Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

Abstract

In a theory where the cosmological constant $\Lambda$ 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 $dE=TdS+{\Omega }_{i}d{J}_i+{\Phi }_{\alpha }d{Q}_{\alpha }+\Theta d\Lambda$, where $E$ is now the enthalpy of the spacetime, and $\Theta$, the thermodynamic conjugate of $\Lambda$, is proportional to an effective volume $V=-\frac{16\pi \Theta }{D-2}$ "inside the event horizon." Here we calculate $\Theta$ 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 conformal Killing-Yano tensors. We conjecture that the volume $V$ and the horizon area $A$ satisfy the inequality $R\equiv ((D-1)V/{\mathcal{A}}_{D-2}{)}^{1/(D-1)}({\mathcal{A}}_{D-2}/A{)}^{1/(D-2)}\ge 1$, where ${\mathcal{A}}_{D-2}$ 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 $R\le 1$. Our conjectured {\it 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 $\Lambda$ or $g$ goes to zero, providing a definition of $V$ even for asymptotically-flat black holes.

29 pages, minor corrections

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• Thermodynamic Phase Transition of Generalized Ayon-Beato Garcia Black Holes
• Lifshitz and hyperscaling violated Yang-Mills-dilaton black holes
• Bulk-Boundary Thermodynamics of Charged Black Holes in Higher Derivative Theory
• Magnetic black holes within Einstein-AdS gravity coupled to nonlinear electrodynamics, extended phase space thermodynamics and Joule-Thomson expansion
• General mass formulas for charged Kerr-AdS black holes
• Volume dependent extension of Kerr-Newman black hole thermodynamics
• Reduced thermodynamics and an exact phase transition of five-dimensional Kerr-AdS black holes with equal spin parameters
• Thermodynamics of a static electric-magnetic black hole in Einstein-Born-Infeld-AdS theory with different horizon geometries
• Thermodynamics and topological classifications of static non-extremal four-charge AdS black hole in the five-dimensional $\mathcal{N}=2$, $STU-W^{2}U$ gauged supergravity
• Event Horizon Wrinklification
• Microstructure of Charged AdS Black Hole with Minimal Length Effects
• Geometrothermodynamics of black holes with a nonlinear source
• Thermodynamics of BTZ Black Holes in Nonlinear Electrodynamics
• Thermodynamics of Inhomogeneously Mass-deformed ABJM Model and Pressure Anisotropy
• Energy-dependent topological anti-de Sitter black holes in Gauss-Bonnet Born-Infeld gravity
• Statistical ensembles and logarithmic corrections to black hole entropy
• Thermodynamics of 5D Charged Rotating Black Holes: A Counterterms Treatment
• Symmetry Breaking Vacua in Lovelock Gravity
• Quasi-topological Electromagnetism: Dark Energy, Dyonic Black Holes, Stable Photon Spheres and Hidden Electromagnetic Duality
• Einstein-AdS gravity coupled to nonlinear electrodynamics, magnetic black holes, thermodynamics in an extended phase space and Joule--Thomson expansion
• Stability of Schwarzschild (Anti)de Sitter black holes in Conformal Gravity
• Extended Phase Space Thermodynamics of Black Holes in Massive Gravity
• $\lambda$ phase transition in Horava gravity
• Thermodynamics with pressure and volume of black holes based on two assumptions under scalar field scattering
• Shadowless rapidly rotating yet not ultraspinning Kerr-AdS${}_4$ and Kerr-Newman-AdS${}_4$ black holes
• On thermodynamics of compact objects
• Black Hole: The Interior Spacetime
• Super-entropic black holes in gravity's rainbow and determining constraints on rainbow functions
• Holographic pressure and volume for black holes
• Joule-Thomson expansion of Reissner-Nordström-Anti-de Sitter black holes with cloud of strings and quintessence
• Magnetic black hole thermodynamics in an extended phase space with nonlinear electrodynamics
• Joule-Thomson expansion of the torus-like black hole
• Generating Kerr-anti-de Sitter thermodynamics
• Topological confinement in Skyrme holography
• Evidence for NO violation of the second law in extended black hole thermodynamics
• Complexity growth rate, grand potential and partition function
• Extended phase space thermodynamics of magnetized black holes with nonlinear electrodynamics
• Universal entropy relations: entropy formulae and entropy bound
• Phase transition of the higher dimensional charged Gauss-Bonnet black hole in de Sitter spacetime
• Thermodynamics of spinning AdS4 black holes in gauged supergravity
• Coexistence curve and molecule number density of AdS topological charged black hole in massive gravity
• Regular black holes and reductions of thermodynamic phase spaces
• First Law and a Variational Principle for Static Asymptotically Randall-Sundrum Black Holes
• Topological Signatures and Geometrothermodynamics of Critical Phenomena in Regularized Maxwell Black Holes
• Global and local thermodynamics of the (2+1)-dimensional rotating Gauss-Bonnet black hole
• Phase transition and entropic force in Reissner-Nordström-de Sitter spacetime
• The role of the cosmological constant as a pressure in the (2+1)-dimensional black string
• Maximal hypersurface in a D-dimensional dynamical spacetime
• The CR Volume for Black Holes and the Corresponding Entropy Variation: A Review
• Refined Thermodynamics of Black Holes with Proper Conserved Charges
• Absence of isolated critical points with nonstandard critical exponents in the four-dimensional regularization of Lovelock gravity
• Holographic complexity of rotating black holes
• Thermodynamics and Phase Transitions of NUTty Dyons
• Generalized thermodynamic identity and new Maxwell's law for charged AdS black hole
• Volumetric Maxima to be Attained by a Nonstatic Black Hole
• Black-Hole Thermodynamics from Gauge Freedom in Extended Iyer-Wald Formalism
• Dimensional structure of thermodynamic topology in ultraspinning Kerr-AdS black holes
• Notes on sums over horizons
• Area products for ${\mathcal{H}}^{\pm }$ in AdS space
• Thermodynamics and overcharging problem in the extended phase spaces of charged AdS black holes with cloud of strings and quintessence under charged particle absorption
• The Thermodynamic Properties of Warped Taub-NUT AdS Black String
• Maximal hypersurfaces and aspects of volume of the Kerr family of black holes
• Null hypersurface caustics and super-entropic black holes
• Degenerate Black Rings in $D=5$ Minimal Supergravity
• Black hole images as probes of thermodynamic evolution
• Statistical Physics of 3D Hairy Black Holes
• Surface tension of the horizon
• The Smarr formula within the Geroch-Held-Penrose formalism
• A new rotating axionic AdS${}_4$ black hole dressed with a scalar field
• P-V criticality of conformal anomaly corrected AdS black holes
• Thermodynamics and the Joule-Thomson expansion of dilaton black holes in 2+1 dimensions
• Ultra-spinning Chow's black holes in six-dimensional gauged supergravity and their properties
• Extended black hole thermodynamics in a DGP braneworld
• Holographic Extended Thermodynamics of deformed AdS-Schwarzschild black hole
• Hawking Radiation from Boundary Scalar Field
• Revisiting a family of five-dimensional charged, rotating black holes
• Radially Excited AdS${}_5$ Black Holes in Einstein--Maxwell--Chern--Simons Theory
• Thermodynamic Volume Product in Spherically Symmetric and Axisymmetric Spacetime
• Critical points in Lovelock Black Holes
• Planckian charged black holes and their cosmological ramifications
• Revisiting particle circular orbits as probes of black hole phase transitions
• Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes
• On Thermodynamic Stability of Black Holes. Part II: AdS Family of Solutions
• Phase transition structure and breaking of universal nature of central charge criticality in a Born-Infeld $AdS$ black hole
• Extended symmetry of the Maxwell theory with a gauge coupling constant as a conserved charge
• Finite temperature R-squared quantum gravity