The validity of Birkhoff’s theorem in modified teleparallel equivalent theories of gravity

Abstract

Birkhoff’s theorem is a vitally important theorem in studying local systems in theories of gravity. In this paper we provide a straightforward proof of the validity of Birkhoff’s theorem in modified teleparallel equivalent theories of gravity, which states that if the matter distribution is spherically symmetric, then the spacetime is static in a region in which t remains timelike and (r,theta ,phi ) stay spacelike. This version of Birkhoff’s theorem is rather trivial. The theorem also reveals the fact that the gravitational field of a spherically symmetric distribution of matter and of the spherically symmetric vacuum spacetime are both static in modified teleparallel equivalent theories of gravity.