The Schwarzschild-de Sitter solution in five-dimensional general relativity briefly revisited

TL;DR

This paper revisits five-dimensional Schwarzschild-de Sitter solutions in general relativity, showing they can embed four-dimensional solutions and behave well as the cosmological constant approaches zero, challenging previous assumptions about their rigidity.

Contribution

It introduces a class of five-dimensional solutions embedding four-dimensional Schwarzschild-de Sitter space, demonstrating their well-behaved limits and independence from the cosmological constant.

Findings

Five-dimensional solutions can embed four-dimensional Schwarzschild-de Sitter space.

These solutions are well-behaved as the cosmological constant approaches zero.

The presence of a non-zero cosmological constant does not rigidly relate 4D and 5D spacetimes.

Abstract

We briefly revisit the Schwarzschild-de Sitter solution in the context of five-dimensional general relativity. We obtain a class of five-dimensional solutions of Einstein vacuum field equations into which the four-dimensional Schwarzschild-de Sitter space can be locally and isometrically embedded. We show that this class of solutions is well-behaved in the limit of lambda approaching zero. Applying the same procedure to the de Sitter cosmological model in five dimensions we obtain a class of embedding spaces which are similarly well-behaved in this limit. These examples demonstrate that the presence of a non-zero cosmological constant does not in general impose a rigid relation between the (3+1) and (4+1)-dimensional spacetimes, with degenerate limiting behaviour.