Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Sergiu Klainerman, Jérémie Szeftel

• Publisher: Princeton University Press
• Year: 2020
• Print ISBN:
  9780691212425
  9780691212432
• eBook ISBN:
  9780691218526
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

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.

Table of contents

• Cover
• Title
• Copyright
• Contents
• List of Figures
• Acknowledgments
• 1 Introduction
• 2 Preliminaries
• 3 Main Theorem
• 4 Consequences of the Bootstrap Assumptions
• 5 Decay Estimates for q (Theorem M1)
• 6 Decay Estimates for α and α (Theorems M2, M3)
• 7 Decay Estimates (Theorems M4, M5)
• 8 Initialization and Extension (Theorems M6, M7, M8)
• 9 GCM Procedure
• 10 Regge-Wheeler Type Equations
• A Appendix to Chapter 2
• B Appendix to Chapter 8
• C Appendix to Chapter 9
• D Appendix to Chapter 10
• Bibliography