Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Abstract

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. This book takes an 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, the book introduces 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, the book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.