Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

TL;DR

This paper explores whether higher-dimensional spatially homogeneous vacuum solutions of Einstein's equations are future geodesically complete, extending known four-dimensional results using Kaluza-Klein reduction and a general completeness criterion.

Contribution

It demonstrates that a broad class of higher-dimensional models are future geodesically complete, generalizing four-dimensional findings through new analytical techniques.

Findings

Higher-dimensional models are future geodesically complete under certain conditions.

Kaluza-Klein reduction links higher-dimensional solutions to four-dimensional Einstein-scalar field solutions.

A new criterion for geodesic completeness applies to various spatially homogeneous models.

Abstract

It is known that all spatially homogeneous solutions of the vacuum Einstein equations in four dimensions which exist for an infinite proper time towards the future are future geodesically complete. This paper investigates whether the analogous statement holds in higher dimensions. A positive answer to this question is obtained for a large class of models which can be studied with the help of Kaluza-Klein reduction to solutions of the Einstein-scalar field equations in four dimensions. The proof of this result makes use of a criterion for geodesic completeness which is applicable to more general spatially homogeneous models.