Future asymptotic expansions of Bianchi VIII vacuum metrics

TL;DR

This paper analyzes the long-term behavior of Bianchi VIII vacuum solutions to Einstein's equations, providing asymptotic expansions of the metrics and relating these to the topology of spatial hypersurfaces.

Contribution

It offers a detailed asymptotic expansion of Bianchi VIII vacuum metrics and connects these expansions to the topology of the spatial hypersurfaces, extending previous analyses.

Findings

In NUT Bianchi VIII spacetimes, the circle fiber length converges to a positive constant.

In general Bianchi VIII solutions, the fiber length tends to infinity at a specific rate.

The paper provides an explicit asymptotic expansion for the metric in the future direction.

Abstract

Bianchi VIII vacuum solutions to Einstein's equations are causally geodesically complete to the future, given an appropriate time orientation, and the objective of this article is to analyze the asymptotic behaviour of solutions in this time direction. For the Bianchi class A spacetimes, there is a formulation of the field equations that was presented in an article by Wainwright and Hsu, and in a previous article we analyzed the asymptotic behaviour of solutions in these variables. One objective of this paper is to give an asymptotic expansion for the metric. Furthermore, we relate this expansion to the topology of the compactified spatial hypersurfaces of homogeneity. The compactified spatial hypersurfaces have the topology of Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII spacetimes, the length of a circle fibre converges to a positive constant but that in the case of general Bianchi VIII solutions, the length tends to infinity at a rate we determine.