Scaling of curvature in sub-critical gravitational collapse

TL;DR

This paper investigates how the maximum curvature behaves near the critical point in spherically symmetric scalar field collapse, revealing a scaling relation and periodic features similar to black hole mass scaling.

Contribution

It introduces a new scaling relation for maximum curvature in near-critical collapse, analogous to black hole mass scaling, with observed periodic wiggles.

Findings

Maximum curvature scales with distance from critical solution.

A periodic wiggle is observed in the scaling exponent.

Scaling behavior is similar to black hole mass scaling.

Abstract

We perform numerical simulations of the gravitational collapse of a spherically symmetric scalar field. For those data that just barely do not form black holes we find the maximum curvature at the position of the central observer. We find a scaling relation between this maximum curvature and distance from the critical solution. The scaling relation is analogous to that found by Choptuik for black hole mass for those data that do collapse to form black holes. We also find a periodic wiggle in the scaling exponent.