Nonspherical perturbations: Stability and angular momentum

3.7 Nonspherical perturbations: Stability and angular momentum

Critical collapse is really relevant for cosmic censorship only if it is not restricted to spherical symmetry. Martín-García and Gundlach [154

] have analysed all nonspherical perturbations of the scalar field critical solution by solving a linear eigenvalue problem with an ansatz of regularity at the centre and the SSH. They find that the only growing mode is the known spherical one, while all other spherical modes and all non-spherical modes decay. This strongly suggests that the critical solution is an attractor of codimension one not only in the space of spherically symmetric data but (modulo linearisation stability) of all data in a finite neighbourhood of spherical symmetry.

More recently, Choptuik and collaborators [54 ] have carried out axisymmetric time evolutions for the massless scalar field using adaptive mesh refinement. They find that in the limit of fine-tuning generic axisymmetric initial data the spherically symmetric critical solution is approached at first but then deviates from spherical symmetry and eventually develops two centres, each of which approaches the critical solution and bifurcates again in a universal way. This suggests that the critical solution has non-spherical growing perturbation modes, possibly a single l = 2 even parity mode (in axisymmetry, only m = 0 is allowed). There appears to be a conflict between the time evolution results [54 ] and the perturbative results [154 ], which needs to be resolved by more work (see Section 5.2).

Perturbing the scalar field around spherical symmetry, angular momentum comes in to second order in perturbation theory. All angular momentum perturbations were found to decay, and a critical exponent $μ ≃ 0.76$ for the angular momentum was derived for the massless scalar field in [87 ]. This prediction has not yet been tested in nonlinear collapse simulations.