Gravitational energy, gravitational pressure, and the thermodynamics of a charged black hole in teleparallel gravity

We investigate, in the case of a Reissner-Nordstrom black hole, the definitions of gravitational energy and gravitational pressure that naturally arise in the framework of the teleparallel equivalent of general relativity. In particular, we calculate the gravitational energy enclosed by the event horizon of the black hole, E, and the radial pressure over it, p. With these quantities, we then analyze the thermodynamic relation dE + pdV (as p turns out to be a density, and dV is actually given by dV = drd theta d phi, in spherical-type coordinates). We compare the latter with the standard first law of black hole dynamics. Also, by identifying TdS = dE + pdV, we comment on a possible modification of the standard, Bekenstein-Hawking entropy-area relation due to the gravitational energy and gravitational pressure of the black hole. The infinitesimal variations in question refer to the Penrose process for a Reissner-Nordstrom black hole.