Geometrothermodynamics of Myers-Perry Black Holes

Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 549808, 11 pages http://dx.doi.org/10.1155/2013/549808

Research Article Geometrothermodynamics of Myers-Perry Black Holes

Alessandro Bravetti,1,2 Davood Momeni,3 Ratbay Myrzakulov,3 and Aziza Altaibayeva3

1 Dipartimento di Fisica and ICRA, “Sapienza” Universita` di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy 2 Instituto de Ciencias Nucleares, Universidad Nacional Autonoma´ de Mexico,´ AP 70543, 04510 Mexico,´ DF, Mexico 3 Eurasian International Center for Theoretical Physics, L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan

Correspondence should be addressed to Alessandro Bravetti; [email protected]

Received 13 March 2013; Revised 4 June 2013; Accepted 5 June 2013

Academic Editor: Rong-Gen Cai

Copyright © 2013 Alessandro Bravetti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 hard to verify Davies approach with the usual technique of calculating the corresponding critical exponents (although Black holes are very special thermodynamic systems. They very interesting works on this subject exist, see, e.g., [4, 6– are thermodynamic system since they have a temperature, 18]). In fact, the main drawback of this approach is that one the celebrated Hawking temperature [1], and a definition of has to choose arbitrarily the order parameter for black holes. entropy via the Bekenstein area law [2], from which one Apossibleresolutiontothissituationcanthencomefrom can prove that the laws of thermodynamics apply to black the use of thermodynamic geometry. Since the pioneering holes [3]. On the other side, they are very special thermo- works of Gibbs [19]andCaratheodory´ [20], techniques dynamic systems, and since, for instance, the entropy is not from geometry have been introduced into the analysis of extensive, they cannot be separated into small subsystems, thermodynamics. In particular, Fisher [21]andRadhakr- and perhaps the worst fact, their thermodynamics does not ishna Rao [22]proposedametricstructureinthespaceof possess a microscopic description yet (see, e.g., [4]foraclear probability distributions which has been extensively used description of these problems). both in statistical physics and in economics (for a recent In this puzzling situation, one of the most successful and review see [23]). Later, Weinhold [24]introducedaninner atthesametimediscussedapproachtothestudyofblack product in the equilibrium space of thermodynamics based holes phase transitions is the work of Davies [5]. According on the stability conditions of the internal energy, taken as the to Davies, black holes can be regarded as ordinary systems, thermodynamic potential. The work of Weinhold was then showing phase transitions right at those points where the developed by Ruppeiner [25] from a different perspective. generalized susceptibilities, that is, second-order derivatives Ruppeinermovedfromtheanalysisoffluctuationsaround ofthepotential,changesignmostnotablythroughan equilibrium and from the gaussian approximation of the infinite discontinuity. Since there is no statistical mechanical probability of fluctuations and found a thermodynamic description of black holes as thermodynamic systems, it is metric which is defined as (minus) the hessian of the entropy 2 Advances in High Energy Physics of the system. Remarkably, Ruppeiner geometry was found give special emphasis on the relation between divergences to be conformally related to the one proposed by Weinhold. of the generalized susceptibilities and curvature singularities Moreover, Ruppeiner metric is intrinsically related to the of the metric from GTD. For example, we do not consider underlying statistical model, and in fact the scalar curvature here possible phase transitions related to change in the of the Riemannian manifold representing the system using topology of the event horizon, an intriguing question which Ruppeiner metric happens to have exactly the same exponent was addressed, for example, in [4]. We find out that the as the correlation volume for ordinary systems (see, e.g., [26] GTD thermodynamic geometry is always curved for the for a review). considered cases, showing the presence of thermodynamic All these approaches have been widely used to study interaction and that its singularities always correspond to ordinary systems, and in particular Ruppeiner metric has divergences of the susceptibilities or to points where there is a been also used to study many black holes configurations (see change from a stable to an unstable phase. This will allow us to [27] and refernces therein). This is because one can argue that infer new results on the physical meaning of the equilibrium being Ruppeiner metric defined only from thermodynamic manifold of GTD. quantities and on the other side giving information about the The work is organized as follows. In Section 2,we statistical model, then it can provide some hints towards the present the basic aspects of GTD and introduce all the resolution of the long standing problem of understanding the mathematical concepts that are needed. In Section 3,we microscopic properties of black holes (see, e.g., [27]). perform the parallel between the thermodynamic quantities On the other side, the problem with the use of thermo- and the Geometrothermodynamic description of the five- dynamic geometries to study black holes thermodynamics dimensional Myers-Perry black holes for three different cases, is that black holes are not ordinary systems, as we argued depending on the values of the angular momenta. Finally, in previous. For instance, Ruppeiner metric in many cases Section 4, we comment on the results and discuss possible gives exactly the same results of Davies approach (which is developments. based upon ordinary thermodynamics), while in some other important cases it does not converge to the same results, as 2. Basics of Geometrothermodynamics it happens for example, in the Reissner-Nordstrom¨ and Kerr cases (see, e.g., [27–29]). One can argue either that Davies Geometrothermodynamics (GTD) is a geometric theory approach is inaccurate or that the application of Ruppeiner of thermodynamics recently proposed by Quevedo [31]. It metric to black holes may be imperfect, due to the strange makes use of contact geometry in order to describe the nature of black holes as thermodynamic systems. In fact, phase space T of thermodynamic systems and express the there is still an open debate on this topic (see, e.g., the first law of thermodynamics in a coordinate-free fashion. discussion in [27, 30]). Furthermore, GTD adds a Riemannian structure 𝐺 to the Furthermore,intherecentyears,anewapproachinthe phase space and requests 𝐺 to be invariant under Legendre context of thermodynamic geometry has been proposed by transformations, in order to give it the same properties which Quevedo [31], known as geometrothermodynamics (GTD). one expects for ordinary thermodynamics. Moreover, GTD According to this approach, the Riemannian structure to be introduces the manifold of the equilibrium space E as the introduced in the equilibrium space should be consistent maximum integral submanifold of the contact structure of with the property of Legendre invariance, a property which T, characterized by the validity of the first law of thermo- isatthecoreofordinarythermodynamics.InGTDsome dynamics [31].Atthesametime,GTDprescribesalsotopull families of metrics have been found that share the Legendre back the Riemannian structure 𝐺 to the equilibrium space. invariance property, and they have also been proven to This results in a naturally induced Riemannian structure 𝑔 interpret in a consistent manner the thermodynamic prop- in E, which is supposed to be the geometric counterpart erties of ordinary systems, chemical interactions, black holes of the thermodynamic system. Such a description has been configurations, and cosmological solutions (see31 [ –39]). In proposed in order to give thermodynamic geometry a new particular, the correspondence between the divergences of symmetry which was not present in previous approaches, that the scalar curvature of the equilibrium manifold of GTD and is, the Legendre invariance. the phase transition points signaled by the divergences of the Let us see now the mathematical definitions of the GTD heat capacity (i.e., phase transitions alaDavies` )seemstobe objects that we shall use in this work. If we are given a system a general fact, according to the variety of systems analyzed so with 𝑛 thermodynamic degrees of freedom, we introduce 𝐴 far and to the general expressions given in [40]. In addition, first a (2𝑛+1)-dimensional space T with coordinates 𝑍 = 𝑎 𝑎 arecentstudy[41] of the thermodynamics of the Reissner- {Φ, 𝐸 ,𝐼 },with𝐴 = 0,...,2𝑛 and 𝑎 = 1,...,𝑛,whichis Nordstrom¨ and Kerr black holes in any dimensions suggested known as the thermodynamic phase space [31]. We make use that the GTD approach can detect not only the points of of the phase space T in order to correctly handle both the phase transitions due to singularities of the heat capacities Legendre transformations and the first law of thermodynam- but also divergences of the full spectrum of the generalized ics. In fact, in classical thermodynamics, we can change the susceptibilities. thermodynamic potential using aLegendretransformation, On the other side, the thermodynamic properties of which is defined in T asthechangeofcoordinatesgivenby the Myers-Perry black holes in five dimensions have been [47]: extensively studied in the literature from completely different 𝑎 𝑎 ̃ ̃𝑎 ̃𝑎 points of view (see, e.g., [4, 29, 42–46]). In this work, we {Φ, 𝐸 ,𝐼 }󳨀→{Φ, 𝐸 , 𝐼 }, Advances in High Energy Physics 3

̃ ̃𝑘 ̃𝑙 𝑖 ̃𝑖 𝑗 ̃𝑗 Φ=Φ−𝛿𝑘𝑙𝐸 𝐼 ,𝐸=−𝐼 ,𝐸= 𝐸 , geometric properties of the thermodynamic system, which is paralleled by the equilibrium space E.Thus,wepullbackthe 𝑖 𝑖 𝑗 𝑗 𝐼 = 𝐸̃ ,𝐼= 𝐼̃ , metric 𝐺 onto E and obtain a Riemannian structure for the (1) equilibrium space which reads 2 𝑖∪𝑗 𝐼𝐼 ∗ 𝑎 𝜕Φ 𝑐 𝜕 Φ 𝑏 𝑑 where canbeanydisjointdecompositionofthesetof 𝑔 ≡𝜑 (𝐺) =(𝐸 )(𝜂 𝑑𝐸 𝑑𝐸 ), (5) indices {1,...,𝑛}and 𝑘,𝑙 = 1,...,𝑖. We remark that Legen- 𝜕𝐸𝑎 𝑏 𝜕𝐸𝑐𝜕𝐸𝑑 dre transformations are change of coordinates in T and that ∗ 𝑐 theyarenotdefinedintheequilibriumspace.Moreover,the where 𝜑 is the pullback of 𝜑 as in (3)and𝜂𝑏 = 𝐼𝐼 phase space T is equipped with a canonical contact structure diag(−1,1,...,1).Weremarkthat𝑔 is (by definition) called the Gibbs 1-form defined as invariant under (total) Legendre transformations (see, e.g., 𝐼𝐼 𝑏 𝑎 𝑎 [33]). Moreover, we also note that 𝑔 can be calculated Θ=𝑑Φ−𝛿 𝐼 𝑑𝐸 =𝑑Φ−𝐼𝑑𝐸 , 𝑎 𝑎𝑏 𝑎 (2) explicitly once the fundamental equation Φ=Φ(𝐸) is which extremely resembles the first law of thermodynamics known. and hence will be the starting point to define the equilibrium The main thesis of GTD is that the thermodynamic properties of a system described by a fundamental equation space. 𝑎 E 𝑛 Φ(𝐸 ) can be translated into geometrical features of the Furthermore, the equilibrium space is the -dimension- E al submanifold of T defined by the embedding 𝜑:E → T equilibrium manifold , which in our case is described by 𝑔𝐼𝐼 E under the condition the metric . For example, the scalar curvature of should give information about the thermodynamic interaction. This 𝜕Φ ∗ 𝑎 means that systems without interaction shall correspond 𝜑 (Θ) =0, that is,𝑑Φ=𝐼𝑎𝑑𝐸 ,𝐼𝑎 = , (3) 𝜕𝐸𝑎 to flat geometries and systems showing interaction and ∗ phase transitions should correspond to curved equilibrium where 𝜑 is the pullback of 𝜑.Itfollowsimmediatelyfrom manifolds having curvature singularities. It has been tested (2)that(3)representsboththefirstlawandtheequilibrium in a number of works (see, e.g. [31–35]) that indeed such conditions for the thermodynamic system under analysis, so correspondence works. Furthermore, a previous work [41] that E results to be (by definition) the submanifold of points studying the thermodynamics and GTD of the Reissner- where the first law and the equilibrium conditions hold, that Nordstrom¨ and of the Kerr black holes in any dimensions, is, the geometric counterpart of the thermodynamic system. 𝐼𝐼 𝐴 highlighted that curvature singularities of 𝑔 are exactly It also follows that the coordinates {𝑍 } of T assume a phys- 𝑎 at the same points where the generalized susceptibilities ical meaning in E.Infact,theset{𝐸 },with𝑎=1,...,𝑛,can diverge. be identified with the extensive thermodynamic variables, 𝑎 In this work, we extend the work in [41]tothecaseof while Φ=Φ(𝐸) with the fundamental equation for the 𝑎 Myers-Perry black holes in five dimensions, with the aim of thermodynamic potential, and finally the coordinates {𝐼 }= 𝑎 𝑎 both to analyze their thermodynamic geometry from a new {𝐼 (𝐸 )} ≡ {𝜕𝐸𝑎 Φ}, 𝑎 = 1,...,𝑛 represent the intensive 𝑎 perspective and to focus on the idea of checking what happens quantities corresponding to the extensive set {𝐸 } (see, e.g., with a change of the potential from Φ=𝑀to Φ=𝑆 [48] for these definitions). in the GTD analysis and when the equilibrium manifold is Now,letusaddtheRiemannianstructure.Sincewewant 3 dimensional. The investigation of the phase structure of the Riemannian structure to share the same properties of the Myers-Perry black holes in five dimensions is thus a matter first law and since the first law is invariant under Legendre which is interesting by itself and that will provide us with transformations, we introduce in the phase space T ametric 𝐺 the necessary ground for a new test of the correspondence which is invariant under Legendre transformations. In 𝑔𝐼𝐼 GTD, there are several families of metrics which have this between the thermodynamic geometry of GTD and black property (for a recent work on this topic see [49]). Among holes thermodynamics. them, one has been proven particularly successful to describe systems with second-order phase transitions, as black holes 3. Myers-Perry Black Holes aresupposedtobe.Thus,thecandidatemetricweshallusein this work is The Kerr black hole can be generalized to the case of arbitrary dimensions and arbitrary number of spins. It turns out 𝑎 2 𝑎 𝑏 𝑐 𝑑 𝐺=(𝑑Φ−𝐼𝑎𝑑𝐸 ) +Λ(𝜉𝑎𝑏𝐸 𝐼 )(𝜒𝑐𝑑𝑑𝐸 𝑑𝐼 ), (4) that, provided, 𝑑 is the number of spacetime dimensions, that the maximum number of possible independent spins where 𝜉𝑎𝑏 and 𝜒𝑎𝑏 are diagonal constant tensors and Λ is is (𝑑 − 1)/2 if 𝑑 is odd and (𝑑 − 2)/2 if 𝑑 is even [50]. an arbitrary Legendre invariant function of the coordinates Such general configurations are called Myers-Perry black 𝐴 {𝑍 }.Inparticular,wechoosetofixΛ=1,𝜉𝑎𝑏 =𝛿𝑎𝑏 ≡ holes. They deserve a special interest because they are the diag(1,...,1)and 𝜒𝑎𝑏 =𝜂𝑎𝑏 ≡ diag(−1,1,...,1)in order to natural generalization of the well-known Kerr black hole get the exact expression for the metric describing black holes to higher number of spins and because they are shown to phase transitions (see also [33]). coexist with the Emparan-Reall black ring solution for some On the other side, we are not interested in the geometric values of the parameters, thus providing the first explicit representation of the phase space, while we care about the example of a violation in a dimension higher than four of 4 Advances in High Energy Physics

the uniqueness theorem (see, e.g., [51] for more details). 3.1. The Case 𝐽2=0. If either 𝐽1 =0or 𝐽2 =0,we The line element of the Myers-Perry black hole with an obtain the Kerr black hole in 5 dimensions, which has been arbitrary number of independent angular momenta in Boyer- analyzed in [41]. We briefly review here some of the results Lindquist coordinates for 𝑑=2𝑛+1(i.e., odd 𝑑) reads [50] presented there and improve the analysis, including the

2 investigation of the response functions defined in the total 𝜇𝑟2 𝑛 Π𝐹 Legendre transformation of the mass 𝑀, which we will call 𝑑𝑠2 =−𝑑𝑡2 + (𝑑𝑡 + ∑𝑎 𝜇2𝑑𝜙 ) + 𝑑𝑟2 Π𝐹 𝑖 𝑖 𝑖 Π−𝜇𝑟2 theGibbsianresponsefunctions,inanalogywithstandard 𝑖=1 𝐽 = (6) thermodynamics [48]. Therefore, let us suppose that 2 𝑛 0 2 2 2 2 2 . According to our previous results [41], we know that the + ∑ (𝑟 +𝑎𝑖 ) (𝑑𝜇𝑖 +𝜇𝑖 𝑑𝜙𝑖 ), response functions defined in the mass representation read 𝑖=1 5/3 3𝑆 2(𝑆 +4𝐽2) 3(𝑆2 +4𝐽2) with 1 1 𝐶𝐽 =− ,𝜅𝑆 = , 1 𝑆2 − 12𝐽2 2𝐽 (3𝑆2 −4𝐽2) 𝑛 𝑎2𝜇2 𝑛 1 1 1 𝐹≡1−∑ 𝑖 𝑖 ,Π≡∏ (𝑟2 +𝑎2), (11) 𝑟2 +𝑎2 𝑖 2 2 5/3 𝑖=1 𝑖 𝑖=1 3 (𝑆 +4𝐽 ) (7) 𝛼 =− 1 , 𝑆 8 𝐽2𝑆 16𝜋𝐺𝑀 (𝑑−2) 𝐽𝑖 1 𝜇≡ ,𝑎𝑖 ≡ , (𝑑−2) Ω(𝑑−2) 2 𝑀 𝑎 wherewemakeuseofthenotation𝑀𝐸𝑎 ≡𝜕𝑀/𝜕𝐸and 2 𝑎 𝑏 𝑖 𝑛 𝑀 𝑎 𝑏 ≡𝜕𝑀/𝜕𝐸 𝜕𝐸 𝐸 =𝑆,𝐽 where Ω(𝑑−2) =2𝜋/Γ(𝑛), 𝑀 is the mass of the black 𝐸 𝐸 ,for 1.Itfollowsthat 𝛼 hole, 𝐽𝑖 =𝐽1,...,𝐽𝑛 are the (𝑑 − 1)/2 independent angular 𝑆 does not show any singularity (apart from the extremal 𝑛 2 𝑆=0 𝐶 𝑆2 = momenta, and the constraint ∑𝑖=1 𝜇𝑖 =1holds. Solving the limit ), while 𝐽1 diverges at the Davies point 𝑟𝑟 2 equation 𝑔 =1/𝑔𝑟𝑟 =0, one finds the radius of the event 12𝐽1 and 𝜅𝑆 shows an additional possible phase transition at 2 2 horizon (in any dimensions) and thus derives the area and 3𝑆 =4𝐽1 . As it was pointed out in [41], both singularities the corresponding entropy, using Bekenstein area law [29]. of the heat capacity and of the compressibility are in the Inparticular,inthiswork,weareinterestedinthefive blackholeregionandhencearephysicallyrelevant.Itwas dimensional case, that is, when 𝑑=5. Myers-Perry black also shown that the GTD geometry (5)withfundamental holes in five-dimensions can have up to 2 independent equation (10)(with𝐽2 =0) is curved, indicating the presence angular momenta, and the general equation for the area reads of thermodynamic interaction, and that the singularities of [29] the scalar curvature are situated exactly at the same points 𝐶 𝜅 2 where the response functions 𝐽1 ,and 𝑆 diverge, both in the 2𝜋 2 2 2 2 𝐴= (𝑟+ +𝑎1 )(𝑟+ +𝑎2 ) , (8) mass and in the entropy representations. Furthermore, it was 𝑟+ also commented that Weinhold geometry is flat in this case 𝑟 and Ruppeiner thermodynamic metric diverges only in the where + is the radius of the event horizon. From the previous extremal limit 𝑆=0(see, e.g., [29] for a complete analysis expression the entropy can be calculated, being using these metrics). 𝑘 𝐴 1 Moreover,sincethethermodynamicsofblackholescan 𝑆= 𝐵 = (𝑟2 +𝑎2)(𝑟2 +𝑎2), 4𝐺 𝑟 + 1 + 2 (9) 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. where we choose 𝑘𝐵 and 𝐺 such that 𝑆 simplifies as in the second equality in (9). Using the relations between thermodynamic derivatives (see, [48]), we find out that the expressions for such response Since it is rather complicated to calculate explicitly the (𝑆, 𝐽 ) previous expression for the entropy, we will use the 𝑀 functions in the coordinates 1 used here are representation throughout the paper. This is possible since the 𝑆 (3𝑆2 −4𝐽2) 𝑆2 − 12𝐽2 mass can be written in terms of 𝑆, 𝐽1,and𝐽2 as [29] 1 1 𝐶Ω =− ,𝜅𝑇 =− , 1 2 2 2 2 1/3 𝑆 +4𝐽1 2𝐽 (𝑆 +4𝐽 ) 2 1/3 2 1/3 1 1 3 2/3 𝐽 𝐽 (12) 𝑀(𝑆,𝐽 ,𝐽 )= 𝑆 (1 + 4 1 ) (1 + 4 2 ) . (10) 8𝑆 1 2 4 𝑆2 𝑆2 𝛼Ω =− . 1 2 2 1/3 (𝑆 +4𝐽1 ) Equation (10)thusrepresentsthefundamentalequation 𝐶 for the Myers-Perry black hole in five dimensions as a It is immediate to see that Ω1 never diverges and it thermodynamic system. Starting from (10), we can analyze vanishes exactly at the same points where 𝜅𝑆 diverges. On the both the thermodynamic properties and their geometrother- other side, 𝜅𝑇 is never divergent and it vanishes exactly where 𝐶 𝛼 modynamic counterparts. We will split the work in order to 𝐽1 diverges, while Ω1 is always finite. It follows that the consider the three most interesting cases, that is, when one Gibbsian response functions do not add any information to of the two angular momenta is zero, when they are both the knowledge of the phase structure of this configuration, as nonzero but equal, and finally when they are both nonzero they change the sign exactly at the points that we have already and different among each other. analyzed; therefore, we conclude that the divergences of the Advances in High Energy Physics 5 scalar curvature of the metric (5) match exactly the points of Again, making use of the relations between thermodynamic second-order phase transitions. derivatives, we find out that they read Let us now add a second spin parameter and show that there is still a concrete correlation between the geometric 2 2 2 2 𝐼𝐼 𝑆(𝑆 −4𝐽 )(3𝑆 +4𝐽 ) description performed with 𝑔 and the thermodynamic 𝐶 =− , Ω 2 properties. To do so, we first focus on the special case of (10) (𝑆2 +4𝐽2) in which 𝐽1 =𝐽2 =𝐽, and afterwards we will consider the 2/3 completely general case, that is, with 𝐽1 and 𝐽2 both different 𝑆 D𝐶 𝜅𝑇 =− , (17) from zero and from each other. In particular, in the latter 4𝐽 (𝑆2 +4𝐽2)5/3 case, we will get a 3-dimensional thermodynamic manifold 1 2 3 5/3 2 2 labelled by (𝐸 =𝑆,𝐸 =𝐽1,𝐸 =𝐽2), and hence we will 8𝑆 (𝑆 +2𝐽 ) consider the 3-dimensional version of the metric (5). 𝛼Ω = . (𝑆2 +4𝐽2)5/3

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 In this case, we observe that the only divergence of the arefixedtobeequal,thatis,𝐽1 =𝐽2 ≡𝐽. This is interesting response functions in (14), that is, the divergence of 𝐶𝐽,is from the mathematical and physical point of view since it is again controlled by the vanishing of 𝜅𝑇.Furthermore,both theonlycaseinwhichtheangularmomentaarebothdifferent 𝐶𝐽 and 𝐶Ω vanish at the extremal limit, but this does not fromzero,andatthesametimethethermodynamicmanifold correspond to any divergence of 𝜅𝑆, and hence we expect the is 2-dimensional. In fact, the mass fundamental equation (10) curvature of the thermodynamic metric to diverge only at the in this case is given by points where D𝐶 =0. From the point of view of Geometrothermodynamics, 3 𝐽2 2/3 given the fundamental equation (13) and the general metric 𝑀 (𝑆,) 𝐽 = 𝑆2/3(1 + 4 ) , (13) 4 𝑆2 (5), we can calculate the particular metric and the scalar curvature for the equilibrium manifold of the MP black hole and the response functions can then be accordingly calcu- with two equal angular momenta, both in the mass and in the latedtogive entropy representations. The metric in the 𝑀 representation reads 4/3 3𝑆 4(𝑆 − 16𝐽4) 3𝑆2/3(𝑆2 +4𝐽2) 𝐶 =− ,𝜅= , 𝐼𝐼 1 𝐽 4 2 2 4 𝑆 𝑆 − 32𝐽 𝑆 − 80𝐽 4𝐽 (3𝑆2 +4𝐽2) 𝑔𝑀 = 𝑆4/3(𝑆2 +4𝐽2)2/3 5/3 2 2 4/3 (18) 3 𝑆 (𝑆 +4𝐽 ) 2 (3𝑆2 +4𝐽2) 𝛼 =− . 1 D𝐶 2 2 𝑆 2 2 2 ×{ 𝑑𝑆 + 𝑑𝐽 }. 16 𝐽 (𝑆 +2𝐽 ) 12 𝑆2 3 (14)

From (14), it follows that in this case 𝛼𝑆 and 𝜅𝑆 do not Therefore, its scalar curvature is show any singularity, while 𝐶𝐽 diverges at the roots of the D =𝑆4 − 32𝐽2𝑆2 − 80𝐽4 2/3 denominator 𝐶 .Wealsoobservethat 𝑅 = 24𝑆10/3(𝑆2 +4𝐽2) the temperature of this black hole is given by 𝑀 × (5𝑆6 + 48𝐽2𝑆4 − 368𝐽4𝑆2 − 896𝐽6) 𝜕𝑀 1 𝑆2 −4𝐽2 (19) 𝑇≡( ) = , 1/3 (15) −1 𝜕𝑆 𝐽 2 𝑆5/3(𝑆2 +4𝐽2) 2 2 2 2 ×(D𝐶(3𝑆 +4𝐽 ) ) . 2 2 therefore, the extremal limit 𝑇=0is reached when 𝐽 /𝑆 = 1/4. The numerator is a not very illuminating function that never Solving D𝐶 =0, we find that the singularities of the heat vanishes when the denominator is zero, and D𝐶 is exactly 𝑆 𝐶 capacity are situated at a value critical for the entropy such that the denominator of the heat capacity 𝐽. Therefore, the singularities of 𝑅𝑀 correspond exactly to those of 𝐶𝐽 (resp., 2 󵄨 𝐽 󵄨 √21 − 4 𝜅 ) 3𝑆2 +4𝐽2 󵄨 = , to the vanishing of 𝑇 .Weremarkthatthefactor in 2 󵄨 (16) 𝑅 𝑆 󵄨𝑆=𝑆 20 the denominator of 𝑀, despite being always different from critical 0 (thus not indicating any phase transition in this case), is which is less than the extremal limit. Therefore Davies point exactly the denominator of the compressibility 𝜅𝑆 (resp., a of phase transition belongs to the black hole region and we factor in the numerator of 𝐶Ω). shall investigate it. To continue with the analysis, in [49], a general relation 𝐼𝐼 It is convenient also in this case to write the full set of ther- was presented (see (34) therein) to express 𝑔 with Φ=𝑆 modynamic response functions, including the Gibbsian ones. (i.e., in the 𝑆 representation) in the coordinates of the 𝑀 6 Advances in High Energy Physics

𝑎 representation (i.e., {𝐸 } = (𝑆, 𝐽)).Suchrelationinthepresent the thermodynamic manifold is 3 dimensional and the mass case reads fundamental equation is given by (10). The generalized susceptibilities can then be accordingly 𝐼𝐼 𝑀−𝐽Ω 𝑔 = calculated. The heat capacity at constant angular momenta 𝐽1 𝑆 𝑇3 and 𝐽2 reads 2 ×[𝑇𝑀𝑆𝑆𝑑𝑆 +2Ω𝑀𝑆𝑆𝑑𝑆 𝑑𝐽 (20) 𝑀𝑆 𝐶𝐽 ,𝐽 = 2 1 2 𝑀 +(2Ω𝑀𝑆𝐽 −𝑇𝑀𝐽𝐽)𝑑𝐽 ], 𝑆𝑆 (23) 3𝑆 2(𝑆 +4𝐽2)(𝑆2 +4𝐽2)(𝑆4 − 16𝐽2𝐽2) where 𝑇≡𝜕𝑀/𝜕𝑆is the temperature, Ω≡𝜕𝑀/𝜕𝐽is the 1 2 1 2 2 𝑎 𝑏 =− , angular velocity at the horizon and 𝑀𝐸𝑎𝐸𝑏 ≡𝜕𝑀/𝜕𝐸 𝜕𝐸 , D𝐶 𝑖 for 𝐸 =𝑆,𝐽.Using(20)and(13)forthemassinterms 𝐼𝐼 where of 𝑆 and 𝐽,wecancalculatetheexpressionformetric𝑔𝑆 in the coordinates (𝑆, 𝐽), which reads 8 2 2 6 2 2 4 D𝐶 =𝑆 −12(𝐽1 +𝐽2 )𝑆 − 320𝐽1 𝐽2 𝑆 𝐼𝐼 1 (24) 𝑔 = 2 2 2 2 2 4 4 𝑆 2 2 3(𝑆2 +4𝐽2) (𝑆+2𝐽) (𝑆−2𝐽) − 576𝐽1 𝐽2 (𝐽1 +𝐽2 )𝑆 − 1280𝐽1 𝐽2 .

3𝑆2 −4𝐽2 ×{− D 𝑑𝑆2+ Furthermore, one can define the 3 analogues of the adiabatic 2 𝐶 compressibility as

2 2 (21) 8𝑆𝐽 (3𝑆 −4𝐽 ) 𝜕𝐽1 2 (𝜅𝑆) ≡( ) + D𝐶𝑑𝑆𝑑𝐽 −4𝑆 11 𝜕Ω (𝑆+2𝐽)(𝑆−2𝐽) 1 𝑆

6 4 2 2 4 6 2/3 2 2 5/3 9𝑆 + 156𝑆 𝐽 + 112𝑆 𝐽 − 448𝐽 2 3𝑆 (𝑆 +4𝐽 ) × 𝑑𝐽 }. = 1 , (𝑆+2𝐽)(𝑆−2𝐽) 2 2 1/3 2 2 2(𝑆 +4𝐽2 ) (3𝑆 −4𝐽1 )

Consequently, the scalar curvature is 𝜕𝐽2 (𝜅𝑆)22 ≡( ) N 𝜕Ω2 𝑆 𝑅 = 𝑆 , 𝑆 3 2 (22) (3𝑆2 −4𝐽2) (𝑆2 +4𝐽2) D2 5/3 (25) 𝐶 3𝑆2/3(𝑆2 +4𝐽2) = 2 , N 2 2 1/3 2 2 where 𝑆 is again a function which never vanishes at the 2(𝑆 +4𝐽1 ) (3𝑆 −4𝐽2 ) points where the denominator is zero. From (22), we see that the denominator of 𝐶𝐽 is present in the denominator of 𝑅𝑆. 𝜕𝐽1 2 2 (𝜅𝑆)12 ≡( ) Furthermore, the factor 𝑆 +4𝐽 is never zero, and hence it 𝜕Ω2 𝑆 does not give any additional singularity. On the other hand, 2 2 2 2 2/3 2 2 2/3 2 2 2/3 the factor 3𝑆 −4𝐽 is clearly vanishing when 𝐽 /𝑆 =3/4, 3 𝑆 (𝑆 +4𝐽1 ) (𝑆 +4𝐽2 ) 𝐽2/𝑆2 =1/4 = . which is readily greater than the extremal limit , 16 𝐽1𝐽2 and hence it has no physical relevance in our analysis. We thus conclude that also in this case the GTD 𝐼𝐼 Finally, the analogues of the expansion are given by geometry 𝑔 exactly reproduces the phase transition structure of the Myers-Perry black holes both in the mass and in 𝜕𝐽 𝛼 ≡( 1 ) 𝑆,𝐽2 the entropy representation. We comment that in the entropy 𝜕𝑇 𝑆 representation there is an additional singularity which does not correspond to any singularity of the response functions. 5/3 2 2 5/3 2 2 2/3 3 𝑆 (𝑆 +4𝐽1 ) (𝑆 +4𝐽2 ) However, such singularity is situated out of the black hole =− , 8 𝐽 (𝑆4 +6𝑆2𝐽2 +8𝐽2𝐽2) region, and thus it is not to be considered here. We also 1 2 1 2 remark that Ruppeiner curvature in this case reads 𝑅= (26) 2 2 4 4 𝜕𝐽1 −𝑆(𝑆 + 12𝐽 )/(𝑆 − 16𝐽 ), and hence it diverges only in the 𝛼𝑆,𝐽 ≡( ) 1 𝜕𝑇 extremal limit, while Weinhold metric is flat. 𝑆 In the next subsection, we will analyze the general case of 5/3 2 2 5/3 2 2 2/3 3 𝑆 (𝑆 +4𝐽 ) (𝑆 +4𝐽 ) the Myers-Perry black hole in five dimensions, that is, when =− 2 1 . 4 2 2 2 2 the two angular momenta are allowed to vary freely. 8 𝐽2 (𝑆 +6𝑆 𝐽1 +8𝐽1 𝐽2 ) (𝜅 ) 3.3. The General Case in Which 𝐽1 =𝐽̸ 2 =0̸ . Perhaps the most In this case, neither 𝑆 12 nor the expansions show any 𝐶 D =0 interesting case is the most general one, in which the two singularity, while 𝐽1,𝐽2 diverges when 𝐶 and the 2 2 angular momenta are allowed to vary freely. In this case, compressibilities (𝜅𝑆)11 and (𝜅𝑆)22 diverge when 3𝑆 −4𝐽1 =0 Advances in High Energy Physics 7

complete. The heat capacity at constant angular velocities read 𝜕𝑆 𝐶Ω ,Ω ≡𝑇( ) 1 2 𝜕𝑇 Ω1,Ω2 0.250 =−3𝑆(𝑆4 − 16𝐽2𝐽2) (3𝑆2 −4𝐽2) 1 2 1 (29) 0.245 × (3𝑆2 −4𝐽2)(𝑆2 +4𝐽2)(𝑆2 +4𝐽2) 0.240 2 1 2 0.235 −1 ×(D (𝑆,1 𝐽 ,𝐽2)) , 0.230 where the denominator is given by 0.225 12 2 2 10 D (𝑆,1 𝐽 ,𝐽2)=9𝑆 +72(𝐽1 +𝐽2 )𝑆 0 0 2 4 2 2 4 8 2 + 16 (9𝐽1 + 95𝐽1 𝐽2 +9𝐽2 )𝑆 4 4 2 2 2 2 6 J + 5376𝐽 𝐽 (𝐽 +𝐽 )𝑆 b 6 6 1 2 1 2 (30) Ja 8 8 2 2 4 2 2 4 4 − 256𝐽1 𝐽2 (9𝐽1 − 101𝐽1 𝐽2 +9𝐽2 )𝑆 10 4 4 2 2 2 6 6 2 − 6144𝐽1 𝐽2 (𝐽1 +𝐽2 )𝑆 − 53248𝐽1 𝐽2 . Figure 1: The difference between the extremal limit(𝐽1𝐽2/𝑆 =1/4) 𝐽 𝐽 /𝑆2 and the value of 1 2 at the critical point of the heat capacity, Furthermore, one can define three generalized susceptibili- plotted for values of 𝐽1 and 𝐽2 in the interval [0, 10]. ties, analogous to the isothermal compressibility, as 𝜕𝐽 𝜕𝐽 2 2 (𝜅 ) ≡( 1 ) ,(𝜅) ≡( 1 ) , and 3𝑆 −4𝐽2 =0, respectively. Furthermore, the temperature 𝑇 11 𝑇 12 𝜕Ω1 𝑇 𝜕Ω2 𝑇 reads (31) 𝜕𝐽 1 𝑆4 − 16𝐽2𝐽2 (𝜅 ) ≡( 2 ) . 𝑇= 1 2 . 𝑇 22 𝜕Ω 2𝑆5/3 2 2 2/3 2 2 2/3 (27) 2 𝑇 (𝑆 +4𝐽1 ) (𝑆 +4𝐽2 ) 2 For the Myers-Perry black hole it can be written as Hence; the extremal limit is reached for 𝐽1𝐽2/𝑆 =1/4.The D =0 heat capacity diverges when 𝐶 , which is an algebraic (𝜅𝑇)11 equation of degree 8 in 𝑆. We can solve numerically such 2/3 𝑆= 𝑆 1/3 1/3 equation and obtain the critical value Scritical in terms of 2 2 2 2 =− (D𝐶 ((𝑆 +4𝐽1 ) (𝑆 +4𝐽2 ) 𝐽1 and 𝐽2. Taking only the roots which are real and positive, 2 we can compare them with the extremal limit by doing −1 󵄨 󵄨 󵄨 6 2 4 2 2 2 2 4 𝐽 𝐽 󵄨 𝐽 𝐽 󵄨 1 𝐽 𝐽 󵄨 ×(𝑆 − 12𝐽2 𝑆 + 48𝐽1 𝐽2 𝑆 + 192𝐽2 𝐽2 )) ), 1 2 󵄨 − 1 2 󵄨 = − 1 2 󵄨 . 𝑆2 󵄨 𝑆2 󵄨 4 𝑆2 󵄨 (28) 󵄨𝑆=Sextremal 󵄨𝑆=Scritical 󵄨𝑆=Scritical (𝜅𝑇) The plot of the result is given in Figure 1 for some values 22 𝐽 𝐽 2/3 of 1 and 2.AswecanseefromFigure 1,thedifferencein 𝑆 1/3 1/3 =− (D ((𝑆2 +4𝐽2) (𝑆2 +4𝐽2) (28)isalwayspositive,andhencethepointofphasetransition 2 𝐶 1 2 signaled by the divergence of the heat capacity is always in the −1 black hole region. 6 2 4 2 2 2 2 4 2 2 ×(𝑆 − 12𝐽1 𝑆 + 48𝐽1 𝐽2 𝑆 + 192𝐽2 𝐽1 )) ), Inthesameway,wecansolve3𝑆 −4𝐽1 =0and see (𝜅 ) whether the divergence of 𝑆 11 lies in the black hole region (32) or not. It turns out that the denominator of (𝜅𝑆)11 vanishes for 2 2 2 values of 𝑆 such that 𝐽1 /𝑆 =3/4,whichmeansthat 𝐽1𝐽2/𝑆 = while (𝜅𝑇)12 has a more cumbersome expression and we (3/4)(𝐽2/𝐽1). Therefore, we have that (1/4) − (3/4)(𝐽2/𝐽1) is will not write it here, since it has the same properties positive provided that 𝐽1 >3𝐽2 for 𝐽1 >0or 𝐽1 <3𝐽2 of (𝜅𝑇)11 and (𝜅𝑇)22 as regards to our analysis; that is, it is 𝐽 <0 (𝜅 ) 𝐶 for 1 . Summing up, the divergences of 𝑆 11 can be proportional to the denominator of 𝐽1,𝐽2 defined in (24)and in the black hole region for appropriate values of 𝐽1 and 𝐽2. it has a nontrivial denominator (one can also introduce the Analogously, the divergences of (𝜅𝑆)22 canalsobeintheblack two analogues of the thermal expansion, but for the sake of hole region. simplicity, we are not going to write them here, since they do As in the preceding sections, we will now focus on the not show any singularities, and hence they do not play any Gibbsian response functions, in order to make the analysis role in our analysis). 8 Advances in High Energy Physics

Therefore, from the thermodynamic point of view, we We can thus calculate the difference between the extremal 𝐶 𝐽 𝐽 /𝑆2 =1/4 remark that the divergences of 𝐽1,𝐽2 are matched by the limit 1 2 and the critical value (35). The result is vanishing of the three quantities (𝜅𝑇)𝑖𝑗 , while the divergences of (𝜅𝑆)11 and (𝜅𝑆)22 are reproduced as zeroes of (−𝐽2 −𝐽2 + √𝐽4 + 14𝐽2𝐽2 +𝐽4) the heat capacity 𝐶Ω ,Ω . This behavior is in agreement 1 1 1 2 1 1 2 2 1 2 − with the analysis of the preceding sections. Furthermore, 4 8 𝐽 𝐽 𝐶 1 2 in this case, the heat capacity Ω1,Ω2 and the generalized (𝜅 ) 1 compressibilities 𝑇 𝑖𝑗 possibly show additional phase tran- =− (((𝐽2 +𝐽2 −6𝐽𝐽 + √𝐽4 + 14𝐽2𝐽2 +𝐽4) sitions, which is a further indication of the fact that black 48 1 2 1 2 1 1 2 2 holes exhibit different thermodynamic behavior in different 2 2 √ 4 2 2 4 potentials. ×(𝐽1 +𝐽2 − 𝐽1 + 14𝐽1 𝐽2 +𝐽2 )) Now, let us turn to the GTD analysis. Given the fundamental equation (10) and the general metric (5), we can −1 ×(𝐽2𝐽2) ), calculate the particular metric and the scalar curvature for 1 2 the MP black hole with two free angular momenta, both in (36) the mass and in the entropy representations. The metric in the 𝑀 representation reads which can be positive for appropriate values of 𝐽1 and 𝐽2. Therefore, such points of divergence of 𝑅𝑀 are in the black 𝐼𝐼 1 𝑔𝑀 = hole region for some values of the parameters. Hence, we 3𝑆4/3(𝑆2 +4𝐽2)1/3(𝑆2 +4𝐽2)1/3 1 2 conclude that the behavior of 𝑅𝑀 perfectly matches the behavior of 𝐶𝐽 ,𝐽 , but in this case, it does not reproduce 1 D 1 2 ×{− 𝐶 𝑑𝑆2 the additional possible phase transitions indicated by the 4 2 2 2 2 2 𝑆 (𝑆 +4𝐽1 )(𝑆 +4𝐽2 ) singularities of the compressibilities (𝜅𝑆)11 and (𝜅𝑆)22 and possibly shows some additional unexpected singularities. (3𝑆2 −4𝐽2)(𝑆2 +4𝐽2) + 1 2 𝑑𝐽2 However, we can give a precise physical meaning to such 2 2 1 (33) 𝑆 +4𝐽1 additional singularities. In fact, if we evaluate the determinant of the Hessian of the mass with respect to the angular 2 2 2 2 (3𝑆 −4𝐽 )(𝑆 +4𝐽 ) momenta 𝐽1 and 𝐽2,weget + 2 1 𝑑𝐽2 2 2 2 𝑆 +4𝐽2 ( (𝑀) )≡𝑀 𝑀 −𝑀2 det Hess 𝐽1𝐽2 𝐽1𝐽1 𝐽2𝐽2 𝐽1𝐽2 +16𝐽 𝐽 𝑑𝐽 𝑑𝐽 }. 1 2 1 2 4 2 2 2 2 2 4 3𝑆 −4𝑆 (𝐽 +𝐽 ) − 16𝐽 𝐽 (37) = 1 2 1 2 , 3 𝑆4/3(𝑆2 +4𝐽2)4/3(𝑆2 +4𝐽2)4/3 Hence, its scalar curvature is 1 2

𝑅𝑀 = N𝑀 from which we can see that the numerator is exactly the factor in the denominator of 𝑅𝑀 whose roots give the 2 4 2 2 2 2 2 2 ×(D𝐶 [3𝑆 −4𝑆 (𝐽1 +𝐽2 ) − 16𝐽1 𝐽2 ] additional singularities. Since the Hessians of the energy (34) in thermodynamics are related to the stability conditions, −1 we suggest that the physical meaning of such additional 2 2 2/3 2 2 2/3 ×(𝑆 +4𝐽1 ) (𝑆 +4𝐽2 ) ) , divergences of 𝑅𝑀 is to be found in a change of stability of the system, for example, from a stable phase to an unstable one. D 𝐶 𝐼𝐼 where 𝐶 is as usual the denominator of 𝐽1,𝐽2 defined in 𝑔 N On the other side, using the relation (20)for between (24). Since there is no term in the numerator 𝑀 which the 𝑀 and the 𝑆 representations, naturally extended to the cancels out the divergences that happen when D𝐶 =0,we 3-dimensional case with coordinates (𝑆,1 𝐽 ,𝐽2),thatis, canconcludethateveryphasetransitionrelatedtotheheat 𝐶 capacity 𝐽1,𝐽2 is properly reproduced by the scalar curvature 4 2 2 𝐼𝐼 𝑀−𝐽1Ω1 −𝐽2Ω2 𝑅𝑀. In addition, in this case, the factor 3𝑆 −4𝑆(𝐽1 + 𝑔 = 2 2 2 𝑆 𝑇3 𝐽2 ) − 16𝐽1 𝐽2 can also vanish, possibly giving an additional singularity which does not correspond to the ones shown by ×[𝑇𝑀 𝑑𝑆2 +2Ω 𝑀 𝑑𝑆 𝑑𝐽 +2Ω 𝑀 𝑑𝑆 𝑑𝐽 4 2 2 𝑆𝑆 1 𝑆𝑆 1 2 𝑆𝑆 2 theresponsefunctions.Itiseasytocalculatethat3𝑆 −4𝑆 (𝐽1 + 2 2 2 𝐽 ) − 16𝐽 𝐽 =0for values of 𝑆 such that + (2Ω 𝑀 −𝑇𝑀 )𝑑𝐽2 (38) 2 1 2 1 𝑆𝐽1 𝐽1𝐽1 1

2 2 2 + (2Ω2𝑀𝑆𝐽 −𝑇𝑀𝐽 𝐽 )𝑑𝐽 (−𝐽 −𝐽 + √𝐽4 + 14𝐽2𝐽2 +𝐽4) 2 2 2 2 𝐽 𝐽 1 1 2 1 1 2 2 1 2 = . (35) 2 −2 (𝑇𝑀 −Ω 𝑀 −Ω 𝑀 )𝑑𝐽 𝑑𝐽 ], 𝑆 8 𝐽1𝐽2 𝐽1𝐽2 1 𝑆𝐽2 2 𝑆𝐽1 1 2 Advances in High Energy Physics 9

we can now calculate the metric in the 𝑆 representation, which entropy 𝑆 with respect to the angular momenta 𝐽1 and 𝐽2. reads Therefore, they still indicate the points where the Hessian vanishes, although they are not situated in the black hole [3𝑆4 +4𝑆2 (𝐽2 +𝐽2) − 16𝐽2𝐽2] D 𝑔𝐼𝐼 = 1 2 1 2 𝐶 region in this case. We infer from these results that the 𝑆 3(𝑆2 +4𝐽2)(𝑆2 +4𝐽2)(𝑆2 −4𝐽𝐽 )2(𝑆2 +4𝐽𝐽 )2 physical meaning of the divergences of the scalar curvature of 1 2 1 2 1 2 𝐼𝐼 the metric 𝑔 for such a 3-dimensional equilibrium manifold 2 2 1 𝑆𝐽 (𝑆 +4𝐽 ) is related to the divergences of the heat capacity at constant ×{ 𝑑𝑆2 +4 1 2 𝑑𝑆𝑑𝐽 2 (𝑆2 −4𝐽𝐽 )(𝑆2 +4𝐽𝐽 ) 1 angular momenta and to the zeroes of the Hessian of the 1 2 1 2 potential with respect to those momenta, both in the mass 𝑆𝐽 (𝑆2 +4𝐽2) and in the entropy representations. On the other side, from +4 2 1 𝑑𝑆𝑑𝐽 the full analysis of the divergences of the generalized response 2 2 2 (𝑆 −4𝐽1𝐽2)(𝑆 +4𝐽1𝐽2) functions, we see that there are other possible points of phase transitions related to divergences of the compressibilities, 2 2 2 2 −2(𝑆 (𝑆 +4𝐽2 ) whichappeartobenotenclosedbytheanalysisgivenwith 𝐼𝐼 𝑔 .Wealsocommentthatwecouldhaveusedthepotential 6 4 2 2 2 2 4 2 ×[3𝑆 + 26𝑆 𝐽1 + 144𝑆 𝐽1 𝐽2 + 320𝐽1 𝐽2 ] Φ=𝐺≡𝑀−𝑇𝑆−𝐽1Ω1 −𝐽2Ω2 in writing the metric (5) to study the GTD analysis in the 𝐺 representation, but 2 2 −1 2 ×(D𝐶 (𝑆 −4𝐽1𝐽2)(𝑆 +4𝐽1𝐽2)) )𝑑𝐽1 such investigation would have led to exactly the same results, as it has to be, since the metric (5) is invariant under total 2 −2(𝑆2(𝑆2 +4𝐽2) Legendre transformations. 1 To conclude, we observe that in [29]thecaseofthe full Myers-Perry black hole thermodynamics has been ×[3𝑆6 + 26𝑆4𝐽2 + 144𝑆2𝐽2𝐽2 + 320𝐽4𝐽2] 2 1 2 2 1 investigated using Weinhold and Ruppeiner thermodynamic 2 2 −1 2 geometries. The authors proved that both Weinhold and ×(D𝐶 (𝑆 −4𝐽1𝐽2)(𝑆 +4𝐽1𝐽2)) )𝑑𝐽2 Ruppeiner scalar curvatures only diverge in the extremal limit. 2 2 2 2 2 −32(𝑆 𝐽1𝐽2 (𝑆 +4𝐽1 )(𝑆 +4𝐽2 ) 4 2 2 2 2 2 4. Conclusions ×[5𝑆 + 12𝑆 (𝐽1 +𝐽2 ) + 16𝐽1 𝐽2 ] Inthiswork,wehaveanalyzedthethermodynamicsand ×(D (𝑆2 −4𝐽𝐽 ) 𝐶 1 2 thermodynamic geometry of different Myers-Perry black holes configurations in five dimensions, classifying them −1 according to the values of the two possible independent ×(𝑆2 +4𝐽𝐽 )) )) 𝑑𝐽 𝑑𝐽 }. 1 2 1 2 angular momenta. To this end, we followed the approach of Davies for (39) the standard analysis of the thermodynamic properties in The scalar curvature can thus be calculated to obtain different potentials and used the approach of GTD for the thermodynamic geometric investigation. The present work 2 4 2 2 2 2 2 3 has been carried out with the twofold aim of understand- 𝑅𝑆 = N𝑆 ×(D𝐶[3𝑆 +4𝑆 (𝐽1 +𝐽2 ) − 16𝐽1 𝐽2 ] ing the phase structure of Myers-Perry black holes in five −1 (40) 2 2 2 2 2 dimensions and inferring new conclusions on the physical ×𝑆 (𝑆 +4𝐽 )(𝑆 +4𝐽 )) . 𝐼𝐼 1 2 meaning of the metric 𝑔 ,bothinthemassandintheentropy representations. In this case, we see again that the denominator of the Our results indicate that the Myers-Perry black holes D 𝑅 heat capacity 𝐶 is present in the denominator of 𝑆. infivedimensionshaveanontrivialphasestructureinthe Furthermore, the second factor, which is slightly different sense of Davies. In particular, the analysis of the response 𝑅 from the factor in the denominator of 𝑀,vanishesforvalues functions indicates that both the heat capacities and the 𝑆 of such that compressibilitiesdefinedinthe𝑀 potential diverge at some points, which is usually interpreted as the hallmark of a phase 2 2 √ 4 2 2 4 𝐽 𝐽 1 𝐽1 +𝐽2 + 𝐽1 + 14𝐽1 𝐽2 +𝐽2 1 2 = . (41) transition. Interestingly, such a behavior is matched by the 2 𝑆 8 𝐽1𝐽2 vanishing of the corresponding Gibbsian response functions in all the cases studied here. Moreover, in the most general The earlier discussion for the additional singularity of 𝑅𝑀 case when the two angular momenta vary freely, we have does not apply in this case, since one can easily show that shown that the Gibbsian response functions provide some the points described by (41) do not belong to the black hole additional singularities, indicating that the analysis in the 𝑀 region for any values of 𝐽1 and 𝐽2.However,wecomment potential is different from that performed in the 𝐺 potential. in passing that such additional singularities are still related In all the cases studied in this work, the phase transi- to the vanishing of the determinant of the Hessian of the tions are well reproduced by the GTD analysis, while they 10 Advances in High Energy Physics are not reproduced by the thermodynamic geometries of Acknowledgments Weinhold and Ruppeiner, whose analysis has been observed to correspond to other approaches (see e.g., [4]). We have The authors want to thank Professor H. Quevedo for insight- 𝐼𝐼 also found that the scalar curvature of the metric 𝑔 shows ful suggestions. Alessandro Bravetti wants to thank ICRA for a very similar behavior in the 𝑀 representation to that of financial support. the 𝑆 representation. In particular, for the cases in which we have only two degrees of freedom we argue that no physical References difference has been detected and we have shown that not only the phase transitions indicated by 𝐶𝐽 are reproduced, [1] S. W. Hawking, “Particle creation by black holes,” Communica- tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. but also the ones indicated by divergences of 𝜅𝑆.Moreover,a detailed analysis of the Gibbsian response functions showed [2] J. D. Bekenstein, “Black holes and entropy,” Physical Review D, that such divergences correspond to points where 𝜅𝑇 and vol. 7, pp. 2333–2346, 1973. 𝐶Ω vanish and change their character. We therefore conclude [3] J. M. Bardeen, B. Carter, and S. W. 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