A simple proof of Birkhoff's theorem for cosmological constant

TL;DR

This paper presents a straightforward proof of Birkhoff's theorem in the presence of a cosmological constant, clarifying its local nature and implications for spacetime extensions, especially in de Sitter contexts.

Contribution

It offers a simplified, unified proof of Birkhoff's theorem with a cosmological constant and clarifies misconceptions about staticity in these spacetimes.

Findings

Maximal extensions of extremal Schwarzschild-de Sitter spacetimes lack static regions.

Birkhoff's theorem is a local, not global, uniqueness theorem.

Locally spherically symmetric solutions have an additional local Killing vector.

Abstract

We provide a simple, unified proof of Birkhoff's theorem for the vacuum and cosmological constant case, emphasizing its local nature. We discuss its implications for the maximal analytic extensions of Schwarzschild, Schwarzschild(-anti)-de Sitter and Nariai spacetimes. In particular, we note that the maximal analytic extensions of extremal and over-extremal Schwarzschild-de Sitter spacetimes exhibit no static region. Hence the common belief that Birkhoff's theorem implies staticity is false for the case of positive cosmological constant. Instead, the correct point of view is that generalized Birkhoff's theorems are local uniqueness theorems whose corollary is that locally spherically symmetric solutions of Einstein's equations exhibit an additional local killing vector field.