Black hole enthalpy and an entropy inequality for the thermodynamic volume

In a theory where the cosmological constant $\Lambda$ 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 $dE=TdS+{\Omega }_{i}d{J}_i+{\Phi }_{\alpha }d{Q}_{\alpha }+\Theta d\Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V=-\frac{16\pi \Theta }{D-2}$ “inside the event horizon.” Here we calculate $\Theta$ and $V$ for a wide variety of $D$-dimensional charged rotating asymptotically anti-de Sitter (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 $R\equiv ((D-1)V/{\mathcal{A}}_{D-2}{)}^{1/(D-1)}({\mathcal{A}}_{D-2}/A{)}^{1/(D-2)}\ge 1$, where ${\mathcal{A}}_{D-2}$ 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 reverse isoperimetric inequality can be interpreted as the statement that the entropy inside a horizon of a given ”volume” $V$ is maximized for Schwarzschild-AdS. The thermodynamic definition of $V$ requires a cosmological constant (or gauge coupling constant). However, except in seven dimensions, a smooth limit exists where $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically flat black holes.

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 parameters, 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.