Critical gravitational collapse of a perfect fluid: nonspherical perturbations

TL;DR

This paper constructs and analyzes self-similar solutions for gravitational collapse of perfect fluids, revealing the stability properties and critical exponents for black hole formation across different equations of state, including nonspherical perturbations.

Contribution

It numerically constructs CSS solutions for perfect fluid collapse and investigates their linear perturbations, including analytical treatment of axial modes, providing new insights into critical phenomena and stability.

Findings

CSS solutions have one spherical growing mode for 1/9<kappa<0.49.

No nonspherical growing modes in this range, indicating stability against nonspherical perturbations.

Critical exponent for black hole angular momentum is derived as a function of kappa.

Abstract

Continuously self-similar (CSS) solutions for the gravitational collapse of a spherically symmetric perfect fluid, with the equation of state p=kappa rho, with 0<kappa<1 a constant, are constructed numerically and their linear perturbations, both spherical and nonspherical, are investigated. The l=1 axial perturbations admit an analytical treatment. All others are studied numerically. For intermediate equations of state, with 1/9<kappa<0.49, the CSS solution has one spherical growing mode, but no nonspherical growing modes. That suggests that it is a critical solution even in (slightly) nonspherical collapse. For this range of kappa we predict the critical exponent for the black hole angular momentum to be 5(1+3kappa)/3(1+kappa) times the critical exponent for the black hole mass. For kappa=1/3 this gives an angular momentum critical exponent of mu=0.898, correcting a previous result. For stiff equations of state, 0.49<kappa<1, the CSS solution has one spherical and several nonspherical growing modes. For soft equations of state, 0<kappa<1/9, the CSS solution has 1+3 growing modes: a spherical one, and an l=1 axial mode (with m=-1,0,1).