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Galaxies 2014, 2(1), 89-159; doi:10.3390/galaxies2010089
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 validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state is studied for various black holes both numerically and analytically—in the ultraspinning and slow rotation regimes.
The subject of black hole thermodynamics continues to be one of import in gravitational physics. Despite long-established notions that black holes have a temperature proportional to their surface gravity, an entropy proportional to their horizon area, and obey a version of the first law of thermodynamics, the subject is still not fully understood. Investigations in asymptotically anti-de Sitter (AdS) spacetimes have been carried out in considerable detail for three decades since it was pointed out that radiation/large AdS black hole phase transitions can take place . Thermodynamic equilibrium can be defined, and many additional corroborative results have been obtained that provide strong evidence that such objects indeed behave as thermodynamic systems.
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  that suggested the cosmological constant should be considered as a thermodynamic variable  analogous to pressure in the first law . This has led to a number of investigations of black hole thermodynamics in this extended phase space [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Extensions of this idea to asymptotically de Sitter space times have also been recently explored, with the resultant thermodynamic notions for black holes remaining intact  (see also [41,42,43,44,45,46,47,48,49]).
The purpose of this paper is to provide a detailed overview and investigation of the thermodynamics of rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. Despite the intensive study such objects have been subject to , we find a variety of novel features emerge. In particular, we shall show that the thermodynamics of multiply-rotating Kerr-AdS black holes are, under certain circumstances, 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. We also find that some of these phenomena, such as reentrant phase transitions, also occur for multiply-spinning asymptotically flat Myers–Perry black holes . We compute the thermodynamic volume (conjugate to the pressure), and demonstrate that in all examples we consider it satisfies the reverse isoperimetric inequality. In particular, 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.
1.1. Canonical Ensemble and Phase Transitions
Concentrating entirely on asymptotically flat and AdS black holes, we consider the proposal that thermodynamic pressure is given by:
We shall study the thermodynamics in a canonical ensemble, that is for fixed black hole angular momenta , and/or, in the case of charged black holes for fixed charge Q. The stability of supersymmetric AdS vacua  ensures that these kinds of thermodynamic ensembles can in principle be prepared by coupling an asymptotically AdS space-time to a bath that allows transfer of energy but not angular momentum or charge by coupling only to operators with , . Hence within this context one can prepare a canonical ensemble with any fixed and Q.
The equilibrium thermodynamics is governed by the Gibbs free energy, , whose global minimum yields the state of a system for a fixed . Note that there are states of matter, for example supercooled water, which do not correspond to the global minimum of G but rather are “metastable”. Since mass of the black hole M is interpreted as the enthalpy, we have the following thermodynamic relation:
Understanding the behavior of G is essential for uncovering possible thermodynamic phase transitions. We shall see that it is often impossible to write the relation explicitly. For this reason we fix the pressure and angular momenta (and charge)—usually expressed parametrically through the horizon radius —to particular values and plot G numerically. In this way we obtain diagrams which incorporate the information about possible phase transitions. An interesting alternative proposal which attracted attention in past few years is to consider an “effective thermodynamic geometry” and study its curvature singularities [53,54,55,56,57,58]. These then provide an information about presence of critical points and hence possible phase transitions in the given spacetime. However, in this paper we do not follow this strategy and concentrate entirely on analyzing the Gibbs free energy.
The local thermodynamic stability of a canonical ensemble is characterized by positivity of the specific heat at constant pressure:
Once the behavior of the Gibbs free energy is known, we construct the associated phase diagrams. These are usually drawn in the plane and display the coexistence lines of various black hole phases, inter-related by the first-order phase transitions, as well as reveal possible critical points where the coexistence lines terminate/merge together.
A renowned example of a transition in black hole spacetimes is the radiation/black hole first-order phase transition of Hawking and Page observed for Schwarzschild-AdS black holes immersed in a bath of radiation . Such a phenomenon (discussed further in the next section) has a dual interpretation for a boundary quantum field theory via the AdS/CFT correspondence and is related to a confinement/deconfinement phase transition in the dual quark gluon plasma . For charged or rotating black holes one likewise observes a small/large black hole first order phase transition reminiscent of the liquid/gas transition of the van der Waals fluid [4,6,8,15,16,17,21,22,27,28,33,39,62,63,64,65,66,67,68,69,70,71].
Interesting new phenomena appear in higher dimensions. Recently it has been shown  that in all dimensions singly spinning Kerr-AdS black holes demonstrate the peculiar behavior of large/small/large black hole transitions reminiscent of reentrant phase transitions (RPT) observed for multicomponent fluid systems, gels, ferroelectrics, liquid crystals, and binary gases, e.g., . A system undergoes an RPT if a monotonic variation of any thermodynamic quantity results in two (or more) phase transitions such that the final state is macroscopically similar to the initial state. For singly spinning Kerr-AdS black holes it was found  that for a fixed pressure within a certain range (and for a given angular momentum) a monotonic lowering of the temperature yields a large/small/large black hole transition. The situation is accompanied by a discontinuity in the global minimum of the Gibbs free energy, referred to as a zeroth-order phase transition, a phenomenon seen in superfluidity and superconductivity . The RPT was also found for the four-dimensional Born–Infeld-AdS black hole spacetimes, deep in the non-linear regime of the Born–Infeld theory . Remarkably, here we show (see Section 4) that a similar phenomenon is observed for the asymptotically flat doubly-spinning Myers–Perry black holes of vacuum Einstein gravity. Hence, neither exotic matter nor a cosmological constant (and hence the AdS/CFT correspondence) are required for this phenomena to occur in black hole spacetimes.
More intriguingly, other qualitatively different interesting phenomena known from the “every day thermodynamics” emerge when multiply rotating Kerr-AdS black holes are considered . Depending on the ratios of the angular momenta, we find an analogue of a solid/liquid phase transition, multiple first-order small/intermediate/large black hole phase transitions with one tricritical (triple) and two critical points reminiscent of the solid/liquid/gas phase transition, and the ‘standard’ liquid/gas behavior of a van der Waals fluid. We review and provide more detail concerning these phenomena in the following sections.
1.2. Thermodynamic Volume
A conjugate quantity to the thermodynamic pressure is the thermodynamic volume V. For AdS black hole spacetimes this can formally be obtained from the first law of black hole thermodynamics by allowing the pressure, given by Equation (1), (and hence the cosmological constant) to vary:
Euler’s theorem enables one to derive the Smarr–Gibbs–Duhem relation . This theorem states that if a function obeys the scaling relation , then the function and its partial derivatives satisfy:
Either of these relations can be used for calculating the thermodynamic volume V of AdS black holes. As we shall see in each case of interest, the final expression for V has a smooth limit for , which defines a thermodynamic volume for asymptotically flat black holes. Alternatively, one can calculate V using the method of Killing co-potentials [2,10]. This consists of finding a co-potential 2-form whose divergence gives the Killing vector that is a horizon generator of the black hole spacetime under consideration. Once known, such a co-potential is integrated to give the thermodynamic volume. However finding the co-potential is not easy. For spherical rotating black hole spacetimes this was possible due to the presence of a special hidden symmetry of a Killing–Yano tensor . This symmetry does not exist for black rings or black saturns.
The thermodynamic volume has dimensions of (length) and describes a spatial volume characterizing the black hole spacetime. For alternative definitions of various volumes associated with black hole spacetimes, see [42,75,76,77]. In particular, for the Schwarzschild black hole in one obtains the intriguing relation:
In Euclidean space , the isoperimetric inequality for the volume of a connected domain whose area is states that the ratio:
1.3. Equation of State
Once the thermodynamic volume is known and the cosmological constant identified with the thermodynamic pressure, one can, for a given black hole, write down the corresponding “fluid” equation of state relating the pressure, temperature, volume and other external parameters characterizing the black hole, . Similar to references [15,16], instead of V we employ a new quantity v, corresponding to the “specific volume” of the fluid, defined by:
We pause to note an interesting coincidence pointed to us by Jennie Traschen in private communications. Similar to the specific volume of the fluid which as per usual is taken to be the total volume of the fluid divided by the number of particles, for non-rotating AdS black holes one may define their specific volume to be:
The equation of state allows one to employ standard thermodynamic machinery and for instance calculate the critical exponents associated with the critical point. For example, the “” criticality of charged AdS black holes in four dimensions was studied in . The system exhibits a first-order phase transition à la van der Waals, with the coexistence line terminating at a critical point where the phase transition becomes second-order  and is characterized by the following (mean field theory) critical exponents:
The plan of our paper is as follows. In the next section we recapitulate the relevant thermodynamic results for four-dimensional black holes. Section 3 deals with the thermodynamics and various phase transitions of higher-dimensional Kerr-AdS black holes, whereas Section 4 studies their asymptotically flat cousins. Section 5 is devoted to analytically known 5-dimensional vacuum black holes with non-spherical horizon topology. In Section 6 AdS black rings in all dimensions are studied in the thin (ultraspinning) approximation. The aforementioned thermodynamic phase transitions are put in context with various classical instabilities in Section 7. Section 8 is devoted to the final discussion.
2. Black Holes in 4d
To set the stage for higher-dimensional results in this section we review some interesting features about the thermodynamics of four-dimensional black holes. We concentrate our attention on plotting the Gibbs free energy and discussing the associated phase diagrams. For asymptotically flat black holes we have , whereas from Equation (1), for asymptotically AdS black holes.
2.1. Asymptotically Flat Black Holes
2.1.1. Schwarzschild Solution
To start with let us consider a spherically symmetric (four-dimensional) metric element:
In particular, for the Schwarzschild black hole one has:
2.1.2. Charged Black Hole: Reissner–Nordström Solution
When the charge Q is added to the Schwarzschild black hole, we obtain the Reissner–Nordström (RN) solution, for which the metric is Equation (16) with:
2.1.3. Rotating Black Hole: Kerr Solution
The rotating black hole metric described by the Kerr solution is written:
The behavior of the Gibbs free energy is qualitatively very similar to that of the charged black hole , given by Figure 2, with fixed J replacing fixed Q. In fact the thermodynamic behavior remains qualitatively same even in the charged rotating Kerr–Newman case. As for the RN black hole we observe two branches of the Gibbs free energy. The lower one occurs for:
We conclude that in the canonical ensemble of fixed J, the nearly extremal fast spinning black holes are thermodynamically preferred to the slowly rotating ones.
2.2. AdS Black Holes
The simplest (spherical) asymptotically AdS black hole is described by the Schwarzschild-AdS spacetime. The metric takes the form Equation (16) with:
We observe a qualitatively different thermodynamic behavior when compared to the asymptotically flat Schwarzschild case. Specifically is no longer always negative: it becomes positive for large (when compared to the AdS radius) black holes:
The behavior of the Gibbs free energy G is displayed in Figure 3. We observe a minimum temperature , corresponding to , below which no black holes can exist. Above this temperature we have two branches of black holes. The upper one describes small (Schwarzschild-like) black holes with negative specific heat; these are thermodynamically unstable and cannot be in a thermal equilibrium with a thermal bath of radiation. The large () black holes at lower branch have positive specific heat and hence are locally thermodynamically stable. However, just above the Gibbs free energy of such black holes is positive and the thermal AdS space with approximately zero Gibbs free energy represents a globally preferred thermodynamic state. We refer to a thermal AdS space as a (global) AdS space coupled to a bath of thermal radiation. Since the number of particle quanta representing the radiation is relatively small, the corresponding Gibbs free energy approximately equals zero. This continues until temperature for which the black hole Gibbs free energy becomes negative, with the corresponding black hole radius given by:
Considering an extended phase space, the coexistence line of thermal radiation/large black hole phases, determined from , reads:
The behavior of this equation is displayed in the diagram in Figure 5. For each isotherm there is a maximum which occurs for . For a given temperature this precisely corresponds to ; the dashed blue curves with positive slope (and possibly negative pressures) to the left of the maximum correspond to small black holes with negative which are thermodynamically unstable, whereas solid red curves correspond to large black holes with positive that are locally thermodynamically stable.
Let us compare the fluid equation of state of the Schwarzschild-AdS black hole Equation (33) with the famous van der Waals (VdW) equation, e.g., , which is a popular two parameter closed form modification of the ideal gas law that approximates the behavior of real fluids. The VdW equation takes into account the nonzero size of fluid molecules (described by a constant and the attraction between them (described by a constant ) and is often used to capture basic qualitative features of the liquid–gas phase transition. The equation reads (setting the Boltzmann constant ):
We pause to consider topological Schwarzschild-AdS black holes. Making use of the metric Equation (16) for arbitrary k, the formulae (29) take the following more general form:
2.2.2. Charged AdS Black Hole
The charged AdS black hole metric is described by Equation (16) with:
It was first noticed in [62,63] that in a canonical (fixed charge) ensemble, charged AdS black holes allow for a first order small-black-hole/large-black-hole phase (SBH/LBH) transition which is in many ways reminiscent of the liquid/gas transition of a van der Waals fluid. This analogy becomes more complete in extended phase space . Namely, the equation of state:
We remark that the Hawking–Page phase transition as well as the SBH/LBH phase transition à la van der Waals can be also observed for asymptotically flat or de Sitter charged black holes provided these are placed in a finite cavity .
We close this subsection by noting that strongly charged small AdS black holes are subject to superradiant instabilities. For a charged scalar field coupled to Einstein–Maxwell theory a similar situation occurs: at small temperatures charged scalar hair forms and the resulting “hairy black holes” are thermodynamically preferred [90,91,92,93,94,95,96,97]. This means that in the presence of a charged scalar field, the left-most branch of small black holes in the Gibbs free diagram Figure 8, although locally thermodynamically stable, does not globally minimize the Gibbs free energy. Rather, there is another branch of hairy black holes with lower Gibbs free energy. The hair/no-hair black hole phase transition is of the second order and underlies the theory of holographic superconductors.
The thermodynamics of four-dimensional rotating AdS black holes is qualitatively similar to the charged AdS case. In fact qualitatively similar behavior occurs for charged rotating AdS black holes, see . The metric is:
The Gibbs free energy, equation of state, and the phase diagram are qualitatively similar to Figure 7, Figure 8 and Figure 9—with fixed J replacing fixed Q. For any fixed J, there is a critical point, characterized by , that can be determined numerically. For , the Gibbs free energy displays swallowtail behavior and there is a corresponding SBH/LBH first order phase transition with a phase diagram similar to Figure 9. It will be shown in the next section that in the limit of slow rotation, , the equation of state can be approximated by:
We pause to remark that, similar to the charged AdS case, the rotating AdS black holes close to extremality are unstable with respect to superradiant instabilities; the resultant objects (hairy black hole, soliton, or boson star) are expected to globally minimize the Gibbs free energy [98,99,100,101].
3. Higher-Dimensional Kerr-AdS Black Hole Spacetimes
3.1. General Metrics
The thermodynamic quantities associated with Kerr-AdS black holes were first calculated in . The mass M, the angular momenta , and the angular velocities of the horizon read:
3.2. Classical Swallowtail
The behavior of the Gibbs free energy G, Equation (61), depends crucially on the number of spacetime dimensions d and the number and ratios of the nontrivial angular momenta.
The dimension is an exception. The Gibbs free energy is in principle a function of two possible angular momenta and . However, as long as at least one of them is nontrivial, there exists a critical pressure below which we observe the qualitatively same characteristic swallowtail behavior indicating the small/large black hole phase transition, as shown in Figure 10. The situation is in fact very similar to what happens for Kerr-AdS in . The corresponding and diagrams (not shown) are reminiscent of what was observed for charged or rotating black holes in and are analogous to the van der Waals and diagrams.
The classical swallowtail behavior is also observed for equal-spinning () Kerr-AdS black holes in all dimensions.
3.3. Reentrant Phase Transition
It was observed in  that in all dimensions the singly spinning Kerr-AdS black holes admit a reentrant large/small/large black hole phase transition. This is quite exciting as similar reentrant phase transitions are commonly observed for multicomponent fluid systems, gels, ferroelectrics, liquid crystals, and binary gases, e.g., , and the observed feature could provide a toy gravitational model for such ‘everyday’ thermodynamic phenomena.
In general, a system undergoes a reentrant phase transition if a monotonic variation of one thermodynamic quantity results in two (or more) phase transitions such that the final state is macroscopically similar to the initial state. For singly spinning Kerr-AdS black holes in a certain range of pressures (and a given angular momentum) it was found  that a monotonic lowering of the black hole temperature yields a large/small/large black hole transition. The situation is accompanied by a zeroth-order phase transition, a discontinuity in the global minimum of the Gibbs free energy, a phenomenon seen in superfluidity and superconductivity , and recently in four-dimensional Born–Infeld black holes . Interestingly, such reentrant phase transitions were recently found to be absent in higher-dimensional Born–Infeld-AdS black holes, giving the four-dimensional result a special significance .
In this subsection we provide more detail about this interesting transition. We start with a review of singly spinning Kerr-AdS black holes.
For singly spinning ( and other ) Kerr-AdS black holes the metric Equation (49) significantly simplifies and reduces to:
To analyze the behaviour of for fixed J and P we proceed as follows. We first express from the (quartic) J Equation (64). Inserting the (well-behaved) solution into the expressions for G and T, consequently expressed as functions of P and , we can then plot G vs. T parametrically. Similarly, we obtain an expression for the specific heat and plot this function for fixed P and J (not displayed) to see for which regions of it becomes negative, thereby finding which branches of G are thermodynamically unstable.
The resulting Gibbs free energy in is illustrated in Figure 11; the behaviour in higher dimensions is qualitatively similar. As with the Schwarzschild-AdS case discussed in Section 2, for a given pressure P there is a minimum temperature for which the black holes can exist. Above this temperature we observe two (or more) branches of black holes. The branch corresponding to the largest black holes has positive and is locally thermodynamically stable. Its Gibbs free energy becomes negative above a certain temperature, analogous to . The new feature present in six and higher dimensions is that between and the branch of largest black holes may not correspond to a global minimum of the Gibbs free energy.
The complicated behaviour of G between and stems from the presence of a critical point occurring in the large black hole branch at .
The complicated behavior of G between and stems from the presence of a critical point occurring in the large black hole branch at . For , G resembles the curve characteristic for the Schwarzschild-AdS black hole seen in Section 2, Figure 3. However, below the situation is more subtle, as displayed in Figure 12. The reentrant phase transition occurs for a range of pressures . The pressure , see Figure 12b, is a pressure for which occurs for two locally thermodynamically stable branches of black holes. Slightly below this pressure we observe a reentrant phase transition, Figure 12c. It is connected with the discontinuity of the global minimum of the Gibbs free energy at , where we observe a zeroth order phase transition between small and large black holes. The reentrant phase transition terminates at below which the global minimum of the Gibbs free energy is again continuous and occurs for a branch of large black holes. We also note a presence of the first order SBH/LBH phase transition at in the range of pressures . As displayed in Figure 12d, at temperatures and coincide to give . At both the zeroth order and the first order phase transitions terminate and we observe a “virtual triple point”, characterized by , where three black hole phases (small, large and intermediate) coexist together. We use the term “virtual triple point” to stress a distinction from the standard triple point where three first-order phase transition coexistence lines join together. In a virtual triple point a zeroth-order phase transition coexistence line joins the first order phase transition coexistence lines where “three types” of black holes meet.
Figure 12c illustrates in detail the typical behavior of G when the reentrant phase transition is present, . Black arrows indicate increasing . If we start decreasing the temperature from, say , the system follows the lower vertical solid red curve of large stable black holes until it joins the upper horizontal solid red curve of small stable black holes—this corresponds to a first order LBH/SBH phase transition at . As T continues to decrease the system follows this upper curve until , where G has a discontinuity at its global minimum. Further decreasing T, the system jumps to the uppermost vertical red line of large stable black hole—this corresponds to the zeroth order phase transition between small and large black holes. In other words, as T continuously decreases, we observe LBH/SBH/LBH reentrant phase transition. We note that the blue dashed curve with the smallest admits black holes subject to the ultraspinning instabilities (Section 7), whereas all other branches are stable with respect to this instability.
This interesting situation is clearly illustrated in the diagram in Figure 13. There is the expected SBH/LBH line of coexistence corresponding to the liquid/gas Van der Walls case, ending in a critical point at . On the other end this line terminates at , where there is a virtual triple point where the large, small, and large again black holes coexist. For a new LBH/SBH line of coexistence emerges (see inset of Figure 13) that terminates in another “critical point" at . We note that the range for the reentrant phase transition is quite narrow and must be determined numerically. For example for and we obtain and .
Several remarks are in order:
It is well known that in dimensions there is no “kinematic” limit on how fast the singly spinning Kerr-AdS black holes can rotate. However, fast spinning black holes are subject to various dynamical instabilities, such as ultraspinning instability, superradiant instability, or bar mode instability; these will be discussed in greater detail in Section 7. It turns out that black holes which participate in the reentrant phase transition are stable with respect to the ultraspinning instability: in Figure 12 c only the blue dashed curve with the smallest admits black holes subject to this instability. Unfortunately, this is no longer true for the superradiant and bar mode instability, which “compete” with the reentrant phase transition.
One may wonder why the reentrant phase transition, which is characteristic for multicomponent systems where various phenomena compete among each other to result in reentrance, should occur at all in a “homogeneous” system of one black hole. What are the competing phenomena in our case? A possible explanation is related to the ultraspinning regime. If so, this would also explain why we see reentrance in dimensions but not in or 5 where such a regime does not exist. It is well known that as we spin the spherical black hole faster and faster, its horizon flattens and the resulting object is in many respects similar to a black brane, see the next subsection. However, the thermodynamic behaviour of black branes is completely different from that of spherical black holes. It happens that small black holes that participate in the reentrant phase transition are “almost ultraspinning” and hence possess almost black brane behavior. For this reason it may be the competition between the black brane thermodynamic behavior and the black hole thermodynamic behavior which causes the ‘multicomponency’ and results in the reentrant phase transition.
We note that all the interesting behaviour leading to the reentrant phase transition occurs for a positive Gibbs free energy, i.e., below temperature . For this reason, one may expect that the thermal AdS (see Section 2) is actually preferred thermodynamic state in this region and the various black holes participating in the reentrant phase transition are actually metastable. If so, the reentrant phase transition may actually be destroyed and one would simply observe a Hawking–Page transition between thermal radiation and large black holes at . We stress that similar arguments also apply to the four-dimensional charged AdS black hole discussed in Section 2 and the corresponding “van der Waals” phase transition.
The observed reentrant phase transition is well suited for the AdS/CFT interpretation. Although first observed  in the context of extended phase space thermodynamics, the existence of the reentrant phase transition does not require a variable cosmological constant. For any fixed value of Λ within the allowed range of pressure, the reentrant phase transition will take place. This opens up a possibility for an AdS/CFT interpretation—in particular in the dual CFT there will be a corresponding reentrant phase transition within the allowed range of N. In fact, we can fix the pressure and construct a phase diagram plotting J vs. T (Figure 15) showing that reentrant phase behaviour occurs. Hence in the dual CFT at this fixed pressure there will be a corresponding reentrant transition as the relative values of the quantities dual to the angular momenta are adjusted.
The existence of reentrant phase transitions in the context of black hole thermodynamics seems quite general. Similar phenomena have been observed in Born–Infeld black hole spacetimes . We shall also see in Section 4, that reentrant phase transitions are observed for the asymptotically flat doubly-spinning Myers–Perry black holes of vacuum Einstein gravity. Hence, neither exotic matter nor a cosmological constant (and hence AdS/CFT correspondence) are required for this phenomenon to occur in black hole spacetimes.
3.4. Equation of State
Let us now look at the equation of state. For simplicity we shall first study the case of singly spinning Kerr-AdS black holes, given by Equations (62)–(64); the case of equal spinning Kerr-AdS black holes is discussed towards the end.
The aim is to express the pressure P as a function of the specific volume v, the black hole temperature T, and the angular momentum J:
In two cases of physical interest—the slowly rotating case and the ultraspinning regime—the equation of state Equation (65) can be approximated analytically. In the former case, as with , we consider an expansion. This slow rotation expansion allows us to approximate the equation of state demonstrating the van der Waals behavior and to study the corresponding critical point. In the ultraspinning regime, , black hole geometry approaches that of a black membrane , and is characterized by completely different thermodynamic behavior. We also construct the Kerr-AdS equation of state at the very ultraspinning limit and show in Section 6 that it coincides with the equation of state for ultraspinning black rings.
3.4.1. Slow Rotation Expansion
Here we demonstrate that in the regime of slow rotation, where the equation of state can be well approximated, a van der Waals type of phase transition takes place. Within the validity of the slow rotation expansion we can analytically study the properties of the critical point as well as the critical exponents.
The slow rotation expansion is an expansion in the parameter:
To obtain the expansion for the equation of state and the Gibbs free energy we proceed as follows. First we solve the equation:
Inserting the expansion (68) and the solution (69) into the expressions for J, T and V, in Equation (64), we find:
In order to get we proceed perturbatively in ϵ, successively eliminating and ϵ from the resultant expressions. To zeroth order we find:
3.4.2. Critical Point
The critical point of the truncated equation of state Equation (72) is obtained from solving:
Introducing new variables:
3.4.3. Remark on Exact Critical Exponents in
So far we have studied the properties of the critical point using the truncated equation of state Equation (72). Although this provides a good approximation to the exact equation of state, the exact critical point will be shifted from the preceding values and one may wonder whether the critical exponents will be modified. We argue that this is unlikely to happen. For simplicity, we limit ourselves to .
For this purpose we approximate the exact equation of state by the slow rotating expansion to a higher order than Equation (72), writing:
3.4.4. Ultraspinning Expansion
In singly spinning spherical black holes admit an interesting feature, the so called ultraspinning regime, which is present already for asymptotically flat Myers–Perry black holes . This limit consists of rotating the black hole faster and faster, while keeping its mass and the cosmological constant fixed. Consequently, the black hole horizon flattens, becomes pancake-like, and the black hole enters the regime of black membrane-like behavior. For asymptotically flat Myers–Perry black holes, the ultraspinning limit is achieved when the rotation parameter . In the presence of a negative cosmological constant one instead considers (for an alternative see ), or, equivalently:
Similar to the slow rotating case discussed in the previous section, in the ultraspinning expansion we expand in small parameter ϵ, Equation (84), while we keep the pressure P and the mass M of the black hole fixed.
We again solve the equation:
To find the equation of state , as well as other quantities, to any order in ϵ, we proceed algorithmically, as in the case of the slow rotating expansion. In particular, to the zeroth order we find:
For this reason, we give up on finding a general expression and provide rather the expanded equations of state and the Gibbs free energies in selected dimensions and . The first three non-trivial terms in the expansion of equation of state read:
We display the two expansions, the slow rotating one and the ultraspinning one together with the exact quantities in Figure 17.
3.4.5. Ultraspinning Limit: Black Membranes
The full ultraspinning limit, in which ϵ given by Equation (84) actually equals zero, is obtained by keeping only the dominant terms in the ultraspinning expansion Equations (87) and (88). For Kerr-AdS black holes this was studied in  (see also  for an alternative approach). As noted above, in this limit the black hole horizon radius shrinks to zero, the black hole temperature becomes infinite, the angular momentum and the (specific) thermodynamic volume remain finite, and the horizon area (and hence also the entropy) shrinks to zero; the angular velocity of the horizon goes as .
These ultraspinning black holes share many properties with black membranes, including the Gregory–Laflamme instability . In fact, when the limit is accompanied with the following change of coordinates:
3.4.6. Equal Spinning AdS Black Holes
Let us briefly mention how the equation of state gets modified for the equal spinning Kerr-AdS black holes. Equal spinning Kerr-AdS black holes are characterized by equal angular momenta:
Repeating the procedure for the slow rotation expansion, , as we did for the singly spinning case, we recover:
3.5. An Analogue of Triple Point and Solid/Liquid/Gas Phase Transition
We now turn to a study of the thermodynamic behaviour of multi-spinning Kerr-anti de Sitter black holes with two fixed (non-trivial) angular momenta and . Following , depending on the ratio:
Specifying to , we consider the Gibbs free energy , obtained from the expression (61) by eliminating parameters in favor of using Equations (1), (53) and (54). An analytic solution is not possible since higher-order polynomials are encountered and so we proceed numerically: for a given and , we solve equations in Equation (53) for and and thence calculate the values of T and G using Equations (54) and (61), yielding a diagram. Once the behavior of G is known, we compute the corresponding phase diagram, coexistence lines, and critical points in the plane. We display our results in Figure 18, Figure 19 and Figure 20. Since the qualitative behavior of the system depends only on we set everywhere .
3.5.1. Solid/Liquid Analogue
For the behaviour of G, displayed for in Figure 18, completely changes to the behaviour discussed in the previous subsection, cf. Figure 11 and Figure 12. Namely the unstable branch of tiny hot black holes on the right of the diagram in Figure 11 disappears and a new branch of (locally) stable tiny cold black holes appears to the left. The “no black hole region” is eliminated and the situation is very similar to what happens when a small charge is added to a Schwarzschild black hole, cf. Section 2. The zeroth-order phase transition is “replaced” by a “solid/liquid”-like phase transition of small to large black holes. Although in this range of angular momenta G admits two critical points (one at and another at ), only the one at occurs for stable black holes that minimize G globally. Consequently we observe one phase transition between small and large black holes. The corresponding coexistence line (not displayed) terminates at a critical point characterized by and is qualitatively similar to Figure 9. As q decreases, the critical point occurs for larger and larger pressures and temperatures and in the limit we find ‘an infinite’ coexistence line, similar to what happens for a solid/liquid phase transition. This is indicated in Figure 21.
3.5.2. Triple Point and Solid/Liquid/Gas Analogue
For fixed a new phenomenon occurs: for a certain range of pressures we observe two swallowtails in the stable branch of black holes, see Figure 19, corresponding to a small/intermediate/large black hole phase transition. As pressure decreases, the two swallowtails move closer together until, at , the corresponding first order phase transitions coincide—giving a triple point. Below , only one of the first order phase transitions continues to occur in a stable branch. The whole situation is reminiscent of the solid/liquid/gas phase transition.
Let us describe in more detail what happens for fixed as pressure decreases, illustrated in Figure 19 and Figure 20. For very large pressures there are no swallowtails and only one branch of locally stable black holes with positive exists. Decreasing the pressure, at a critical point occurs; slightly below we observe a first order phase transition between large and small black holes. This continues to happen until at we observe another critical point. Slightly below the system exhibits two swallowtails in the stable branch of black holes and thence two first order phase transitions between various size black holes are present: from small to intermediate and from intermediate to large. As P decreases even further, the two swallowtails move closer to each other, until at the corresponding first order phase transitions coincide—we observe a triple point. Below only one of the first order phase transitions continues to occur in the stable branch of black holes. The corresponding phase diagram displaying all the features is depicted in Figure 20; in many respects it is reminiscent of a solid/liquid/gas phase diagram.
So far we considered a fixed parameter . Let us now look at what happens as we allow q to vary, see Figure 21. We already know that for we observe one first order phase transition terminating at a critical point characterized by , as described in the previous subsection. Qualitatively new feature occurs at : a triple point and a second critical point emerge from the coexistence line at and . As q increases, the triple point moves away from the second critical point; the values of and decrease, while the values of and almost do not change. A small/intermediate/large black hole phase transition may occur and the situation resembles that of the solid/liquid/gas phase transition. At the same time as q increases, the first critical point moves towards the triple point ( rapidly decreases). At both critical points occur at the same pressure , whereas . Increasing q even further, the first critical point moves closer and closer to the triple point and finally for the two merge at . Above only the second critical point remains.
We remark that, similar to the reentrant phase transition, the analogue of a solid/liquid/gas phase transition as well as of the triple point occurs for any fixed P in the allowed range. Once again, the possibility for an AdS/CFT interpretation emerges: a corresponding phase transition may occur in the dual CFT within the allowed range of N. To stress this point we plot in Figure 22 a phase diagram, showing the existence of small/intermediate/large black hole phase transition and the corresponding triple point for a fixed pressure.
3.5.3. Van Der Waals Behavior
For only one small/large black hole coexistence line exists and terminates at a corresponding critical point. We observe an analogue of a van der Waals “liquid/gas” phase transition.
In fact, the situation is a little more subtle. For , similar to the ‘solid/liquid analogue’, the second critical point still exists, but occurs for the locally unstable branch of black holes with negative heat capacity that does not globally minimize the Gibbs free energy. Hence, only one phase transition is physical. For this second critical point completely disappears. In both cases we observe a “standard liquid/gas” van der Waals phase transition, qualitatively similar to case and the case of equal spinning Kerr-AdS black holes in any dimension .
4. Myers–Perry Solutions
We now turn to consider the d-dimensional multi-rotating black holes with spherical horizon topology for which the cosmological constant and hence the pressure vanishes, described by the Myers–Perry solution . The most interesting observation is that even in this asymptotically flat case we find that reentrant phase transitions can take place.
The Myers-Perry (MP) solutions are the most general rotating black hole solutions (with zero NUT charge) to the vacuum Einstein equations of spherical horizon topology in d-dimensions . The metric as well as its associated thermodynamic quantities, particularly thermodynamic volume, can be obtained as the (or ) limit of the general Kerr-AdS case discussed in previous section. In Boyer–Lindquist coordinates the metric reads:
4.1. Five-Dimensional Case
Let us first consider the MP black hole solution, which in general admits two angular momenta and . As with the Kerr-AdS case, we consider the ratio:
4.2. Reentrant Phase Transition
Next we consider the MP solution with two angular momenta and . As before, we consider the ratio Equation (107) and without loss of generality set . We now observe various phenomena reflected in the behavior of the Gibbs free energy for various values of q, as shown in Figure 24. For we observe a singly branch of thermodynamically unstable black holes with negative ; qualitatively same behavior occurs for singly spinning MP black holes in all . For larger values of q more interesting phase behavior occurs.
For a new branch of small thermodynamically stable black holes appears, similar to the Kerr case in , except there is a swallowtail in the upper unstable branch of large black holes. In this range of q’s the global minimum of G corresponds to small fast spinning black holes that exist up to some maximum temperature . Above this temperature no black holes are possible; this “no black hole region” is shown in Figure 25. At , Figure 24c, the swallowtail in the upper unstable branch reaches the lower stable branch and we observe the entrance of reentrant phase transition.
The typical behavior of G for is displayed in Figure 24d (enlarged in Figure 26) and indicates the presence of the reentrant phase transition. If we start increasing the temperature from, say , the system follows the lower left horizontal solid red curve, describing small black holes. At it undergoes a first order phase transition and follows the lowest stable branch of large black holes until, at , there is a cusp in G and this branch terminates. Beyond the cusp, as the temperature increases even further, there is a discontinuity in the global minimum of G, yielding a zeroth-order phase transition to the upper red branch of small black holes. In other words, we observe a SBH/LBH/SBH reentrant phase transition. This reentrant phase transition, however, stands in contrast to the situation in the Kerr-AdS case discussed in the previous section, cf. Figure 12, Figure 13 and Figure 15. In the Kerr-AdS case the black holes undergoing the reentrant phase transition were singly spinning, the phase transition was of the LBH/SBH/LBH type with and the ‘no black hole region’ occurred for small temperatures, whereas in the MP case the black holes must be doubly spinning, the phase transition is of the SBH/LBH/SBH type with , and the “no black hole region” occurs for high temperatures.
The reentrant phase transition terminates at as shown in Figure 24e. In between and there is a first order SBH/LBH phase transition illustrated in Figure 24f. This transition terminates at a critical point at , Figure 24g. Above the system displays “Kerr-like” behavior with no phase transitions, as seen in Figure 24h,i. The overall situation is summarized in the phase diagram in Figure 25.
The important lesson of this section is that we showed that the reentrant phase transition can take place in black hole spacetimes without the cosmological constant, that is, when the system has zero pressure. A gauge/gravity interpretation of this phenomenon, similar to the asymptotically AdS case, remains to be explored.
5. Five-Dimensional Black Rings and Black Saturns
In this section we look at the thermodynamics of other analytically known black hole objects with horizon topologies more complicated than spherical, specifically singly spinning black rings and black saturns. Unfortunately, due to the lack of a generating technique for , only vacuum solutions are known. For this reason, there is no effective thermodynamic way to calculate the thermodynamic volume and the fluid equation of state is trivial; we limit ourselves to studying the Gibbs free energy for these objects and comparing it to the Gibbs free energy of the MP solution. In the next section we shall try to circumvent the lack of knowledge of exact asymptotically AdS black ring solutions by considering the approximate thin AdS black rings constructed via the blackfold approach. In the given (fast spinning approximation) this will allow us to calculate the thermodynamic volume, verify the reverse isoperimetric inequality, and construct the corresponding equation of state.
5.1. Singly Spinning Black Ring
The exact solution for a singly spinning black ring in five dimensions was constructed by Emparan and Reall  and reads:
The thermodynamic quantities are given by:
It is not by any means obvious that the two kinds of black objects, the spherical MP black hole and the toroidal black ring, may be treated as “two phases” of the same system. Notwithstanding this concern, we plot the Gibbs free energy for the black ring:
Our considerations imply that spherical black holes are thermodynamically preferable to fat black rings. This is supported by the study of the horizon areas, e.g., [110,111,112,113,114,115]. It is an open question as to whether there actually might be a phase transition from fat black rings to small MP black holes. Likewise one can ask if all thin rings, besides being thermodynamically unstable, are also classically unstable, see . If so then are the small MP solutions the only stable vacuum black holes in five dimensions?
5.2. Black Saturn
This exact multi-black hole solution was constructed by Elvang and Figueras  and its “phases” and the first law were studied in . The solution consists of a central rotating spherical black hole surrounded by a black ring. The solution depends on one dimensionful parameter L, and the dimensionless parameters . The thermodynamic quantities associated with the two horizons are (using an abbreviation ):
To ensure thermodynamic and mechanical equilibrium, we require:
6. Thin Black Rings in AdS
6.1. Review of the Construction
Following  we briefly recapitulate the perturbative construction of an asymptotically AdS singly spinning thin black ring with horizon topology using the blackfold approach [78,114]. Such a ring is characterized by the two radii R, corresponding to , and , associated with . The idea of the construction is to “bend” a straight thin boosted black string of width into a circular shape of radius R. In the far region the ring is described by a distributional source of energy-momentum of a boosted string in the given (global) AdS background:
In terms of the dimensionless parameter:
The thermodynamic quantities of asymptotically flat thin black rings in higher dimensions  are obtained from these expressions by setting .
To plot the Gibbs free energy:
The specific heat of AdS black rings can be analyzed in the same way as for black holes. We find that in a given approximation is always negative. Hence higher-dimensional AdS thin black rings in this (fast spinning) approximation are thermodynamically unstable. We expect that this may not be longer true for the full, not necessarily thin, exact black ring solution, and perhaps “fat AdS ring analogues” with positive exist in higher dimensions, cf. discussion of exact asymptotically flat black rings in the previous section.
6.3. Thermodynamic Volume
Let us briefly comment on the structure of formula for , Equation (126). We first note that a naive geometric volume of a string “bent to” a radius R goes as:
6.4. Isoperimetric Inequality
It was conjectured in  that a reverse isoperimetric inequality holds for a thermodynamic volume of any asymptotically AdS black holes, the statement being verified for a variety of (charged rotating) black holes with the horizon of spherical topology. Here we show that this conjecture remains true for asymptotically AdS black rings, at least in the thin ring (fast spinning) approximation.
6.5. Equation of State
In order to find the equation of state for our black rings, we start from the expressions for J, T, and V in Equation (122). Expressing from the J equation as in (124), the V equation becomes a quadratic equation for , and gives:
The T Equation (122):
Note that, in the region of validity of our approximation we must have:
The resulting diagram is “trivial”, as depicted in Figure 31 for . In higher dimensions the situation is similar. The critical point, if it exists for AdS black rings, is outside of the domain of validity of our approximation. To see this we will have to wait until the exact AdS black ring solution is known.
Similar to the spherical black hole case we shall now study the ultraspinning regime. (The regime of slow rotation is excluded by the blackfold approximation).
6.6. Ultraspinning Expansion
By inspection, the ultraspinning limit, for which , corresponds to the limit where we keep the mass of the black ring M fixed and let:
To write down the equation of state in this limit we proceed similar to the black hole case. Namely, we express from the M Equation (122) :
Note the interesting coincidence that the first term, , in the expansion coincides with the ultraspinning limit of the Kerr-AdS black hole equation of state. Therefore both the ultraspinning Kerr-AdS black holes and AdS black rings have infinite temperature and share the “same equation of state”.
7. Beyond Thermodynamic Instabilities
Thus far we have considered only the thermodynamic behavior of black holes, using the global minimum of the Gibbs free energy (and the positivity of the specific heat) as a criterion for thermodynamic stability. However rotating black holes are subject to other kinds of (classical) instabilities. Ultraspinning instabilities can lead to bifurcations to new stationary black hole families [107,119,120,121,122,123]. Superradiant instabilities [124,125,126] amplify waves that scatter off of the AdS black hole, extracting its rotational energy and driving it to a new state.
A third instability is the so called (non-axisymmetric) bar-mode instability [107,127,128,129], which possibly occurs for both MP and Kerr-AdS black holes. However, nothing is known about the latter case, and even in the former case the mechanism as well as the endpoint of such instability (apart from the slowly spinning case which results in a slowly spinning MP black hole) is far from being well understood. It may result in the irradiation of the excessive angular momenta or even a fragmentation into a multiple black hole configuration . We shall not consider this instability in further discussion.
7.1. Ultraspinning Instability
Ultraspinning (axisymmetric) instabilities occur for both asymptotically flat MP and Kerr-AdS black holes in all dimensions . It is related to the fact that in the angular momentum of the black hole is not limited from above and in principle can grow all the way to infinity. In fact, even when the angular momenta are limited (for instance in the equal spinning MP case) the black hole can enter the ultraspining regime. Consequently as the angular momentum grows larger, the black hole horizon flattens, becomes pancake like, and the black hole enters the region of black membrane like behavior (see also Section 3). The resulting object is subject to the Gregory–Laflamme instability and the axisymmetric perturbations can lead to bifurcations to new stationary black hole families. In the absence of a decoupled master equation for gravitational perturbations in higher dimensions, the final endpoint as well as the exact onset of ultraspinning instabilities have to be investigated numerically [119,120,121,122,123]. However, for our purposes we just need to know when such an instability approximately “kicks in” and may have some effect on the thermodynamic considerations studied in previous sections.
The method for investigating ultraspinning instabilities consists of finding zero modes for stationary perturbations that preserve rotational symmetries. The onset of such instabilities can be assigned a “harmonic number” k. The modes respectively correspond to metric perturbations with nodes. These are the so called thermodynamic zero modes and can be determined from the thermodynamic Hessian, as we shall discuss below. However, these modes do not correspond to bifurcations to new families of stationary black holes; they can only change the black hole into another black hole of the same family, for example an MP black hole into another MP black hole.
The actual bifurcations occur for and higher; the case of corresponds to a new branch of black holes with a central pinch. For higher k further axisymmetric pinches appear. These cannot be obtained from thermodynamic considerations. However it has been conjectured  that the zero modes with can only occur for rotations higher than that of the modes. The surface, calculable from thermodynamic considerations, is called the ultra-spinning surface: it encloses the region where the thermodynamic Hessian has less than two negative eigenvalues. For higher rotations the black holes are subject to ultraspinning instabilities leading to bifurcations to new black hole families .
Let us first illustrate this for a singly spinning MP black hole.
7.1.1. Bifurcations of Singly Spinning MP Black Holes
A simple criterion  for the onset of ultraspinning instabilities of singly spinning MP black holes involves a consideration of the behaviour of the temperature T for a fixed mass M. For slowly spinning black holes this decreases, as is familiar for the Kerr solution, and in reaches zero. However, for the temperature instead reaches its minimum and diverges in the ultraspinning limit. This temperature growth is typical for black membranes.
Let us define a dimensionless parameter , in terms of which we have, cf. J Equation (106):
The actual onsets of instabilities for perturbations preserving the original Killing symmetries of singly spinning MP black holes in have been studied  and, in terms of a new parameter , where , were found to be:
7.1.2. Thermodynamic Argument and Other Examples
For more complicated black hole spacetimes, such as doubly spinning Kerr-AdS black holes, the numeric results are not available (see, however,  for the multiply spinning MP case). However, the ultraspinning surface can be obtained from the following thermodynamic argument, e.g., [60,120,121,122] (see also ).
Let I be the (grand-canonical ensemble) Euclidean action, dependent on parameters that completely characterize the black hole in this ensemble: . Then we consider the Hessian of the action, given by  (keeping and T fixed):
For every asymptotically flat vacuum black hole possesses at least one negative eigenvalue and so all such black holes are locally thermodynamically unstable . For singly spinning MP black holes, as their rotation increases, we obtain an additional thermodynamic instability at precisely , as given by Equation (146) above. This determines the ultraspinning surface .
For singly spinning Kerr-AdS black holes in and , one of the eigenvalues of the Hessian matrix is always positive and the second one changes sign. We therefore have the thermodynamic zero mode at , given by Equation (151) below; the zero mode does not exist and the black holes are ultraspinning stable.
For the singly spinning Kerr-AdS case one can show that two eigenvalues of the Hessian matrix change sign, giving two thermodynamic zero modes, and . Solving for the zero eigenvalues of in Equation (149), taking , we obtain :
We demonstrated in Section 4 that an RPT also occurs for the doubly spinning MP black holes. Such black holes always possess a negative mode. The thermodynamic argument gives the following formula for the appearance of the zero mode:
Turning finally to the doubly spinning Kerr-AdS black holes and the corresponding ‘solid/liquid/gas’ phase transition studied in Section 3, no simple analytic formula for the appearance of thermodynamic zero modes exists, and we proceed numerically as displayed in Figure 34. We find that the part of the Gibbs diagram important for the existence of multiple first order phase transitions and possibly a triple point occurs in the ultraspinning region and is subject to the superradiant instability. Hence the observed thermodynamic phenomena have to “compete” with both the ultraspinning and the superradiant classical instabilities.
We pause here to note another suggestion for characterizing ultraspinning instabilities , based on the relation for the adiabatic compressibility κ. For MP black holes κ can be entirely expressed in terms of quadratic and quartic Casimir invariants of . For non-rotating black holes, κ vanishes and the system is incompressible. However if some of the are kept small while others diverge, then κ diverges and the black hole equation of state becomes soft. The suggestion is that large compressibility signals the onset of ultraspinning instability.