Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Name: Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations | (AMS-210)
Brand: Princeton University Press
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(AMS-210)

Jérémie Szeftel Sergiu Klainerman(Autor*in)

Princeton University Press

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Erschienen am 15. Dezember 2020

856 Seiten

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Inhalt

Cover
Title
Copyright
Contents
List of Figures
Acknowledgments
1 Introduction
1.1 Basic notions in general relativity
1.1.1 Spacetime and causality
1.1.2 The initial value formulation for Einstein equations
1.1.3 Special solutions
1.1.4 Stability of Minkowski space
1.1.5 Cosmic censorship
1.2 Stability of Kerr conjecture
1.2.1 Formal mode analysis
1.2.2 Vectorfield method
1.3 Nonlinear stability of Schwarzschild under polarized perturbations
1.3.1 Bare-bones version of our theorem
1.3.2 Linear stability of the Schwarzschild spacetime
1.3.3 Main ideas in the proof of Theorem 1.6
1.3.4 Beyond polarization
1.3.5 Note added in proof
1.4 Organization
2 Preliminaries
2.1 Axially symmetric polarized spacetimes
2.1.1 Axial symmetry
2.1.2 Z-frames
2.1.3 Axis of symmetry
2.1.4 Z-polarized S-surfaces
2.1.5 Invariant S-foliations
2.1.6 Schwarzschild spacetime
2.2 Main equations
2.2.1 Main equations for general S-foliations
2.2.2 Null Bianchi identities
2.2.3 Hawking mass
2.2.4 Outgoing geodesic foliations
2.2.5 Additional equations
2.2.6 Ingoing geodesic foliation
2.2.7 Adapted coordinates systems
2.3 Perturbations of Schwarzschild and invariant quantities
2.3.1 Null frame transformations
2.3.2 Schematic notation Gg and Gb
2.3.3 The invariant quantity q
2.3.4 Several identities for q
2.4 Invariant wave equations
2.4.1 Preliminaries
2.4.2 Wave equations for a, a, and q
3 Main Theorem
3.1 General covariant modulated admissible spacetimes
3.1.1 Initial data layer
3.1.2 Main definition
3.1.3 Renormalized curvature components and Ricci coefficients
3.2 Main norms
3.2.1 Main norms in ^(ext)M
3.2.2 Main norms in ^(int)M
3.2.3 Combined norms
3.2.4 Initial layer norm
3.3 Main theorem
3.3.1 Smallness constants
3.3.2 Statement of the main theorem
3.4 Bootstrap assumptions and first consequences
3.4.1 Main bootstrap assumptions
3.4.2 Control of the initial data
3.4.3 Control of averages and of the Hawking mass
3.4.4 Control of coordinates system
3.4.5 Pointwise bounds for higher order derivatives
3.4.6 Construction of a second frame in ^(ext)M
3.5 Global null frames
3.5.1 Extension of frames
3.5.2 Construction of the first global frame
3.5.3 Construction of the second global frame
3.6 Proof of the main theorem
3.6.1 Main intermediate results
3.6.2 End of the proof of the main theorem
3.6.3 Conclusions
3.7 The general covariant modulation procedure
3.7.1 Spacetime assumptions for the GCM procedure
3.7.2 Deformations of surfaces
3.7.3 Adapted frame transformations
3.7.4 GCM results
3.7.5 Main ideas
3.8 Overview of the proof of Theorems M0-M8
3.8.1 Discussion of Theorem M0
3.8.2 Discussion of Theorem M1
3.8.3 Discussion of Theorem M2
3.8.4 Discussion of Theorem M3
3.8.5 Discussion of Theorem M4
3.8.6 Discussion of Theorem M5
3.8.7 Discussion of Theorem M6
3.8.8 Discussion of Theorem M7
3.8.9 Discussion of Theorem M8
3.9 Structure of the rest of the book
4 Consequences of the Bootstrap Assumptions
4.1 Proof of Theorem M0
4.2 Control of averages and of the Hawking mass
4.2.1 Proof of Lemma 3.15
4.2.2 Proof of Lemma 3.16
4.3 Control of coordinates systems
4.4 Pointwise bounds for higher order derivatives
4.5 Proof of Proposition 3.20
4.6 Existence and control of the global frames
4.6.1 Proof of Proposition 3.23
4.6.2 Proof of Lemma 4.16
4.6.3 Proof of Proposition 3.26
5 Decay Estimates for q (Theorem M1)
5.1 Preliminaries
5.1.1 The foliation of M by t
5.1.2 Assumptions for Ricci coefficients and curvature
5.1.3 Structure of nonlinear terms
5.1.4 Main quantities
5.2 Proof of Theorem M1
5.2.1 Flux decay estimates for q
5.2.2 Proof of Theorem M1
5.2.3 Proof of Proposition 5.10
5.3 Improved weighted estimates
5.3.1 Basic and higher weighted estimates for wave equations
5.3.2 Proof of Theorem 5.14
5.3.3 Proof of Theorem 5.15
5.4 Decay estimates
5.4.1 First flux decay estimates
5.4.2 Flux decay estimates for q?
5.4.3 Proof of Theorem 5.9
5.4.4 Proof of Proposition 5.12
5.4.5 Proof of Proposition 5.13
6 Decay Estimates for a and a (Theorems M2, M3)
6.1 Proof of Theorem M2
6.1.1 A renormalized frame on ^(ext)M
6.1.2 A transport equation for a
6.1.3 Estimates for transport equations in e3
6.1.4 Decay estimates for a
6.1.5 End of the proof of Theorem M2
6.2 Proof of Theorem M3
6.2.1 Estimate for a in ^(int)M
6.2.2 Estimate for a on S*
6.2.3 Proof of Proposition 6.10
6.2.4 Proof of Lemma 6.12
6.2.5 Proof of Proposition 6.14
6.2.6 Proof of Lemma 6.16
7 Decay Estimates (Theorems M4, M5)
7.1 Preliminaries to the proof of Theorem M4
7.1.1 Geometric structure of S*
7.1.2 Main assumptions
7.1.3 Basic lemmas
7.1.4 Main equations
7.1.5 Equations involving q
7.1.6 Additional equations
7.2 Structure of the proof of Theorem M4
7.3 Decay estimates on the last slice S*
7.3.1 Preliminaries
7.3.2 Differential identities involving GCM conditions on S*
7.3.3 Control of the flux of some quantities on S*
7.3.4 Estimates for some l = 1 modes on S*
7.3.5 Decay of Ricci and curvature components on S*
7.4 Control in ^(ext)M, Part I
7.4.1 Preliminaries
7.4.2 Proposition 7.33
7.4.3 Estimates for K?, µ? in ^(ext)M
7.4.4 Estimates for the l = 1 modes in ^(ext)M
7.4.5 Completion of the proof of Proposition 7.33
7.5 Control in ^(ext)M, Part II
7.5.1 Estimate for ?
7.5.2 Crucial lemmas
7.5.3 Proof of Proposition 7.35, Part I
7.5.4 Proof of Proposition 7.35, Part II
7.6 Conclusion of the proof of Theorem M4
7.7 Proof of Theorem M5
8 Initialization and Extension (Theorems M6, M7, M8)
8.1 Proof of Theorem M6
8.2 Proof of Theorem M7
8.3 Proof of Theorem M8
8.3.1 Main norms
8.3.2 Control of the global frame
8.3.3 Iterative procedure
8.3.4 End of the proof of Theorem M8
8.4 Proof of Proposition 8.7
8.4.1 A wave equation for ?~
8.4.2 Control of ?g(r)
8.4.3 End of the proof of Proposition 8.7
8.5 Proof of Proposition 8.8
8.5.1 A wave equation for a + ?^2a
8.5.2 End of the proof of Proposition 8.8
8.6 Proof of Proposition 8.9
8.6.1 Control of a and ?^2a
8.6.2 Control of a
8.6.3 End of the proof of Proposition 8.9
8.7 Proof of Proposition 8.10
8.7.1 r-weighted divergence identities for Bianchi pairs
8.7.2 End of the proof of Proposition 8.10
8.7.3 Proof of (8.3.12)
8.8 Proof of Proposition 8.11
8.8.1 Proof of Proposition 8.31
8.8.2 Weighted estimates for transport equations along e4 in ^(ext)M
8.8.3 Several identities
8.8.4 Proof of Proposition 8.32
8.8.5 Proof of Proposition 8.33
8.9 Proof of Proposition 8.12
8.9.1 Weighted estimates for transport equations along e3 in ^(int)M
8.9.2 Proof of Proposition 8.42
8.10 Proof of Proposition 8.13
9 GCM Procedure
9.1 Preliminaries
9.1.1 Main assumptions
9.1.2 Elliptic Hodge lemma
9.2 Deformations of S surfaces
9.2.1 Deformations
9.2.2 Pullback map
9.2.3 Comparison of norms between deformations
9.2.4 Adapted frame transformations
9.3 Frame transformations
9.3.1 Main GCM equations
9.3.2 Equation for the average of a
9.3.3 Transversality conditions
9.4 Existence of GCM spheres
9.4.1 The linearized GCM system
9.4.2 Comparison of the Hawking mass
9.4.3 Iteration procedure for Theorem 9.32
9.4.4 Existence and boundedness of the iterates
9.4.5 Convergence of the iterates
9.5 Proof of Proposition 9.37 and of Corollary 9.38
9.5.1 Proof of Proposition 9.37
9.5.2 Proof of Corollary 9.38
9.6 Proof of Proposition 9.43
9.6.1 Pullback of the main equations
9.6.2 Basic lemmas
9.6.3 Proof of the estimates (9.6.5), (9.6.6), (9.6.7)
9.7 A corollary to Theorem 9.32
9.8 Construction of GCM hypersurfaces
9.8.1 Definition of S0
9.8.2 Extrinsic properties of S0
9.8.3 Construction of S0
10 Regge-Wheeler Type Equations
10.1 Basic Morawetz estimates
10.1.1 Structure of the proof of Theorem 10.1
10.1.2 A simplified set of assumptions
10.1.3 Functions depending on m and r
10.1.4 Deformation tensors of the vectorfields R, T, X
10.1.5 Basic integral identities
10.1.6 Main Morawetz identity
10.1.7 A first estimate
10.1.8 Improved lower bound in ^(ext)M
10.1.9 Cut-off correction in ^(int)M
10.1.10 The redshift vectorfield
10.1.11Combined estimate
10.1.12 Lower bounds for Q
10.1.13 First Morawetz estimate
10.1.14 Analysis of the error term e?
10.1.15 Proof of Theorem 10.1
10.2 Dafermos-Rodnianski r^p-weighted estimates
10.2.1 Vectorfield X = f(r)e4
10.2.2 Energy densities for X = f(r)e4
10.2.3 Proof of Theorem 10.37
10.3 Higher weighted estimates
10.3.1 Wave equation for ?
10.3.2 The r^p-weighted estimates for ?
10.4 Higher derivative estimates
10.4.1 Basic assumptions
10.4.2 Strategy for recovering higher order derivatives
10.4.3 Commutation formulas with the wave equation
10.4.4 Some weighted estimates for wave equations
10.4.5 Proof of Theorem 5.17
10.4.6 Proof of Theorem 5.18
10.5 More weighted estimates for wave equations
A Appendix to Chapter 2
A.1 Proof of Proposition 2.64
A.2 Proof of Proposition 2.71
A.3 Proof of Lemma 2.72
A.4 Proof of Proposition 2.73
A.5 Proof of Proposition 2.74
A.6 Proof of Proposition 2.90
A.7 Proof of Lemma 2.92
A.8 Proof of Corollary 2.93
A.9 Proof of Lemma 2.91
A.10 Proof of Proposition 2.99
A.11 Proof of Proposition 2.100
A.12 Proof of the Teukolsky-Starobinsky identity
A.13 Proof of Proposition 2.107
A.14 Proof of Theorem 2.108
A.14.1 The Teukolsky equation for a
A.14.2 Commutation lemmas
A.14.3 Main commutation
A.14.4 Proof of Theorem 2.108
B Appendix to Chapter 8
B.1 Proof of Proposition 8.14
C Appendix to Chapter 9
C.1 Proof of Lemma 9.11
D Appendix to Chapter 10
D.1 Horizontal S-tensors
D.1.1 Mixed tensors
D.1.2 Invariant Lagrangian
D.1.3 Comparison of the Lagrangians
D.1.4 Energy-momentum tensor
D.2 Standard calculation
D.3 Vector eld Xf
D.4 Proof of Proposition 10.47
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