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

Abstract

A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state $P=kρ$ 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.