2010/12/31 by Mirjam Cvetič, M. Cvetic, G. W. Gibbons +4 · 15 citations
Mathematics · Physics and Astronomy · #Baryon #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Event horizon #Geometry #Inverse #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Omega #Particle physics #Physics #Quantum mechanics #Spacetime #gr-qc #hep-th
paper · pdf · doi:10.1103/physrevd.84.024037
published as Phys.Rev.D84:024037,2011 · 29 pages, minor corrections
arxiv created 2011/04/20 · openalex publication_date 2011/07/20 · arxiv updated 2011/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In a theory where the cosmological constant \ensuremathΛ 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+\ensuremathΩidJi+\ensuremathΦ_\ensuremathαdQ_\ensuremathα+\ensuremathΘd\ensuremathΛ, where E is now the enthalpy of the spacetime, and \ensuremathΘ, the thermodynamic conjugate of \ensuremathΛ, is proportional to an effective volume V=\ensuremath-\frac16\ensuremathπ\ensuremathΘD\ensuremath-2 ``inside the event horizon.'' Here we calculate \ensuremathΘ 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≡\phantom\rule0ex0ex((D\ensuremath-1)V/A_D\ensuremath-2)^1/(D\ensuremath-1)(A_D\ensuremath-2/A)^1/(D\ensuremath-2)\ensuremath≥1, where 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≤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 \ensuremathΛ or g goes to zero, providing a definition of V even for asymptotically flat black holes.