Enthalpy and the Mechanics of AdS Black Holes

TL;DR

This paper derives geometric formulas for the thermodynamics of AdS black holes, incorporating variations in the cosmological constant as an effective volume, and interprets black hole mass as enthalpy.

Contribution

It introduces a geometric derivation of the Smarr formula and an expanded first law for AdS black holes, including the cosmological constant variation.

Findings

Effective volume outside the horizon is proportional to the variation coefficient.

The mass of AdS black holes can be interpreted as enthalpy.

The new formulas relate surface integrals to thermodynamic quantities.

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.