A new proof of Birkhoff's theorem

TL;DR

This paper presents a novel proof of Birkhoff's theorem using a reduction of Einstein's equations under spherical symmetry to a 2D gravity framework, avoiding coordinate transformations and generalizing to multiple dimensions.

Contribution

It introduces a new proof of Birkhoff's theorem based on 2D gravity equations, extending the theorem to arbitrary dimensions and signatures without coordinate dependence.

Findings

Provides a coordinate-free proof of Birkhoff's theorem.

Generalizes Birkhoff's theorem to higher dimensions and arbitrary signatures.

Connects spherical symmetry in Einstein equations to 2D gravity models.

Abstract

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.