Contact geometry and thermodynamics of black holes in AdS spacetimes

TL;DR

This paper applies contact geometry to formulate the thermodynamics of black holes in AdS spacetimes, revealing new mappings between different black hole solutions and modeling thermodynamic processes via Hamilton-Jacobi formalism.

Contribution

It introduces a contact geometric framework for black hole thermodynamics in AdS, including mappings between solutions and a Hamilton-Jacobi approach to thermodynamic processes.

Findings

Mapped Schwarzschild to Reissner-Nordström black holes using contact deformations

Derived equations of state for general AdS black holes from high-temperature limits

Modeled black hole thermodynamic processes with characteristic contact vector field curves

Abstract

In this paper we discuss a formulation of extended phase space thermodynamics of black holes in Anti de Sitter (AdS) spacetimes from the contact geometry point of view. Thermodynamics of black holes can be understood within the framework of contact geometry as flows of vector fields generated by Hamiltonian functions on equilibrium submanifolds in the extended phase space that naturally incorporates the structure of a contact manifold. Deformations induced by the contact vector fields are used to construct various maps among thermodynamic quantities. Thermodynamic variables and equations of state of Schwarzschild black holes are mapped to that of Reissner-Nordstr\"{o}m black holes in AdS, with charge as the deformation parameter. In addition, the equations of state of general black holes in AdS are shown to emerge from the high-temperature ideal gas limit equations via suitable deformations induced by contact vector fields. The Hamilton-Jacobi formalism analogous to mechanics is set up, and the corresponding characteristic curves of contact vector fields are explicitly obtained to model thermodynamic processes of black holes. Extension to thermodynamic cycles in this framework is also discussed.