GEMS Embeddings of Schwarzschild and RN Black Holes in Painlevé-Gullstrand Spacetimes

Abstract

Making use of the higher dimensional global embedding Minkowski spacetime (GEMS), we embed (3 + 1)-dimensional Schwarzschild and Reissner-Nordström (RN) black holes written by the Painlevé-Gullstrand (PG) spacetimes, which have off-diagonal components in metrics, into (5 + 1)- and (5 + 2)-dimensional flat ones, respectively. As a result, we have shown the equivalence of the GEMS embeddings of the spacetimes with the diagonal and off-diagonal terms in metrics. Moreover, with the aid of their geodesic equations satisfying various boundary conditions in the flat embedded spacetimes, we directly obtain freely falling temperatures. We also show that freely falling temperatures in the PG spacetimes are well-defined beyond the event horizons, while they are equivalent to the Hawking temperatures, which are obtained in the original curved ones in the ranges between the horizon and the infinity. These will be helpful to study GEMS embeddings of more realistic Kerr, or rotating BTZ black holes.

GEMS Embeddings of Hayward Regular Black Holes in Massless and Massive Gravities

“…Later, Brynjolfsson and Thorlacius [139] introduced a local temperature measured by a freely falling observer in the GEMS method. We have also studied various interesting curved spacetimes [140,141] to investigate local temperatures and their equivalences to Hawking ones.…”