Spherical Gravitating Systems of Arbitrary Dimension

TL;DR

This paper extends solutions of Einstein's equations to arbitrary dimensions, exploring spherically symmetric systems, their known analogues, and conditions for Birkhoff's theorem validity in higher-dimensional space-times.

Contribution

It generalizes the classical solutions and Birkhoff's theorem to arbitrary dimensions, providing a framework for higher-dimensional gravitational systems.

Findings

Extended Synge's solutions to any number of spatial dimensions.

Presented higher-dimensional analogues of known four-dimensional solutions.

Analyzed conditions under which Birkhoff's theorem holds or fails in arbitrary dimensions.

Abstract

We study spherically symmetric solutions to the Einstein field equations under the assumption that the space-time may possess an arbitrary number of spatial dimensions. The general solution of Synge is extended to describe systems of any dimension. Arbitrary dimension analogues of known four dimensional solutions are also presented, derived using the above scheme. Finally, we discuss the requirements for the existence of Birkhoff's theorems in space-times of arbitrary dimension with or without matter fields present. Cases are discussed where the assumptions of the theorem are considerably weakened yet the theorem still holds. We also discuss where the weakening of certain conditions may cause the theorem to fail.