Newman–Penrose scalars and black hole equations of state

In this work we explore the connections between Newman–Penrose scalars, including the Penrose–Rindler K-curvature, with the equation of state of asymptotically Anti-de Sitter Reissner–Nordström black holes. After briefly reviewing the equation of state for these black holes from the point of view of both the Extended Phase Space and the Horizon Thermodynamics approaches, a geometric splitting is given for such an equation in terms of the non vanishing Newman–Penrose scalars which define the K-curvature at the horizon. From this splitting, a possible thermodynamical interpretation is developed for such scalars in the context of the black hole thermodynamics approaches initially discussed. Afterwards, the square root of the Bel–Robinson tensor is employed to propose conditions at the horizons in terms of pressures or energy densities, which can be understood as alternative thermodynamical definitions of these surfaces.