The Penrose-Rindler equation and horizon thermodynamics of stationary black holes

TL;DR

This paper develops a geometric framework using NP and GHP formalisms to relate horizon conditions of stationary black holes to thermodynamic laws, including pressure and volume concepts, extending black hole thermodynamics beyond spherical symmetry.

Contribution

It introduces a geometric reformulation of the Penrose-Rindler equation at the horizon and derives a Smarr-like formula incorporating horizon rotation and matter pressure.

Findings

Reformulation of horizon conditions as the Penrose-Rindler equation.

Derivation of a Smarr-like formula from horizon geometry.

Introduction of horizon-averaged matter pressure and conjugate volume.

Abstract

Black holes are the natural arena for exploring the interplay between gravity and thermodynamics. Although the association between black hole mechanics and black hole thermodynamics is well-established, the comprehensive geometric formulation of thermodynamic variables deserves further investigation. In this work, both Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms are considered within the framework of horizon thermodynamics. We show that the NP formalism reformulates the horizon condition as the Penrose-Rindler equation. In this context, a Smarr-like formula for stationary black holes is recovered from the Penrose-Rindler equation reinterpreted as a horizon equilibrium of pressures, which includes a pressure associated with the horizon rotation. A complete geometric reformulation of this reinterpretation of the Penrose-Rindler equation evaluated at the horizon is developed within the GHP formalism. The GHP approach further inspires the introduction of the horizon-averaged matter pressure and its conjugate volume, thereby enabling a quasi-local realization of the Smarr-like formula for stationary black holes. This geometric formulation clarifies the connection between horizon dynamics and thermodynamics and offers a unified setting for extending black hole thermodynamics beyond spherical symmetry.