Enthalpy and the Mechanics of AdS Black Holes

TL;DR

The paper establishes a geometric framework for AdS black hole thermodynamics with a variable cosmological constant by introducing a Killing-potential two-form ω^{ab}. Through a generalized Komar relation and Hamiltonian perturbation theory, it derives the AdS Smarr formula and an extended first law that includes δΛ, identifying the Λ-variation coefficient Θ with a finite effective volume outside the horizon and interpreting the black hole mass as the spacetime enthalpy. The work unifies the Smarr relation and first law via Euler scaling and provides a general expression for Θ that remains valid under broad fall-off conditions, while also detailing the near-horizon structure of the Killing potential. This framework clarifies the thermodynamic role of the cosmological constant and has potential implications for holography and Lovelock gravity generalizations.

Abstract

We present geometric derivations of the Smarr formula for static AdS black holes and an expanded first law that includes variations in the cosmological constant. These two results are further related by a scaling argument based on Euler's theorem. The key new ingredient in the constructions is a two-form potential for the static Killing field. Surface integrals of the Killing potential determine the coefficient of the variation of the cosmological constant in the first law. This coefficient is proportional to a finite, effective volume for the region outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded from a spatial slice by the black hole horizon. This effective volume also contributes to the Smarr formula. Since the cosmological constant is naturally thought of as a pressure, the new term in the first law has the form of effective volume times change in pressure that arises in the variation of the enthalpy in classical thermodynamics. This and related arguments suggest that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime.