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

Understand

We investigate, in the case of a Reissner-Nordstr\"om 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 analyse the thermodynamic relation dE + pdV (as p turns out to be a density, dV is actually given by dV = dr d\theta d\phi, in spherically-type coordinates).
• We compare the latter with the standard first law of black hole dynamics.