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A simple proof of Birkhoff's theorem for cosmological constant

### 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 overextremal 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.

© 2010 American Institute of Physics

Received 15 July 2010
Accepted 27 September 2010
Published online 24 November 2010

Acknowledgments:
The work was supported by NSERC. In addition, the authors would like to thank the Perimeter Institute for its hospitality.