Black hole enthalpy and an entropy inequality for the thermodynamic volume

By M. Cvetič, G. W. Gibbons, D. Kubizňák, C. Pope. 2010

680 citations recorded by the source corpus.

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 $dE= TdS + \Omega_i dJ_i + \Phi_\alpha d Q_\alpha + \Theta d \Lambda$, 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 $R\equiv ((D-1)V/{\cal A}_{D-2})^{1/(D-1)}\, ({\cal A}_{D-2}/A)^{1/(D-2)}\ge1$, where ${\cal A}_{D-2}$ 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8fc1f4f32d948a52a2d0c9340bd27cff3eb81ef0

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Citations in the corpus

Construction of Regular Black Holes in General Relativity (2016, neutral): “For the electrically charged case (38), the solution reads ds(2) = −fdt(2) + dr 2 f + r(2)dΩ(2)k , f = r (2)/`(2) + k − 2M r − 2α −1q3rμ−1 ( rν + qν )μ/ν , A = q √ 2α (( 3− (μ− 3)( qr ) ν )( 1 + ( qr ) ν )−μ+ν ν − 3 ) dt , (54) Treating the cosmological constant as well as the parameter α as a therm”
Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization (2012, background): “• Thermodynamic volume has been studied for a wide variety of black holes and is conjectured to satisfy the reverse isoperimetric inequality [9].”
P − V criticality of charged AdS black holes (2012, background): “More generally, the thermodynamic volume for a wide variety of known black hole solutions was studied in [11].”
Noether charge, black hole volume, and complexity (2016, methodology): “The paper [36] asks the interesting question of what integral over the black hole interior would give rise to the volume (2.”
AdS and Lifshitz black holes in conformal and Einstein-Weyl gravities (2012, background): “It is also worth remarking that the Smarr formula E = 2TS − 2ΘΛ in Einstein gravity [74] now becomes E = 2ΘΛ in conformal gravity.”
Holographic black hole chemistry (2015, neutral): “Note that the notion of thermodynamic volume can differ from the geometric volume that would be calculated from simply integrating the volume form from the origin to the outer horizon of the black hole [9].”
Black Hole Chemistry (2014, background): “rotation parameters, where the quantities Ω∞ allow for the possibility of a rotating frame at infinity [7, 8].”
Holographic entanglement entropy and the extended phase structure of STU black holes (2015, background): “Rather, the thermodynamical volume agrees with the integral of the scalar potential over the black hole interior [40].”
Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume (2014, methodology): “Alternatively, one can calculate V using the method of Killing co-potentials [2, 10].”
Multiple reentrant phase transitions and triple points in Lovelock thermodynamics (2014, background): “Furthermore, the use of an extended thermodynamic phase space is consistent with considering more fundamental theories of physics that admit variation of physical constants [2, 4, 9].”
Holographic heat engines (2014, neutral): “[5–13], including the reviews in refs.”
AdS dyonic black hole and its thermodynamics (2013, background): “We find that Υ, which can be interpreted as a volume (see, for example, [22] and references therein), is given by”