Black hole enthalpy and an entropy inequality for the thermodynamic volume

Abstract

In a theory where the cosmological constant $\ensuremath{\Lambda}$ 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 $dE=TdS+{\ensuremath{\Omega}}_{i}d{J}_{i}+{\ensuremath{\Phi}}_{\ensuremath{\alpha}}d{Q}_{\ensuremath{\alpha}}+\ensuremath{\Theta}d\ensuremath{\Lambda}$, where $E$ is now the enthalpy of the spacetime, and $\ensuremath{\Theta}$, the thermodynamic conjugate of $\ensuremath{\Lambda}$, is proportional to an effective volume $V=\ensuremath{-}\frac{16\ensuremath{\pi}\ensuremath{\Theta}}{D\ensuremath{-}2}$ ``inside the event horizon.'' Here we calculate $\ensuremath{\Theta}$ 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 $R\ensuremath{\equiv}\phantom{\rule{0ex}{0ex}}((D\ensuremath{-}1)V/{\mathcal{A}}_{D\ensuremath{-}2}{)}^{1/(D\ensuremath{-}1)}({\mathcal{A}}_{D\ensuremath{-}2}/A{)}^{1/(D\ensuremath{-}2)}\ensuremath{\ge}1$, where ${\mathcal{A}}_{D\ensuremath{-}2}$ is the volume of the unit ($D\ensuremath{-}2$) 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 ($D\ensuremath{-}1$) space bounded by a surface of area $A$, for which $R\ensuremath{\le}1$. 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 $\ensuremath{\Lambda}$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically flat black holes.