Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

TL;DR

This work promotes the cosmological constant (or gauge coupling) to a thermodynamic variable in black hole thermodynamics, introducing the enthalpy $E$ and its conjugate $\Theta$, linked to a thermodynamic volume via $\Theta= -\frac{D-2}{16\pi} V$. By applying the generalized first law and Smarr relations to static and rotating AdS black holes in diverse dimensions, the authors compute $\Theta$ and define a physically meaningful interior volume, sometimes distinguishing it from a purely geometric $V'$. They demonstrate a universal Reverse Isoperimetric Inequality, $R\ge1$, for the thermodynamic volume, implying entropy maximization by Schwarzschild–AdS for fixed $V$, and explore the limit to asymptotically flat spacetimes (except in $D=7$). A complementary Komar/Killing-potential analysis via CKY tensors confirms the thermodynamic results and illuminates the geometric underpinnings of the interior volume, tying horizon data to global charges. The paper thus unifies black hole thermodynamics with geometric and symmetry-based constructions across dimensions.

Abstract

In a theory where the cosmological constant $Λ$ or the gauge coupling constant $g$ arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes $dE= TdS + Ω_i dJ_i + Φ_αd Q_α+ Θd Λ$, where $E$ is now the enthalpy of the spacetime, and $Θ$, the thermodynamic conjugate of $Λ$, is proportional to an effective volume $V = -\frac{16 πΘ}{D-2}$ "inside the event horizon." Here we calculate $Θ$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically AdS black hole spacetimes, using the first law or the Smarr relation. We compare our expressions with those obtained by implementing a suggestion of Kastor, Ray and Traschen, involving Komar integrals and Killing potentials, which we construct from conformal Killing-Yano tensors. We conjecture that the volume $V$ and the horizon area $A$ satisfy the inequality $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ is the volume of the unit $(D-2)$-sphere, and we show that this is obeyed for a wide variety of black holes, and saturated for Schwarzschild-AdS. Intriguingly, this inequality is the "inverse" of the isoperimetric inequality for a volume $V$ in Euclidean $(D-1)$ space bounded by a surface of area $A$, for which $R\le 1$. Our conjectured {\it Reverse Isoperimetric Inequality} can be interpreted as the statement that the entropy inside a horizon of a given "volume" $V$ is maximised for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in 7 dimensions, a smooth limit exists where $Λ$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.

Figures (1)

• Figure 1: Thermodynamic volume of the Kerr-AdS black hole. The graph displays the dependence of $V$ on gauge coupling $g$, $\Lambda=-3g^2$, for various rotation parameteres, while we keep the total gravitational enthalpy fixed, $E=1$$(J=a)$. The upper curve represents Schwarzschild-AdS $(a=0$), the lower curves, in descending order, correspond to Kerr-AdS with $a=0.5$, $a=0.7$, $a=0.9$ and $a=0.99$, respectively. Obviously, the smooth limit exists for $g\to 0$, the volume is smooth also in the transition between large and small black holes.