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. The applications are the following: the expanding flat Friedmann solution for 1/3 ≤ k < 1 and the collapsing one for 0 < k ≤ 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 ≲ 0.036, while it is unstable for 0.036 ≲ k; the Evans–Coleman (critical) solution is stable for this mode for 0 < k ≲ 0.89, while it is unstable for 0.89 ≲ k. The last application suggests that the Evans–Coleman solution for 0.89 ≲ k is not critical because it has at least two unstable modes.