The Evolution Problem in General Relativity

393 Pages·2002·9.308 MB·English

by  Sergiu Klainerman, Francesco Nicolò (auth.)

Progress in Mathematical Physics
Volume 25
Editors-in-Chief
Anne Boutet de Monvel, Universite Paris VII Denis Diderot
Gerald Kaiser, The Virginia Center for Signals and Waves
Editorial Board
D. Bao, University of Houston
C. Berenstein, University of Maryland, College Park
P. Blanchard, Universitiit Bielefeld
A.S. Fokas, Imperial College of Science, Technology and Medicine
C. Tracy, University of California, Davis
H. van den Berg, Wageningen University
Sergiu Klainerman
Francesco Nicolo
The Evolution Problem
in General Relativity
Birkhauser
Boston • Basel· Berlin
Sergiu Klainennan Francesco Nicolo
Princeton University Universita degli studi di Roma "Tor Vergata"
Department of Mathematics Dipartimento di Matematica
Princeton, NJ 08544 Facolta di Scienze, M.F.N.
U.S.A. Roma,OOIOO
Italy
Library of Congress Cataloging·in·Publication Data
Klainerman, Sergiu, 1950
The evolution problem in general relativity / Sergiu Klainerman and Francesco Nicolo.
p. cm.-(Progress in mathematical physics; v. 25)
Includes bibliographical references and index.
ISBN·13: 978·1·4612·7408·7 e·ISBN·13: 978·1-4612·2084·8
DOl: 10.1007/978·1·4612·2084·8
1. General relativity (Physics) 2. Evolution equations. 3. Mathematical physics. I.
Nicolo, Francesco, 1943-II. Title. III. Series.
QC173.6 .K57 2002
530.l1-dc21 2002074351
CIP
AMS Subject Classifications: Primary: 83C05; Secondary: 83C20, 35L70, 35L15
05)®
Printed on acid-free paper.
©2003 Birkhauser Boston Birkhiiuser H()?J
Softcover reprint of the hardcover I st edition 2003
All rights reserved. This work may not be translated or copied in whole or in part without the written
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SPIN 10842860
Reformatted from authors' files by TEXniques, Cambridge, MA.
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Contents
Preface xi
1 Introduction 1
1.1 Generalities about Lorentz manifolds.
1.1.1 Lorentz metric, vector and tensor fields,
covariant derivative, Lie derivative . . .
1.1.2 Riemann curvature tensor, Ricci tensor, Bianchi identities 7
1.1.3 Isometries and conformal isometries, Killing and conformal
Killing vector fields. . . . . . . . . . . . . . . . . . . . . . 9
1.2 The Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 The initial value problem, initial data sets and constraint equations 13
1.3 Local existence for Einstein's vacuum equations ..... . 14
1.3.1 Reduction to the nonlinear wave equations ... . 14
1.3.2 Local existence for the Einstein vacuum equations
using wave coordinates . . . . . . . . . . . . 17
1.3.3 General foliations of the Einstein spacetime . . . . 19
1.3.4 Maximal foliations of Einstein spacetime ..... 21
1.3.5 A proof of local existence using the maximal foliation 21
1.3.6 Maximal Cauchy developments . . . . . . . . . . . . 23
1.3.7 Hawking-Penrose singularities, the cosmic censorship 23
1.3.8 The C-K Theorem and the Main Theorem 25
1.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Analytic Methods in the Study of the Initial Value Problem 31
2.1 Local and global existence for systems of nonlinear wave equations 31
2.1.1 Local existence for nonlinear wave equations . . . . . . . . 31
vi Contents
2.1.2 Global existence for nonlinear wave equations .......... 37
2.2 Weyl fields and Bianchi equations in Minkowski spacetime. . . . . . .. 41
2.2.1 Asymptotic behavior of the Weyl fields in Minkowski spacetime. 43
2.3 Global nonlinear stability of Minkowski spacetime 52
2.4 Structure of the work . . . . . . . . . . . . . . . . 53
3 Definitions and Results 55
3.1 Connection coefficients . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
3.1.1 Null second fundamental forms and torsion of a spacelike 2-surface 55
3.1.2 Null decomposition of the curvature tensor 60
3.1.3 Null structure equations of a 2-surface S . . . . . . 62
3.1.4 Integrable S -foliations of the spacetime . . . . . . 62
3.1.5 Null structure equations of a double null foliation . 67
3.1.6 The Einstein equations relative to a double null foliation 69
3.1.7 The characteristic initial value problem for the Einstein equations 72
3.2 Bianchi equations in an Einstein vacuum spacetime . 75
3.3 Canonical double null foliation of the spacetime . . . 78
3.3.1 Canonical foliation of the initial hypersurface 78
3.3.2 Foliations on the last slice . . . . . 79
3.3.3 Canonical foliation of the last slice. 80
3.3.4 Initial layer foliation . . . . . . . . 82
3.4 Deformation tensors. . . . . . . . . . . . . 84
3.4.1 Approximate Killing and conformal Killing vector fields. . 84
3.4.2 Deformation tensors of the vector fields T, S, Ko 85
3.4.3 Rotation deformation tensors. . . 87
3.5 The definitions of the fundamental norms ... . . 87
3.5.1 Q integral norms . . . . . . . . . . . . . . 88
n
3.5.2 norms for the Riemann null components 90
3.5.3 0 norms for the connection coefficients . . 93
3.5.4 Norms on the initial layer region. . . . . . 96
3.5.5 0 norms on the initial and final hypersurfaces . 96
3.5.6 V norms for the rotation deformation tensors 97
3.6 The initial data ..... . . . . . . . 98
3.6.1 Global initial data conditions. . . . . . 98
3.7 The Main Theorem . . . . . . . . . . . . . . . 101
3.7.1 Estimates for the initial layer foliation . 102
3.7.2 Estimates for the 0 norms in K . . . . 102
3.7.3 Estimates for the V norms in K .... 102
3.7.4 Estimates for the 0 norms on the initial hypersurface . 103
3.7.5 Estimates for the 0 norms and the V norms on the last slice 103
3.7.6 Estimates for the n norms ..... 104
3.7.7 Estimates for the Q integral norms. 105
3.7.8 Extension theorem ..... 105
3.7.9 Proof of the Main Theorem. 105
3.8 Appendix . . . . . . . . . . . . . . 109
Contents vii
3.8.1 Proof of Proposition 3.1.1 ................. 109
3.8.2 Derivation of the structure equations. . . . . . . . . . . . 110
3.8.3 Some remarks on the definition of the adapted null frame. 111
3.8.4 Proof of Proposition 3.3.1 ................. 112
4 Estimates for the Connection Coefficients 115
4.1 Preliminary results ......... . 115
4.1.1 Elliptic estimates for Hodge systems . 115
4.1.2 Global Sobolev inequalities ..... 117
4.1.3 The initial layer foliation . . . . . . . 125
4.1.4 Comparison estimates for the function rCu,!i) . 128
4.2 Proof of Theorem Ml . . . . . . . . . . . . . . . . . . 130
4.3 Proof of Theorem 4.2.1 and estimates for the zero and first derivatives of
the connection coefficents. . . . . . . . . . . . . . . . . . . . . 134
4.3.1 Estimate for 0b}Ctrx) and 0b}CX) with p E [2,4] .. 134
4.3.2 Estimates for Ir2-% (trx - trx)lp.s and Ir3-2/pptrx Ip,s,
with P E [2,4] . . . . . . . . . . . . . . . . . . . . 137
2 I
4.3.3 Estimates for Ir2-pr2(trx - trx)lp.s with p E [2,4] 137
4.3.4 Estimate for Ir2-% (ntrX - ;) Ip,s with p E [2,4] 137
4.3.5 Estimates for 0b} (tr~) and 0b} (i) with p E [2,4] 137
4.3.6 Estimates for Ir2-% (tr,! - tr,!)lp.s and Ir3-2/pptr,!lp,s,
with p E [2,4] . . . . . . . . . . . . . . . . . . . . . 139
, I
4.3.7 Estimate for Ir2-~r2 (tr,! - tr,!)lp,s with p E [2,4] . 140
4.3.8 Estimate for Ir1-% L (ntr,! + ;) Ip,s with p E [2,4] . 140
4.3.9 Estimates for og}(I1) and og}(!l) with P E [2,4] 140
4.3.10 Estimates for 0b's (w) and 0b's (fQ) with p E [2,4] 146
4.3.11 Estimate for sup Ir(n - ~)I ............ 147
4.3.12 Completion of the estimates for trX and tr,! . . . . 149
4.3.13 Estimates for ors (w) and 0i'\fQ) with P E [2,4] 151
4.3.14 Estimates for Ob,s(D4w) and 0{S(D3fQ) with p E [2,4] 153
4.3.15 Estimate for 01 (fQ) with P E [2,4] ........ ,.. 155
4.3.16 Improved estimates under stronger assumptions on LO and £. 155
4.4 Proof of Theorem 4.2.2 and estimates for the second derivatives
of the connection coefficients . . . . . . . . . . . . . . . . 161
4.4.1 Estimates for 0i'sCw) and Ors(fQ) with p E [2,4] 161
4.4.2 Estimate for 02CfQ) with p E [2,4] ...... 164
4.5 Proof of Theorem 4.2.3 and control of third derivatives
of the connection coefficients . 168
4.6 Rotation tensor estimates . . . . . . . . . . . . . . . . 172
4.6.1 Technical aspects . . . . . . . . . . . . . . . . 172
4.6.2 Derivatives of the rotation deformation tensors 175
viii Contents
4.7 Proof of Theorem M2 and estimates for the 'D norms
of the rotation deformation tensors . . . 177
4.8 Appendix . . . . . . . . . . . . . . . . . . 183
4.8.1 Some commutation relations . 183
4.8.2 Proof of Lemma 4.3.5 .. 186
4.8.3 Proof of Lemma 4.4.1 .. 188
4.8.4 Proof of Proposition 4.6.2 188
4.8.5 Proof of Proposition 4.6.3 190
4.8.6 Proof of the Oscillation Lemma 192
4.8.7 Proof of Lemma 4.1.7 ..... 200
5 Estimates for the Riemann Curvature Tensor 203
5.1 Preliminary tools . . . . . . . . . . . . . . 205
5.1.1 L 2 estimates for the zero derivatives 208
5.1.2 L 2 estimates for the first derivatives 213
5.1.3 Auxiliary L 2 norms for the zero and first derivatives
of the Riemann components . . . . . . . . . . . . . . 220
5.1.4 The asymptotic behavior ofp and 0' ......... . 223
5.1.5 Asymptotic behavior of the null Riemann components 226
5.2 Appendix . . . . . . . . . . . .. ....... . 226
5.2.1 Proof of Proposition 5.1.4 226
5.2.2 Proof of Proposition 5.1.5 230
6 The Error Estimates 241
6.1 Definitions and prerequisites . . . . . . . . . . . . . . 243
6.1.1 Estimates for the T, S, j( deformation tensors 250
6.1.2 Estimates for the rotation deformation tensors . 257
6.2 The error terms £1 . . . . . . . . . . . . . . . . . . . . 259
1
6.2.1 Estimate of Ie DivQ(L:T W),8yo(j(,8, j(Y, j(O) 259
(u,,!!)
6.2.2 Estimate of IV(u,g) Q(L:T W)a,8yo«Kl,ra,8 j(y j(O) 268
lie
6.2.3 Estimate of DivQ(L:o W),8yo(j(,8 j(YTO) . 269
Iv, (u,!D
6.2.4 Estimate of Q(L:o W)a,8yo «K)]l'a,8 j(Y TO) 273
(u.!:!)
1"
6.2.5 Estimate of Q(L:o W)a,8yo«T)]l'a,8 j(Y j(O) 275
'(U,u)
6.3 The error terms £2 . . . -. . . . . . . . . . . . . . . . . 276
Iv, 2 - -
6.3.1 Estimate of DivQ(.Ac o W),8yo(K,8 KYTO) . 277
(u,g)
6.3.2 Proof of Lemma 6.3.1 and Lemma 6.3.2 . . . . 283
2 --
6.3.3 Estimate of IV(u,g) Q(.cA o W)a,8yo«K)]l'a,8 KYTO) 285
lie 2 - -
6.3.4 Estimate of Q(£A o W)a,8yo «T)]l'a,8 KY KO) 285
lie (u,~)
6.3.5 Estimate of DivQ(L:OL:T W),8yoj(,8 j(y j(o . 285
(U,!!)
6.3.6 Estimate of Iv(u,u) Q(L:OL:T W)a,8yo«K)]l'a,8 j(y j(0) 287
1" -,8 - -0
6.3.7 Estimate of DivQ(£A s.cA TW),8yo(K KY K) ........ . 287
'(u,1{)
Contents IX
6.3.8 Estimate of Iv{",,) Q(CsCr W)"ilyo((K)n"il KY KO) • • • • 292
604 Appendix . . . . . . . ~. . . . . . . . . . . . . . . . . . . . . . 293
604.1 The third~order derivatives of the connection coefficients 293
7 The Initial Hypersurface and the Last Slice 295
7.1 Initial hypersurface foliations ............ . 295
7.1.1 Some general properties of a foliation of bQ . 295
7.1.2 The structure equations on bQ . . . . . . . . 296
7.1.3 The construction of the background foliation of bQ 298
7.104 The construction of the canonical foliation of bO 299
7.1.5 Proof of Theorem M3 ....... . 300
7.2 The initial hypersurface connection estimates 300
7.2.1 Proof of Lemma 3.7.1 ....... . 303
7.3 The last slice foliation. . . . . . . . . . . . . 304
7.3.1 Construction of the canonical foliation of ~* 304
7 A The last slice connection estimates . . . . . 305
704.1 0 norms on the last slice . . . . . . . 305
704.2 Implementation of Proposition 704.1 . 308
704.3 Implementation of Proposition 704.2 . 313
7.5 The last slice rotation deformation estimates. 315
7.6 The extension argument. . . . . . . . . . . . 320
7.7 Appendix . . . . . . . . . . . . . . . . . . . 323
7.7.1 Comparison between different foliations. 323
7.7.2 Proof of the local existence part of Theorem M6 327
7.7.3 Proof of Propositions 70404, 704.5 ........ 333
8 Conclusions 347
8.1 The spacetime null infinity . . . . . . . . . . . . 349
8.1.1 The existence of a global optical function 349
8.1.2 The null-outgoing infinity.J+ . . . . . . 351
8.1.3 The null-outgoing limit of the metric .. 353
8.104 The null-outgoing infinite limit of the SeA, v)-orthonormal frame 354
8.2 The behavior of the curvature tensor at the null-outgoing infinity . . .. 355
8.3 The behavior of the connection coefficients at the null-outgoing infinity 358
804 The null-outgoing infinity limit of the structure equations 365
8.5 The Bondi mass . . . . . . . . . . . . . . . . . . . . 366
8.6 Asymptotic behavior of null-outgoing hypersurfaces 370
References 375
Index 381
Preface
The main goal of this work is to revisit the proof of the global stability of Minkowski space
by D. Christodoulou and S. Klainerman, [Ch-KI]. We provide a new self-contained proof
of the main part of that result, which concerns the full solution of the radiation problem in
vacuum, for arbitrary asymptotically flat initial data sets. This can also be interpreted as a
proof of the global stability of the external region of Schwarzschild spacetime.
The proof, which is a significant modification of the arguments in [Ch-Kl], is based on
a double null foliation of spacetime instead of the mixed null-maximal foliation used in
[Ch-Kl]. This approach is more naturally adapted to the radiation features of the Einstein
equations and leads to important technical simplifications.
In the first chapter we review some basic notions of differential geometry that are sys
tematically used in all the remaining chapters. We then introduce the Einstein equations
and the initial data sets and discuss some of the basic features of the initial value problem
in general relativity. We shall review, without proofs, well-established results concerning
local and global existence and uniqueness and formulate our main result.
The second chapter provides the technical motivation for the proof of our main theorem.
We start by reviewing the standard proof of local existence and uniqueness for systems of
nonlinear wave equations. We then discuss methods for proving global existence results,
by stressing the importance of symmetries. We also emphasize the importance of a struc
tural condition, called the null condition in establishing global results in 3 + I dimensions.
The cancellation that results when this formal condition is adopted illustrates the advan
tage of working with null frames. An essential result is the derivation of uniform decay
estimates for linearized equations using only energy inequalities and the symmetries of
Minkowski spacetime.
We proceed to show how the same method can be used to derive full decay estimates
for the Weyl fields which satisfy the linear Bianchi equations in flat spacetime. The lat
ter provides a crucial stepping stone to the Einstein equations. Finally we provide the
reader with a detailed discussion of the basic ideas in the proof of the main theorem. All