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

We embed (3+1)-dimensional Schwarzschild and RN black holes in the Painlev\'e-Gullstrand (PG) spacetimes into (5+1)- and (5+2)-dimensional flat spacetimes, respectively. Making use of embedding coordinates with the aid of their geodesic equations, we directly obtain freely falling temperatures in drip, rain, and hail frames, which are the Unruh temperatures in higher-dimensional spacetimes. We also show that using the GEMS embedding method the freely falling temperatures in the PG spacetimes are equivalent to the Hawking temperatures in the original curved ones.Comment: 15 pages, 2 figure

GUP corrected entropy of the Schwarzschild black hole in massive gravity

Considering corrections to all orders in the Planck length on the density of quantum states from a generalized uncertainty principle (GUP), we calculate the statistical entropy of a scalar field on the background of the Schwarzschild black hole in massive gravity. As a result, we newly obtain the generalized Bekenstein-Hawking entropy depending on a gravitational mass without any artificial cutoff.Comment: 11 pages, 3 figure