Black hole interior in Painleve-Gullstrand coordinates

TL;DR

This paper presents a new nonstatic Schwarzschild black hole interior solution in Painleve-Gullstrand coordinates, deriving exact geodesic equations by exploiting the cyclic nature of the spatial coordinate.

Contribution

It introduces a novel interior solution in Painleve-Gullstrand coordinates and derives exact geodesic equations using coordinate transformation techniques.

Findings

Exact timelike and null geodesic equations obtained

Coordinate transformation preserves the timelike coordinate

Spatial coordinate becomes cyclic, simplifying calculations

Abstract

A nonstatic Schwarzschild black hole interior solution in Painleve-Gullstrand coordinates is proposed in this paper, by means of a coordinate transformation that changes the spatial coordinate but the timelike one is preserved. The timelike and null geodesic equations are obtained exactly, taking advantage that the spatial z-coordinate is cyclic.