On 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 solution to Einstein vacuum equations using a generalized wave gauge, showing solutions remain bounded and decay, which is a step toward full nonlinear stability.

Contribution

It demonstrates linear stability of Schwarzschild in a generalized wave gauge, extending previous null gauge results and aiding the goal of nonlinear stability analysis.

Findings

Solutions remain uniformly bounded in the exterior region.

Solutions decay at an inverse polynomial rate.

Results support future nonlinear stability proofs.

Abstract

In a recent seminal paper \cite{D-H-R} of Dafermos, Holzegel and Rodnianski the linear stability of the Schwarzschild family of black hole solutions to the Einstein vacuum equations was established by imposing a double null gauge. In this paper we shall prove that the Schwarzschild family is linearly stable as solutions to the Einstein vacuum equations by imposing instead a generalised wave gauge: all sufficiently regular solutions to the system of equations that result from linearising the Einstein vacuum equations, as expressed in a generalised wave gauge, about a fixed Schwarzschild solution remain uniformly bounded on the Schwarzschild exterior region and in fact decay to a member of the linearised Kerr family. The dispersion is at an inverse polynomial rate and therefore in principle sufficient for future nonlinear applications. The result thus fits into the wider goal of establishing the full nonlinear stability of the exterior Kerr family as solutions to the Einstein vacuum equations by employing a generalised wave gauge and therefore complements \cite{D-H-R} in a similar vein as the pioneering work \cite{L-R} of Lindblad and Rodnianski complemented the monumental achievement of Christodoulou and Klainerman in \cite{C-K} whereby the global nonlinear stability of the Minkowski space was established.