Painlevé–Gullstrand coordinates for Schwarzschild–de Sitter spacetime

Abstract

The Painlevé–Gullstrand coordinates are extended to describe the black hole in the cosmological environment: the Schwarzschild–de Sitter black hole, which has two horizons. The extension is made using the Arnowitt–Deser–Misner formalism. In this extension, which describes the metric in the whole range of radial coordinates 0<r<∞, there is the point r=r0 at which the shift function (velocity) changes sign. At this point the observer is at rest, while the observers at r<r0 are free falling to the black hole and the observers at r>r0 are free falling towards the cosmological horizon. The existence of the stationary observer allows to determine the temperature of Hawking radiation, which is in agreement with Bousso and Hawking (1996). It is the red-shifted modification of the conventional Hawking temperature determined by the gravity at the horizon. We also consider the Painlevé–Gullstrand coordinates and their extension for such configurations as Schwarzschild–de Sitter white hole, where the sign of the shift function is everywhere positive; the black hole in the environment of the contracting de Sitter spacetime, where the sign of the shift function is everywhere negative; and the white hole in the contracting de Sitter spacetime, where the shift velocity changes sign at r=r0.