Generalizing Birkhoff

TL;DR

This paper extends Birkhoff's Theorem to more general spherically symmetric space-times by establishing conditions based on the stress-energy tensor's eigenvalues, including new equations and examples.

Contribution

It generalizes Birkhoff's Theorem and the TOV equation, providing necessary and sufficient conditions for staticity in spherically symmetric space-times based on stress-energy eigenvalues.

Findings

Derived conditions for staticity in terms of stress-energy eigenvalues

Generalized the TOV equation under weaker pressure conditions

Identified asymptotically flat static spherically symmetric solutions

Abstract

We generalize Birkhoff's Theorem in the following fashion. We find necessary and sufficient conditions for any spherically symmetric space-time to be static in terms of the eigenvalues of the stress-energy tensor. In particular, we generalize the Tolman-Oppenheimer-Volkoff equation and prove that Birkhoff's theorem holds under the weaker hypothesis of no pressure (with respect to an appropriate frame.) We provide equations that show how the coefficients of the metric relate to the eigenvalues of the stress-energy tensor. These involve integrals that are simple functions of those eigenvalues. We also determine among all static spherically symmetric space-times those that are asymptotically flat. A few examples are presented taking advantage of the results. The calculations are done by viewing the space-times as warped products and the computations are done using Cartan's moving frames approach.