Global structure of Choptuik's critical solution in scalar field collapse

TL;DR

This paper analyzes the global spacetime structure of Choptuik's critical solution in scalar field collapse, revealing its self-similarity, regularity properties, and the nature of its Cauchy horizon and continuations.

Contribution

It provides a detailed global analysis of the critical solution, including its regularity, self-similarity, and the structure of continuations beyond the Cauchy horizon.

Findings

The solution is discretely self-similar and spherically symmetric.

The curvature is finite and continuous at the past lightcone of the singularity.

Beyond the Cauchy horizon, the solution depends on free null data, with a unique regular continuation.

Abstract

At the threshold of black hole formation in the gravitational collapse of a scalar field a naked singularity is formed through a universal critical solution that is discretely self-similar. We study the global spacetime structure of this solution. It is spherically symmetric, discretely self-similar, regular at the center to the past of the singularity, and regular at the past lightcone of the singularity. At the future lightcone of the singularity, which is also a Cauchy horizon, the curvature is finite and continuous but not differentiable. To the future of the Cauchy horizon the solution is not unique, but depends on a free function (the null data coming out of the naked singularity). There is a unique continuation with a regular center (which is self-similar). All other self-similar continuations have a central timelike singularity with negative mass.