Birkhoff’s theorem for Einstein gravity and beyond

Abstract

Birkhoff’s theorem establishes that any spherically symmetric vacuum solution of Einstein’s equations is necessarily static and uniquely given by the Schwarzschild metric. In this TFG we prove this theorem for Einstein gravity and several of its extensions, which include the presence of electric charge as well as higher-curvature terms in the action. More precisely, we explicitly prove Birkhofflike theorems for: Einstein, Einstein-Maxwell, Einstein-Gauss-Bonnet and Einstein-Maxwell-Gauss-Bonnet theories in general dimensions. In all cases, starting from a general spherically symmetric ansatz, we are able to show that the equations of motion impose the staticity condition, and fully constrain the metric to take the form of the unique static and spherically symmetric spacetime of each theory. We also comment on the validity of the theorem for general Lovelock theories and its current status for more general higher-curvature theories of gravity