Stability criterion for self-similar solutions with perfect fluids in general relativity

TL;DR

This paper derives a stability criterion for self-similar perfect fluid solutions in general relativity, revealing which solutions are stable or unstable against kink modes based on sonic point classification, with implications for collapse and expansion scenarios.

Contribution

It introduces a direct stability criterion linked to sonic point classification, providing new insights into the stability of various self-similar solutions in relativistic fluid dynamics.

Findings

Most self-similar solutions are unstable against kink modes.

Stability depends on the type of sonic point and solution parameters.

The Evans-Coleman solution is not critical for certain parameter ranges.

Abstract

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state $P=k\rho$ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. Applications are the following: the expanding flat Friedmann solution for $1/3 \le k < 1$ and the collapsing one for $0< k \le 1/3$ are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson-Penston (attractor) solution is stable for this mode for $0<k\alt 0.036$, while it is unstable for $0.036\alt k $; the Evans-Coleman (critical) solution is stable for this mode for $0<k\alt 0.89$, while it is unstable for $0.89\alt k$. The last application suggests that the Evans-Coleman solution for $0.89\alt k $ is {\em not critical} because it has at least two unstable modes.