Gauge invariant perturbations of self-similar Lema\^itre-Tolman-Bondi spacetime: even parity modes with $l\geq 2$

TL;DR

This paper analyzes gauge invariant linear perturbations of self-similar Lemaître-Tolman-Bondi spacetime with naked singularities, showing certain scalars remain finite at the Cauchy horizon, suggesting linear stability.

Contribution

It introduces a method to study gauge invariant perturbations of self-similar LTB spacetime and demonstrates the stability of the Cauchy horizon under these perturbations.

Findings

Certain scalars from perturbation modes remain finite at the Cauchy horizon.

The perturbation equations reduce to ODEs with singular points.

The analysis supports linear stability of the Cauchy horizon.

Abstract

In this paper we consider gauge invariant linear perturbations of the metric and matter tensors describing the self-similar Lema\^itre-Tolman-Bondi (timelike dust) spacetime containing a naked singularity. We decompose the angular part of the perturbation in terms of spherical harmonics and perform a Mellin transform to reduce the perturbation equations to a set of ordinary differential equations with singular points. We fix initial data so the perturbation is finite on the axis and the past null cone of the singularity, and follow the perturbation modes up to the Cauchy horizon. There we argue that certain scalars formed from the modes of the perturbation remain finite, indicating linear stability of the Cauchy horizon.