When Painlev\'e-Gullstrand coordinates fail

TL;DR

This paper investigates the limitations of Painlevé-Gullstrand coordinates in various spacetimes, identifying specific conditions and regions where these coordinates fail, and explores the applicability of the Martel-Poisson family of charts.

Contribution

It reveals the failure of PG coordinates in anti-de Sitter space and inner Reissner-Nordström regions due to negative quasilocal mass, and discusses the use of Martel-Poisson charts in static cosmologies.

Findings

PG coordinates fail in anti-de Sitter space and Reissner-Nordström inner regions.

Failure linked to negative Misner-Sharp-Hernandez mass and repulsive gravity.

Martel-Poisson charts extend analysis to static cosmologies.

Abstract

Painlev\'e-Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner-Nordstr\"om. We predict this breakdown to occur in any region containing negative Misner-Sharp-Hernandez quasilocal mass because of repulsive gravity stopping the motion of PG observers in radial free fall with zero initial velocity. PG coordinates break down also in the static Einstein universe for completely different reasons. The more general Martel-Poisson family of charts, which normally has PG coordinates as a limit, is reported for static cosmologies (de Sitter, anti-de Sitter and the static Einstein universe).