Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

by Sergiu Klainerman, Jérémie Szeftel

Overview

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.

Product Details

ISBN-13:	9780691218526
Publisher:	Princeton University Press
Publication date:	12/15/2020
Series:	Annals of Mathematics Studies , #210
Sold by:	Barnes & Noble
Format:	eBook
Pages:	856
File size:	6 MB

About the Author

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.

Table of Contents

List of Figures xiii

Acknowledgments xv

1 Introduction 1

1.1 Basic notions in general relativity 1

1.1.1 Spacetime and causality 1

1.1.2 The initial value formulation for Einstein equations 2

1.1.3 Special solutions 3

1.1.4 Stability of Minkowski space 10

1.1.5 Cosmic censorship 11

1.2 Stability of Kerr conjecture 13

1.2.1 Formal mode analysis 15

1.2.2 Vectorfield method 16

1.3 Nonlinear stability of Schwarzschild under polarized perturbations 17

1.3.1 Bare-bones version of our theorem 17

1.3.2 Linear stability of the Schwarzschild spacetime 17

1.3.3 Main ideas in the proof of Theorem 1.6 18

1.3.4 Beyond polarization 21

1.3.5 Note added in proof 22

1.4 Organization 22

2 Preliminaries 24

2.1 Axially symmetric polarized spacetimes 24

2.1.1 Axial symmetry 24

2.1.2 Z-frames 25

2.1.3 Axis of symmetry 26

2.1.4 Z-polarized S-surfaces 28

2.1.5 Invariant S-foliations 45

2.1.6 Schwarzschild spacetime 50

2.2 Main equations 51

2.2.1 Main equations for general S-foliations 51

2.2.2 Null Bianchi identities 54

2.2.3 Hawking mass 56

2.2.4 Outgoing geodesic foliations 57

2.2.5 Additional equations 70

2.2.6 Ingoing geodesic foliation 71

2.2.7 Adapted coordinates systems 71

2.3 Perturbations of Schwarzschild and invariant quantities 78

2.3.1 Null frame transformations 78

2.3.2 Schematic notation Γ_g and Γ_b 82

2.3.3 The invariant quantity q 83

2.3.4 Several identities for q 84

2.4 Invariant wave equations 84

2.4.1 Preliminaries 85

2.4.2 Wave equations for α, α, and q 87

3 Main Theorem 89

3.1 General covariant modulated admissible spacetimes 89

3.1.1 Initial data layer 89

3.1.2 Main definition 91

3.1.3 Renormalized curvature components and Ricci coefficients 95

3.2 Main norms 96

3.2.1 Main norms in ^(ext)M 96

3.2.2 Main norms in ^(ext)M 99

3.2.3 Combined norms 100

3.2.4 Initial layer norm 100

3.3 Main theorem 101

3.3.1 Smallness constants 101

3.3.2 Statement of the main theorem 102

3.4 Bootstrap assumptions and first consequences 105

3.4.1 Main bootstrap assumptions 105

3.4.2 Control of the initial data 105

3.4.3 Control of averages and of the Hawking mass 106

3.4.4 Control of coordinates system 107

3.4.5 Pointwise bounds for higher order derivatives 109

3.4.6 Construction of a second frame in ^(ext)M 109

3.5 Global null frames 111

3.5.1 Extension of frames 111

3.5.2 Construction of the first global frame 112

3.5.3 Construction of the second global frame 113

3.6 Proof of the main theorem 114

3.6.1 Main intermediate results 114

3.6.2 End of the proof of the main theorem 115

3.6.3 Conclusions 116

3.7 The general covariant modulation procedure 125

3.7.1 Spacetime assumptions for the GCM procedure 125

3.7.2 Deformations of surfaces 128

3.7.3 Adapted frame transformations 128

3.7.4 GCM results 129

3.7.5 Main ideas 131

3.8 Overview of the proof of Theorems M0-M8 133

3.8.1 Discussion of Theorem M0 133

3.8.2 Discussion of Theorem M1 134

3.8.3 Discussion of Theorem M2 135

3.8.4 Discussion of Theorem M3 136

3.8.5 Discussion of Theorem M4 137

3.8.6 Discussion of Theorem M5 138

3.8.7 Discussion of Theorem M6 138

3.8.8 Discussion of Theorem M7 139

3.8.9 Discussion of Theorem M8 140

3.9 Structure of the rest of the book 143

4 Consequences of the Bootstrap Assumptions 145

4.1 Proof of Theorem M0 145

4.2 Control of averages and of the Hawking mass 164

4.2.1 Proof of Lemma 3.15 164

4.2.2 Proof of Lemma 3.16 172

4.3 Control of coordinates systems 174

4.4 Pointwise bounds for higher order derivatives 183

4.5 Proof of Proposition 3.20 188

4.6 Existence and control of the global frames 197

4.6.1 Proof of Proposition 3.23 197

4.6.2 Proof of Lemma 4.16 200

4.6.3 Proof of Proposition 3.26 208

5 Decay Estimates for q (Theorem M1) 213

5.1 Preliminaries 213

5.1.1 The foliation of M by τ 214

5.1.2 Assumptions for Ricci coefficients and curvature 215

5.1.3 Structure of nonlinear terms 216

5.1.4 Main quantities 218

5.2 Proof of Theorem M1 223

5.2.1 Flux decay estimates for q 223

5.2.2 Proof of Theorem M1 224

5.2.3 Proof of Proposition 5.10 226

5.3 Improved weighted estimates 230

5.3.1 Basic and higher weighted estimates for wave equations 231

5.3.2 Proof of Theorem 5.14 233

5.3.3 Proof of Theorem 5.15 244

5.4 Decay estimates 249

5.4.1 First flux decay estimates 249

5.4.2 Flux decay estimates for q 253

5.4.3 Proof of Theorem 5.9 255

5.4.4 Proof of Proposition 5.12 259

5.4.5 Proof of Proposition 5.13 260

6 Decay Estimates for α and α (Theorems M2, M3) 264

6.1 Proof of Theorem M2 264

6.1.1 A renormalized frame on ^(ext)M 264

6.1.2 A transport equation for α 264

6.1.3 Estimates for transport equations in e₃ 267

6.1.4 Decay estimates for α 271

6.1.5 End of the proof of Theorem M2 278

6.2 Proof of Theorem M3 279

6.2.1 Estimate for α in ^(int)M 279

6.2.2 Estimate for α on Σ_* 281

6.2.3 Proof of Proposition 6.10 282

6.2.4 Proof of Lemma 6.12 286

6.2.5 Proof of Proposition 6.14 289

6.2.6 Proof of Lemma 6.16 292

7 Decay Estimates (Theorems M4, M5) 295

7.1 Preliminaries to the proof of Theorem M4 295

7.1.1 Geometric structure of Σ_* 295

7.1.2 Main assumptions 296

7.1.3 Basic lemmas 299

7.1.4 Main equations 301

7.1.5 Equations involving q 302

7.1.6 Additional equations 305

7.2 Structure of the proof of Theorem M4 308

7.3 Decay estimates on the last slice Σ_* 311

7.3.1 Preliminaries 311

7.3.2 Differential identities involving GCM conditions on Σ_* 314

7.3.3 Control of the flux of some quantities on Σ_* 315

7.3.4 Estimates for some l = 1 modes on Σ_* 322

7.3.5 Decay of Ricci and curvature components on Σ_* 332

7.4 Control in ^extM, Part I 336

7.4.1 Preliminaries 336

7.4.2 Proposition 7.33 338

7.4.3 Estimates for κ, μ in ^(ext)M 339

7.4.4 Estimates for the l = 1 modes in ^(ext)M 340

7.4.5 Completion of the proof of Proposition 7.33 343

7.5 Control in ^(ext)M, Part II 346

7.5.1 Estimate for η 347

7.5.2 Crucial lemmas 347

7.5.3 Proof of Proposition 7.35, Part I 355

7.5.4 Proof of Proposition 7.35, Part II 359

7.6 Conclusion of the proof of Theorem M4 362

7.7 Proof of Theorem M5 366

8 Initialization and Extension (Theorems M6, M7, M8) 372

8.1 Proof of Theorem M6 372

8.2 Proof of Theorem M7 376

8.3 Proof of Theorem M8 387

8.3.1 Main norms 389

8.3.2 Control of the global frame 391

8.3.3 Iterative procedure 393

8.3.4 End of the proof of Theorem M8 396

8.4 Proof of Proposition 8.7 399

8.4.1 A wave equation for ρ 399

8.4.2 Control of $$$_g(r) 400

8.4.3 End of the proof of Proposition 8.7 405

8.5 Proof of Proposition 8.8 408

8.5.1 A wave equation for α + ϒ²α 408

8.5.2 End of the proof of Proposition 8.8 417

8.6 Proof of Proposition 8.9 418

8.6.1 Control of α and ϒ²α 418

8.6.2 Control of α 420

8.6.3 End of the proof of Proposition 8.9 424

8.7 Proof of Proposition 8.10 424

8.7.1 r-weighted divergence identities for Bianchi pairs 425

8.7.2 End of the proof of Proposition 8.10 435

8.7.3 Proof of (8.3.12) 440

8.8 Proof of Proposition 8.11 442

8.8.1 Proof of Proposition 8.31 444

8.8.2 Weighted estimates for transport equations along 64 in ^(ext)M 454

8.8.3 Several identities 460

8.8.4 Proof of Proposition 8.32 464

8.8.5 Proof of Proposition 8.33 471

8.9 Proof of Proposition 8.12 479

8.9.1 Weighted estimates for transport equations along e₃ in ^(int)M 480

8.9.2 Proof of Proposition 8.42 482

8.10 Proof of Proposition 8.13 485

9 GCM Procedure 486

9.1 Preliminaries 486

9.1.1 Main assumptions 488

9.1.2 Elliptic Hodge lemma 489

9.2 Deformations of S surfaces 489

9.2.1 Deformations 489

9.2.2 Fullback map 490

9.2.3 Comparison of norms between deformations 492

9.2.4 Adapted frame transformations 496

9.3 Frame transformations 504

9.3.1 Main GCM equations 513

9.3.2 Equation for the average of a 518

9.3.3 Transversality conditions 519

9.4 Existence of GCM spheres 520

9.4.1 The linearized GCM system 524

9.4.2 Comparison of the Hawking mass 526

9.4.3 Iteration procedure for Theorem 9.32 527

9.4.4 Existence and boundedness of the iterates 530

9.4.5 Convergence of the iterates 535

9.5 Proof of Proposition 9.37 and of Corollary 9.38 538

9.5.1 Proof of Proposition 9.37 538

9.5.2 Proof of Corollary 9.38 542

9.6 Proof of Proposition 9.43 545

9.6.1 Fullback of the main equations 545

9.6.2 Basic lemmas 548

9.6.3 Proof of the estimates (9.6.5), (9.6.6), (9.6.7) 556

9.7 A corollary to Theorem 9.32 559

9.8 Construction of GCM hypersurfaces 566

9.8.1 Definition of Σ₀ 569

9.8.2 Extrinsic properties of Σ₀ 570

9.8.3 Construction of Σ₀ 583

10 Regge-Wheeler Type Equations 600

10.1 Basic Morawetz estimates 600

10.1.1 Structure of the proof of Theorem 10.1 601

10.1.2 A simplified set of assumptions 602

10.1.3 Functions depending on m and r 602

10.1.4 Deformation tensors of the vectorfields R, T, X 603

10.1.5 Basic integral identities 607

10.1.6 Main Morawetz identity 609

10.1.7 A first estimate 613

10.1.8 Improved lower bound in ^(ext)M 618

10.1.9 Cut-off correction in ^(int)M 625

10.1.10 The redshift vectorfield 632

10.1.11 Combined estimate 636

10.1.12 Lower bounds for Q 642

10.1.13 First Morawetz estimate 644

10.1.14 Analysis of the error term E_ε 651

10.1.15 Proof of Theorem 10.1 653

10.2 Dafermos-Rodnianski r^p-weighted estimates 656

10.2.1 Vectorfield X = f(r)e₄ 659

10.2.2 Energy densities for X = f(r)e₄ 659

10.2.3 Proof of Theorem 10.37 668

10.3 Higher weighted estimates 675

10.3.1 Wave equation for ψ 675

10.3.2 The r^p-weighted estimates for ψ 676

10.4 Higher derivative estimates 682

10.4.1 Basic assumptions 682

10.4.2 Strategy for recovering higher order derivatives 682

10.4.3 Commutation formulas with the wave equation 683

10.4.4 Some weighted estimates for wave equations 696

10.4.5 Proof of Theorem 5.17 701

10.4.6 Proof of Theorem 5.18 706

10.5 More weighted estimates for wave equations 711

A Appendix to Chapter 2 719

A.1 Proof of Proposition 2.64 719

A.2 Proof of Proposition 2.71 721

A.3 Proof of Lemma 2.72 725

A.4 Proof of Proposition 2.73 728

A.5 Proof of Proposition 2.74 733

A.6 Proof of Proposition 2.90 737

A.7 Proof of Lemma 2.92 750

A.8 Proof of Corollary 2.93 753

A.9 Proof of Lemma 2.91 755

A.10 Proof of Proposition 2.99 757

A.11 Proof of Proposition 2.100 760

A.12 Proof of the Teukolsky-Starobinsky identity 765

A.13 Proof of Proposition 2.107 773

A.14 Proof of Theorem 2.108 776

A.14.1 The Teukolsky equation for α 779

A.14.2 Commutation lemmas 781

A.14.3 Main commutation 788

A.14.4 Proof of Theorem 2.108 796

B Appendix to Chapter 8 799

B.1 Proof of Proposition 8.14 799

C Appendix to Chapter 9 806

C.1 Proof of Lemma 9.11 806

D Appendix to Chapter 10 819

D.1 Horizontal S-tensors 819

D.1.1 Mixed tensors 820

D.1.2 Invariant Lagrangian 820

D.1.3 Comparison of the Lagrangians 821

D.1.4 Energy-momentum tensor 822

D.2 Standard calculation 823

D.3 Vectorfield X_f 824

D.4 Proof of Proposition 10.47 827

Bibliography 836