Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume

TL;DR

This review develops the extended (P,V) thermodynamics of rotating black holes and black rings, focusing on canonical ensembles with fixed angular momenta and charges. It demonstrates a rich landscape of phase behavior—including reentrant transitions, multiple first-order and zeroth-order transitions, triple points, and Van der Waals–like SBH/LBH transitions—across Kerr–AdS, Myers–Perry, and black ring geometries, with both asymptotically flat and AdS spacetimes. A central theme is the thermodynamic volume and the reverse isoperimetric inequality, extended to non-spherical horizons, and the analysis of the equation of state in slow-rotation and ultraspinning regimes. The work also discusses ultraspinning and superradiant instabilities and contemplates AdS/CFT interpretations of these gravitational phase transitions, highlighting the interplay between thermodynamics and dynamics in high-dimensional rotating black objects.

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 behaviour 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. 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.

Figures (34)

• Figure 1: Gibbs free energy: Schwarzschild black hole. The dashed blue line corresponds to a negative specific heat; for an asymptotically flat Schwarzschild black hole this quantity is negative for any temperature.
• Figure 2: Gibbs free energy: RN black hole. The Gibbs free energy of $Q=1$ RN black hole is displayed. The horizon radius $r_+$ increases from left to right and then up; $T=0$ corresponds to the extremal black hole with $r_+=M=Q=1$. For a fixed temperature there are two branches of RN black holes. The lower thermodynamically preferred branch corresponds to small strongly charged nearly extremal black holes with positive $C_P$. The upper branch of weakly charged RN (almost Schwarzschild-like) black holes has higher Gibbs free energy and negative specific heat and hence is thermodynamically unstable. Its Euclidean action also possesses a negative zero mode. The situation for the Kerr-AdS black hole is qualitatively similar, with fixed $J$ replacing fixed $Q$.
• Figure 3: Gibbs free energy: Schwarzchild-AdS black hole. When compared to the asymptotically flat Schwarzschild case (fig. \ref{['Fig:Gschflat']}) for $P>0$ the Gibbs free energy acquires a new thermodynamically stable branch of large black holes. For $T>T_{\hbox{\tiny HP}}$ this branch has negative Gibbs free energy and the corresponding black holes represent the globally thermodynamically preferred state.
• Figure 4: Hawking--Page transition is a first-order phase transition between thermal radiation in AdS and large stable Schwarzschild-AdS black hole. It occurs when $G$ of the Schwarzschild-AdS black hole approximately vanishes. Considering various pressures $P$ gives the radiation/large black hole coexistence line \ref{['HPcoexistence']} displayed in this figure. Similar to a "solid/liquid" phase transition, this line continues all the way to infinite pressure and temperature.
• Figure 5: Equation of state: Schwarzschild-AdS black hole. The equation of state \ref{['HPstate']} is displayed for various temperatures. For a given temperature the maximum occurs at $v=2r_0$. The dashed blue curves correspond to small unstable black holes. The red curves depict the stable large black hole branch; we observe 'ideal gas' behaviour for large temperatures.