A Universal Smarr Formula via Coupling Constants

TL;DR

The paper addresses inconsistencies in the Smarr relation when gravity theories include matter fields and dimensionful couplings. It introduces a universal framework that promotes each coupling to a dynamical solution parameter via auxiliary scalar and (D-1)-form gauge fields, recasts couplings as conserved charges with horizon potentials as conjugates, and derives extended first laws and Smarr formulas by scaling. The approach is validated across multiple black hole solutions (NMG BTZ, Horndeski BTZ-like and branes, MTZ, and higher-curvature AdS-Schwarzschild), showing that including all couplings restores a universal Smarr relation and consistent thermodynamics, with some models requiring a modified physical temperature. The results establish a universal, geometrically grounded formulation of black hole thermodynamics that extends beyond General Relativity to include arbitrary dimensionful couplings.

Abstract

In gravitational theories containing matter fields and higher-derivative corrections, the standard Smarr formula often fails unless all dimensionful couplings are incorporated consistently. Traditionally, parameters such as the cosmological constant or the coefficients of higher-derivative terms are regarded as immutable features of the theory and therefore excluded from the thermodynamic phase space. In our recent work, we developed a fully general framework that promotes every such coupling to a dynamical, freely varying parameter of black hole solutions. This is accomplished by introducing, for each coupling, an auxiliary scalar and gauge field, through which the coupling appears as a conserved charge associated with the global sector of an emergent gauge symmetry. The corresponding conjugate variables naturally arise as electric potentials evaluated at the black hole horizon. As a result, the first law and the Smarr relation acquire additional, systematically determined contributions, yielding a consistent and universal extension of black hole thermodynamics. We illustrate the validity of this construction by revisiting several black hole examples in the literature where the Smarr formula remains inconsistent even after treating the cosmological constant as a thermodynamic variable. Our analysis shows that only by including all dimensionful couplings in this generalized manner can one obtain an internally consistent Smarr relation, thereby providing the foundation for a truly universal formulation of black hole thermodynamics.