Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume

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 der Waals 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 Galaxies 2014, 2 90 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.

Van Der Waals black holes in d dimensions

“…The first law of thermodynamics then reads 10) where the momentum V Λ conjugated to P Λ is interpreted as a volume, given by 11) and M is the enthalpy,…”

“…However the corresponding equations of state for the black hole and the VdW fluid, though qualitatively mathematically similar, are not identical. Consequently neither charge Q nor rotation J (or other parameters for more complicated black holes [11]) can be directly identified with the fluid parameters a and b. In the following, we will denote the pressure associated with the cosmological constant P Λ to avoid confusion with the pressure of the fluid sourcing the Einstein equations.…”

Bose condensation and branes

“…This picture has been extensively investigated recently, for a review see [4], with most work focusing on the case of negative Λ (an exception being [5]). A natural question to ask in this context is what the role of the cosmological constant might be on the boundary CFT, at finite temperature, in the AdS/CFT correspondence.…”

Analytical and exact critical phenomena ofd-dimensional singly spinning Kerr-AdS black holes

“…Through the Gibbs free energy, the four-dimensional Kerr-AdS black hole was found to possess a small/large black hole phase transition [44], which is similar to the charged AdS black hole. The five-dimensional singly rotating Kerr-AdS black hole also has such a phenomenon [49]. However, when the dimension d increases or the multiply rotating parameters are presented, the case becomes much more exciting.…”

“…In Refs. [44,49], the authors first expanded the state equation and all quantities in the small-J limit and obtained an approximate critical point for the d-dimensional singly rotating Kerr-AdS black hole. As is known, the appearance of the reentrant phase transition requires more than one critical point.…”