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E-raamat: Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

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Global Nonlinear Stability of Schwarzschild Spacetime under Polarized PerturbationsSuurem pilt
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Essential mathematical insights into one of the most important and challenging open problems in general relativity—the stability of black holes

One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. In this book, Sergiu Klainerman and Jérémie Szeftel take a first important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes—or Schwarzschild spacetimes—under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, Klainerman and Szeftel introduce a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, this book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.

List of Figures
xiii
Acknowledgments xv
1 Introduction
1(23)
1.1 Basic notions in general relativity
1(12)
1.1.1 Spacetime and causality
1(1)
1.1.2 The initial value formulation for Einstein equations
2(1)
1.1.3 Special solutions
3(7)
1.1.4 Stability of Minkowski space
10(1)
1.1.5 Cosmic censorship
11(2)
1.2 Stability of Kerr conjecture
13(4)
1.2.1 Formal mode analysis
15(1)
1.2.2 Vectorfield method
16(1)
1.3 Nonlinear stability of Schwarzschild under polarized perturbations
17(5)
1.3.1 Bare-bones version of our theorem
17(1)
1.3.2 Linear stability of the Schwarzschild spacetime
17(1)
1.3.3 Main ideas in the proof of Theorem 1.6
18(3)
1.3.4 Beyond polarization
21(1)
1.3.5 Note added in proof
22(1)
1.4 Organization
22(2)
2 Preliminaries
24(65)
2.1 Axially symmetric polarized spacetimes
24(27)
2.1.1 Axial symmetry
24(1)
2.1.2 Z-frames
25(1)
2.1.3 Axis of symmetry
26(2)
2.1.4 Z-polarized S-surfaces
28(17)
2.1.5 Invariant S-foliations
45(5)
2.1.6 Schwarzschild spacetime
50(1)
2.2 Main equations
51(27)
2.2.1 Main equations for general 5-foliations
51(3)
2.2.2 Null Bianchi identities
54(2)
2.2.3 Hawking mass
56(1)
2.2.4 Outgoing geodesic foliations
57(13)
2.2.5 Additional equations
70(1)
2.2.6 Ingoing geodesic foliation
71(1)
2.2.7 Adapted coordinates systems
71(7)
2.3 Perturbations of Schwarzschild and invariant quantities
78(6)
2.3.1 Null frame transformations
78(4)
2.3.2 Schematic notation Γg and Γg
82(1)
2.3.3 The invariant quantity q
83(1)
2.3.4 Several identities for q
84(1)
2.4 Invariant wave equations
84(5)
2.4.1 Preliminaries
85(2)
2.4.2 Wave equations for α, α, and q
87(2)
3 Main Theorem
89(56)
3.1 General covariant modulated admissible spacetimes
89(7)
3.1.1 Initial data layer
89(2)
3.1.2 Main definition
91(4)
3.1.3 Renormalized curvature components and Ricci coefficients
95(1)
3.2 Main norms
96(5)
3.2.1 Main norms in (ext)M
96(3)
3.2.2 Main norms in (int)M
99(1)
3.2.3 Combined norms
100(1)
3.2.4 Initial layer norm
100(1)
3.3 Main theorem
101(4)
3.3.1 Smallness constants
101(1)
3.3.2 Statement of the main theorem
102(3)
3.4 Bootstrap assumptions and first consequences
105(6)
3.4.1 Main bootstrap assumptions
105(1)
3.4.2 Control of the initial data
105(1)
3.4.3 Control of averages and of the Hawking mass
106(1)
3.4.4 Control of coordinates system
107(2)
3.4.5 Pointwise bounds for higher order derivatives
109(1)
3.4.6 Construction of a second frame in (ext)M
109(2)
3.5 Global null frames
111(3)
3.5.1 Extension of frames
111(1)
3.5.2 Construction of the first global frame
112(1)
3.5.3 Construction of the second global frame
113(1)
3.6 Proof of the main theorem
114(11)
3.6.1 Main intermediate results
114(1)
3.6.2 End of the proof of the main theorem
115(1)
3.6.3 Conclusions
116(9)
3.7 The general covariant modulation procedure
125(8)
3.7.1 Spacetime assumptions for the GCM procedure
125(3)
3.7.2 Deformations of surfaces
128(1)
3.7.3 Adapted frame transformations
128(1)
3.7.4 GCM results
129(2)
3.7.5 Main ideas
131(2)
3.8 Overview of the proof of Theorems M0--M8
133(10)
3.8.1 Discussion of Theorem M0
133(1)
3.8.2 Discussion of Theorem M1
134(1)
3.8.3 Discussion of Theorem M2
135(1)
3.8.4 Discussion of Theorem M3
136(1)
3.8.5 Discussion of Theorem M4
137(1)
3.8.6 Discussion of Theorem M5
138(1)
3.8.7 Discussion of Theorem M6
138(1)
3.8.8 Discussion of Theorem M7
139(1)
3.8.9 Discussion of Theorem M8
140(3)
3.9 Structure of the rest of the book
143(2)
4 Consequences of the Bootstrap Assumptions
145(68)
4.1 Proof of Theorem M0
145(19)
4.2 Control of averages and of the Hawking mass
164(10)
4.2.1 Proof of Lemma 3.15
164(8)
4.2.2 Proof of Lemma 3.16
172(2)
4.3 Control of coordinates systems
174(9)
4.4 Pointwise bounds for higher order derivatives
183(5)
4.5 Proof of Proposition 3.20
188(9)
4.6 Existence and control of the global frames
197(16)
4.6.1 Proof of Proposition 3.23
197(3)
4.6.2 Proof of Lemma 4.16
200(8)
4.6.3 Proof of Proposition 3.26
208(5)
5 Decay Estimates for q (Theorem M1)
213(51)
5.1 Preliminaries
213(10)
5.1.1 The foliation of M by τ
214(1)
5.1.2 Assumptions for Ricci coefficients and curvature
215(1)
5.1.3 Structure of nonlinear terms
216(2)
5.1.4 Main quantities
218(5)
5.2 Proof of Theorem M1
223(7)
5.2.1 Flux decay estimates for q
223(1)
5.2.2 Proof of Theorem M1
224(2)
5.2.3 Proof of Proposition 5.10
226(4)
5.3 Improved weighted estimates
230(19)
5.3.1 Basic and higher weighted estimates for wave equations
231(2)
5.3.2 Proof of Theorem 5.14
233(11)
5.3.3 Proof of Theorem 5.15
244(5)
5.4 Decay estimates
249(15)
5.4.1 First flux decay estimates
249(4)
5.4.2 Flux decay estimates for q
253(2)
5.4.3 Proof of Theorem 5.9
255(4)
5.4.4 Proof of Proposition 5.12
259(1)
5.4.5 Proof of Proposition 5.13
260(4)
6 Decay Estimates for a and a (Theorems M2, M3)
264(31)
6.1 Proof of Theorem M2
264(15)
6.1.1 A renormalized frame on (ext)M
264(1)
6.1.2 A transport equation for α
264(3)
6.1.3 Estimates for transport equations in e3
267(4)
6.1.4 Decay estimates for α
271(7)
6.1.5 End of the proof of Theorem M2
278(1)
6.2 Proof of Theorem M3
279(16)
6.2.1 Estimate for α in (int)M
279(2)
6.2.2 Estimate for α on Σ*
281(1)
6.2.3 Proof of Proposition 6.10
282(4)
6.2.4 Proof of Lemma 6.12
286(3)
6.2.5 Proof of Proposition 6.14
289(3)
6.2.6 Proof of Lemma 6.16
292(3)
7 Decay Estimates (Theorems M4, M5)
295(77)
7.1 Preliminaries to the proof of Theorem M4
295(13)
7.1.1 Geometric structure of Σ*
295(1)
7.1.2 Main assumptions
296(3)
7.1.3 Basic lemmas
299(2)
7.1.4 Main equations
301(1)
7.1.5 Equations involving q
302(3)
7.1.6 Additional equations
305(3)
7.2 Structure of the proof of Theorem M4
308(3)
7.3 Decay estimates on the last slice Σ*
311(25)
7.3.1 Preliminaries
311(3)
7.3.2 Differential identities involving GCM conditions on Σ*
314(1)
7.3.3 Control of the flux of some quantities on Σ*
315(7)
7.3.4 Estimates for some = 1 modes on Σ*
322(10)
7.3.5 Decay of Ricci and curvature components on Σ*
332(4)
7.4 Control in (ext)M, Part I
336(10)
7.4.1 Preliminaries
336(2)
7.4.2 Proposition 7.33
338(1)
7.4.3 Estimates for k, μ in (ext)M
339(1)
7.4.4 Estimates for the = 1 modes in (ext)M
340(3)
7.4.5 Completion of the proof of Proposition 7.33
343(3)
7.5 Control in (ext)M, Part II
346(16)
7.5.1 Estimate for η
347(1)
7.5.2 Crucial lemmas
347(8)
7.5.3 Proof of Proposition 7.35, Part I
355(4)
7.5.4 Proof of Proposition 7.35, Part II
359(3)
7.6 Conclusion of the proof of Theorem M4
362(4)
7.7 Proof of Theorem M5
366(6)
8 Initialization and Extension (Theorems M6, M7, M8)
372(114)
8.1 Proof of Theorem M6
372(4)
8.2 Proof of Theorem M7
376(11)
8.3 Proof of Theorem M8
387(12)
8.3.1 Main norms
389(2)
8.3.2 Control of the global frame
391(2)
8.3.3 Iterative procedure
393(3)
8.3.4 End of the proof of Theorem M8
396(3)
8.4 Proof of Proposition 8.7
399(9)
8.4.1 A wave equation for p
399(1)
8.4.2 Control of g(r)
400(5)
8.4.3 End of the proof of Proposition 8.7
405(3)
8.5 Proof of Proposition 8.8
408(10)
8.5.1 A wave equation for α + Υ2α
408(9)
8.5.2 End of the proof of Proposition 8.8
417(1)
8.6 Proof of Proposition 8.9
418(6)
8.6.1 Control of a and Υ2α
418(2)
8.6.2 Control of α
420(4)
8.6.3 End of the proof of Proposition 8.9
424(1)
8.7 Proof of Proposition 8.10
424(18)
8.7.1 τ-weighted divergence identities for Bianchi pairs
425(10)
8.7.2 End of the proof of Proposition 8.10
435(5)
8.7.3 Proof of (8.3.12)
440(2)
8.8 Proof of Proposition 8.11
442(37)
8.8.1 Proof of Proposition 8.31
444(10)
8.8.2 Weighted estimates for transport equations along e4 in (ext)M
454(6)
8.8.3 Several identities
460(4)
8.8.4 Proof of Proposition 8.32
464(7)
8.8.5 Proof of Proposition 8.33
471(8)
8.9 Proof of Proposition 8.12
479(6)
8.9.1 Weighted estimates for transport equations along e3 in (int)M
480(2)
8.9.2 Proof of Proposition 8.42
482(3)
8.10 Proof of Proposition 8.13
485(1)
9 GCM Procedure
486(114)
9.1 Preliminaries
486(3)
9.1.1 Main assumptions
488(1)
9.1.2 Elliptic Hodge lemma
489(1)
9.2 Deformations of S surfaces
489(15)
9.2.1 Deformations
489(1)
9.2.2 Pullback map
490(2)
9.2.3 Comparison of norms between deformations
492(4)
9.2.4 Adapted frame transformations
496(8)
9.3 Frame transformations
504(16)
9.3.1 Main GCM equations
513(5)
9.3.2 Equation for the average of a
518(1)
9.3.3 Transversality conditions
519(1)
9.4 Existence of GCM spheres
520(18)
9.4.1 The linearized GCM system
524(2)
9.4.2 Comparison of the Hawking mass
526(1)
9.4.3 Iteration procedure for Theorem 9.32
527(3)
9.4.4 Existence and boundedness of the iterates
530(5)
9.4.5 Convergence of the iterates
535(3)
9.5 Proof of Proposition 9.37 and of Corollary 9.38
538(7)
9.5.1 Proof of Proposition 9.37
538(4)
9.5.2 Proof of Corollary 9.38
542(3)
9.6 Proof of Proposition 9.43
545(14)
9.6.1 Pullback of the main equations
545(3)
9.6.2 Basic lemmas
548(8)
9.6.3 Proof of the estimates (9.6.5), (9.6.6), (9.6.7)
556(3)
9.7 A corollary to Theorem 9.32
559(7)
9.8 Construction of GCM hypersurfaces
566(34)
9.8.1 Definition of Σ0
569(1)
9.8.2 Extrinsic properties of Σ0
570(13)
9.8.3 Construction of Σ0
583(17)
10 Regge-Wheeler Type Equations
600(119)
10.1 Basic Morawetz estimates
600(56)
10.1.1 Structure of the proof of Theorem 10.1
601(1)
10.1.2 A simplified set of assumptions
602(1)
10.1.3 Functions depending on m and r
602(1)
10.1.4 Deformation tensors of the vectorfields R, T, X
603(4)
10.1.5 Basic integral identities
607(2)
10.1.6 Main Morawetz identity
609(4)
10.1.7 A first estimate
613(5)
10.1.8 Improved lower bound in (ext)M
618(7)
10.1.9 Cut-off correction in (int)M
625(7)
10.1.10 The redshift vectorfield
632(4)
10.1.11 Combined estimate
636(6)
10.1.12 Lower bounds for Q
642(2)
10.1.13 First Morawetz estimate
644(7)
10.1.14 Analysis of the error term ε
651(2)
10.1.15 Proof of Theorem 10.1
653(3)
10.2 Dafermos-Rodnianski rp-weighted estimates
656(19)
10.2.1 Vectorfield X = ƒ(r)e4
659(1)
10.2.2 Energy densities for X = ƒ(r)e4
659(9)
10.2.3 Proof of Theorem 10.37
668(7)
10.3 Higher weighted estimates
675(7)
10.3.1 Wave equation for Ψ
675(1)
10.3.2 The τp-weighted estimates for Ψ
676(6)
10.4 Higher derivative estimates
682(29)
10.4.1 Basic assumptions
682(1)
10.4.2 Strategy for recovering higher order derivatives
682(1)
10.4.3 Commutation formulas with the wave equation
683(13)
10.4.4 Some weighted estimates for wave equations
696(5)
10.4.5 Proof of Theorem 5.17
701(5)
10.4.6 Proof of Theorem 5.18
706(5)
10.5 More weighted estimates for wave equations
711(8)
A Appendix to
Chapter 2
719(80)
A.1 Proof of Proposition 2.64
719(2)
A.2 Proof of Proposition 2.71
721(4)
A.3 Proof of Lemma 2.72
725(3)
A.4 Proof of Proposition 2.73
728(5)
A.5 Proof of Proposition 2.74
733(4)
A.6 Proof of Proposition 2.90
737(13)
A.7 Proof of Lemma 2.92
750(3)
A.8 Proof of Corollary 2.93
753(2)
A.9 Proof of Lemma 2.91
755(2)
A.10 Proof of Proposition 2.99
757(3)
A.11 Proof of Proposition 2.100
760(5)
A.12 Proof of the Teukolsky-Starobinsky identity
765(8)
A.13 Proof of Proposition 2.107
773(3)
A.14 Proof of Theorem 2.108
776(23)
A.14.1 The Teukolsky equation for α
779(2)
A.14.2 Commutation lemmas
781(7)
A.14.3 Main commutation
788(8)
A.14.4 Proof of Theorem 2.108
796(3)
B Appendix to
Chapter 8
799(7)
B.1 Proof of Proposition 8.14
799(7)
C Appendix to
Chapter 9
806(13)
C.1 Proof of Lemma 9.11
806(13)
D Appendix to
Chapter 10
819(17)
D.1 Horizontal S-tensors
819(4)
D.1.1 Mixed tensors
820(1)
D.1.2 Invariant Lagrangian
820(1)
D.1.3 Comparison of the Lagrangians
821(1)
D.1.4 Energy-momentum tensor
822(1)
D.2 Standard calculation
823(1)
D.3 Vectorfield Xf
824(3)
D.4 Proof of Proposition 10.47
827(9)
Bibliography 836
Sergiu Klainerman is Eugene Higgins Professor of Mathematics at Princeton University. His books include The Global Nonlinear Stability of the Minkowski Space (Princeton). Jérémie Szeftel is a CNRS senior researcher in mathematics at the Laboratoire Jacques-Louis Lions of Sorbonne Université in Paris.
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