Perturbations of Spatially Closed Bianchi III Spacetimes

TL;DR

This paper analyzes linear perturbations of spatially closed Bianchi III vacuum spacetimes with nontrivial topology, developing mode functions and gauge-invariant wave equations to study stability and perturbation behavior.

Contribution

It introduces a method to decouple and analyze perturbations in topologically nontrivial Bianchi III spacetimes, extending previous spherical symmetry results.

Findings

Decoupling of perturbation equations into gauge-invariant wave equations

Analysis of stability properties of Bianchi III spacetimes

Development of mode functions for nontrivial topology

Abstract

Motivated by the recent interest in dynamical properties of topologically nontrivial spacetimes, we study linear perturbations of spatially closed Bianchi III vacuum spacetimes, whose spatial topology is the direct product of a higher genus surface and the circle. We first develop necessary mode functions, vectors, and tensors, and then perform separations of (perturbation) variables. The perturbation equations decouple in a way that is similar to but a generalization of those of the Regge--Wheeler spherically symmetric case. We further achieve a decoupling of each set of perturbation equations into gauge-dependent and independent parts, by which we obtain wave equations for the gauge-invariant variables. We then discuss choices of gauge and stability properties. Details of the compactification of Bianchi III manifolds and spacetimes are presented in an appendix. In the other appendices we study scalar field and electromagnetic equations on the same background to compare asymptotic properties.