Simulations of gravitational collapse in null coordinates. II. Critical collapse of an axisymmetric scalar field

We present the first numerical simulations in null coordinates of the collapse of nonspherical regular initial data to a black hole.We restrict to twist-free axisymmetry, and re-investigate the critical collapse of a non-spherical massless scalar field.We find that the Choptuik solution governing scalar field critical collapse in spherical symmetry persists when fine-tuning moderately non-spherical initial data to the threshold of black hole formation.The non-sphericity evolves as an almost-linear perturbation until the end of the self-similar phase, and becomes dominant only in the final collapse to a black hole.We compare with numerical results of Choptuik et al, Baumgarte, and Marouda et al, and conclude that they have been able to evolve somewhat more non-spherical solutions.Future work with larger deviations from spherical symmetry, and in particular vacuum collapse, will require a different choice of radial coordinate that allows the null generators to reconverge locally. CONTENTSI. Introduction 1 II.Setup 2 A. Metric and field equations 2 B. Gauge choice 3 C. Diagnostics 3 D. Initial data 4 E. Bisection to the threshold of black hole formation 4 III.Results in spherical symmetry 5 A. Fine structure of the mass and curvature scaling laws 5 B. Self-similarity of the critical solution 6 C. Comparison with gsB gauge 7 IV.Results for nonspherical initial data 8 A. The final collapse phase 8 B. Scaling and echoing 8 C. Self-similarity and evolution of the nonsphericity 10 D. Comparison with the evolutions of Choptuik et al, Baumgarte, and Marouda et al 13 E. Numerical problems at larger non-sphericity 14 V. Conclusions 15 Acknowledgments 16 A. Family-dependent parameters in the fine structure of the scaling laws 16 B. Convergence tests in the strong subcritical regime 17 C. Convergence tests with initial data that are regular at the origin 19