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We use the global embedding Minkowski space (GEMS) geometries of a(3+1)-dimensional curved Reissner-Nordstr\"om(RN)-AdS black hole spacetime intoa (5+2)-dimensional flat spacetime to define a proper local temperature, whichremains finite at the event horizon, for freely falling observers outside astatic black hole. Our extended results include the known limiting cases of theRN, Schwarzschild--AdS, and Schwarzschild black holes.

Local free-fall temperature of a RN-AdS black hole