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.
Highlights
Any low dimensional Riemannian manifold can be locally isometrically embedded in a higher dimensional flat one [1,2,3]
Following the Brynjolfsson and Thorlacius (BT)’s approach, we have extended the methods to various interesting curved spacetimes [30,31,32,33,34,35] to investigate local temperatures of corresponding spacetimes and their equivalence to Hawking ones
We have globally embedded the (3 + 1)-dimensional Schwarzschild and RN black holes in the PG spacetimes into (5 + 1)- and (5 + 2)-dimensional flat ones, respectively, which were made by the introduction of the PG time
Summary
Any low dimensional Riemannian manifold can be locally isometrically embedded in a higher dimensional flat one [1,2,3]. Following the BT’s approach, we have extended the methods to various interesting curved spacetimes [30,31,32,33,34,35] to investigate local temperatures of corresponding spacetimes and their equivalence to Hawking ones. In this classification scheme, freely falling local temperature of the BT’s method is the one seen by an observer in a drip frame. By following BT’s method and full geodesic equations satisfying various boundary conditions, we will find freely falling temperatures seen by observers in the PG-embedded Schwarzschild and RN black holes, which can be extended smoothly through the future event horizon.
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