Black hole enthalpy and an entropy inequality for the thermodynamic volume

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 = T dSwhere E is now the enthalpy of the spacetime, and Θ, the thermodynamic conjugate of Λ, is proportional to an effective volume V = − 16πΘ 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 1 conformal Killing-Yano tensors. We conjecture that the volume V and the horizon areaA 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 ≤ 1. Our conjectured 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.2

Van Der Waals black holes in d dimensions

“…for the isoperimetric ratio R. This will obey the reverse isoperimetric inequality [18] provided C ℓ r h + b > 0, or in other words if C ℓ is not too negative.…”

P − V criticality of charged AdS black holes

“…Recently, the idea of including the variation of the cosmological constant Λ in the first law of black hole thermodynamics has attained increasing attention [6][7][8][9][10][11][12]. 1 From a general relativistic point of view, such a variation is a slightly awkward thing to do, as the cosmological constant should be treated as a fixed external parameter of the theory.…”

First law and a variational principle for static asymptotically Randall-Sundrum black holes

“…The coefficient of δP defines a thermodynamic volume V eff in a black hole first law [4,5,27]. For a static asymptotically AdS black hole, it was found in [4] that V eff > 0 is the volume removed by the black hole (the volume of pure AdS space minus the volume outside the black hole).…”

“…For this purpose, we provide a new, intrinsic, mass definition. Our first law defines a thermodynamic volume and a gravitational tension that are braneworld analogs of thermodynamic volume in asymptotically AdS black hole spacetimes [4,5] and gravitational tension in asymptotically flat compactifications [6].…”