Black-Hole Thermodynamics from Gauge Freedom in Extended Iyer-Wald Formalism

TL;DR

This paper develops an extended Iyer-Wald formalism incorporating gauge freedom to unify and clarify the thermodynamics of black holes, especially Kerr-anti de Sitter spacetimes, addressing previous inconsistencies in mass definitions.

Contribution

It introduces exact isohomogeneous transformations within the extended Iyer-Wald formalism, providing a unified framework for black hole thermodynamics with gauge-dependent mass and volume.

Findings

Reconciles different formulations of KadS thermodynamics.

Identifies gauge choices compatible with EITs for consistent first laws.

Demonstrates how conventional KadS thermodynamics is a special case.

Abstract

Thermodynamic systems admit multiple equivalent descriptions related by transformations that preserve their fundamental structure. This work focuses on exact isohomogeneous transformations (EITs), a class of mappings that keep fixed the set of independent variables of the thermodynamic potential, while preserving both the original homogeneity and the validity of a first law. Our investigation explores EITs within the extended Iyer--Wald formalism for theories containing free parameters (e.g., the cosmological constant). EITs provide a unifying framework for reconciling the diverse formulations of Kerr-anti de Sitter (KadS) thermodynamics found in the literature. While the Iyer--Wald formalism is a powerful tool for deriving first laws for black holes, it typically yields a non-integrable mass variation that prevents its identification as a proper thermodynamic potential. To address this issue, we investigate an extended Iyer--Wald formalism where mass and thermodynamic volume become gauge dependent. Within this framework, we identify the gauge choices and Killing vector normalizations that are compatible with EITs, ensuring consistent first laws. As a key application, we demonstrate how conventional KadS thermodynamics emerges as a special case of our generalized approach.