Stability analysis of (a class of) anisotropic spacetimes

TL;DR

This paper investigates the stability of spherically symmetric anisotropic spacetimes under axial perturbations using gauge-invariant methods, revealing stability in both stellar and naked singularity cases, and extending to anisotropic cosmology.

Contribution

It provides a comprehensive stability analysis of anisotropic spacetimes, including stellar interiors, naked singularities, and anisotropic cosmologies, using gauge-invariant perturbation theory.

Findings

Stellar interior is stable under axial perturbations.

Naked singularity remains stable under axial perturbations.

Anisotropic cosmology perturbations involve anisotropy direction changes.

Abstract

We consider spherically symmetric spacetimes sourced by a fluid with pressure anisotropy in the radial direction. We use gauge-invariant perturbation theory to study the stability of this class of spacetimes under axial perturbations. We apply our results to three diverse examples. Two examples arise as endpoints of collapse of a ball of fluid --- one describes a well-behaved stellar interior and the other has a naked singularity. We prove the stability of the stellar interior both with respect to Dirichlet and quasinormal mode boundary conditions on the perturbation. Surprisingly, the naked singularity is also stable under axial perturbations. Lastly, we take the example of anisotropic cosmology to show that in this case, the relevant perturbations are those in which the direction of anisotropy is also perturbed.