A simple proof of Birkhoff's theorem for cosmological constant

Details A simple proof of Birkhoff's theorem for cosmological constant A simple proof of Birkhoff's theorem for cosmological constant

2009-08-27

arxiv.org/abs/0908.4110v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

10 pages, 5 figures References added; typo in eqn. 12 fixed

Scientific paper

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.

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