Numerical study of black-hole formation in Painlevé-Gullstrand coordinates

We study numerically black-hole formation from a collapsing massless scalar field. The use of Painlevé-Gullstrand coordinates allows the evolution to proceed until singularity formation. We generate spacetime maps of the collapse, illustrating the evolution of apparent horizons for various initial data. A study of the Choptuik scaling reveals the expected universal values for the critical exponent and echoing period. The periodic oscillations in the supercritical horizon scaling relation, while universal with respect to initial Painlevé-Gullstrand data, show unexpected structure with large amplitude cusps.

Figure 1

Spacetime diagram for Gaussian initial data with solid (black) lines mapping out the null geodesics and a dashed (red) line indicating the trapping horizon. The horizontal (green) line indicates the PG time slice within which an apparent horizon first appears. A different choice of slicings would find the initial apparent at a different location.

Figure 2

A plot of the scalar field at the origin versus $\tau$, illustrating the scale echoing in the critical solution with $\Delta r/r_0={10}^{-5}$ near the origin.

Figure 3

The maximum curvature scaling relation for a variation in the parameter $A$ in Gaussian initial data with $\Delta r/r_0={10}^{-5}$ near the origin. The straight line is a linear fit that corresponds to the given equation. The flat bottoms of the periodic component were also observed by Garfinkle and Duncan [5].

Figure 4

The mass scaling relation for a variation in the parameter $A$ in Gaussian initial data with $\Delta r/r_0={10}^{-5}$ near the origin.