Abstract
In this review we summarize, expand, and set in context recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. We specifically consider the thermodynamics of higher-dimensional rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. We plot the associated thermodynamic potential—the Gibbs free energy—and study its behavior to uncover possible thermodynamic phase transitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdS black holes exhibit a rich set of interesting thermodynamic phenomena analogous to the “every day thermodynamics” of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van derWaals type. Furthermore, the reentrant phase transitions also occur for multiply-spinning asymptotically flat Myers–Perry black holes. These phenomena do not require a variable cosmological constant, though they are more naturally understood in the context of the extended phase space. The thermodynamic volume, a quantity conjugate to the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy the reverse isoperimetric inequality; this provides a first example of calculation confirming the validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state P = P(V,T) is studied for various black holes both numerically and analytically—in the ultraspinning and slow rotation regimes.
Highlights
The subject of black hole thermodynamics continues to be one of import in gravitational physics
It emerged from geometric derivations of the Smarr formula for anti-de Sitter (AdS) black holes [2] that suggested the cosmological constant should be considered as a thermodynamic variable [3] analogous to pressure in the first law [4]
Counter-intuitively in a fixed charge canonical ensemble strongly charged small RN black holes are thermodynamically preferred to weakly charged large black holes. (Despite the fact that the entropy is larger on the upper branch in Figure 2, since it increases quadratically with r+, the Gibbs free energy in the lower branch is the global minimum at fixed Q due to the larger enthalpy contribution in the upper branch.) A directly related result is that in the range Equation√(23) there are no negative modes in the Euclidean action, whereas the negative mode appears for r+ ≥ 3|Q|, indicating the onset of Schwarzschild-like behavior [59]
Summary
The subject of black hole thermodynamics continues to be one of import in gravitational physics. The proposal that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime represents an interesting new development in this subject It emerged from geometric derivations of the Smarr formula for AdS black holes [2] that suggested the cosmological constant should be considered as a thermodynamic variable [3] analogous to pressure in the first law [4]. We find that this inequality remains true even for the thin asymptotically AdS black rings—providing a first confirmation of the reverse isoperimetric inequality conjecture for a black hole with non-spherical horizon topology
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