Aspherical deformations of the Choptuik spacetime

We perform dynamical and nonlinear numerical simulations to study critical phenomena in the gravitational collapse of massless scalar fields in the absence of spherical symmetry. We evolve axisymmetric sets of initial data and examine the effects of deviation from spherical symmetry. For small deviations we find values for the critical exponent and echoing period of the discretely self-similar critical solution that agree well with established values; moreover we find that such small deformations behave like damped oscillations whose damping coefficient and oscillation frequencies are consistent with those predicted in the linear perturbation calculations of Mart\'n-Garc\'a and Gundlach. However, we also find that the critical exponent and echoing period appear to decrease with increasing departure from sphericity and that, for sufficiently large departures from spherical symmetry, the deviations become unstable and grow, confirming earlier results by Choptuik et al. We find some evidence that these growing modes lead to a bifurcation, similar to those reported by Choptuik et al. , with two centers of collapse forming on the symmetry axis above and below the origin. These findings suggest that nonlinear perturbations of the critical solution lead to changes in the effective values of the critical exponent, echoing period and damping coefficient and may even change the sign of the latter, so that perturbations that are stable in the linear regime can become unstable in the nonlinear regime.