Physical reinterpretation of heat capacity discontinuities for static black holes

A generic characteristic of self-gravitating systems is that they have a negative heat capacity. An important example of this behavior is given by the Schwarzschild black hole. The case of charged and rotating black holes is even more interesting since a change of sign of the specific heat takes place through an infinite discontinuity. This has been usually associated with a black hole thermodynamic phase transition appearing at the points where the heat capacity diverges, the so-called Davies points. This aspect of black hole thermodynamics has been addressed from different perspectives, motivating different interpretations since its discovery in the 1970s. In this paper, a physical reinterpretation of the heat capacity is provided for spherically symmetric and static black holes. Our analysis is partially based on a reformulation of the black hole heat capacity using the Newman-Penrose formalism. The application to the Reissner-Nordstr\"om-de Sitter black hole case reveals a clear physical interpretation of the Newman-Penrose scalars evaluated at the event horizon. This allows us to write the heat capacity as a balance of pressures defined at the horizon, in particular, a matter pressure (coming from the energy-momentum tensor) and a thermal pressure (coming from the holographic energy equipartition of the horizon). The Davies point is identified with the point where the Komar thermal energy density matches the matter pressure at the horizon. We also compare the black hole case with the case of self-gravitating objects and their corresponding thermal evolutions. We conclude that the heat capacity of black holes and self-gravitating systems can be understood qualitatively in similar terms.