GEMS embeddings of Schwarzschild and RN black holes in Painlev\'e-Gullstrand spacetimes

TL;DR

This paper constructs higher-dimensional flat spacetime embeddings of Schwarzschild and Reissner-Nordström black holes in Painlevé-Gullstrand coordinates, enabling analysis of freely falling temperatures beyond horizons, which match Hawking temperatures between the horizon and infinity.

Contribution

It demonstrates GEMS embeddings for black holes with off-diagonal metric components and derives freely falling temperatures, extending previous embeddings to more realistic black hole models.

Findings

Successfully embedded Schwarzschild and RN black holes in higher-dimensional flat spacetimes.

Derived freely falling temperatures that are well-defined beyond horizons.

Showed equivalence of freely falling and Hawking temperatures in specified ranges.

Abstract

Making use of the higher dimensional global embedding Minkowski spacetime (GEMS), we embed (3+1)-dimensional Schwarzschild and Reissner-Nordstr\"om (RN) black holes written by the Painlev\'e-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.