Stability criterion for self-similar solutions with a scalar field and those with a stiff fluid in general relativity

TL;DR

This paper derives a stability criterion for self-similar solutions in general relativity with scalar fields and stiff fluids, revealing that many such solutions are unstable against kink mode perturbations, impacting models of gravitational collapse.

Contribution

It introduces a new stability criterion for self-similar solutions with scalar fields and stiff fluids, showing their instability against kink mode perturbations.

Findings

Many self-similar solutions are unstable against kink perturbations.

The Evans-Coleman stiff-fluid solution is unstable and not a critical solution.

Both flat Friedmann universes with scalar fields and stiff fluids are kink unstable.

Abstract

A stability criterion is derived in general relativity for self-similar solutions with a scalar field and those with a stiff fluid, which is a perfect fluid with the equation of state $P=\rho$. A wide class of self-similar solutions turn out to be unstable against kink mode perturbation. According to the criterion, the Evans-Coleman stiff-fluid solution is unstable and cannot be a critical solution for the spherical collapse of a stiff fluid if we allow sufficiently small discontinuity in the density gradient field in the initial data sets. The self-similar scalar-field solution, which was recently found numerically by Brady {\it et al.} (2002 {\it Class. Quantum. Grav.} {\bf 19} 6359), is also unstable. Both the flat Friedmann universe with a scalar field and that with a stiff fluid suffer from kink instability at the particle horizon scale.