Is Birkhoff's theorem valid in Einstein-Aether theory?

Abstract

We attempt to answer whether Birkhoff's theorem (BT) is valid in the Einstein-Aether (EA) theory. The BT states that any spherically symmetric solution of the vacuum field equations must be static, unique, and asymptotically flat. For a general spherically symmetric metric with metric functions A(r,t) & B(r,t), and aether components a(r,t) & b(r,t), we prove the conditions for the staticity of spacetime using two different methods. We point out that BT is valid in EA theory only for special values of c1+c3, c1+c4, and c2, where we can show that all these special cases are asymptotically flat. In particular, when the aether has only a temporal component i.e., b(r,t)=0, the c14≠0 case gives us spherically symmetric static solutions with singularities without Killing nor universal horizons, at least for special values of c14. However, when we have an aether vector with temporal and radial components, we prove that the staticity and the flatness at infinity hold for only a special metric and a particular combination of the aether parameters. These solutions have universal horizons.