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

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## Abstract

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## 1. Introduction

#### 1.1. Canonical Ensemble and Phase Transitions

#### 1.2. Thermodynamic Volume

#### 1.3. Equation of State

## 2. Black Holes in 4d

#### 2.1. Asymptotically Flat Black Holes

#### 2.1.1. Schwarzschild Solution

**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.

#### 2.1.2. Charged Black Hole: Reissner–Nordström Solution

**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.

#### 2.1.3. Rotating Black Hole: Kerr Solution

#### 2.2. AdS Black Holes

#### 2.2.1. Schwarzschild-AdS

**Figure 3.**Gibbs free energy: Schwarzchild-AdS black hole. When compared to the asymptotically flat Schwarzschild case (Figure 1) for $P>0$ the Gibbs free energy acquires a new thermodynamically stable branch of large black holes. For $T>{T}_{\mathrm{HP}}$ 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 Equation (32) 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 Equation (33) is displayed for various temperatures. For a given temperature the maximum occurs at $v=2{r}_{0}$. The dashed blue curves correspond to small unstable black holes. The red curves depict the stable large black hole branch; we observe ‘ideal gas’ behavior for large temperatures.

**Figure 6.**$P-v$ diagram of van der Waals fluid. The temperature of isotherms decreases from top to bottom. The two upper dashed lines correspond to the “ideal gas” phase for $T>{T}_{c}$, the critical isotherm $T={T}_{c}$ is denoted by the thick solid line, lower solid lines correspond to temperatures smaller than the critical temperature; for $T<{T}_{c}$ parts of the isotherms are actually unphysical, and must be replaced by a constant pressure line according to the Maxwell’s equal area prescription [15]. Constants a and b were set equal to one.

#### 2.2.2. Charged AdS Black Hole

**Figure 7.**Equation of state: charged AdS black hole. The temperature of isotherms decreases from top to bottom. The two upper dashed lines correspond to the “ideal gas” one-phase behavior for $T>{T}_{c}$, the critical isotherm $T={T}_{c}$ is denoted by the thick solid line, lower solid lines correspond to temperatures smaller than the critical temperature. We have set $Q=1$. $P-v$ diagram for the Kerr-AdS black hole with $J=1$ is qualitatively similar.

**Figure 8.**Gibbs free energy: charged AdS black hole. Characteristic swallowtail behavior is observed for $P<{P}_{c}$, corresponding to a small/large black hole phase transition. An unstable branch of the Gibbs free energy is displayed in dashed blue line. We have set $Q=1$. The behavior of G for Kerr-AdS black hole with $J=1$ is qualitatively similar.

**Figure 9.**Phase diagram: charged AdS black hole. The coexistence line of the SBH/LBH phase transition of the charged AdS black hole system in $(P,T)$-plane is displayed. The critical point is highlighted by a small circle at the end of the coexistence line. The phase diagram for a Kerr-AdS black hole with $J=1$ is qualitatively similar.

#### 2.2.3. Kerr-AdS

## 3. Higher-Dimensional Kerr-AdS Black Hole Spacetimes

#### 3.1. General Metrics

#### 3.2. Classical Swallowtail

**Figure 10.**Gibbs free energy: Kerr-AdS in $d=5$. The Gibbs free energy of equal spinning Kerr-AdS black hole is displayed for ${J}_{1}={J}_{2}=1$. The situation is very similar to what happens for charged or rotating black holes in four dimensions. A qualitatively similar swallowtail behavior, indicating a SBH/LBH phase transition, is observed for any ratio of the two angular momenta, as long as, at least one of them is non-trivial. Equal spinning Kerr-AdS black holes in all $d\ge 5$ demonstrate the qualitatively same feature.

#### 3.3. Reentrant Phase Transition

**Figure 11.**Gibbs free energy: singly spinning Kerr-AdS in $d=6$. The behavior of G for $d\ge 6$ is completely different from that of $d<6$, cf., Figure 8 and Figure 10. The Gibbs free energy for a singly spinning black hole is displayed for increasing pressure (from left to right). As with Schwarzschild-AdS black holes, for $P\ge {P}_{c}$, the (lower) large black hole branch is thermodynamically stable whereas the upper branch is unstable. For $P={P}_{c}$ we observe critical behavior. In a range of pressures $P\in ({P}_{t},{P}_{z})$, we observe a discontinuity in the global minimum of G or zeroth-order phase transition, signifying the presence of a reentrant phase transition. This feature is qualitatively similar and present for all $d\ge 6$ singly spinning Kerr-AdS black holes. For $P<{P}_{t}$ only one branch of stable large black holes exists. Note that the specific heat is positive in zones where the Gibbs free energy reaches a global minimum.

**Figure 12.**Reentrant phase transition: singly spinning Kerr-AdS black holes in $d=6$. The figure (a close up of Figure 11) illustrates the behaviour of the Gibbs free energy for various pressures, $P=\{0.073,Pz=0.0579,0.0564,{P}_{t}=0.0553\}$ (from top to bottom), when the reentrant phase transition is present. We have set $J=1$, for which ${P}_{c}=0.0958$. Solid-red/dashed-blue lines correspond to ${C}_{P}$ positive/negative respectively. (a) Describes a LBH/SBH first order phase transition at temperature ${T}_{1}$ as indicated by a dashed vertical line. (b) Displays the behaviour of G for $P={P}_{z}$ for which the ‘upper peak’ occurs at the same temperature $T={T}_{z}$ as the lower peak—indicating the ‘entrance’ of the reentrant phase transition. The LBH/SBH first order phase transition is still present and occurs at $T={T}_{1}>{T}_{z}$. (c) Displays a typical behaviour of G when the reentrant phase transition is present, $P\in ({P}_{t},{P}_{z})$. Black arrows indicate increasing ${r}_{+}$. (d) Shows the behaviour for $P={P}_{t}$ when the lower peak merges with the vertical red line. For this pressure temperature of the zeroth-order phase transition ${T}_{0}$ coincides with the temperature of the first-order phase transition ${T}_{1}$, we call it ${T}_{t}={T}_{0}={T}_{1}$.

**Figure 13.**Reentrant phase transition in $P-T$ plane. The coexistence line of the first order phase transition between small and large black holes is depicted by a thick black solid line for $J=1$ and $d=6$. It initiates from the critical point $({P}_{c},{T}_{c})$ and terminates at virtual triple point at $({P}_{t},{T}_{t})$. The red solid line (inset; shown in detail in figure 14) indicates the “coexistence line” of small and intermediate black holes, separated by a finite gap in G, indicating the reentrant phase transition. It commences from $({T}_{z},{P}_{z})$ and terminates at $({P}_{t},{T}_{t}$). The “No BH region", given by ${T}_{\mathrm{min}}$, is to the left of the dashed oblique curve, containing the $({T}_{z},{P}_{z})$ point. A similar figure is valid for any $d\ge 6$.

- (a)
- It is well known that in $d\ge 6$ 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 ${r}_{+}$ 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.
- (b)
- 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 $d\ge 6$ dimensions but not in $d=4$ 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.
- (c)
- We note that all the interesting behaviour leading to the reentrant phase transition occurs for a positive Gibbs free energy, i.e., below temperature ${T}_{\mathrm{HP}}$. 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 $T={T}_{\mathrm{HP}}$. 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.
- (d)
- The observed reentrant phase transition is well suited for the AdS/CFT interpretation. Although first observed [25] 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.
- (e)
- 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 [16]. 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.

**Figure 15.**Reentrant phase transition in $J-T$ plane. The coexistence line of the first order phase transition between small and large black holes is depicted by a thick black solid line. The red curve denotes a zeroth-order phase transition between small and intermediate (large) black holes. The dashed line outlines the ‘no black hole region’. The diagram is displayed for fixed $l\approx 2.656$ and $d=6$.

#### 3.4. Equation of State

**Figure 16.**Exact $P-v$ diagram: Kerr-AdS black holes. In $d=5$ (inset) we observe standard van der Waals like behavior, similar to $d=4$ in Figure 7. In $d\ge 6$ the $P-v$ diagram is more complex than that of the standard van der Waals, reflecting the interesting behaviour of the Gibbs free energy and a possible reentrant phase transition. Namely, whereas for large v the isotherms approximate the van der Waals behavior, for small v, similar to Schwarzschild-AdS in Figure 5, the isotherms turn and lead back to a region with negative pressures.

#### 3.4.1. Slow Rotation Expansion

**Figure 17.**The two approximations. Exact critical and subcritical isotherms, depicted by black crosses, are compared to the slow spinning expansion (72) denoted by red curves and the ultraspinning expansion (91) denoted by a blue curve; value of $r+$ decreases right to left. Obviously, the slow rotation approximation is valid for ${r}_{+}\to \infty $ whereas the ultraspinning one for ${r}_{+}\to 0$. In the regime of very small P (not displayed), v grows arbitrarily large for any temperature T, according to Equation (89). Note that the ultraspinning black holes correspond to the upper branch in the Gibbs free energy in Figure 11, which is unstable. We have set $d=6$ and $J=1$.

#### 3.4.2. Critical Point

#### 3.4.3. Remark on Exact Critical Exponents in $d=4$

#### 3.4.4. Ultraspinning Expansion

#### 3.4.5. Ultraspinning Limit: Black Membranes

#### 3.4.6. Equal Spinning AdS Black Holes

#### 3.5. An Analogue of Triple Point and Solid/Liquid/Gas Phase Transition

#### 3.5.1. Solid/Liquid Analogue

**Figure 18.**Gibbs free energy for $q=0.005$ Kerr-AdS black hole in $d=6$ is displayed for decreasing pressures (from top to bottom). The horizon radius ${r}_{+}$ increases from left to right. The uppermost isobar corresponds to $P={P}_{c}=4.051$; for higher pressures only one branch of (stable) black holes with positive ${C}_{P}$ exists. The second uppermost isobar displays the swallowtail behaviour and implies the existence of a first order phase transition. For $P={P}_{v}\approx 0.0958$ another critical point emerges but occurs for a branch that does not minimize G globally. Consequently, out of the two swallowtails for $P<{P}_{v}$ only one occurs in the branch globally minimizing G and describes a “physical” first order phase transition.

#### 3.5.2. Triple Point and Solid/Liquid/Gas Analogue

**Figure 19.**Gibbs free energy for $q=0.05$ Kerr-AdS black hole in $d=6$ is displayed for various pressures (from top to bottom) $P=0.260,0.170,0.087,0.0642,0.015$. The horizon radius ${r}_{+}$ increases from left to right. The uppermost isobar corresponds to $P={P}_{{c}_{1}}=0.260$; for higher pressures only one branch of stable black hole with positive ${C}_{P}$ exists. The second uppermost isobar displays the swallowtail behavior, implying a first order phase transition. The third isobar corresponds to P between ${P}_{{c}_{2}}=0.0957$ and ${P}_{{c}_{1}}$ for which we have “two swallowtails”. For such pressures there are two first order phase transitions. The fourth isobar displays the tricritical pressure ${P}_{tr}=0.0642$ where the two swallowtails “merge” and the triple point occurs. Finally the lower-most isobar corresponds to $P<{P}_{tr}$.

**Figure 20.**$P-T$ diagram for $q=0.05$. The diagram is analogous to the solid/liquid/gas phase diagram. Note however that there are two critical points. That is, small/intermediate black hole coexistence line does not extend to infinity but rather terminates, similar to the “liquid/gas” coexistence line, in a critical point.

**Figure 21.**Critical pressures and variable q. Depending on the value of q, we observe one critical point or two critical points and one triple point. This figure displays the corresponding ${P}_{{c}_{1}}$, ${P}_{{c}_{2}}$, and ${P}_{tr}$. For $q<{q}_{1}$ only one critical point occurs: as $q\to 0$, ${P}_{{c}_{1}}$ rapidly diverges to infinity and the coexistence line becomes infinitely long—this is an analogue of a solid/liquid phase transition. Between ${q}_{1}$ and ${q}_{2}$ we observe two critical points at ${P}_{{c}_{1}}$ and ${P}_{{c}_{2}}$, displayed by solid thick black line and thin dashed line respectively, and a triple point displayed by the red solid curve. As q increases, ${P}_{{c}_{1}}$ rapidly decreases until it meets ${P}_{tr}$ at $q={q}_{2}$ where it terminates; in between there exists $q={q}_{eq}$ where ${P}_{{c}_{1}}={P}_{{c}_{2}}$. Above ${q}_{2}$ only the second critical point at ${P}_{{c}_{2}}$ exists.

**Figure 22.**Kerr-AdS analogue of solid/liquid/gas phase transition in $q-T$ plane. The diagram is displayed for fixed pressure $l\approx 2.491$.

#### 3.5.3. Van Der Waals Behavior

## 4. Myers–Perry Solutions

#### 4.1. Five-Dimensional Case

**Figure 23.**Gibbs free energy: $d=5$ MP black holes. The lower curve corresponds to the singly-spinning $(q=0)$ MP black hole whereas the upper curve is for equal spinning ($q=1$) one. The qualitative behavior remains the same irrespective of the value of q and is similar to $d=4$ Kerr case, cf. Figure 2.

#### 4.2. Reentrant Phase Transition

**Figure 24.**Reentrant phase transition: $d=6$ doubly spinning MP black hole. The Gibbs free energy is displayed for various ratios of angular momenta $q={J}_{2}/{J}_{1}=\{0,0.05,0.1082,0.112,0.11715,0.122,0.1278,0.5,1\}$ (from top to bottom). Solid-red/dashed-blue lines correspond to ${C}_{P}$ positive/negative respectively. (

**a**) Shows the thermodynamically unstable branch of $q=0$ black holes with negative ${C}_{P}$; qualitatively similar behaviour occurs for singly spinning MP black holes in all $d\ge 6$. When a second angular momentum is switched on; (

**b**) a new branch of small thermodynamically stable black holes appears, similar to the $d=4$ Kerr case. At $q={q}_{t}\approx 0.1082$; (

**c**) the swallowtail in the upper unstable branch touches the lower stable branch, and we observe the appearance of a reentrant phase transition; (

**d**) displays typical behavior of G when the reentrant phase transition is present, $q\in ({q}_{t},{q}_{z})$, ${q}_{z}\approx 0.11715$. Black arrows indicate increasing ${r}_{+}$; (

**e**) shows $q={q}_{z}$ at which the reentrant phase transition terminates, while the first order phase transition at $T={T}_{1}$ is still present. The first order phase transition continues to occur; (

**f**) and (

**g**) until at $q={q}_{c}\approx 0.1278$ we observe a critical point at ${T}_{c}\approx 0.18577$. Above ${q}_{c}$ the system displays “Kerr-like” behavior with no phase transitions.

**Figure 25.**$q-T$ phase diagram: $d=6$ MP black holes. The first order phase transition between small and large black holes is displayed by solid black curve; it terminates at a critical point characterized by $({T}_{c},{q}_{c})$. The solid red curve corresponds to the 0th-order phase transition between large and intermediate (small) black holes; it occurs for $q\in ({q}_{t},{q}_{z})$ and $T\in ({T}_{t},{T}_{z})$. In this range of q’s reentrant phase transition is possible. The “no BH region” is outlined by thin dashed black curve.

**Figure 26.**Enlargement of Figure 23d to better illustrate the typical behavior of G when the reentrant phase transition is present.

## 5. Five-Dimensional Black Rings and Black Saturns

#### 5.1. Singly Spinning Black Ring

**Figure 27.**Gibbs free energy: black ring vs. black hole. The Gibbs free energy of black rings is compared to the Gibbs free energy of an MP black hole for $J=1$. For temperatures below ${T}_{r}$ we observe three branches of black holes: small MP black holes displayed by thick solid red curve have positive ${C}_{P}$ and the lowest Gibbs free energy, fat black rings (depicted by the thick black solid curve slightly above the red curve) have positive ${C}_{P}$ and are locally thermodynamically stable, and large MP black holes (upper branch depicted by the thin blue dashed curve) with negative ${C}_{P}$. Above ${T}_{r}$ black holes are no longer possible (G has a cusp there) and only a branch of thin black rings (displayed by a thick solid blue curve) with ${C}_{P}<0$ exists. The approximate thin black ring solution, constructed using the blackfold approach in the next section, is displayed by the thin dashed black curve. We observe that the approximation works well for high temperatures, whereas it is completely off for low temperatures. We also note that the exact Gibbs free energy is actually lower than the one given by the blackfold approximation.

#### 5.2. Black Saturn

**Figure 28.**Gibbs free energy: $5d$ vacuum black holes. The Gibbs free energy of black saturn in thermodynamic and mechanical equilibrium (red curve) is compared to the Gibbs free energy of black ring (blue curve) and the MP spherical black hole (black curve). For temperatures below ${T}_{r}$, all three branches are possible, however, the spherical black hole branch globally minimizes the Gibbs free energy. We have set $J=1$.

## 6. Thin Black Rings in AdS

#### 6.1. Review of the Construction

#### 6.2. Thermodynamics

**Figure 29.**Gibbs free energy: thin AdS black rings vs. Kerr-AdS black holes in $d=6$. The Gibbs free energy of thin black rings in the blackfold approximation is displayed for $P=\{0.2,0.073,0.0564,0.04\}$ (the upper blue, red, black, and green curves that emerge from the left top corner) and is compared to the corresponding exact Gibbs free energy of Kerr-AdS black holes for the same pressures, cf. Figure 11. For large temperatures the branch of large Kerr-AdS black holes is thermodynamically preferred, whereas in the regime of small temperatures we cannot trust our blackfold approximation.

#### 6.3. Thermodynamic Volume

**Figure 30.**Onsets of ultraspinning instabilities: $d=6$ MP black holes. The Gibbs free energy of a singly spinning MP black hole with $J=1$ (displayed by solid colored curves) is compared to the Gibbs free energy of a thin black ring constructed in the blackfold approximation (displayed by a black thin dashed curve); ${r}_{+}$ decreases from left to right. The onsets of ultraspinning instabilities Equation (147) are displayed by various colors as follows. The black solid line corresponds to MP black holes outside of the ultraspinning region (with only the $k=0$ mode present). The $k=1$ zero mode appears for ${r}_{+}={r}_{1}\approx 0.605$ indicated by a point where solid red and black curves join together; the appearance of this zero mode can be predicted from thermodynamic considerations. The actual onsets of ultraspinning instabilities occur for $k=2,3,4,\cdots $, and are displayed by onsets of green, blue, brown,$\cdots ,$ curves; these indicate possible bifurcations to new families of stationary black holes. It is reasonable to conjecture that the actual G of an exact black ring is slightly smaller than that of the displayed approximation and would cross the G of an MP black hole close to ${r}_{+}={r}_{2}$, indicating a possible first order phase transition between the two kinds of black holes.

#### 6.4. Isoperimetric Inequality

#### 6.5. Equation of State

#### 6.6. Ultraspinning Expansion

**Figure 31.**Equation of state: thin AdS black ring. The “exact” equation of state, given implicitly by Equation (137), for $T\approx 0.133$ is displayed by the thick solid black curve, and compared to the ultraspinning equation of state (136), displayed by the thick dashed blue curve, and the first correction equation of state (142), displayed by the thick dashed green curve. The two conditions (138) are respectively displayed by thin dashed red and black curves. They almost coincide: setting $R/{r}_{0}=N=l/{r}_{0}$, the approximation is valid when $N\gg 1$. As in the AdS/CFT correspondence, we set $N=3\gg 1$, which gives the curves; the approximation is valid above these curves. This means that we can trust our “exact” equation of state in the top left corner of the figure. In this region all three equations of state basically coincide.

## 7. Beyond Thermodynamic Instabilities

#### 7.1. Ultraspinning Instability

#### 7.1.1. Bifurcations of Singly Spinning MP Black Holes

#### 7.1.2. Thermodynamic Argument and Other Examples

**Figure 32.**RPT and classical instabilities: $d=6$ singly spinning Kerr-AdS black holes. The Gibbs free energy for the $d=6$ singly spinning Kerr-AdS black hole is displayed for $J=1$ and $P=0.0564\in \left({P}_{t}{P}_{z}\right)$ for which the RPT is present, cf. Figure 12c. Black holes displayed by the solid black curve are classically stable (potentially subject to the bar-mode instability). The red curve indicates the branch of black holes subject to the superradiant instability, ${r}_{+}<1.11$. The blue dotted curve displays black holes in the ultraspinning region ($k\ge 1$ zero modes) derived from the thermodynamic argument, ${r}_{+}<0.55$. Such black holes are also superradiant unstable. We observe that black holes that play a role in the RPT are ultraspinning stable but some of them are subject to the superradiant instability.

**Figure 33.**RPT and ultraspinning instability: $d=6$ doubly spinning MP black holes. The Gibbs free energy for the $d=6$ doubly spinning MP black hole is displayed for $q=0.112\in ({q}_{t},{q}_{z})$ for which the RPT is present, cf. Figure 24d. The black curve is classically stable (potentially subject to the bar-mode instability). The blue dotted curve indicates the branch of black holes subject to the ultraspinning instability, ${r}_{+}<0.613$. Hence, contrary to the Kerr-AdS case the MP black holes that play the role in the RPT are ultraspinning unstable.

**Figure 34.**Classical instabilities: doubly spinning $d=6$ Kerr-AdS black hole. The Gibbs free energy for the $d=6$ doubly spinning Kerr-AdS black holes is displayed for $q=0.05$ and various pressures (from top to bottom) $P=0.260,0.170,0.087,0.0642,0.015$, for which we observe the SBH/IBH/LBH phase transition, cf. Figure 19. The black solid curves denote classically stable black holes (potentially subject to the bar-mode instability). The blue color indicates branches of black holes subject to both the ultraspinning and superradiant instabilities. The red curves correspond to black holes that are superradiant unstable and ultraspinning stable. We conclude that the Kerr-AdS black holes that play the role in the SBH/IBH/LBH phase transitions are potentially subject to both classical instabilities.

#### 7.2. Superradiant Instabilities

**Figure 35.**Superradiant instability: $d=4$ Kerr-AdS black hole. The Gibbs free energy for the Kerr-AdS black hole in $d=4$ is displayed for various pressures, $P/{P}_{c}=\{1.6,1,0.6,0.2\}$, ${P}_{c}\approx 0.002856$, and fixed $J=1$. The red curves display black holes that are subject to the superradiant instability, the black ones are superradiant stable. The figure clearly demonstrates that even the van der Waals like phase transition occurring for $d=4$ Kerr-AdS black holes, discussed in Section 2, is subject to the superradiant instability.

## 8. Conclusions

**Figure 36.**Grand-canonical thermodynamic potential. The grand-canonical thermodynamic potential ${G}_{\Omega}$ is displayed for $d=6$ singly spinning Kerr-AdS black hole for fixed $\Omega =1$ and various pressures (from left to right) $P=0.5,0.8,1$. Its behaviour is qualitatively similar to the behavior of G for the Schwarzschild-AdS black hole. Namely, although we may observe an analogue of the Hawking–Page transition, there is no swallow tail present and hence no first order phase transition of the van der Waals type.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Altamirano, N.; Kubizňák, D.; Mann, R.B.; Sherkatghanad, Z.
Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume. *Galaxies* **2014**, *2*, 89-159.
https://doi.org/10.3390/galaxies2010089

**AMA Style**

Altamirano N, Kubizňák D, Mann RB, Sherkatghanad Z.
Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume. *Galaxies*. 2014; 2(1):89-159.
https://doi.org/10.3390/galaxies2010089

**Chicago/Turabian Style**

Altamirano, Natacha, David Kubizňák, Robert B. Mann, and Zeinab Sherkatghanad.
2014. "Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume" *Galaxies* 2, no. 1: 89-159.
https://doi.org/10.3390/galaxies2010089