When Painlev\'e-Gullstrand coordinates fail

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).