The Birkhoff Theorem in Multidimensional Gravity

TL;DR

This paper explores the conditions under which the extended Birkhoff theorem holds in multidimensional gravity theories with multiple internal spaces, providing examples and discussing potential generalizations.

Contribution

It formulates the validity conditions for the extended Birkhoff theorem in multidimensional gravity without restrictions on dimensionality or signature.

Findings

Identifies specific matter sources and geometries where the theorem applies

Provides examples illustrating the theorem's validity

Discusses possible generalizations of the theorem

Abstract

The validity conditions for the extended Birkhoff theorem in multidimensional gravity with $n$ internal spaces are formulated, with no restriction on space-time dimensionality and signature. Examples of matter sources and geometries for which the theorem is valid are given. Further generalization of the theorem is discussed.