Abstract
Painlevé–Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner–Nordström. 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, which are 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).
Highlights
Black hole thermodynamics is an important area of modern theoretical physics linking quantum processes and classical gravity
The PG coordinates originally introduced for the Schwarzschild geometry [3,4] have proved very useful in the study of the thermodynamics of black holes and other horizons, especially in the context of the tunneling formalism of Parikh and Wilczek [2,32]
It is rather unfortunate that this coordinate chart cannot be introduced for the most important space of string theories, anti-de Sitter space associated with a negative cosmological constant, and for the Schwarzschild-anti de Sitter geometry obtained by embedding the Schwarzschild black hole into it
Summary
Black hole thermodynamics is an important area of modern theoretical physics linking quantum processes and classical gravity. Other very useful coordinates in the literature (e.g., those of [24] for the Reissner–Nordström spacetime) recast a spherically symmetric line element in a form close to the Painlevé– Gullstrand one, but do not have flat spatial sections. These situations include the Schwarzschild-anti de Sitter family of black holes. 4 that, whenever this happens, PG coordinates cannot be introduced and we highlight the physical reason: since gravity becomes repulsive, a massive test particle with zero initial velocity cannot overcome this repulsion and it cannot even begin to travel radially along a radial timelike geodesic. We refer to the case of the Schwarschild naked singularity (which is illuminating as usual) and revert to anti-de Sitter space
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