Stability analysis of a class of anisotropic spacetimes

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. Last, 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.