A new proof of Birkhoff's theorem

Details A new proof of Birkhoff's theorem A new proof of Birkhoff's theorem

1997-09-26

arxiv.org/abs/gr-qc/9709071v1

Grav.Cosmol. 3 (1997) 185-190

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages, LaTeX, no figures, Grav. and Cosm. in print

Scientific paper

Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation reduces to an equation conformally related to the field equation for 2-dimensional gravity following from the Lagrangian L = R^(1/3). Solutions for 2-dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoff's theorem; contrary to other proofs, no coordinates must be introduced. The SO(m)-spherically symmetric solutions of the (m+1)-dimensional Einstein equation can be found by considering L = R^(1/m) in two dimensions. This yields several generalizations of Birkhoff's theorem in an arbitrary number of dimensions, and to an arbitrary signature of the metric.

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