On Birkhoff's theorem in ghost-free bimetric theory

TL;DR

This paper derives a class of nonstatic, spherically symmetric vacuum solutions in ghost-free bimetric theory, showing that Birkhoff's theorem does not hold in this context due to the solutions' nonstationary nature.

Contribution

It provides explicit nonstatic vacuum solutions in ghost-free bimetric theory, challenging the applicability of Birkhoff's theorem in this framework.

Findings

Solutions are non-asymptotically flat and nonstationary.

Solutions admit only three global spacelike Killing vectors.

Birkhoff's theorem does not hold in ghost-free bimetric theory.

Abstract

We consider the Hassan-Rosen bimetric field equations in vacuum when the two metrics share a single common null direction in a spherically symmetric configuration. By solving these equations, we obtain a class of exact solutions of the generalized Vaidya type parametrized by an arbitrary function. Besides not being asymptotically flat, the found solutions are nonstationary admitting only three global spacelike Killing vector fields which are the generators of spatial rotations. Hence, these are spherically symmetric bimetric vacuum solutions with the minimal number of isometries. The absence of staticity formally disproves an analogue statement to Birkhoff's theorem in the ghost-free bimetric theory which would state that a spherically symmetric solution is necessarily static in empty space.