# Partial Differential Equations Seminar

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

The Schwarzschild solution in general relativity, discovered in 1916, describes a spacetime that contains a non-rotating black hole. A recent (2016) breakthrough paper of Dafermos, Holzegel and Rodnianski showed that the Schwarzschild exterior is linearly stable (in an appropriate sense) as a solution to the vacuum Einstein equations. Their method of proof involved an analysis of the (linearised) Bianchi equations for the Weyl curvature. In this talk, we shall present our proof that the Schwarzschild solution is in fact linearly stable in a generalised harmonic gauge. In particular, we focus on the (linearised) Einstein equations for the metric directly.

The result relies upon insights gained for the scalar wave equation by Dafermos and Rodnianski and a fortiori includes a decay statement for solutions to the Zerilli equation. Moreover, the issue of gauge plays a very important role in the problem and shall be discussed.