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

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 = αρ. 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 (iω log |t|), and present some sufficient conditions for the absence of nonoscillatory unstable normal modes with purely imaginary ω.

Breakdown of self-similar evolution in homogeneous perfect fluid collapse

“…In this paper, we have studied the stability problem for the self-similar behavior in homogeneous collapse of a perfect fluid with the equation of state P = αρ, using the perturbation theory developed in [11]. We have derived the single ordinary differential equation (2.17) governing spherically symmetric non-self-similar perturbations with the time dependence exp {iω log (−t)} in the flat Friedmann background and set up the eigenvalue problem to determine the value of the spectral parameter ω.…”

“…(3.7) is positive, namely, there exists one unstable normal mode, at least for sufficiently small values of α. The proof concerning the absence of unstable normal modes shown in [11] is not applicable to the normal mode φ 3 with the small growth rate Im(ω (0)…”

“…Although the master equation (2.17) for the perturbations in the flat Friedmann background is given by a simpler form (if compared with the form for any other self-similar backgrounds), it is still a difficult task to solve the eigenvalue problem, and in [11] the absence of the unstable normal modes was clearly proven only in a limited range of ω. Such a difficulty may be overcome if we consider the low pressure limit α → 0, keeping the variable x finite in the range 0 ≤ x ≤ 1 and expanding the solution φ(x, ω, α) analytic at x = 0 as follows,…”

“…In this section, we briefly illustrate our analytical scheme to treat spherically symmetric non-self-similar perturbations in homogeneous self-similar perfect fluid collapse described by the flat Friedmann solution (see [11] for the details). The line element considered throughout this paper is given by…”

“…Recently, we have developed an analytical scheme to treat the stability problem by constructing the single wave equation governing non-self-similar spherically symmetric perturbations [11], which is reduced to the ordinary differential equation if we assume the perturbations to have the time dependence given by exp (iω log |t|). In this paper, using this analytical scheme, we study the stability problem for the flat Friedmann solution in the low pressure limit, i.e., 0 < α ≪ 1.…”