Spherically Symmetric Black Hole Formation in Painlev\'e-Gullstrand Coordinates

TL;DR

This paper numerically investigates spherically symmetric black hole formation using Painlevé-Gullstrand coordinates, confirming critical phenomena and revealing unique oscillatory features in the scaling relations.

Contribution

It demonstrates the effectiveness of PG coordinates for simulating black hole collapse and uncovers novel non-sinusoidal oscillations in supercritical mass scaling.

Findings

Confirmed Choptuik scaling with universal critical exponent and echoing period.

Generated detailed spacetime maps of collapse and horizon evolution.

Discovered non-standard oscillatory behavior in supercritical mass scaling.

Abstract

We perform a numerical study of black hole formation from the spherically symmetric collapse of a massless scalar field. The calculations are done in Painlev\'e-Gullstrand (PG) coordinates that extend across apparent horizons and allow the numerical evolution to proceed until the onset of singularity formation. We generate spacetime maps of the collapse and illustrate the evolution of apparent horizons and trapping surfaces for various initial data. We also study the critical behaviour and find the expected Choptuik scaling with universal values for the critical exponent and echoing period consistent with the accepted values of $\gamma=0.374$ and $\Delta = 3.44$, respectively. The subcritical curvature scaling exhibits the expected oscillatory behaviour but the form of the periodic oscillations in the supercritical mass scaling relation, while universal with respect to initial PG data, is non-standard: it is non-sinusoidal with large amplitude cusps.