Classifying self-gravitating radiations

TL;DR

This paper investigates static self-gravitating radiation spheres using numerical and analytical methods, introducing an approximate horizon concept and analyzing solutions near black hole formation.

Contribution

It presents a novel analysis of self-gravitating radiation solutions, including an analytic solution at the black hole formation threshold and a new parameterization method.

Findings

Identification of a unique approximate horizon in solutions

Analytic formulas for naked singularity mass

Solution parameterization via geometric and boundary data

Abstract

We study a static system of self-gravitating radiations confined in a sphere by using numerical and analytical calculations. Due to the scaling symmetry of radiations, most of main properties of a solution can be represented as a segment of a solution curve on a plane of two-dimensional scale invariant variables. We define an `approximate horizon' (AH) from the analogy with an apparent horizon. Any solution curve contains a unique point which corresponds to the AH. A given solution is uniquely labelled by three parameters representing the solution curve, the size of the AH, and the sphere size, which are an alternative of the data at the outer boundary. Various geometrical properties including the existence of an AH and the behaviors around the center can be identified from the parameters. We additionally present an analytic solution of the radiations on the verge of forming a blackhole. Analytic formulae for the central mass of the naked singularity are given.