Analytic approximations to the spacetime of a critical gravitational collapse

TL;DR

This paper develops analytic approximations for the spacetime structure during critical gravitational collapse, dividing it into regions and matching solutions to replicate Choptuik's discretely self-similar behavior.

Contribution

It introduces a novel analytic framework that approximates the critical spacetime by dividing it into regions and matching solutions, providing insight into the collapse dynamics.

Findings

Approximate solutions for quiescent and oscillatory regions.

Identification of a transition edge separating different spacetime behaviors.

Qualitative match with Choptuik's critical solution oscillations.

Abstract

We present analytic expressions that approximate the behavior of the spacetime of a collapsing spherically symmetric scalar field in the critical regime first discovered by Choptuik. We find that the critical region of spacetime can usefully be divided into a ``quiescent'' region and an ``oscillatory'' region, and a moving ``transition edge'' that separates the two regions. We find that in each region the critical solution can be well approximated by a flat spacetime scalar field solution. A qualitative nonlinear matching of the solutions across the edge yields the right order of magnitude for the oscillations of the discretely self-similar critical solution found by Choptuik.