Thermodynamic Geodesics of a Reissner Nordstr\"om Black Hole

TL;DR

This paper analyzes the thermodynamic geodesics of a Reissner Nordström black hole using geometrothermodynamics, revealing how these paths relate to black hole entropy, charge, and processes like Hawking radiation and pair production.

Contribution

It introduces a numerical analysis of thermodynamic geodesics for black holes, incorporating Hawking radiation and Schwinger pair-production constraints, and finds exact solutions for extremal black hole entropy.

Findings

Geodesics relate entropy and charge, extremizing mass changes.

Entropy of extremal black holes is an exact geodesic solution.

Geodesics can model black hole evolution due to Hawking radiation.

Abstract

Starting from a Geometrothermodynamics metric for the space of thermodynamic equilibrium states in the mass representation, we use numerical techniques to analyse the thermodynamic geodesics of a supermassive Reissner Nordstr\"{o}m black hole in isolation. Appropriate constraints are obtained by taking into account the processes of Hawking radiation and Schwinger pair--production. We model the black hole in line with the work of Hiscock and Weems. It can be deduced that the relation which the geodesics establish between the entropy $S$ and electric charge $Q$ of the black hole extremises changes in the black hole's mass. Indeed, the expression for the entropy of an extremal black hole is an exact solution to the geodesic equation. We also find that in certain cases, the geodesics describe the evolution brought about by the constant emission of Hawking radiation and charged-particle pairs.