Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

Thumbnail Image
Penn collection
Department of Physics Papers
Degree type
Physical Sciences and Mathematics
Grant number
Copyright date
Related resources
Gibbons, G. W.
Kubizňák, D.
Pope, C. N.

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+ΩidJi+ΦαdQα+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/AD-2)1/(D-1)(AD-2/A)1/(D-2)≥1, where AD-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.

Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
Journal title
Volume number
Issue number
Publisher DOI
Journal Issue
Cvetič, M., Gibbons, G.W., Kubizňák, D. & Pope, C.N. (2011). Black hole enthalpy and an entropy inequality for the thermodynamic volume. Phys. Rev. D 84, 024037. © 2011 American Physical Society
Recommended citation