Even perturbations of self-similar Vaidya space-time

TL;DR

This paper analyzes the stability of self-similar Vaidya space-time with naked singularities under even parity perturbations, finding that the Cauchy horizon remains finite while the second future similarity horizon is unstable.

Contribution

It provides a detailed mode analysis of perturbations in self-similar Vaidya space-time, revealing stability at the Cauchy horizon and instability at the second future horizon.

Findings

Perturbations remain finite at the Cauchy horizon.

Divergence occurs at the second future similarity horizon.

Full perturbation may remain finite after mode resummation.

Abstract

We study even parity metric and matter perturbations of all angular modes in self-similar Vaidya space-time. We focus on the case where the background contains a naked singularity. Initial conditions are imposed describing a finite perturbation emerging from the portion of flat space-time preceding the matter-filled region of space-time. The most general perturbation satisfying the initial conditions is allowed impinge upon the Cauchy horizon (CH), whereat the perturbation remains finite: there is no ``blue-sheet'' instability. However when the perturbation evolves through the CH and onto the second future similarity horizon of the naked singularity, divergence necessarily occurs: this surface is found to be unstable. The analysis is based on the study of individual modes following a Mellin transform of the perturbation. We present an argument that the full perturbation remains finite after resummation of the (possibly infinite number of) modes.