Explicit and universal formula for thermodynamic volume in extended black hole thermodynamics

TL;DR

This work resolves the long-standing ambiguity in extended black hole thermodynamics by providing a universal, geometry-based formula for the thermodynamic volume $\mathcal{V}$. Using an extended Iyer-Wald formalism that includes variations of couplings $\alpha_i$, the authors show that each $\mathcal{V}_i$ decomposes as $\mathcal{V}_i = \mathcal{V}_i^{(1)} + \mathcal{V}_i^{(2)}$, where $\mathcal{V}_i^{(1)}$ encodes the explicit Lagrangian dependence $L_i$ and $\mathcal{V}_i^{(2)}$ captures the dynamical-field response via the boundary term $F$. They derive explicit expressions for these two contributions and demonstrate them in Kerr-AdS and rotating BTZ black holes in New Massive Gravity, recovering known results such as $\mathcal{V} = V + \frac{4\pi}{3} M a^2$ and validating the framework with higher-derivative couplings. This clarifies the physical meaning of $\mathcal{V}$, places it on par with other thermodynamic quantities, and opens avenues for applications to charged AdS black holes and holographic contexts. The approach also clarifies the role of background regularization and the $F$ term in encoding dynamical-field information relevant to thermodynamic variations.

Abstract

In extended black hole thermodynamics, the cosmological constant and other couplings are treated as thermodynamic variables, yielding a first law $\tildeδM=T\tildeδS+Ω\tildeδJ+\mathcal{V}\tildeδP+\cdots$, where $P\equiv -\fracΛ{8π}$. A long-standing conceptual gap in this framework is that, unlike $M$, $T$, $S$, $Ω$, and $J$, the thermodynamic volume $\mathcal{V}$ lacks a first-principles definition and can only be deduced from other thermodynamic quantities. This deficiency indicates that the underlying origin of $\mathcal{V}$ has remained poorly understood. In this work, we resolve this issue and provide an explicit universal formula for $\mathcal{V}$. We demonstrate that it universally decomposes into two contributions, one arising from the explicit dependence of the action on the couplings and the other from the response of the fundamental dynamical fields. This clarifies the physical meaning of thermodynamic volume and places it on the same footing as other intrinsic thermodynamic quantities.