Self-Similar Collapse in Painleve-Gullstrand Coordinates

TL;DR

This paper presents a family of exact self-similar solutions in General Relativity describing gravitational collapse with diverse outcomes, including bounces, re-collapse, and horizon formation, highlighting energy condition violations and wormhole-like regions.

Contribution

It introduces new self-similar solutions in Painleve-Gullstrand coordinates, analyzing collapse scenarios, energy conditions, and null surface formations, including potential wormhole regions.

Findings

Multiple collapse outcomes depending on parameters

Null energy condition violations during bounces

Formation of multiple apparent horizons

Abstract

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space : (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve-Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.