Birkhoff's theorem in Lovelock gravity

TL;DR

This paper proves that in Lovelock gravity, all spherically, planar, or hyperbolically symmetric solutions are locally equivalent to static black holes, extending to charged cases and establishing local staticity.

Contribution

It establishes a Lovelock gravity analogue of Birkhoff's theorem, showing all symmetric solutions are locally static or isometric to known static black holes.

Findings

Solutions are locally isometric to static Lovelock black holes.

The result extends to solutions with an abelian gauge field, including charged black holes.

All symmetric solutions are locally static, admitting an additional Killing vector.

Abstract

We show that the generic solutions of the Lovelock equations with spherical, planar or hyperbolic symmetry are locally isometric to the corresponding static Lovelock black hole. As a consequence, these solutions are locally static: they admit an additional Killing vector that can either be space-like or time-like, depending on the position. This result also holds in the presence of an abelian gauge field, in which case the solutions are locally isometric to a charged static black hole.