Odd Parity Perturbations of the Self-Similar LTB Spacetime

TL;DR

This paper analyzes odd-parity perturbations of self-similar LTB spacetimes with naked singularities, showing these perturbations remain finite at the Cauchy horizon, but does not establish overall stability due to unexamined even-parity modes.

Contribution

It demonstrates the finiteness of odd-parity perturbations at the Cauchy horizon in self-similar LTB spacetimes with naked singularities, providing insights into their linear stability.

Findings

Perturbations remain finite on the Cauchy horizon.

Finiteness holds for general initial data.

Results do not confirm overall stability of the spacetime.

Abstract

We consider the behaviour of odd-parity perturbations of those self-similar Lema\^{i}tre-Tolman-Bondi spacetimes which admit a naked singularity. We find that a perturbation which evolves from initially regular data remains finite on the Cauchy horizon. Finiteness is demonstrated by considering the behaviour of suitable energy norms of the perturbation (and pointwise values of these quantities) on natural spacelike hypersurfaces. This result holds for a general choice of initial data and initial data surface. Finally, we examine the perturbed Weyl scalars in order to provide a physical interpretation of our results. Taken on its own, this result does not support cosmic censorship; however a full perturbation of this spacetime would include even parity perturbations, so we cannot conclude that this spacetime is stable to all linear perturbations.