Geometrothermodynamics of Myers-Perry Black Holes

Abstract

We consider the thermodynamics and geometrothermodynamics of the Myers-Perry black holes in five dimensions for three different cases, depending on the values of the angular momenta. We follow Davies approach to study the thermodynamics of black holes and find a nontrivial thermodynamic structure in all cases, which is fully reproduced by the analysis performed with the techniques of Geometrothermodynamics. Moreover, we observe that in the cases when only one angular momentum is present or the two angular momenta are fixed to be equal, that is, when the thermodynamic system is two dimensional, there is a complete agreement between the divergences of the generalized susceptibilities and the singularities of the equilibrium manifold, whereas when the two angular momenta are fully independent, that is, when the thermodynamic system is three dimensional, additional singularities in the curvature appear. However, we prove that such singularities are due to the changing from a stable phase to an unstable one.

1. Introduction

Black holes are very special thermodynamic systems. They are thermodynamic system since they have a temperature, the celebrated Hawking temperature [1], and a definition of entropy via the Bekenstein area law [2], from which one can prove that the laws of thermodynamics apply to black holes [3]. On the other side, they are very special thermodynamic systems, and since, for instance, the entropy is not extensive, they cannot be separated into small subsystems, and perhaps the worst fact, their thermodynamics does not possess a microscopic description yet (see, e.g., [4] for a clear description of these problems).

In this puzzling situation, one of the most successful and at the same time discussed approach to the study of black holes phase transitions is the work of Davies [5]. According to Davies, black holes can be regarded as ordinary systems, showing phase transitions right at those points where the generalized susceptibilities, that is, second-order derivatives of the potential, change sign most notably through an infinite discontinuity. Since there is no statistical mechanical description of black holes as thermodynamic systems, it is hard to verify Davies approach with the usual technique of calculating the corresponding critical exponents (although very interesting works on this subject exist, see, e.g., [4, 6–18]). In fact, the main drawback of this approach is that one has to choose arbitrarily the order parameter for black holes.

A possible resolution to this situation can then come from the use of thermodynamic geometry. Since the pioneering works of Gibbs [19] and Carathéodory [20], techniques from geometry have been introduced into the analysis of thermodynamics. In particular, Fisher [21] and Radhakrishna Rao [22] proposed a metric structure in the space of probability distributions which has been extensively used both in statistical physics and in economics (for a recent review see [23]). Later, Weinhold [24] introduced an inner product in the equilibrium space of thermodynamics based on the stability conditions of the internal energy, taken as the thermodynamic potential. The work of Weinhold was then developed by Ruppeiner [25] from a different perspective. Ruppeiner moved from the analysis of fluctuations around equilibrium and from the gaussian approximation of the probability of fluctuations and found a thermodynamic metric which is defined as (minus) the hessian of the entropy of the system. Remarkably, Ruppeiner geometry was found to be conformally related to the one proposed by Weinhold. Moreover, Ruppeiner metric is intrinsically related to the underlying statistical model, and in fact the scalar curvature of the Riemannian manifold representing the system using Ruppeiner metric happens to have exactly the same exponent as the correlation volume for ordinary systems (see, e.g., [26] for a review).

All these approaches have been widely used to study ordinary systems, and in particular Ruppeiner metric has been also used to study many black holes configurations (see [27] and refernces therein). This is because one can argue that being Ruppeiner metric defined only from thermodynamic quantities and on the other side giving information about the statistical model, then it can provide some hints towards the resolution of the long standing problem of understanding the microscopic properties of black holes (see, e.g., [27]).

On the other side, the problem with the use of thermodynamic geometries to study black holes thermodynamics is that black holes are not ordinary systems, as we argued previous. For instance, Ruppeiner metric in many cases gives exactly the same results of Davies approach (which is based upon ordinary thermodynamics), while in some other important cases it does not converge to the same results, as it happens for example, in the Reissner-Nordström and Kerr cases (see, e.g., [27–29]). One can argue either that Davies approach is inaccurate or that the application of Ruppeiner metric to black holes may be imperfect, due to the strange nature of black holes as thermodynamic systems. In fact, there is still an open debate on this topic (see, e.g., the discussion in [27, 30]).

Furthermore, in the recent years, a new approach in the context of thermodynamic geometry has been proposed by Quevedo [31], known as geometrothermodynamics (GTD). According to this approach, the Riemannian structure to be introduced in the equilibrium space should be consistent with the property of Legendre invariance, a property which is at the core of ordinary thermodynamics. In GTD some families of metrics have been found that share the Legendre invariance property, and they have also been proven to interpret in a consistent manner the thermodynamic properties of ordinary systems, chemical interactions, black holes configurations, and cosmological solutions (see [31–39]). In particular, the correspondence between the divergences of the scalar curvature of the equilibrium manifold of GTD and the phase transition points signaled by the divergences of the heat capacity (i.e., phase transitions à la Davies) seems to be a general fact, according to the variety of systems analyzed so far and to the general expressions given in [40]. In addition, a recent study [41] of the thermodynamics of the Reissner-Nordström and Kerr black holes in any dimensions suggested that the GTD approach can detect not only the points of phase transitions due to singularities of the heat capacities but also divergences of the full spectrum of the generalized susceptibilities.

On the other side, the thermodynamic properties of the Myers-Perry black holes in five dimensions have been extensively studied in the literature from completely different points of view (see, e.g., [4, 29, 42–46]). In this work, we give special emphasis on the relation between divergences of the generalized susceptibilities and curvature singularities of the metric from GTD. For example, we do not consider here possible phase transitions related to change in the topology of the event horizon, an intriguing question which was addressed, for example, in [4]. We find out that the GTD thermodynamic geometry is always curved for the considered cases, showing the presence of thermodynamic interaction and that its singularities always correspond to divergences of the susceptibilities or to points where there is a change from a stable to an unstable phase. This will allow us to infer new results on the physical meaning of the equilibrium manifold of GTD.

The work is organized as follows. In Section 2, we present the basic aspects of GTD and introduce all the mathematical concepts that are needed. In Section 3, we perform the parallel between the thermodynamic quantities and the Geometrothermodynamic description of the five-dimensional Myers-Perry black holes for three different cases, depending on the values of the angular momenta. Finally, in Section 4, we comment on the results and discuss possible developments.

2. Basics of Geometrothermodynamics

Geometrothermodynamics (GTD) is a geometric theory of thermodynamics recently proposed by Quevedo [31]. It makes use of contact geometry in order to describe the phase space 𝒯 of thermodynamic systems and express the first law of thermodynamics in a coordinate-free fashion. Furthermore, GTD adds a Riemannian structure𝐺to the phase space and requests𝐺to be invariant under Legendre transformations, in order to give it the same properties which one expects for ordinary thermodynamics. Moreover, GTD introduces the manifold of the equilibrium space ℰ as the maximum integral submanifold of the contact structure of 𝒯, characterized by the validity of the first law of thermodynamics [31]. At the same time, GTD prescribes also to pull back the Riemannian structure𝐺to the equilibrium space. This results in a naturally induced Riemannian structure𝑔 in ℰ, which is supposed to be the geometric counterpart of the thermodynamic system. Such a description has been proposed in order to give thermodynamic geometry a new symmetry which was not present in previous approaches, that is, the Legendre invariance.

Let us see now the mathematical definitions of the GTD objects that we shall use in this work. If we are given a system with𝑛thermodynamic degrees of freedom, we introduce first a (2𝑛+1)-dimensional space 𝒯 with coordinates 𝑍^𝐴={Φ,𝐸^𝑎,𝐼^𝑎}, with 𝐴=0,…,2𝑛 and 𝑎=1,…,𝑛, which is known as the thermodynamic phase space [31]. We make use of the phase space𝒯in order to correctly handle both the Legendre transformations and the first law of thermodynamics. In fact, in classical thermodynamics, we can change the thermodynamic potential using a Legendre transformation, which is defined in𝒯as the change of coordinates given by [47]: Φ,𝐸^𝑎,𝐼^𝑎→Φ,𝐸^𝑎,𝐼^𝑎̃,Φ=Φ−𝛿_𝑘𝑙𝐸^𝑘𝐼̃^𝑙,𝐸^𝑖=−𝐼̃^𝑖,𝐸^𝑗=𝐸^𝑗,𝐼^𝑖=𝐸^𝑖,𝐼^𝑗=𝐼^𝑗̃, (1) where 𝑖∪𝑗 can be any disjoint decomposition of the set of indices{1,…,𝑛} and𝑘,𝑙=1,…,𝑖. We remark that Legendre transformations are change of coordinates in𝒯and that they are not defined in the equilibrium space. Moreover, the phase space𝒯is equipped with a canonical contact structure called the Gibbs 1-form defined as Θ=𝑑Φ−𝛿_𝑎𝑏𝐼^𝑏𝑑𝐸^𝑎=𝑑Φ−𝐼_𝑎𝑑𝐸^𝑎, (2) which extremely resembles the first law of thermodynamics and hence will be the starting point to define the equilibrium space.

Furthermore, the equilibrium space ℰ is the𝑛-dimensional submanifold of𝒯defined by the embedding 𝜑∶ℰ→𝒯 under the condition 𝜑^∗(Θ)=0,thatis,𝑑Φ=𝐼_𝑎𝑑𝐸^𝑎,𝐼_𝑎=𝜕Φ𝜕𝐸^𝑎, (3) where 𝜑^∗ is the pullback of 𝜑. It follows immediately from (2) that (3) represents both the first law and the equilibrium conditions for the thermodynamic system under analysis, so that ℰ results to be (by definition) the submanifold of points where the first law and the equilibrium conditions hold, that is, the geometric counterpart of the thermodynamic system. It also follows that the coordinates{𝑍^𝐴} of 𝒯 assume a physical meaning in ℰ. In fact, the set {𝐸^𝑎}, with 𝑎=1,…,𝑛, can be identified with the extensive thermodynamic variables, while Φ=Φ(𝐸^𝑎) with the fundamental equation for the thermodynamic potential, and finally the coordinates{𝐼^𝑎}={𝐼^𝑎(𝐸^𝑎)}≡{𝜕_𝐸^𝑎Φ},𝑎=1,…,𝑛 represent the intensive quantities corresponding to the extensive set{𝐸^𝑎} (see, e.g., [48] for these definitions).

Now, let us add the Riemannian structure. Since we want the Riemannian structure to share the same properties of the first law and since the first law is invariant under Legendre transformations, we introduce in the phase space 𝒯 a metric 𝐺 which is invariant under Legendre transformations. In GTD, there are several families of metrics which have this property (for a recent work on this topic see [49]). Among them, one has been proven particularly successful to describe systems with second-order phase transitions, as black holes are supposed to be. Thus, the candidate metric we shall use in this work is 𝐺=𝑑Φ−𝐼_𝑎𝑑𝐸^𝑎2+Λ𝜉_𝑎𝑏𝐸^𝑎𝐼^𝑏𝜒_𝑐𝑑𝑑𝐸^𝑐𝑑𝐼^𝑑, (4) where 𝜉_𝑎𝑏 and 𝜒_𝑎𝑏 are diagonal constant tensors and Λ is an arbitrary Legendre invariant function of the coordinates {𝑍^𝐴}. In particular, we choose to fix Λ=1,𝜉_𝑎𝑏=𝛿_𝑎𝑏≡diag(1,…,1) and 𝜒_𝑎𝑏=𝜂_𝑎𝑏≡diag(−1,1,…,1) in order to get the exact expression for the metric describing black holes phase transitions (see also [33]).

On the other side, we are not interested in the geometric representation of the phase space, while we care about the geometric properties of the thermodynamic system, which is paralleled by the equilibrium space ℰ. Thus, we pull back the metric𝐺ontoℰand obtain a Riemannian structure for the equilibrium space which reads 𝑔^𝐼𝐼≡𝜑^∗(𝐺)=𝐸^𝑎𝜕Φ𝜕𝐸^𝑎𝜂^𝑐_𝑏𝜕²Φ𝜕𝐸^𝑐𝜕𝐸^𝑑𝑑𝐸^𝑏𝑑𝐸^𝑑, (5) where 𝜑^∗ is the pullback of𝜑as in (3) and 𝜂^𝑐_𝑏=diag(−1,1,…,1). We remark that 𝑔^𝐼𝐼 is (by definition) invariant under (total) Legendre transformations (see, e.g., [33]). Moreover, we also note that 𝑔^𝐼𝐼 can be calculated explicitly once the fundamental equation Φ=Φ(𝐸^𝑎) is known.

The main thesis of GTD is that the thermodynamic properties of a system described by a fundamental equation Φ(𝐸^𝑎) can be translated into geometrical features of the equilibrium manifold ℰ, which in our case is described by the metric 𝑔^𝐼𝐼. For example, the scalar curvature of ℰ should give information about the thermodynamic interaction. This means that systems without interaction shall correspond to flat geometries and systems showing interaction and phase transitions should correspond to curved equilibrium manifolds having curvature singularities. It has been tested in a number of works (see, e.g. [31–35]) that indeed such correspondence works. Furthermore, a previous work [41] studying the thermodynamics and GTD of the Reissner-Nordström and of the Kerr black holes in any dimensions, highlighted that curvature singularities of 𝑔^𝐼𝐼 are exactly at the same points where the generalized susceptibilities diverge.

In this work, we extend the work in [41] to the case of Myers-Perry black holes in five dimensions, with the aim of both to analyze their thermodynamic geometry from a new perspective and to focus on the idea of checking what happens with a change of the potential from Φ=𝑀 to Φ=𝑆 in the GTD analysis and when the equilibrium manifold is 3 dimensional. The investigation of the phase structure of Myers-Perry black holes in five dimensions is thus a matter which is interesting by itself and that will provide us with the necessary ground for a new test of the correspondence between the thermodynamic geometry 𝑔^𝐼𝐼 of GTD and black holes thermodynamics.

3. Myers-Perry Black Holes

The Kerr black hole can be generalized to the case of arbitrary dimensions and arbitrary number of spins. It turns out that, provided, 𝑑is the number of spacetime dimensions, that the maximum number of possible independent spins is (𝑑−1)/2 if 𝑑 is odd and (𝑑−2)/2 if 𝑑 is even [50]. Such general configurations are called Myers-Perry black holes. They deserve a special interest because they are the natural generalization of the well-known Kerr black hole to higher number of spins and because they are shown to coexist with the Emparan-Reall black ring solution for some values of the parameters, thus providing the first explicit example of a violation in a dimension higher than four of the uniqueness theorem (see, e.g., [51] for more details). The line element of the Myers-Perry black hole with an arbitrary number of independent angular momenta in Boyer-Lindquist coordinates for 𝑑=2𝑛+1 (i.e., odd 𝑑) reads [50] 𝑑𝑠²=−𝑑𝑡²+𝜇𝑟²Π𝐹𝑑𝑡+^𝑛_𝑖=1𝑎_𝑖𝜇²_𝑖𝑑𝜙_𝑖²+Π𝐹Π−𝜇𝑟²𝑑𝑟²+^𝑛_𝑖=1𝑟²+𝑎²_𝑖𝑑𝜇²_𝑖+𝜇²_𝑖𝑑𝜙²_𝑖, (6) with 𝐹≡1−^𝑛_𝑖=1𝑎²_𝑖𝜇²_𝑖𝑟²+𝑎²_𝑖,Π≡^𝑛_𝑖=1𝑟²+𝑎²_𝑖,𝜇≡16𝜋𝐺𝑀(𝑑−2)Ω_(𝑑−2),𝑎_𝑖≡(𝑑−2)2𝐽_𝑖𝑀, (7) where Ω_(𝑑−2)=2𝜋^𝑛/Γ(𝑛),𝑀 is the mass of the black hole, 𝐽_𝑖=𝐽₁,…,𝐽_𝑛 are the(𝑑−1)/2 independent angular momenta, and the constraint ∑^𝑛_𝑖=1𝜇²_𝑖=1 holds. Solving the equation 𝑔^𝑟𝑟=1/𝑔_𝑟𝑟=0, one finds the radius of the event horizon (in any dimensions) and thus derives the area and the corresponding entropy, using Bekenstein area law [29].

In particular, in this work, we are interested in the five dimensional case, that is, when 𝑑=5. Myers-Perry black holes in five-dimensions can have up to 2 independent angular momenta, and the general equation for the area reads [29] 𝐴=2𝜋²𝑟₊𝑟²₊+𝑎²₁𝑟²₊+𝑎²₂, (8) where 𝑟₊ is the radius of the event horizon. From the previous expression the entropy can be calculated, being 𝑘𝑆=^𝐵𝐴=14𝐺𝑟₊𝑟²₊+𝑎²₁𝑟²₊+𝑎²₂, (9) where we choose 𝑘_𝐵 and 𝐺 such that 𝑆 simplifies as in the second equality in (9).

Since it is rather complicated to calculate explicitly the previous expression for the entropy, we will use the 𝑀 representation throughout the paper. This is possible since the mass can be written in terms of 𝑆, 𝐽₁, and 𝐽₂ as [29] 𝑀𝑆,𝐽₁,𝐽₂=34𝑆^2/3𝐽1+4²¹𝑆^21/3𝐽1+4²²𝑆^21/3. (10) Equation (10) thus represents the fundamental equation for the Myers-Perry black hole in five dimensions as a thermodynamic system. Starting from (10), we can analyze both the thermodynamic properties and their geometrothermodynamic counterparts. We will split the work in order to consider the three most interesting cases, that is, when one of the two angular momenta is zero, when they are both nonzero but equal, and finally when they are both nonzero and different among each other.

3.1. The Case 𝐽₂=0

If either 𝐽₁=0 or 𝐽₂=0, we obtain the Kerr black hole in 5 dimensions, which has been analyzed in [41]. We briefly review here some of the results presented there and improve the analysis, including the investigation of the response functions defined in the total Legendre transformation of the mass 𝑀, which we will call the Gibbsian response functions, in analogy with standard thermodynamics [48]. Therefore, let us suppose that 𝐽₂=0. According to our previous results [41], we know that the response functions defined in the mass representation read 𝐶_𝐽1𝑆=−3𝑆²+4𝐽²₁𝑆²−12𝐽²₁,𝜅_𝑆=3𝑆²+4𝐽²₁^5/32𝐽₁3𝑆²−4𝐽²₁,𝛼_𝑆3=−8𝑆²+4𝐽²₁^5/3𝐽²₁𝑆, (11) where we make use of the notation𝑀_𝐸^𝑎≡𝜕𝑀/𝜕𝐸^𝑎 and 𝑀_𝐸^𝑎_𝐸^𝑏≡𝜕²𝑀/𝜕𝐸^𝑎𝜕𝐸^𝑏, for 𝐸^𝑖=𝑆,𝐽₁. It follows that 𝛼_𝑆 does not show any singularity (apart from the extremal limit𝑆=0), while𝐶_𝐽1diverges at the Davies point 𝑆²=12𝐽²₁ and 𝜅_𝑆 shows an additional possible phase transition at 3𝑆²=4𝐽²₁. As it was pointed out in [41], both singularities of the heat capacity and of the compressibility are in the black hole region and hence are physically relevant. It was also shown that the GTD geometry (5) with fundamental equation (10) (with𝐽₂=0) is curved, indicating the presence of thermodynamic interaction, and that the singularities of the scalar curvature are situated exactly at the same points where the response functions𝐶_𝐽1,   and 𝜅_𝑆 diverge, both in the mass and in the entropy representations. Furthermore, it was also commented that Weinhold geometry is flat in this case and Ruppeiner thermodynamic metric diverges only in the extremal limit 𝑆=0 (see, e.g., [29] for a complete analysis using these metrics).

Moreover, since the thermodynamics of black holes can depend on the chosen ensemble (see, e.g., [52, 53]), we now proceed to calculate the Gibbsian response functions, which can possibly give new information about the phase structure. Using the relations between thermodynamic derivatives (see, [48]), we find out that the expressions for such response functions in the coordinates (𝑆,𝐽₁) used here are 𝐶_Ω1𝑆=−3𝑆²−4𝐽²₁𝑆²+4𝐽²₁,𝜅_𝑇𝑆=−²−12𝐽²₁2𝐽₁𝑆²+4𝐽²₁^1/3,𝛼_Ω1=−8𝑆𝑆²+4𝐽²₁^1/3. (12)

It is immediate to see that 𝐶_Ω1 never diverges and it vanishes exactly at the same points where 𝜅_𝑆 diverges. On the other side, 𝜅_𝑇 is never divergent and it vanishes exactly where 𝐶_𝐽1 diverges, while 𝛼_Ω1 is always finite. It follows that the Gibbsian response functions do not add any information to the knowledge of the phase structure of this configuration, as they change the sign exactly at the points that we have already analyzed; therefore, we conclude that the divergences of the scalar curvature of the metric (5) match exactly the points of second-order phase transitions.

Let us now add a second spin parameter and show that there is still a concrete correlation between the geometric description performed with 𝑔^𝐼𝐼 and the thermodynamic properties. To do so, we first focus on the special case of (10) in which 𝐽₁=𝐽₂=𝐽, and afterwards we will consider the completely general case, that is, with 𝐽₁ and 𝐽₂ both different from zero and from each other. In particular, in the latter case, we will get a 3-dimensional thermodynamic manifold labelled by (𝐸¹=𝑆,𝐸²=𝐽₁,𝐸³=𝐽₂), and hence we will consider the 3-dimensional version of the metric (5).

3.2. The Case 𝐽₁=𝐽₂≡𝐽

Another special case in (10) which is of interest is the case in which the two angular momenta are fixed to be equal, that is, 𝐽₁=𝐽₂≡𝐽. This is interesting from the mathematical and physical point of view since it is the only case in which the angular momenta are both different from zero, and at the same time the thermodynamic manifold is 2-dimensional. In fact, the mass fundamental equation (10) in this case is given by 𝑀()=3𝑆,𝐽4𝑆^2/3𝐽1+4²𝑆^22/3, (13) and the response functions can then be accordingly calculated to give 𝐶_𝐽𝑆=−3𝑆⁴−16𝐽⁴𝑆⁴−32𝐽²𝑆²−80𝐽⁴,𝜅_𝑆=3𝑆^2/3𝑆²+4𝐽^24/34𝐽3𝑆²+4𝐽²,𝛼_𝑆3=−𝑆16^5/3𝑆²+4𝐽^24/3𝐽²𝑆²+2𝐽². (14) From (14), it follows that in this case 𝛼_𝑆 and 𝜅_𝑆 do not show any singularity, while 𝐶_𝐽 diverges at the roots of the denominator 𝒟_𝐶=𝑆⁴−32𝐽²𝑆²−80𝐽⁴. We also observe that the temperature of this black hole is given by 𝑇≡𝜕𝑀𝜕𝑆_𝐽=12𝑆²−4𝐽²𝑆^5/3𝑆²+4𝐽^21/3, (15) therefore, the extremal limit 𝑇=0 is reached when 𝐽²/𝑆²=1/4.

Solving 𝒟_𝐶=0, we find that the singularities of the heat capacity are situated at a value 𝑆_critical for the entropy such that 𝐽²𝑆²_{𝑆=𝑆critical}=√21−4,20 (16) which is less than the extremal limit. Therefore Davies point of phase transition belongs to the black hole region and we shall investigate it.

It is convenient also in this case to write the full set of thermodynamic response functions, including the Gibbsian ones. Again, making use of the relations between thermodynamic derivatives, we find out that they read 𝐶_Ω𝑆𝑆=−²−4𝐽²3𝑆²+4𝐽²𝑆²+4𝐽²²,𝜅_𝑇𝑆=−^2/3𝒟4𝐽𝑆²+4^𝐶𝐽^25/3,𝛼_Ω=8𝑆^5/3𝑆²+2𝐽²𝑆²+4𝐽^25/3. (17) In this case, we observe that the only divergence of the response functions in (14), that is, the divergence of 𝐶_𝐽, is again controlled by the vanishing of 𝜅_𝑇. Furthermore, both 𝐶_𝐽 and 𝐶_Ω vanish at the extremal limit, but this does not correspond to any divergence of 𝜅_𝑆, and hence we expect the curvature of the thermodynamic metric to diverge only at the points where 𝒟_𝐶=0.

From the point of view of Geometrothermodynamics, given the fundamental equation (13) and the general metric (5), we can calculate the particular metric and the scalar curvature for the equilibrium manifold of the MP black hole with two equal angular momenta, both in the mass and in the entropy representations.

The metric in the 𝑀 representation reads 𝑔_𝑀^𝐼𝐼=1𝑆^4/3𝑆²+4𝐽^22/3×1𝒟12𝑆^𝐶2𝑑𝑆²+23𝑆²+4𝐽²3𝑑𝐽². (18) Therefore, its scalar curvature is 𝑅_𝑀=24𝑆^10/3𝑆²+4𝐽^22/3×5𝑆⁶+48𝐽²𝑆⁴−368𝐽⁴𝑆²−896𝐽⁶×𝒟²_𝐶3𝑆²+4𝐽²²⁻¹. (19) The numerator is a not very illuminating function that never vanishes when the denominator is zero, and 𝒟_𝐶 is exactly the denominator of the heat capacity 𝐶_𝐽. Therefore, the singularities of𝑅_𝑀correspond exactly to those of 𝐶_𝐽 (resp., to the vanishing of 𝜅_𝑇). We remark that the factor 3𝑆²+4𝐽² in the denominator of 𝑅_𝑀, despite being always different from 0 (thus not indicating any phase transition in this case), is exactly the denominator of the compressibility 𝜅_𝑆 (resp., a factor in the numerator of 𝐶_Ω).

To continue with the analysis, in [49], a general relation was presented (see (34) therein) to express𝑔^𝐼𝐼with Φ=𝑆 (i.e., in the𝑆 representation) in the coordinates of the 𝑀 representation (i.e.,{𝐸^𝑎}=(𝑆,𝐽)). Such relation in the present case reads 𝑔_𝑆^𝐼𝐼=𝑀−𝐽Ω𝑇³×𝑇𝑀_𝑆𝑆𝑑𝑆²+2Ω𝑀_𝑆𝑆+𝑑𝑆𝑑𝐽2Ω𝑀_𝑆𝐽−𝑇𝑀_𝐽𝐽𝑑𝐽², (20) where𝑇≡𝜕𝑀/𝜕𝑆 is the temperature,Ω≡𝜕𝑀/𝜕𝐽 is the angular velocity at the horizon and 𝑀_𝐸^𝑎_𝐸^𝑏≡𝜕²𝑀/𝜕𝐸^𝑎𝜕𝐸^𝑏, for 𝐸^𝑖=𝑆,𝐽. Using (20) and (13) for the mass in terms of𝑆and𝐽, we can calculate the expression for metric 𝑔_𝑆^𝐼𝐼 in the coordinates(𝑆,𝐽), which reads 𝑔_𝑆^𝐼𝐼=13𝑆²+4𝐽²(𝑆+2𝐽)²(𝑆−2𝐽)²×−3𝑆²−4𝐽²2𝒟_𝐶𝑑𝑆²++8𝑆𝐽3𝑆²−4𝐽²()𝒟𝑆+2𝐽)(𝑆−2𝐽^𝐶𝑑𝑆𝑑𝐽−4𝑆²×9𝑆⁶+156𝑆⁴𝐽²+112𝑆²𝐽⁴−448𝐽⁶(𝑆+2𝐽)(𝑆−2𝐽)𝑑𝐽². (21) Consequently, the scalar curvature is 𝑅_𝑆=𝒩_𝑆3𝑆²−4𝐽²³𝑆²+4𝐽²²𝒟²_𝐶, (22) where 𝒩_𝑆 is again a function which never vanishes at the points where the denominator is zero. From (22), we see that the denominator of 𝐶_𝐽 is present in the denominator of 𝑅_𝑆. Furthermore, the factor 𝑆²+4𝐽² is never zero, and hence it does not give any additional singularity. On the other hand, the factor 3𝑆²−4𝐽² is clearly vanishing when 𝐽²/𝑆²=3/4, which is readily greater than the extremal limit 𝐽²/𝑆²=1/4, and hence it has no physical relevance in our analysis.

We thus conclude that also in this case the GTD geometry𝑔^𝐼𝐼exactly reproduces the phase transition structure of the Myers-Perry black holes both in the mass and in the entropy representation. We comment that in the entropy representation there is an additional singularity which does not correspond to any singularity of the response functions. However, such singularity is situated out of the black hole region, and thus it is not to be considered here. We also remark that Ruppeiner curvature in this case reads 𝑅=−𝑆(𝑆²+12𝐽²)/(𝑆⁴−16𝐽⁴), and hence it diverges only in the extremal limit, while Weinhold metric is flat.

In the next subsection, we will analyze the general case of the Myers-Perry black hole in five dimensions, that is, when the two angular momenta are allowed to vary freely.

3.3. The General Case in Which 𝐽₁≠𝐽₂≠0

Perhaps the most interesting case is the most general one, in which the two angular momenta are allowed to vary freely. In this case, the thermodynamic manifold is 3 dimensional and the mass fundamental equation is given by (10).

The generalized susceptibilities can then be accordingly calculated. The heat capacity at constant angular momenta 𝐽₁ and 𝐽₂ reads 𝐶_𝐽1,𝐽2=𝑀_𝑆𝑀_𝑆𝑆𝑆=−3𝑆²+4𝐽²₁𝑆²+4𝐽²₂𝑆⁴−16𝐽²₁𝐽²₂𝒟_𝐶, (23) where 𝒟_𝐶=𝑆⁸−12𝐽²₁+𝐽²₂𝑆⁶−320𝐽²₁𝐽²₂𝑆⁴−576𝐽²₁𝐽²₂𝐽²₁+𝐽²₂𝑆²−1280𝐽⁴₁𝐽⁴₂. (24) Furthermore, one can define the 3 analogues of the adiabatic compressibility as 𝜅_𝑆11≡𝜕𝐽₁𝜕Ω_1𝑆=3𝑆^2/3𝑆²+4𝐽²₁^5/32𝑆²+4𝐽²₂^1/33𝑆²−4𝐽²₁,𝜅_𝑆22≡𝜕𝐽₂𝜕Ω_2𝑆=3𝑆^2/3𝑆²+4𝐽²₂^5/32𝑆²+4𝐽²₁^1/33𝑆²−4𝐽²₂,𝜅_𝑆12≡𝜕𝐽₁𝜕Ω_2𝑆=3𝑆16^2/3𝑆²+4𝐽²₁^2/3𝑆²+4𝐽²₂^2/3𝐽₁𝐽₂. (25) Finally, the analogues of the expansion are given by 𝛼_𝑆,𝐽2≡𝜕𝐽₁𝜕𝑇_𝑆3=−8𝑆^5/3𝑆²+4𝐽²₁^5/3𝑆²+4𝐽²₂^2/3𝐽₁𝑆⁴+6𝑆²𝐽²₂+8𝐽²₁𝐽²₂,𝛼_𝑆,𝐽1≡𝜕𝐽₁𝜕𝑇_𝑆3=−8𝑆^5/3𝑆²+4𝐽²₂^5/3𝑆²+4𝐽²₁^2/3𝐽₂𝑆⁴+6𝑆²𝐽²₁+8𝐽²₁𝐽²₂. (26)

In this case, neither (𝜅_𝑆)₁₂ nor the expansions show any singularity, while 𝐶_𝐽1,𝐽2 diverges when 𝒟_𝐶=0 and the compressibilities (𝜅_𝑆)₁₁ and (𝜅_𝑆)₂₂ diverge when 3𝑆²−4𝐽²₁=0 and 3𝑆²−4𝐽²₂=0, respectively. Furthermore, the temperature reads 1𝑇=2𝑆^5/3𝑆⁴−16𝐽²₁𝐽²₂𝑆²+4𝐽²₁^2/3𝑆²+4𝐽²₂^2/3. (27) Hence; the extremal limit is reached for 𝐽₁𝐽₂/𝑆²=1/4. The heat capacity diverges when 𝒟_𝐶=0, which is an algebraic equation of degree 8 in 𝑆. We can solve numerically such equation and obtain the critical value 𝑆=S_critical in terms of 𝐽₁ and 𝐽₂. Taking only the roots which are real and positive, we can compare them with the extremal limit by doing 𝐽₁𝐽₂𝑆²_{𝑆=Sextremal}−𝐽₁𝐽₂𝑆²_{𝑆=Scritical}=14−𝐽₁𝐽₂𝑆²_{𝑆=Scritical}. (28) The plot of the result is given in Figure 1 for some values of𝐽₁and𝐽₂. As we can see from Figure 1, the difference in (28) is always positive, and hence the point of phase transition signaled by the divergence of the heat capacity is always in the black hole region.

Figure 1 

The difference between the extremal limit (𝐽1𝐽2/𝑆2=1/4) and the value of 𝐽1𝐽2/𝑆2 at the critical point of the heat capacity, plotted for values of 𝐽1 and 𝐽2 in the interval [0,10].

In the same way, we can solve 3𝑆²−4𝐽²₁=0 and see whether the divergence of (𝜅_𝑆)₁₁ lies in the black hole region or not. It turns out that the denominator of (𝜅_𝑆)₁₁ vanishes for values of 𝑆 such that 𝐽²₁/𝑆²=3/4, which means that𝐽₁𝐽₂/𝑆²=(3/4)(𝐽₂/𝐽₁). Therefore, we have that (1/4)−(3/4)(𝐽₂/𝐽₁) is positive provided that 𝐽₁>3𝐽₂ for 𝐽₁>0 or 𝐽₁<3𝐽₂ for 𝐽₁<0. Summing up, the divergences of (𝜅_𝑆)₁₁ can be in the black hole region for appropriate values of 𝐽₁ and 𝐽₂. Analogously, the divergences of (𝜅_𝑆)₂₂ can also be in the black hole region.

As in the preceding sections, we will now focus on the Gibbsian response functions, in order to make the analysis complete. The heat capacity at constant angular velocities read 𝐶_Ω1,Ω2≡𝑇𝜕𝑆𝜕𝑇_Ω1,Ω2=−3𝑆𝑆⁴−16𝐽²₁𝐽²₂3𝑆²−4𝐽²₁×3𝑆²−4𝐽²₂𝑆²+4𝐽²₁𝑆²+4𝐽²₂×𝒟𝑆,𝐽₁,𝐽₂⁻¹, (29) where the denominator is given by 𝒟𝑆,𝐽₁,𝐽₂=9𝑆¹²+72𝐽²₁+𝐽²₂𝑆¹⁰+169𝐽⁴₁+95𝐽²₁𝐽²₂+9𝐽⁴₂𝑆⁸+5376𝐽²₁𝐽²₂𝐽²₁+𝐽²₂𝑆⁶−256𝐽²₁𝐽²₂9𝐽⁴₁−101𝐽²₁𝐽²₂+9𝐽⁴₂𝑆⁴−6144𝐽⁴₁𝐽⁴₂𝐽²₁+𝐽²₂𝑆²−53248𝐽⁶₁𝐽⁶₂. (30) Furthermore, one can define three generalized susceptibilities, analogous to the isothermal compressibility, as 𝜅_𝑇11≡𝜕𝐽₁𝜕Ω_1𝑇,𝜅_𝑇12≡𝜕𝐽₁𝜕Ω_2𝑇,𝜅_𝑇22≡𝜕𝐽₂𝜕Ω_2𝑇. (31) For the Myers-Perry black hole itcan be written as 𝜅_𝑇11𝑆=−^2/32𝒟_𝐶𝑆²+4𝐽²₁^1/3𝑆²+4𝐽²₂^1/3×𝑆⁶−12𝐽²₂𝑆⁴+48𝐽²₁𝐽²₂𝑆²+192𝐽²₂𝐽⁴₂⁻¹,𝜅_𝑇22𝑆=−^2/32𝒟_𝐶𝑆²+4𝐽²₁^1/3𝑆²+4𝐽²₂^1/3×𝑆⁶−12𝐽²₁𝑆⁴+48𝐽²₁𝐽²₂𝑆²+192𝐽²₂𝐽⁴₁⁻¹, (32) while(𝜅_𝑇)₁₂has a more cumbersome expression and we will not write it here, since it has the same properties of(𝜅_𝑇)₁₁and(𝜅_𝑇)₂₂ as regards to our analysis; that is, it is proportional to the denominator of𝐶_𝐽1,𝐽2defined in (24) and it has a nontrivial denominator (one can also introduce the two analogues of the thermal expansion, but for the sake of simplicity, we are not going to write them here, since they do not show any singularities, and hence they do not play any role in our analysis).

Therefore, from the thermodynamic point of view, we remark that the divergences of𝐶_𝐽1,𝐽2are matched by the vanishing of the three quantities(𝜅_𝑇)_𝑖𝑗, while the divergences of(𝜅_𝑆)₁₁and(𝜅_𝑆)₂₂are reproduced as zeroes of the heat capacity𝐶_Ω1,Ω2. This behavior is in agreement with the analysis of the preceding sections. Furthermore, in this case, the heat capacity𝐶_Ω1,Ω2and the generalized compressibilities(𝜅_𝑇)_𝑖𝑗 possibly show additional phase transitions, which is a further indication of the fact that black holes exhibit different thermodynamic behavior in different potentials.

Now, let us turn to the GTD analysis. Given the fundamental equation (10) and the general metric (5), we can calculate the particular metric and the scalar curvature for the MP black hole with two free angular momenta, both in the mass and in the entropy representations. The metric in the𝑀representation reads 𝑔_𝑀^𝐼𝐼=13𝑆^4/3𝑆²+4𝐽²₁^1/3𝑆²+4𝐽²₂^1/3×−14𝒟_𝐶𝑆²𝑆²+4𝐽²₁𝑆²+4𝐽²₂𝑑𝑆²+3𝑆²−4𝐽²₁𝑆²+4𝐽²₂𝑆²+4𝐽²₁𝑑𝐽²₁+3𝑆²−4𝐽²₂𝑆²+4𝐽²₁𝑆²+4𝐽²₂𝑑𝐽²₂+16𝐽₁𝐽₂𝑑𝐽₁𝑑𝐽₂. (33) Hence, its scalar curvature is 𝑅_𝑀=𝒩_𝑀×𝒟²_𝐶3𝑆⁴−4𝑆²𝐽²₁+𝐽²₂−16𝐽²₁𝐽²₂²×𝑆²+4𝐽²₁^2/3𝑆²+4𝐽²₂^2/3−1, (34) where𝒟_𝐶is as usual the denominator of 𝐶_𝐽1,𝐽2 defined in (24). Since there is no term in the numerator 𝒩_𝑀 which cancels out the divergences that happen when 𝒟_𝐶=0, we can conclude that every phase transition related to the heat capacity 𝐶_𝐽1,𝐽2 is properly reproduced by the scalar curvature 𝑅_𝑀. In addition, in this case, the factor 3𝑆⁴−4𝑆²(𝐽²₁+𝐽²₂)−16𝐽²₁𝐽²₂ can also vanish, possibly giving an additional singularity which does not correspond to the ones shown by the response functions. It is easy to calculate that 3𝑆⁴−4𝑆²(𝐽²₁+𝐽²₂)−16𝐽²₁𝐽²₂=0 for values of 𝑆 such that 𝐽₁𝐽₂𝑆²=18−𝐽²₁−𝐽²₂+𝐽⁴₁+14𝐽²₁𝐽²₂+𝐽⁴₂𝐽₁𝐽₂. (35) We can thus calculate the difference between the extremal limit 𝐽₁𝐽₂/𝑆²=1/4 and the critical value (35). The result is 14−18−𝐽²₁−𝐽²₂+𝐽⁴₁+14𝐽²₁𝐽²₂+𝐽⁴₂𝐽₁𝐽₂1=−𝐽48²¹+𝐽²₂−6𝐽₁𝐽₂+𝐽⁴₁+14𝐽²₁𝐽²₂+𝐽⁴₂×𝐽²₁+𝐽²₂−𝐽⁴₁+14𝐽²₁𝐽²₂+𝐽⁴₂×𝐽²₁𝐽²₂⁻¹, (36) which can be positive for appropriate values of 𝐽₁ and 𝐽₂. Therefore, such points of divergence of 𝑅_𝑀 are in the black hole region for some values of the parameters. Hence, we conclude that the behavior of 𝑅_𝑀 perfectly matches the behavior of 𝐶_𝐽1,𝐽2, but in this case, it does not reproduce the additional possible phase transitions indicated by the singularities of the compressibilities (𝜅_𝑆)₁₁ and (𝜅_𝑆)₂₂and possibly shows some additional unexpected singularities.

However, we can give a precise physical meaning to such additional singularities. In fact, if we evaluate the determinant of the Hessian of the mass with respect to the angular momenta 𝐽₁ and𝐽₂, we get detHess(𝑀)_𝐽1𝐽2≡𝑀_𝐽1𝐽1𝑀_𝐽2𝐽2−𝑀²_𝐽1𝐽2=433𝑆⁴−4𝑆²𝐽²₁+𝐽²₂−16𝐽²₁𝐽²₂𝑆^4/3𝑆²+4𝐽²₁^4/3𝑆²+4𝐽²₂^4/3, (37) from which we can see that the numerator is exactly the factor in the denominator of 𝑅_𝑀 whose roots give the additional singularities. Since the Hessians of the energy in thermodynamics are related to the stability conditions, we suggest that the physical meaning of such additional divergences of 𝑅_𝑀 is to be found in a change of stability of the system, for example, from a stable phase to an unstable one.

On the other side, using the relation (20) for 𝑔^𝐼𝐼 between the 𝑀 and the 𝑆 representations, naturally extended to the 3-dimensional case with coordinates (𝑆,𝐽₁,𝐽₂), that is, 𝑔_𝑆^𝐼𝐼=𝑀−𝐽₁Ω₁−𝐽₂Ω₂𝑇³×𝑇𝑀_𝑆𝑆𝑑𝑆²+2Ω₁𝑀_𝑆𝑆𝑑𝑆𝑑𝐽₁+2Ω₂𝑀_𝑆𝑆𝑑𝑆𝑑𝐽₂+2Ω₁𝑀_𝑆𝐽1−𝑇𝑀_𝐽1𝐽1𝑑𝐽²₁+2Ω₂𝑀_𝑆𝐽2−𝑇𝑀_𝐽2𝐽2𝑑𝐽²₂−2𝑇𝑀_𝐽1𝐽2−Ω₁𝑀_𝑆𝐽2−Ω₂𝑀_𝑆𝐽1𝑑𝐽₁𝑑𝐽₂, (38) we can now calculate the metric in the 𝑆 representation, which reads 𝑔_𝑆^𝐼𝐼=3𝑆⁴+4𝑆²𝐽²₁+𝐽²₂−16𝐽²₁𝐽²₂𝒟_𝐶3𝑆²+4𝐽²₁𝑆²+4𝐽²₂𝑆²−4𝐽₁𝐽₂²𝑆²+4𝐽₁𝐽₂²×12𝑑𝑆²+4𝑆𝐽₁𝑆²+4𝐽²₂𝑆²−4𝐽₁𝐽₂𝑆²+4𝐽₁𝐽₂𝑑𝑆𝑑𝐽₁+4𝑆𝐽₂𝑆²+4𝐽²₁𝑆²−4𝐽₁𝐽₂𝑆²+4𝐽₁𝐽₂𝑑𝑆𝑑𝐽₂−2𝑆²𝑆²+4𝐽²₂²×3𝑆⁶+26𝑆⁴𝐽²₁+144𝑆²𝐽²₁𝐽²₂+320𝐽⁴₁𝐽²₂×𝒟_𝐶𝑆²−4𝐽₁𝐽₂𝑆²+4𝐽₁𝐽₂⁻¹𝑑𝐽²₁−2𝑆²𝑆²+4𝐽²₁²×3𝑆⁶+26𝑆⁴𝐽²₂+144𝑆²𝐽²₁𝐽²₂+320𝐽⁴₂𝐽²₁×𝒟_𝐶𝑆²−4𝐽₁𝐽₂𝑆²+4𝐽₁𝐽₂⁻¹𝑑𝐽²₂−32𝑆²𝐽₁𝐽₂𝑆²+4𝐽²₁𝑆²+4𝐽²₂×5𝑆⁴+12𝑆²𝐽²₁+𝐽²₂+16𝐽²₁𝐽²₂×𝒟_𝐶𝑆²−4𝐽₁𝐽₂×𝑆²+4𝐽₁𝐽₂⁻¹𝑑𝐽₁𝑑𝐽₂. (39) The scalar curvature can thus be calculated to obtain 𝑅_𝑆=𝒩_𝑆×𝒟²_𝐶3𝑆⁴+4𝑆²𝐽²₁+𝐽²₂−16𝐽²₁𝐽²₂³×𝑆²𝑆²+4𝐽²₁𝑆²+4𝐽²₂⁻¹. (40) In this case, we see again that the denominator of the heat capacity 𝒟_𝐶 is present in the denominator of 𝑅_𝑆. Furthermore, the second factor, which is slightly different from the factor in the denominator of 𝑅_𝑀, vanishes for values of𝑆such that 𝐽₁𝐽₂𝑆²=18𝐽²₁+𝐽²₂+𝐽⁴₁+14𝐽²₁𝐽²₂+𝐽⁴₂𝐽₁𝐽₂. (41) The earlier discussion for the additional singularity of 𝑅_𝑀 does not apply in this case, since one can easily show that the points described by (41) do not belong to the black hole region for any values of 𝐽₁ and 𝐽₂. However, we comment in passing that such additional singularities are still related to the vanishing of the determinant of the Hessian of the entropy𝑆with respect to the angular momenta 𝐽₁ and 𝐽₂. Therefore, they still indicate the points where the Hessian vanishes, although they are not situated in the black hole region in this case. We infer from these results that the physical meaning of the divergences of the scalar curvature of the metric 𝑔^𝐼𝐼 for such a 3-dimensional equilibrium manifold is related to the divergences of the heat capacity at constant angular momenta and to the zeroes of the Hessian of the potential with respect to those momenta, both in the mass and in the entropy representations. On the other side, from the full analysis of the divergences of the generalized response functions, we see that there are other possible points of phase transitions related to divergences of the compressibilities, which appear to be not enclosed by the analysis given with 𝑔^𝐼𝐼. We also comment that we could have used the potential Φ=𝐺≡𝑀−𝑇𝑆−𝐽₁Ω₁−𝐽₂Ω₂ in writing the metric (5) to study the GTD analysis in the𝐺representation, but such investigation would have led to exactly the same results, as it has to be, since the metric (5) is invariant under total Legendre transformations.

To conclude, we observe that in [29] the case of the full Myers-Perry black hole thermodynamics has been investigated using Weinhold and Ruppeiner thermodynamic geometries. The authors proved that both Weinhold and Ruppeiner scalar curvatures only diverge in the extremal limit.

4. Conclusions

In this work, we have analyzed the thermodynamics and thermodynamic geometry of different Myers-Perry black holes configurations in five dimensions, classifying them according to the values of the two possible independent angular momenta.

To this end, we followed the approach of Davies for the standard analysis of the thermodynamic properties in different potentials and used the approach of GTD for the thermodynamic geometric investigation. The present work has been carried out with the twofold aim of understanding the phase structure of Myers-Perry black holes in five dimensions and inferring new conclusions on the physical meaning of the metric 𝑔^𝐼𝐼, both in the mass and in the entropy representations.

Our results indicate that the Myers-Perry black holes in five dimensions have a nontrivial phase structure in the sense of Davies. In particular, the analysis of the response functions indicates that both the heat capacities and the compressibilities defined in the 𝑀 potential diverge at some points, which is usually interpreted as the hallmark of a phase transition. Interestingly, such a behavior is matched by the vanishing of the corresponding Gibbsian response functions in all the cases studied here. Moreover, in the most general case when the two angular momenta vary freely, we have shown that the Gibbsian response functions provide some additional singularities, indicating that the analysis in the 𝑀 potential is different from that performed in the 𝐺 potential.

In all the cases studied in this work, the phase transitions are well reproduced by the GTD analysis, while they are not reproduced by the thermodynamic geometries of Weinhold and Ruppeiner, whose analysis has been observed to correspond to other approaches (see, e.g., [4]). We have also found that the scalar curvature of the metric 𝑔^𝐼𝐼 shows a very similar behavior in the 𝑀 representation to that of the 𝑆 representation. In particular, for the cases in which we have only two degrees of freedom we argue that no physical difference has been detected and we have shown that not only the phase transitions indicated by 𝐶_𝐽 are reproduced, but also the ones indicated by divergences of 𝜅_𝑆. Moreover, a detailed analysis of the Gibbsian response functions showed that such divergences correspond to points where 𝜅_𝑇 and 𝐶_Ω vanish and change their character. We therefore conclude that for such cases the divergences of the scalar curvature of 𝑔^𝐼𝐼 reproduce the full set of second order phase transitions considered here.

On the other side, it seems that analyzing the general case in which both angular momenta are switched on, that is, a thermodynamic system with three degrees of freedom, some differences might appear. In fact, the phase transitions signaled by 𝐶_𝐽1,𝐽2 are still obtained as curvature singularities in both representations. Nevertheless, the scalar curvature has some additional divergences, which for the case of the𝑀representation can be in the black hole region for appropriate values of the angular momenta and that apparently are not directly related to the response functions of the system. However, we claim that such additional divergences are linked to the vanishing of the Hessian determinant of the potential𝑀with respect to the two angular momenta, and therefore they mark the transition from a stable phase to an unstable one. In our opinion, this means that the physical meaning of the scalar curvature of the metric𝑔^𝐼𝐼for thermodynamic systems with three degrees of freedom goes beyond the well-established correspondence with the generalized susceptibilities, that is, second-order derivatives of the potential, encompassing also questions of stability related to their mutual relation, that is, determinants of the Hessian of the potential. This is also supported by the analysis of the scalar curvature in the𝑆representation, which again shows singularities exactly at those points where the Hessian of the entropy with respect to the two angular momenta vanishes, so from the mathematical point of view the situation is basically the same. It is interesting, however, to note that in the𝑆representation such points are not in the black hole region, a direct evidence of the fact that black hole thermodynamics strictly depends on the potential being used. Moreover, in the completely general case, some additional divergences appear when considering the Gibbsian response functions, which are not present in the thermodynamic analysis in the𝑀potential nor are indicated as curvature singularities of𝑔^𝐼𝐼. The study of such additional singularities goes beyond the scope of this work and may be the matter of further investigation. We also expect to extend this work in the nearest future and find a number of further examples which support (or discard) the interpretation of the thermodynamic metric𝑔^𝐼𝐼for thermodynamic systems with3degrees of freedom given here.

Acknowledgments

The authors want to thank Professor H. Quevedo for insightful suggestions. Alessandro Bravetti wants to thank ICRA for financial support.