The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge

TL;DR

This paper proves the linear stability of the Schwarzschild black hole family under gravitational perturbations using a modified generalised wave gauge, advancing understanding of black hole stability in general relativity.

Contribution

It introduces a modified gauge to establish asymptotic flatness and proves linear stability of Schwarzschild solutions, enhancing previous stability analyses.

Findings

Schwarzschild family is linearly stable under gravitational perturbations.

Modified gauge ensures asymptotic flatness of the linearised system.

Results complement and extend prior stability work by Dafermos-Holzegel-Rodnianski.

Abstract

We prove in this paper that the Schwarzschild famiily of black holes are linearly stable as a family of solutions to the system of equations that result from expressing the Einstein vacuum equations in a generalised wave gauge. In particular we improve on our recent work \cite{Johnsonlinstabschwarzold} by modifying the generalised wave gauge employed therein so as to establish asymptotic flatness of the associated linearised system. The result thus complements the seminal work \cite{DHRlinstabschwarz} of Dafermos-Holzegel-Rodnianski in a similar vein as to how the work \cite{LRstabmink} of Lindblad-Rodnianski complemented that of Christodoulou-Klainerman \cite{CKstabmink} in establishing the nonlinear stability of the Minkowski space. This paper is the content of the authors PhD thesis.