2009/08/27 by Kristin Schleich, Donald Witt, Donald M. Witt · 1 voice · 1 citation
Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #gr-qc
paper · pdf · doi:10.1063/1.3503447
arxiv published 2009/08/27 · arxiv updated 2009/10/27 · openalex publication_date 2010/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.