A Universal Smarr Formula via Coupling Constants

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.