## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

7-20-2011

#### 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_{α}dQ_{α}+ΘdΛ, where 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 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≡ ((D-1)V/A_{D-2})^{1/(D-1)}(A_{D-2}/A)^{1/(D-2)}≥1, where 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≤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 Λ or g goes to zero, providing a definition of V even for asymptotically flat black holes.

#### Recommended Citation

Cvetič, M., Gibbons, G. W., Kubizňák, D., & Pope, C. N. (2011). Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume. Retrieved from https://repository.upenn.edu/physics_papers/215

**Date Posted:** 20 January 2012

This document has been peer reviewed.

## Comments

Cvetič, M., Gibbons, G.W., Kubizňák, D. & Pope, C.N. (2011). Black hole enthalpy and an entropy inequality for the thermodynamic volume.

Phys. Rev. D84, 024037. http://dx.doi.org/10.1103/PhysRevD.84.024037© 2011 American Physical Society