Critical gravitational collapse of a perfect fluid: Nonspherical perturbations

Continuously self-similar (CSS) solutions for the gravitational collapse of a spherically symmetric perfect fluid, with the equation of state $p=\kappa \rho ,$ with $0<\mathrm{}\kappa <1\mathrm{}$ a constant, are constructed numerically and their linear perturbations, both spherical and nonspherical, are investigated. The $l=1\mathrm{}$ axial perturbations admit an analytical treatment. All others are studied numerically. For intermediate equations of state, with $1/9<\mathrm{}\kappa \lesssim 0.49,$ 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 $5(1+3\kappa )/3(1+\kappa )$ times the critical exponent for the black hole mass. For $\kappa =1/3\mathrm{}$ this gives an angular momentum critical exponent of $\mu \simeq 0.898,$ correcting a previous result. For stiff equations of state, $0.49\lesssim \kappa <1,$ the CSS solution has one spherical and several nonspherical growing modes. For soft equations of state, $0<\mathrm{}\kappa <1/9,$ the CSS solution has $1+3$ growing modes: a spherical one, and an $l=1\mathrm{}$ axial mode (with $m=-1,0,1).$