Gravitational collapse in Painlev\'e-Gullstrand coordinates

TL;DR

This paper presents an exact solution for spherical gravitational collapse using a generalized Painlevé-Gullstrand coordinate system, enabling analysis inside and outside the event horizon within a single coordinate patch.

Contribution

It introduces a novel coordinate system for modeling collapse that covers both interior and exterior regions in a unified framework.

Findings

Derived exact solutions for collapse from infinity and finite radius.

Coordinates allow analysis inside the event horizon.

Unified description of collapse dynamics in a single patch.

Abstract

We construct an exact solution for the spherical gravitational collapse in a single coordinate patch. To describe the dynamics of collapse, we use a generalized form of the Painlev\'e-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. We show the both solutions corresponding to the gravitational collapse from infinity and from a finite radius.