, 0), the unit normal vectors to With the tangent vector of .∑ as .u ± = (T˙ , R, .∑ are given by ( )| g± ˙ f ± ˙ | μ (10.63) .n ± = R, T , 0, 0, 0 | . ∑ f± g± Notice that due to the interior and exterior characters of the manifolds .M− and + .M , respectively, the sign of the relevant normal vectors is set to be positive. The non-vanishing components of the extrinsic curvature are computed to be [178, 179] .

K τ±τ = −

h(R) dβ± , R dR

K τ±ψ = −

h 3 (R) dΩ± (R) , 2R dR

(10.64)

10.1 Timelike Shells

.

93

h 2 (R)β± dh(R) , R dR

± K ψψ =

.

± K ψφ =

.

1 ± K cos θ, 2 ψψ

± K φφ =

K τ±φ = ± K θθ =

1 ± K cos θ, 2 ψψ

1 h(R)β± , 4

1 ± (K cos2 θ + h(R)β± sin2 θ), 4 ψψ

(10.65) (10.66) (10.67)

√ | 2 ˙ 2| where .β± = f ± 1 + g± R | . These non-zero components of the shell extrinsic ∑ curvature tensor, through the Lanczos equation, dictate the form of the shell energymomentum tensor to be that of an imperfect fluid S = (ρ + p)u i u j + pgi j + ϕu i ξ j + ϕu j ξi + Δp R 2 gˆi j ,

. ij

(10.68)

with .ρ and . p being the shell energy density and pressure, .ϕ denoting the intrinsic momentum of the fluid or heat flow, and .Δp representing pressure anisotropy. In addition, .u = u i ∂i = ∂τ , is the normalized fluid velocity co-rotating with the shell, i −1 .ξ = ξ ∂i = h (R)∂ψ is a unit vector aligned with the direction that effectively incorporates the rotation of spacetime, and .gˆi j dy i dy j = 41 (dθ2 + sin2 θdφ2 ). When .ϕ = Δp = 0, the energy-momentum tensor turns out to be that of a perfect fluid with the heat flow and pressure anisotropy necessarily generated when rotation is present. Writing the Lanczos equation (10.50) for the shell, one obtains [65] (β+ (R) − β− (R)) d (R 2 h(R)), dR κ25 R 3 ) h(R) d ( 2 R (β+ (R) − β− (R) , .p = 2 3 dR κ5 R

ρ=−

.

h 2 (R) d (Ω+ (R) − Ω− (R)), 2κ25 R d R ( ) h(R) (β+ (R) − β− (R) d . .Δp = dR R κ25 ϕ=−

.

(10.69) (10.70) (10.71) (10.72)

The conservation identity (3.14) written as .∇i S i j = 0 yields two independent equations ˙ ' (R) + h(R)(2(ρ + p + Δp) R˙ + R ρ) .(ρ + p)R Rh ˙ = 0, (10.73) ˙ ' (R) + h(R)(2ϕ R˙ + R ϕ) 2ϕR Rh ˙ = 0,

.

(10.74)

The identities (10.73) and (10.74) are automatically satisfied taken the energy density, pressure, heat flow, and pressure anisotropy are given by Eqs. (10.69), (10.70), (10.71), and (10.72), respectively [178].

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Equation (10.74) may be equivalently rewritten as .

) d ( 2 2 ϕR h (R) = 0. dτ

(10.75)

Indeed, it can be shown that the quantity within brackets is proportional to the jump of angular momentum across the shell [179] ϕR 2 h 2 (R) = −

.

Thus we have .

2[ma] . κ25

d [ma] = 0, dτ

(10.76)

(10.77)

indicating the conservation of the shell’s angular momentum during evolution. In addition, Eq. (10.73) can be equivalently rewritten as .

−

d (h R 2 ρ) = dR

(

) 2h(R) + h ' (R) R 2 p + 2Rh(R)Δp. R

(10.78)

This equation has a straightforward interpretation: The left-hand side represents the intrinsic energy gained by the shell as it collapses, while the right-hand side is the work done on the shell by the pressure components.

10.1.6 Cylindrically Thin Shell Wormholes Wormholes are solutions of Einstein field equations which imply a non-trivial topology of spacetime, connecting two regions of the same universe or two separate universes by a traversable throat (see the paper by Morris and Thorne [152]). It requires the presence of an exotic matter concentrated at the throat, violating the null energy condition. Cylindrically symmetric wormholes appearing in the context of local cosmic strings and structures in the universe which are infinitely extended along a certain direction, have also been studied in the literature [47, 78]. Let us use Visser’s cut-and-paste technique [207] to construct wormholes based on the Darmois–Israel junction conditions. Take the general static metric with cylindrical symmetry expressed in coordinates .x μ = (t, r, ϕ, z) in the form ds 2 = f (r )(−dt 2 + dr 2 ) + g(r )dϕ2 + h(r )dz 2 ,

.

(10.79)

where . f , .g and .h are non-negative functions of .r . Take two exterior copies M± = {x μ |r ≥ a} of the spacetime (10.79) with .r ≥ a, and paste them at the identical hypersurface .∑ ≡ ∑± = {x μ |F(r ) = r − a = 0}, to construct a geodesically complete manifold .M = M+ ∪ M− . Values of .a are chosen large enough to avoid

.

10.1 Timelike Shells

95

the presence of singularities and horizons in case that the geometry (10.79) has any of them. The throat of a wormhole is a synchronous timelike hypersurface, in which we adapt coordinates .ξ i = (τ , ϕ, z), with .τ being proper time on the shell. Adapting the orthonormal basis .{eˆτ , eˆϕ , eˆz } on .∑ for which the induced metric .giˆ jˆ has the form .gˆ ˆ = ηˆ ˆ = diag(−1, 1, 1), one gets ij ij eτ , f (r ) eϕ .e ˆϕ = √ , g(r ) ez . .e ˆz = √ h(r ) eˆ = √

(10.80)

. τ

(10.81) (10.82)

The unit normal vectors (.n μ n μ = 1) to .∑ in .M± are calculated as ⎛

n ± = ζ± ⎝ √ 1−

. μ

⎞

f a˙ t˙

f

a˙ 2 t˙2

,√ 1−

a˙ 2 t˙2

, 0, 0⎠ ,

(10.83)

where the overdot represents the derivative with respect to .τ . In order to choose the sign of the normal vector on each side, we note that .r is taken to increase as one moves away from either side of the shell. Hence, the normal vector .n − μ points to decreasing .r on the “-” side, meaning .ζ− = −1, and increasing .r on the “+” side, leading to .ζ+ = +1. The non-vanishing components of the extrinsic curvature are found as [78] 2 f 2 (a)a¨ + f ' (a) + (2 f (a) f ' (a))a˙ 2 ± √ , (10.84) .K = ∓ √ τˆ τˆ 2 f (a) f (a) 1 + f (a)a˙ 2 ± .K ϕˆ ϕˆ

± .K zˆ zˆ

√ g ' (a) 1 + f (a)a˙ 2 =± , √ 2g(a) f (a)

(10.85)

√ h ' (a) 1 + f (a)a˙ 2 =± , √ 2h(a) f (a)

(10.86)

where the prime denotes the derivative with respect to r. Considering the surface energy-momentum tensor in the orthonormal basis as . Siˆ jˆ = diag(σ, pϕ , pz ) with .σ the surface energy density, and . pϕ,z the surface pressures, the Lanczos equation (3.12) yields [78] √ ( ) 1 + f (a)a˙ 2 g ' (a) h ' (a) + , (10.87) .σ = − √ g(a) h(a) 8π f (a)

96 . pϕ

10 Special Applications =

8π

√

( ( ' ) ) h ' (a) 1 h (a) 2 f ' (a) f ' (a) √ 2 f (a)a¨ + f (a) + a˙ 2 + + , h(a) f (a) h(a) f (a) f (a) 1 + f (a)a˙ 2

(10.88)

( ( ' ) ) g (a) g ' (a) 1 2 f ' (a) f ' (a) √ . pz = 2 f (a)a¨ + f (a) + + a˙ 2 + + . √ g(a) f (a) g(a) f (a) 8π f (a) 1 + f (a)a˙ 2

(10.89)

One can easily see that .σ, . pϕ and . pz satisfy the equation .

p z − pϕ =

g(a)h ' (a) − g ' (a)h(a) σ. g(a)h ' (a) + g ' (a)h(a)

(10.90)

These equations explicitly show the time evolution of the wormhole throat. Values of energy density and pressure corresponding to static wormholes are obtained by putting the time derivatives equal to zero in the equations above. According to the areal flare-out condition2 for cylindrical wormholes, the area per unit length is minimal at the wormhole throat [79].√In the present case with the metric (10.79), the area function defined as .A(r ) = 2π g(r )h(r ) should increase at both sides of the throat. Therefore.A' (a) > 0, so that.(gh)' (a) = g ' (a)h(a) + g(a)h ' (a) > 0 implies the energy density (10.87) to be negative at the throat, signaling the presence of exotic matter there. However, as Bronnikov and Lemos [47] pointed out, in the case of infinite cylindrical configurations, it may be more natural to consider the radial flare-out condition, demanding the length of a circumference to increase √ with the radius. This condition requires the circular radius function . R(r ) = g(r ) to have a minimum at the wormhole throat. Hence .g ' (a) > 0, leaving free the sign of .(gh)' (a) in Eq. (10.87). This feature, in principle, allows for the possibility of a positive energy in the cylindrical wormhole throat. Therefore, violation of the energy condition depends on the way that the flare-out condition for minimality is defined.

10.1.7 Collapsing Stars in f(R) Theories of Gravity We have seen in Chap. 8 that for a non-linear . f (R) gravity to be valid on a glued manifold, the trace of the second fundamental form at both sides of the gluing hypersurface has to be continuous. In addition, except for quadratic . f (R), continuity of the scalar curvature . R, is required. As a specific example, let us consider a collapsing star as a spherical ball of perfect fluid with uniform pressure and density. Spacetime inside the ball is described by the Friedmann–Robertson–Walker (FRW) metric, while the outside has to be the Schwarzschild. Could such a spacetimes glued smoothly at .∑ be a solution of non-linear . f (R) gravity? To answer this question, we start with the interior region metric in the form ( ) ds−2 = −dτ 2 + a2 (τ ) dχ2 + r 2 (χ)(dθ2 + sin2 θdϕ2 ) ,

.

2

A geometrical condition one needs to impose about the minimality of wormhole throat.

(10.91)

10.1 Timelike Shells

97

where .τ is the proper time on comoving world lines, .a(τ ) is the scale factor, and r (χ) is defined as in (10.12). The energy-momentum tensor of the inside collapsing star is that of a perfect fluid

.

T − = (ρ˜ + p) ˜ u˜ μ u˜ ν + pg ˜ μν ,

. μν

(10.92)

where .ρ˜ and . p˜ are the corresponding energy density and isotropic pressure, and .u˜ μ is the unit velocity vector field. The hypersurface .∑ coincides with the surface of the collapsing star in the comoving coordinates. For the metric (10.91) to be a solution of . f (R) gravity (8.2) with the energy-momentum tensor of a perfect fluid (10.92), the following relations have to be satisfied [188] ρ˜ =

.

3a˙ 1 1 f R (ρ + 3 p) + f (R) + f R R (ρ˙ − 3 p), ˙ 2 2κ a

(10.93)

1 1 2a˙ f R (ρ − p) − f (R) − f R R (ρ¨ − 3 p¨ + (ρ˙ − 3 p)) ˙ − f R R R (ρ˙ − 3 p) ˙ 2, 2 2κ a (10.94) with .ρ and . p being defined by the general relativistic Friedman equations given by k κ a˙ 2 ρ and .− aa¨ = κ6 (ρ + 3 p), to have a better comparison. The Ricci scalar . 2 + 2 = a a 3 curvature is calculated as ) ( 2 a˙ + k a¨ . R− = 6 + a2 a (10.95) = κ(ρ − 3 p). .

p˜ =

where we have again used the corresponding general relativistic expressions for .ρ and . p. The metric in the exterior region of the star is taken to be the Schwarzschild ds+2 = −(1 − 2M/r )dt 2 + (1 − 2M/r )−1 dr 2 + r 2 dΩ2 ,

.

(10.96)

where . M is the mass parameter of the exterior Schwarzschild region. On the other hand, for the gluing to be smooth, the shell energy-momentum tensor must vanish . Sμν = 0. From identities (8.14) and (8.15), we can then derive n μ [Tμν ] = 0,

.

(10.97)

indicating continuity of the normal components of energy-momentum tensor across ∑ as in general relativity. Taking (10.92) and contracting (10.97) by .u˜ ν , we arrive at

.

ρn ˜ μ u˜ μ |∑ = 0,

.

(10.98)

98

10 Special Applications

whereas contracting (10.97) by .n ν yields .

p| ˜ ∑ = 0.

(10.99)

According to (10.98), either the fluid is tangent to .∑ and therefore .n μ u μ |∑ = 0, or ˜ ∑ = 0. The relation (10.99) is an explicit condition on the perfect there must be .ρ| fluid for smooth gluing at .∑. Now, taking into account that the scalar curvature in the vacuum exterior vanishes, . R+ = 0, the continuity of Ricci scalar across .∑ given by (8.16), as required in the general case . f R R R (R) /= 0, leads to .

R− |∑ = (ρ − 3 p)|∑ = 0.

(10.100)

Since .ρ(t) and . p(t) depend only on .t, and noting that .∑ is timelike, it follows ρ − 3 p = 0.

.

(10.101)

Given (10.93) and (10.94), the energy density .ρ˜ and pressure . p˜ (assuming . f (0) = 0 = Δ, for simplicity) are given by ρ˜ = f R (0)ρ,

.

p˜ = f R (0) p,

(10.102)

showing that the perfect fluid in . f (R) has the same radiation equation of state as in (10.101) .ρ ˜ − 3 p˜ = 0. (10.103) Then, from the matching condition (10.99), one gets . p(t) ˜ = p(t) = 0, and finally (10.103) reduces to .ρ(t) ˜ = 0, (10.104) reducing the inside to a Minkowski spacetime. Therefore, in contrast to general relativity, it is impossible to smoothly glue an FRW interior solution to an exterior vacuum solution in general f(R) with . f R R R (R) /= 0. Hence, the Oppenheimer–Snyder collapse to form a black hole in the closed case.k = 1 [162], or the Einstein-Straus model of a vacuole in an expanding universe [75], in general relativity, has no counterpart in . f (R) theories of gravity (see also [52]).

10.1.8 Gravitational Double Layers in Quadratic f(R) Theories of Gravity According to the results of Chaps. 8 and 9 in the case of quadratic . f (R) theories, i.e. f (R) = R − Δ + αR 2 , on a glued manifold one expects, in general, a thin shell plus

.

10.1 Timelike Shells

99

a double layer at the gluing hypersurface. As discussed in Chap. 8, the presence of a double layer is a result of delta-prime distribution in the formalism. As an example, we consider two different spherically symmetric solutions in quadratic . f (R) gravity with metrics in standard coordinates .x μ = (t± , r, θ, ϕ) [76, 77] 2 2 2 2 2 ds±2 = −A± (r )dt±2 + A−1 ± (r )dr + r (dθ + sin θdϕ ),

.

(10.105)

where . A± (r ) are functions of .r . This general class of static spacetimes in . f (R) theories consists of several exact solutions, including those with the constant curvature scalar . R, corresponding to vacuum [63, 155]. Take two regions .M− and .M+ , defined as the interior .0 ≤ r ≤ a and the exterior + − .r ≥ a of a geodesically complete manifold .M = M ∪ M glued at the hypersurface .∑ situated at .r = a. The intrinsic coordinates after identification at the hypersurface .∑ are .ξi = (τ , θ, ϕ), with .τ the proper time on .∑. The induced metrics from both sides at the gluing hypersurface .∑ read (

2 .ds± |∑

dt±2 = −A± (a) dτ 2

) dτ 2 + a 2 (dθ2 + sin2 θdϕ2 ),

= −dτ 2 + a 2 (dθ2 + sin2 θdϕ2 ).

(10.106)

Continuity of the induced metric then implies √ .

A− (a)

√ dt− dt+ = A+ (a) , dτ dτ

(10.107)

where we have fixed the free signs by choosing all times.t± and.τ running to the future. Adapting the orthonormal basis .{eˆτ , eˆθ , eˆϕ } on .∑, for which the induced metric .giˆ jˆ takes the form .giˆ jˆ = ηiˆ jˆ = diag(−1, 1, 1), one gets eˆ = eτ ,

. τ

eˆθ = (a)−1 eθ , −1 eˆϕ = (a sin θ) eϕ . The unit normals to .∑ in .M± are calculated as ) ( √ A± (a) + a˙ 2 ± , 0, 0 , .n μ = ζ± −a, ˙ A± (a)

(10.108)

(10.109)

where the overdot represents the derivative with respect to .τ . Notice that due to interior and exterior characters of.M− and.M+ , respectively, we must choose.ζ− = 1 and .ζ+ = +1.

100

10 Special Applications

The non-vanishing components of the extrinsic curvature are found as .

.

A' (a) + 2a¨ , K τˆ±τˆ = − √± 2 A± (a) + a˙ 2

K ˆ±ˆ = K ϕ± ˆ ϕˆ = θθ

1√ A± (a) + a˙ 2 , a

(10.110)

(10.111)

where the prime denotes the derivative with respect to .r . By using Eqs. (10.110) and (10.110), the continuity condition for the trace of the second fundamental form, as given in (8.13), takes the form 2a a¨ + a A'− (a) + 4(A− (a) + a˙ 2 ) 2a a¨ + a A'+ (a) + 4(A+ (a) + a˙ 2 ) √ √ + = 0. a A− (a) + a˙ 2 a A+ (a) + a˙ 2 (10.112) Assuming the Ricci scalar curvatures . R− and . R+ are both constant, but may be different, the junction equation (8.39) reduces to .

−

.

¯ i j ] + [R] K¯i j ) = κSi j . − [K i j ] − 2α( R[K

(10.113)

Now, with the surface energy-momentum tensor in the orthonormal basis having the form. Siˆ jˆ = diag(σ, pθˆ , pϕˆ ), where.σ is the surface energy density and. pθˆ = pϕˆ = p the isotropic pressures, (10.113) reduces to σ=−

.

2a¨ + A'− (a) 2a¨ + A'+ (a) √ (1 + 2αR− ) + √ (1 + 2αR+ ) , (10.114) 2κ A− (a) + a˙ 2 2κ A+ (a) + a˙ 2

and √ .

p=

A− (a) + a˙ 2 (1 + 2αR− ) − κa

√

A+ (a) + a˙ 2 (1 + 2αR+ ) . κa

(10.115)

From (8.40), it is easily seen that for the constant Ricci scalar curvatures, the external flux momentum vector turns out to be zero .Tμ = 0. From (8.41), one can compute the external scalar pressure .T .T

=−

2a a¨ + a A'+ (a) + 4(A+ (a) + a˙ 2 ) 2a a¨ + a A'− (a) + 4(A− (a) + a˙ 2 ) √ √ αR− + αR+ . κa A− (a) + a˙ 2 κa A+ (a) + a˙ 2

(10.116) From Eqs. (10.114), (10.115) and (10.116), the following equation of state is derived σ − 2p = T ,

.

(10.117)

which is at the same time the trace of Eq. (10.113). Combining the time derivative of (10.116) with Eqs. (10.114) and (10.115), we arrive at

10.2 Spacelike Shells

101

σ˙ +

.

2a˙ (σ + p) = T˙ , a

(10.118)

which is indeed the generalized conservation identity (9.44). This equation shows a novel fact that in the quadratic f(R) gravity, the energy-momentum tensor . Sμν of the shell surrounding vacuum (10.105) is no longer divergence-free in general. In addition, there is a double layer whose strength is given by (8.44) having the following components in the orthonormal basis .

− Pτˆ τˆ = Pθˆ θˆ = Pϕˆ ϕˆ = 2α[R]/κ,

(10.119)

which are non-zero when there is a jump in the Ricci scalar across .∑. It is also seen that with the double layer strength given in (10.119) and the expressions (10.110) and (10.111) for the non-vanishing components of the extrinsic curvature tensor, the identity (9.43) is satisfied. In the extreme case of a pure double layer including no thin shell, in addition to the junction conditions given in (8.47), we must have . Sμν = 0, .Tμ = 0, .T = 0, while .[R] / = 0. We can easily see that for the present case with constant but different Ricci scalar curvatures, if . Sμν = 0, then all the conditions in Eq. (8.47) are satisfied.

10.2 Spacelike Shells 10.2.1 Emerging a de Sitter Universe Inside a Schwarzschild Black Hole Through a Spacelike Shell Imagine a glued manifold consisting of a black hole inside and a de Sitter spacetime called baby universe (see [1, 5, 43, 49, 64, 74, 83, 99, 209]). As an application of spacelike shells, we consider the model of Frolov, Markov, and Mukhanov [86, 87] in which the Schwarzchild metric inside a black hole is glued to a de Sitter one at some thin spacelike shell representing a short transition layer. The model may be interpreted as the creation scenario of a new universe out of an existing black hole. The spacelike shell.∑ is taken to lie within the Schwarzschild event horizon where the curvature invariant . Rμνσρ R μνσρ is assumed to reach its maximum limiting value (due to the appearance of quantum gravity corrections), and outside the de Sitter horizon. The past of .∑, .M− , is represented by region II of the Kruskal diagram (black hole μ region) given in Schwarzschild coordinates .x− = (t− , r, θ, ϕ) by the metric ds−2 = − f −−1 dr 2 + f − dt−2 + r 2 dΩ2 ,

.

(10.120)

where .t− is a space- and .r a time-coordinate, . f − = 2m/r − 1, and .dΩ2 = dθ2 + sin2 θdϕ2 .

102

10 Special Applications

The future of .∑, .M+ , represented by the bottom quadrant (region IV) of the whole de Sitter diagram in the Gibbons–Hawking coordinate system is given in the μ static coordinates .x+ = (t+ , r, θ, ϕ) by the metric ds+2 = − f +−1 dr 2 + f + dt+2 + r 2 dΩ2 ,

(10.121)

.

2

where .t+ is a space- and .r a time-coordinate, and . f + = rl 2 − 1 with .l denoting the size of de Sitter horizon (see [86] for the casual structure of the model). The induced three-metric on the shell in intrinsic coordinates .ξ i = (z, θ, ϕ) takes the form | 2| 2 2 2 .ds | = dz + R (z)dΩ , (10.122) ∑

indicating the topology of a three-cylinder .(S 2 × R 1 ) whose cross-sections are twospheres having radius . R(z) depending on the proper distance .z along the threecylinder axis. Notice that the total manifold is glued across the spacelike shell at a single moment. Therefore, not its dynamics but the global shape is of interest. Continuity of the induced metric is ensured by the condition .

| | − f ±−1 R˙ 2 + f ± t˙±2 | = 1.

(10.123)

∑

Solving (10.123) for .t˙ yields t˙ =

. ±

1 f±

√

| | f ± + R˙ 2 | ,

(10.124)

∑

where the choice of sign for .t˙± is such that the space coordinate .t increases monotonically with .z. The tangent vector on .∑ viewed from either side is given by μ

.u −

( =

˙ f −1 R, −

√

)| | f − + R˙ 2 , 0, 0, 0 | , ∑

μ

u+ =

(

˙ f −1 R, +

√

)| | f + + R˙ 2 , 0, 0, 0 | , ∑

(10.125) where the overdot denotes differentiation with respect to the proper length .z. From the orthogonality condition .n μ u μ = 0 and .n μ n μ = −1, the unit timelike vector .n μ normal to .∑ as viewed from each side is calculated to be −

.n μ

( = ζ−

f −−1

√

)| ˙ 0, 0 || , f − + R˙ 2 , − R, ∑

n+ μ = ζ+

(

f +−1

√

)| ˙ 0, 0 || . f + + R˙ 2 , − R, ∑

(10.126) Note that the sign of .n μ is chosen such that .n μ is directed from the Schwarzschild past region to the de Sitter future, leading to .ζ± = +1. The spatial components of the extrinsic curvature tensor on each side are then given by

10.2 Spacelike Shells

|

z| .Kz |

1 d = − R dz

103

√

|

| K θθ |

2m − 1 + R˙ 2 , R √ | 1 d R2 z| .Kz | = − 1 + R˙ 2 , + R dz l 2

|

| K ϕϕ | = −

√

2m − 1 + R˙ 2 , − R (10.127) √ | | 2 R 1 | | K θθ | = K ϕϕ | = − 1 + R˙ 2 . + + R l2 (10.128) The surface energy-momentum tensor of the shell in prefect fluid form is written as =

1 R

S = (Pz − P⊥ )u i u j + P⊥ gi j .

. ij

(10.129)

In contrast to surface energy density for the timelike shells, we face here a longitudinal pressure . Pz while . P⊥ is the transverse one. The angular and .zz components of the Lanczos equation (3.12) for the spacelike shell .∑ .(∈ = −1) lead to the following two independent equations √

R2 − 1 + R˙ 2 − l2

√

2m − 1 + R˙ 2 = 4π R Pz , R ) (√ √ R2 2m d 2 2 ˙ − 1 + R˙ = 4π R(2P . − 1 + R˙ − ⊥ − Pz ). dz l2 R .

(10.130)

(10.131)

Combining Eqs. (10.130) and (10.131), we find .

2 R˙ (Pz − P⊥ ) = 0, P˙z + R

(10.132)

which is indeed the conservation identity (3.14) written on the spacelike transition layer .∑. The two Eqs. (10.130) and (10.131) or (10.132) together with an equation of state relating longitudinal pressure . Pz to the transverse one, . P⊥ , describe the global shape, i.e. the spatial structure of the spacelike hypersurface as a three-cylindrical shell for a single instant of time. These equations have been solved in [8] by choosing a linear relation between . Pz and . P⊥ to obtain the shape of spacelike shell. It was shown that a shell with an initial uniform radius independent of .z is stable in the sense that variation of the global parameters, such as the black hole mass m, de Sitter horizon radius .l, as well as the shell’s internal parameters (surface pressures) does not force the three-cylinder shrinking down, but rather induces spatial oscillations in the radius along the cylinder’s axis.

104

10 Special Applications

10.3 Null Shells 10.3.1 Smooth Gluing LTB and Vaidya Spacetimes Through a Null Hypersurface The Lemaitre–Tolman–Bondi (LTB) and Vaidya spacetimes are two solutions to Einstein equations describing inhomogeneous dust and null fluid, respectively. In [88], by constructing a continuous coordinate system which is comoving with both the LTB and Vaidya observers, the possibility of gluing these two solutions in one single spacetime through a null hypersurface was demonstrated. Now, as a first application of null shell formalism, using results in Chap. 5, we look at the possibility of smooth gluing for the configuration above (see also [117]). The LTB metric in the synchronous comoving coordinates is given by ds−2 = −dt 2 +

.

R'2 dr 2 + R 2 (t, r )(dθ2 + sin2 θdϕ2 ), 1 + E(r )

(10.133)

where the overdot and the prime denote partial differentiation with respect to .t and r , respectively. . E(r ) is an arbitrary real function of .r representing local curvature. The corresponding Einstein field equations read

.

2M(r ) , R˙ 2 (t, r ) = E(r ) + R M ' (r ) 4πρ L (t, r ) = 2 ' , R R .

.

(10.134) (10.135)

where .ρ L is the energy density, and . M(r ) is an arbitrary function interpreted as the effective gravitational mass within .r . The incoming Vaidya spacetime describing collapsing null dust in the radiative coordinates .x μ = (v, R, θ, ϕ) is given by [200] ) 2M(v) dv 2 + 2dvd R + R 2 (dθ2 + sin2 θdϕ2 ), =− 1− R (

2 .ds+

(10.136)

with .v representing an advanced time, and . M(v) > 0 an arbitrary function of the ingoing null coordinate .v, representing the mass accreted at time .v. The corresponding energy-momentum tensor for the metric (10.136) is of pure radiation type 1 d M(v) lμ lν , 4π R 2 dv

lμl μ = 0, (10.137) where we will define.ρ R -the radiation density- by the identity.d M(v)/dv = 4π R 2 ρ R . As seen from .M− , the null hypersurface .∑ is described by integrating T

. μν

=

lμ = −δμv ,

10.3 Null Shells

105

R ' (t, r ) dt = √ dr, 1+ E

(10.138)

.

on .∑. Now, hypersurfaces .v = constant within .M+ turn out to be null. The shell induced metric must then be in the form | 2| 2 2 2 2 .ds+ | = R (dθ + sin θdϕ ), (10.139) ∑

being two-dimensional, as expected. The continuity requirement of the induced metric on .∑ yields the condition .

∑

R(t, r ) = R.

(10.140)

∑

where .= means the equality must be evaluated on .∑. For further use, we need .

∑ ˙ + R ' dr = d R. Rdt

(10.141)

Taking .ξ i = (λ, θ, ϕ), with .i = 1, 2, 3 as the intrinsic coordinates on .∑, while identifying . R with the parameter .λ on null generators of the hypersurface, we calculate the tangent basis vectors .ea = ∂/∂ξ a on both sides of .∑. Using Eqs. (10.138) and (10.141), we obtain μ .eλ |+

| = (0, 1, 0, 0) |∑ ,

μ .eλ |−

=

R˙ +

1 √ 1+ E

μ

μ

eθ |+ = δθ , ) √ | 1+ E | , , 0, 0 1, ∑ R'

(

μ

eϕμ |+ = δϕμ ,

(10.142)

μ

(10.143)

eθ |− = δθ ,

eϕμ |− = δϕμ .

Choosing the normal vector.n μ to coincide with the tangent associated with parameter μ μ .λ such that .n = e , the null hypersurface .∑ may be considered to be generated by λ ∂ . The basis is comgeodesic integral curves of the future-directed null vector field . ∂λ pleted by adding the transverse null vector . N μ uniquely defined by four conditions μ μ μ .n μ N = −1, . Nμ e A = 0 .(A = θ, ϕ), and . Nμ N = 0. We then find ) ( | 1 ˙ √ −R ' ( R + 1 + E) −1, √ , 0, 0 |∑ , 2 1+ E ) ) ( ( | 2M(v) 1 1− , −1, 0, 0 |∑ . . N μ |+ = 2 R .

N μ |− =

(10.144) (10.145)

Using (5.55), we compute components of the transverse extrinsic curvature tensor on both sides of .∑. The non-vanishing components on .M− side of .∑ are found as Kθθ |− = sin

.

−2

−R θKϕϕ |− = 2

( ) 2M(r ) || 1− , ∑ R

(10.146)

106

10 Special Applications

| M ' (r ) | √ R R ' ( R˙ + 1 + E)2 ∑ 4π Rρ L (t, r ) || = , √ ( R˙ + 1 + E)2 ∑

Kλλ |− =

.

(10.147)

where (10.135) has been used. The corresponding non-vanishing components on the M+ side of .∑ are ( ) 2M(v) || −R −2 1− .Kθθ |+ = sin θKϕϕ |+ = . (10.148) ∑ 2 R

.

Now, the smooth gluing of two spacetimes requires the continuity of transverse extrinsic curvature tensor across .∑, yielding

.

∑

M(r ) = M(v),

(10.149)

4π Rρ L (t, r ) || = 0, √ ( R˙ + 1 + E)2 ∑

(10.150)

.

where we have used Eqs. (10.146)–(10.148). Condition (10.149) simply expresses the equality of total gravitational mass on .∑ as seen from any side of it. Condition (10.150), however, coming from the continuity of .λλ component of transverse curvature tensor over .∑, is valid only if the energy density of LTB interior region .ρ L (t, r ) goes to zero on hypersurface .∑. The arbitrary function . E(r ) (see [88] for the dependency of . R, . R ' and . R˙ on . E(R)) can remain finite there. The isotropic surface pressure (5.71) given by .

p=−

| 1 Rρ L (t, r ) | , [Kλλ ] = √ 2 ˙ 8π 2( R + 1 + E) ∑

(10.151)

is also vanishing, leading to .[κ] = 0 (see Eq. (5.73) for definition of .κ) for the case of smooth gluing we are considering. As a result, we conclude that the null hypersurface .∑ is affinely conciliable: .λ is an affine parameter on both sides of .∑. Furthermore, from (5.75), it is seen that the normal component of the energy-momentum tensor, μ ν . Tμν n n , is continuous across .∑. Now, the Kretschmann scalar . K = R μνσρ Rμνσρ is computed for the LTB spacetime .

48M 2 32M M ' 12M ' 2 − + , R6 R5 R' R 4 R '2 128π Mρ L (t, r ) 48πρ L (t, r ) 48M 2 − + , = 6 R R3 R2 R'

K |− =

(10.152)

10.3 Null Shells

107

where (10.135) has been used. For the Vaidya spacetime one obtains .

K |+ =

48M 2 (v) . R6

(10.153)

By virtue of the smooth gluing condition .ρ L (t, r )|∑ = 0, as well as (10.149) while R and . R ' remain finite, the continuity of the Kretschmann scalar over the null hypersurface .∑ is fulfilled. Therefore, the LTB and Yaidya solutions can be glued smoothly through a null hypersurface with finite values of . E(r ) provided . M ' (r ), or equivalently the energy density of LTB interior region .ρ L goes to zero on the hypersurface .∑.

.

10.3.2 Coexistence of Matter Shell and Gravitational Wave on a Null Hypersurface As a second application of the null shell formalism, we elucidate an instructive but artificial cosmological model initially presented by Barrabes and Hogan [17] with a simplified version of it studied subsequently by Poisson [171]. In this scenario, the universe initially expanding isotropically is made to expand anisotropically by a sudden explosive event its history being a null hypersurface .∑. The spacetime .M− in the past of .∑ is described by Einstein-de Sitter universe in μ coordinates .x− = (t, r, ϕ, z) given by ) ( ds−2 = −dt 2 + t dr 2 + r 2 dϕ2 + dz 2 ,

.

(10.154)

representing a radiation-dominated universe with isotropic pressure . p− and energy density .ρ− satisfying the equation of state . p− = ρ3− . The spacetime .M+ in the future of .∑ is described by the anisotropic Bianchi type μ I universe [201] having the following metric in coordinates .x+ = (t+ , r+ , ϕ+ , z + ) 1/2

ds+2 = −dt+2 + t+

.

( 2 ) 2 dr+ + r+2 dϕ2+ + t+ dz + .

(10.155)

The corresponding fluid has a density .ρ+ and isotropic pressure . p+ satisfying . p+ = ρ+ . Seen from .M− , the null hypersurface .∑ is defined by .

dr = t −1/2 , dt

(10.156)

whereas seen from .M+ , the null hypersurface .∑ is defined by .

dr+ −1/4 = t+ . dt+

(10.157)

108

10 Special Applications

Using .ξ i = (r, ϕ, z) as intrinsic coordinates on .∑, the continuity of the induced metric is ensured if t = t,

z + = z,

. +

r+ =

4 ( r )3/2 , 3 2

ϕ+ =

3 ϕ, 2

so that the induced metric on .∑ becomes | ) ( 2 2 2| 2 . .ds | = t r dϕ + dz ∑

(10.158)

(10.159)

The intrinsic coordinates .ξ i = (λ, ξ A ) are adapted to the null generators of .∑ by choosing .λ ≡ r as the parameter on them, and the remaining spatial coordinates A .ξ = (ϕ, z) to label individual null generators (see the last paragraph in Section 5.4). Then, the normal vector to .∑, as viewed from .M− is μ

μ

n = e λ |− =

. −

μ d x− || dt dϕ dz || = ( , 1, , ) = (t 1/2 , 1, 0, 0), ∑ dλ dr dr dr ∑

(10.160)

where (10.156) has been used. The transverse null vector . Nμ satisfying . Nμ n μ = −1 μ and . Nμ e A = 0, is found to be .

1 Nμ− = − (t −1/2 , 1, 0, 0). 2

(10.161)

From (5.55), the non-vanishing components of the transverse curvature tensor on the M− side of .∑ are calculated as

.

μ

− Kλλ = = e λ eνλ ∇μ Nν |− μ = n − n ν− ∇μ− Nν−

.

(10.162)

= t −1/2 , − Kzz = eμz eνz ∇μ Nν |− 1 = t −1/2 , 4

.

(10.163)

where (10.156), (10.158), (10.160) and (10.161) have been used. Viewed from .M+ , the normal vector is μ

μ

n = e λ |+ =

. +

μ

d x+ dt+ dr+ dϕ+ dz + =( , , , ) = (t 1/2 , t 1/4 , 0, 0), dλ dr dr dr dr

(10.164)

where (10.156), (10.157) and (10.158) have been used. The transverse null vector is found to be 1 −1/2 −1/4 + ,t , 0, 0). (10.165) . Nμ = − (t 2

10.3 Null Shells

109

The non-vanishing components of the transverse extrinsic curvature tensor on the M+ side of .∑ are calculated as

.

μ

+ Kλλ = e λ eνλ ∇μ Nν |+

.

=

μ n + n ν+ ∇μ+ Nν+

=

3 −1/2 , t 4

+ Kϕϕ = eμϕ eνϕ ∇μ Nν |+

.

= −t

1/2

(10.166)

(10.167)

,

+ Kzz = eμz eνz ∇μ Nν |+ 1 = t −1/2 , 4

.

(10.168)

where (10.157), (10.158), (10.164) and (10.165) have been used. Now, the nonvanishing surface quantities of the shell, namely the energy density .μ and pressure . p are calculated from Eqs. (5.70) and (5.71), respectively, μ=

.

1 −3/2 , t 32π

p=

1 −1/2 . t 32π

(10.169)

It is seen that both the energy density .μ and pressure . p are also positive, indicating a pressure not tension on the shell. For more insight, we calculate the components of tensor .γi j defined in (5.56). The non-vanishing components are γ

. ϕϕ

and

= −2t 1/2 ,

1 γλλ = − t −1/2 , 2

1 γ † = γμν n μ n ν = γi j n i n j = − t −1/2 , 2

.

(10.170)

(10.171)

showing that the geometry induced on .∑ is the type I according to the Penrose classification. In order to examine the possible presence of impulsive gravitational waves on .∑, according to Sect. 5.5.3, we first construct the null tetrad .(n, l, m, m) ¯ based on μ μ coordinates .x− ; it consists of two real null vectors .n − given in (10.160) and .l μ μ given by .l μ = − 2s12 n − + 1s u μ , where .u μ = (1, 0, 0, 0) is a timelike vector such that μ − μ .u n μ = −s < 0, and a complex null vector .m as well as its complex conjugate μ .m ¯ . In addition, all scalar products of tetrad vectors vanish except .m μ m¯ μ = 1 and − μ .n μ l = −1. The result is

110

10 Special Applications

μ

n = (t 1/2 , 1, 0, 0),

. −

1 1 −1/2 μ , −t −1 , 0, 0), m − = √ t −1/2 (0, 0, ir −1 , 1) (t 2 2 1 μ (10.172) m¯ − = √ t −1/2 (0, 0, −ir −1 , 1), 2 μ

l− =

1 with .s = t 2 . A key parameter indicating the existence of wave on .∑ is .ψˆ4 , the Newman–Penrose component of the singular part of the Weyl tensor with respect to null tetrad, obtained from Eq. (5.90)

.

1 μ ψˆ4 = γμν m¯ − m¯ ν− 2 1 = γϕϕ m¯ 2− m¯ 2− 2 1 = t −3/2 8 /= 0,

(10.173)

indicating that .∑ is the history of both an impulsive gravitational wave and a null shell. To see the influence of impulsive wave on the matter crossing null hypersurface .∑, let us look at dynamics of the 4-velocity of cosmic fluid, taking it to be continuous μ μ across .∑: .([u μ ei ] = 0). Therefore, by assuming .u − = (1, 0, 0, 0) in .M− we arrive μ + at the 4-velocity .u + = (1, 0, 0, 0) in .M . The expansion .θ of timelike congruence defined by .θ = ∇μ u μ is then computed for each side θ =

. −

3 , 2t

θ+ =

1 . t

(10.174)

Thus, the jump in expansion across .∑ would be .[θ] = − 2t1 . The shear tensor of the timelike congruence is given by σ

. μν

1 = B(μν) − θh μν , 3

(10.175)

where . Bμν = ∇ν u μ , and .h μν = gμν + u μ u ν . While the cosmic fluid in .M− is found − = 0), in .M+ we encounter the following non-vanishing comto be shear-free (.σμν ponents σ+ = −

. 11

1 −1/2 , t 12

+ σ22 =−

4 t, 27

+ = σ33 μ

1 . 6 μ

Constructing a null tetrad based on coordinates .x+ , with .m + = 3 −1 r , 1), 2

(10.176) √1 t −1/2 (0, 0, 2

the jump in shear tensor of the timelike congruence across .∑ is calcui lated to be

10.3 Null Shells

111

1 8t /= 0,

[σμν m μ m ν ] =

.

(10.177)

Therefore, in case .∑ is the history of an impulsive gravitational wave with nonvanishing Weyl tensor component .ψˆ4 , any timelike congruence crossing .∑ with continuous tangent .u μ has to have a jump in the shear, showing the existence of a direct relationship between the presence of a wave on .∑ and the shear of any timelike congruence crossing it, as expressed in Sect. 5.3.5.

10.3.3 Null Shells Straddling a Common Horizon As an example of null shells on the horizon of a black hole, let us consider a stationary null shell with the history .∑ placed on the common horizon of two spherically symmetric spacetimes [17]. This case will open up the possibility of having different ways of gluing two spacetimes on .∑ arising from the fact that the parametrization of null generators is arbitrary, as mentioned in Sect. 5.3.2. The metric in .M± is given in the general form ( ) ds±2 = −eψ± du ± ( f ± eψ± du ± − 2dr ) + r 2 dθ2 + sin2 θdϕ2 ,

.

(10.178)

where .u ± is the advanced time as .r decreases towards the future along null generators. Functions .ψ± depend on the coordinates .u ± and .r , . f ± , however, are arbitrary functions of .r only. The mass functions .m ± (r ) are conveniently defined as f =1−

. ±

2m ± . r

(10.179)

It is instructive to see gluing through the null hypersurfaces .u ± = constant. Thus, the boundary between past domain .M− and future .M+ is taken to be a particular hypersurface.u ± =constant. Now, a natural choice for the parameter on null generators of the hypersurface.∑ is the radial coordinate.r , namely.λ = −r (Notice that.λ always increases along null generators, leading to decreasing .r ). Then, the induced metric on .∑ is | ( 2 ) 2| 2 dθ + sin2 θdϕ2 . .ds | = λ (10.180) ∑

The induced metric (10.180) depends on .r chosen as a parameter along null generators. Hence, the null shell is non-stationary, and as mentioned in Sect. 5.3.2, the gluing is unique with physical parameters calculated from the null shell formalism [17] 1 [m] [∂r ψ]. , p= (10.181) .μ = 4πr 2 8π

112

10 Special Applications

This shows that for a positive energy density .μ, we must have .[m] > 0, meaning the null shell brings energy to the center while contracting. Now, we consider the null shell situated at the common horizon of the spacetimes (10.178) occurring at .r = r0 where . f ± (r0 ) = 0, or .r0 = 2m ± (r0 ) in term of mass function .m(r ). It is obvious that we can no longer take .r as a parameter along the null generators. Hence, we select .u as the parameter along generators .λ = u, being continuous across .∑. The metric induced from either side on the horizon .∑ located at .r = r0 is | ( 2 ) 2| 2 dθ + sin2 θdϕ2 . .ds | = 4m (10.182) ∑

The induced metric is independent of the parameter along null generators .λ. Therefore, .∑ is stationary. With .ξ i = (u, ξ A ) = (u, θ, φ), being the intrinsic coordinates on .∑ adapted to the null generators, the normal vector which is also tangent to .∑, is given by μ d x± μ = (1, 0, 0, 0). (10.183) .n ± = du The transverse vector . N μ satisfying .n μ Nμ = −1 and . Nμ N μ = 0 is then obtained as .

| | μ N± = (0, −e−ψ± , 0, 0)| . ∑

(10.184)

The non-vanishing components of the transverse extrinsic curvature .Ki j can be computed from (5.70), (5.71), yielding | ± ψ± | Kλλ = ∂u ψ ± + κ ± 0 e | = κ± , ∑ | ± ψ± | .K θθ = −r 0 e | ,

.

∑

(10.185) (10.186)

where.κ0 = 21 ddrf |r =r0 is the surface gravity, and.κ is the acceleration corresponding to the geodesic equation of null generators written as .n μ ∇μ n ν = κn ν vanishing in case the generators are affinely parametrized. Having calculated the transverse curvature tensor above, one can obtain the surface energy .μ and pressure . p as μ=

.

1 [e−ψ ] 4πr0

p=−

1 [κ]. 8π

(10.187)

It turns out that due to the non-vanishing pressure, the parameter .λ is not an affine parameter on each side of .∑. Although the induced metric (10.182) of the shell placed on the horizon .∑ is manifestly invariant under arbitrary reparametrization .λ −→ F(λ, θ, φ) along the null generators, surface quantities of the shell change under such a transformation [171]. The explicit expressions for the energy density .μ, surface pressure . p, and

10.3 Null Shells

113

surface current . J A of the horizon straddling shells with the induced metric (10.182) when the null generators are parametrized with . F(λ, θ, φ) have been derived in [40] 1 (e−F/2m ∇ (2) e F/2m + F,λ − 1), 8πm F,λ ( ) 1 1 F,λλ F,λ − , .p = − + 8π F,λ 4m 4m ( ) F,λB F,B 1 A AB 2 , g + .J = 64πm 2 F,λ 2m μ=−

.

(10.188) (10.189) (10.190)

where .∇ (2) denotes the 2-dimensional covariant derivative associated with .g AB . It can be shown that putting the gluing transformation ∫ .

F(λ) =

u

eψ(u,r0 ) du,

(10.191)

into the expressions (10.188)–(10.190), one can regain the non-vanishing energy density and surface pressure of the horizon straddling shell as given in (10.187). In this way, from (10.188)–(10.190) one can also obtain the Dray ’t Hooft null shell [71] by applying the special gluing . F(λ, ξ A ) = 4m Ln(eλ/4m + b), leading to a null b shell with the constant energy density .μ = 32πm 2 and zero pressure, surface currents and gravitational wave components. Therefore, for any function . F(λ, ξ A ), there is a solution representing a gluing of two geometries (10.178) along a given null hypersurface .∑ placed on the common horizon while maintaining the Darmois junction requirement of induced metric (10.182) being continuous across .∑. This generated null shell will be non-trivial unless the gluing transformation is a pure isometry, i.e. a constant shift of .λ. The resulting horizon shells will, in general, carry an energy density .μ, surface pressure A . p and surface current . J , as well as an impulsive gravitational wave traveling along the shell. The reparametrization freedom on the null generators of .∑ is usually excluded by imposing the null generators to be affinely parametrized [17], as well as in the hybrid black hole solution obtained in [85]. Roughly speaking, the gluing freedom can be attributed to the fact that any null shell placed on the horizon will be infinitely redshifted relative to distant observers and hence have no impact on quantities like the ADM mass, in contrast to the case of matter on a collapsing non-horizon shell.

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