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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 haveoff-diagonal components in metrics, into (5+1)- and (5+2)-dimensional flatones, respectively. As a result, we have shown the equivalence of the GEMSembeddings of the spacetimes with the diagonal and off-diagonal terms inmetrics. Moreover, with the aid of their geodesic equations satisfying variousboundary conditions in the flat embedded spacetimes, we directly obtain freelyfalling temperatures. We also show that freely falling temperatures in the PGspacetimes are well-defined beyond the event horizons, while they areequivalent to the Hawking temperatures, which are obtained in the originalcurved ones in the ranges between the horizon and the infinity. These will behelpful to study GEMS embeddings of more realistic Kerr, or rotating BTZ blackholes.

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