Time (in)dependence in general relativity

TL;DR

This paper clarifies the conditions for time-independence in general relativity, primarily at the linearized level, and illustrates the theorem with Schwarzschild and Kerr solutions, also deriving the equivalence principle.

Contribution

It provides a detailed analysis of Birkhoff's theorem conditions and derives the equivalence principle within the context of linearized general relativity.

Findings

Conditions for Birkhoff's theorem clarified

Time-independent solutions exemplified by Schwarzschild and Kerr

Derivation of the equivalence principle included

Abstract

We clarify the conditions for Birkhoff's theorem, that is, time-independence in general relativity. We work primarily at the linearized level where guidance from electrodynamics is particularly useful. As a bonus, we also derive the equivalence principle. The basic time-independent solutions due to Schwarzschild and Kerr provide concrete illustrations of the theorem. Only familiarity with Maxwell's equations and tensor analysis is required.