Stability analysis of self-similar behaviors in perfect fluid gravitational collapse

TL;DR

This paper investigates the stability of self-similar solutions in perfect fluid gravitational collapse by analyzing non-self-similar perturbations, deriving a master wave equation, and identifying conditions for stability against certain perturbations.

Contribution

It introduces a unified wave equation for perturbations in self-similar backgrounds and applies normal mode analysis to assess stability, providing new analytical tools for gravitational collapse studies.

Findings

Derived a single wave equation for perturbations in self-similar backgrounds.

Identified conditions under which non-oscillatory unstable modes are absent.

Provided a framework for analyzing stability using normal mode analysis.

Abstract

Stability of self-similar solutions for gravitational collapse is an important problem to be investigated from the perspectives of their nature as an attractor, critical phenomena and instability of a naked singularity. In this paper we study spherically symmetric non-self-similar perturbations of matter and metrics in spherically symmetric self-similar backgrounds. The collapsing matter is assumed to be a perfect fluid with the equation of state $P=\alpha\rho$. We construct a single wave equation governing the perturbations, which makes their time evolution in arbitrary self-similar backgrounds analytically tractable. Further we propose an analytical application of this master wave equation to the stability problem by means of the normal mode analysis for the perturbations having the time dependence given by $\exp{(i\omega\log|t|)}$, and present some sufficient conditions for the absence of non-oscillatory unstable normal modes with purely imaginary $\omega$.