  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.
