geometrothermodynamical properties of myers-perry black hole

8/12/2019 Geometrothermodynamical properties of Myers-Perry Black Hole

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Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleGeometrothermodynamics of Myers-Perry Black Holes

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

Dipartimento di Fisica and ICRA, Sapienza Universita di Roma, Piazzale Aldo Moro , Rome, Italy Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, AP , Mexico, DF, Mexico Eurasian International Center for Teoretical Physics, L.N. Gumilyov Eurasian National University, Astana , Kazakhstan

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

Received March ; Revised June ; Accepted June

Academic Editor: Rong-Gen Cai

Copyright Alessandro Bravetti et al. Tis is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We consider the thermodynamics and geometrothermodynamics o the Myers-Perry black holes in ve dimensions or threedifferent cases, depending on the values o the angular momenta. We ollow Davies approach to study the thermodynamics oblack holes and nd a nontrivial thermodynamic structure in all cases, which is ully reproduced by the analysis perormed withthe techniques o Geometrothermodynamics. Moreover, we observe that in the cases when only one angular momentum is presentor the two angular momenta are xed to be equal, that is, when the thermodynamic system is two dimensional, there is a completeagreement between the divergences o the generalized susceptibilities and the singularities o the equilibrium maniold, whereaswhen the two angular momenta are ully 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 rom a stable phase to anunstable one.

1. Introduction

Black holes are very special thermodynamic systems. Teyare thermodynamic system since they have a temperature,the celebrated Hawking temperature [], and a denition oentropy via the Bekenstein area law [], rom which onecan prove that the laws o thermodynamics apply to blackholes []. On the other side, they are very special thermo-dynamic systems, and since, or instance, the entropy is not

extensive, they cannot be separated into small subsystems,and perhaps the worst act, their thermodynamics does notpossess a microscopic description yet (see, e.g., [] or a cleardescription o these problems).

In this puzzling situation, one o the most successul andat the same time discussed approach to the study o blackholes phase transitions is the work o Davies []. Accordingto Davies, black holes can be regarded as ordinary systems,showing phase transitions right at those points where thegeneralized susceptibilities, that is, second-order derivativeso the potential, change sign most notably through aninnite discontinuity. Since there is no statistical mechanicaldescription o black holes as thermodynamic systems, it is

hard to veriy Davies approach with the usual technique ocalculating the corresponding critical exponents (although

very interesting works on this subject exist, see, e.g., [,]). In act, the main drawback o this approach is that onehas to choose arbitrarily the order parameter or black holes.

A possible resolution to this situation can then come romthe use o thermodynamic geometry. Since the pioneeringworks o Gibbs [] and Caratheodory [], techniquesrom geometry have been introduced into the analysis o

thermodynamics. In particular, Fisher [] and Radhakr-ishna Rao [] proposed a metric structure in the space oprobability distributions which has been extensively usedboth in statistical physics and in economics (or a recentreview see []). Later, Weinhold [] introduced an innerproduct in the equilibrium space o thermodynamics basedon the stability conditions o the internal energy, taken as thethermodynamic potential. Te work o Weinhold was thendeveloped by Ruppeiner [] rom a different perspective.Ruppeiner moved rom the analysis o uctuations aroundequilibrium and rom the gaussian approximation o theprobability o uctuations and ound a thermodynamicmetric which is dened as (minus) the hessian o the entropy


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Advances in High Energy Physics

o the system. Remarkably, Ruppeiner geometry was oundto be conormally related to the one proposed by Weinhold.Moreover, Ruppeiner metric is intrinsically related to theunderlying statistical model, and in act the scalar curvatureo the Riemannian maniold representing the system usingRuppeiner metric happens to have exactly the same exponent

as the correlation volume or ordinary systems (see, e.g., []or a review).All these approaches have been widely used to study

ordinary systems, and in particular Ruppeiner metric hasbeen also used to study many black holes congurations (see[] and reernces therein). Tis is because one can argue thatbeing Ruppeiner metric dened only rom thermodynamicquantities and on the other side giving inormation about thestatistical model, then it can provide some hints towards theresolution o the long standing problem o understanding themicroscopic properties o black holes (see, e.g., []).

On the other side, the problem with the use o thermo-dynamic geometries to study black holes thermodynamicsis that black holes are not ordinary systems, as we arguedprevious. For instance, Ruppeiner metric in many casesgives exactly the same results o Davies approach (which isbased upon ordinary thermodynamics), while in some otherimportant cases it does not converge to the same results, asit happens or example, in the Reissner-Nordstrom and Kerrcases (see, e.g., []). One can argue either that Daviesapproach is inaccurate or that the application o Ruppeinermetric to black holes may be imperect, due to the strangenature o black holes as thermodynamic systems. In act,there is still an open debate on this topic (see, e.g., thediscussion in [,]).

Furthermore, in the recent years, a new approach in thecontext o thermodynamic geometry has been proposed byQuevedo [], known as geometrothermodynamics (GD).According to this approach, the Riemannian structure to beintroduced in the equilibrium space should be consistentwith the property o Legendre invariance, a property whichis at the core o ordinary thermodynamics. In GD someamilies o metrics have been ound that share the Legendreinvariance property, and they have also been proven tointerpret in a consistent manner the thermodynamic prop-erties o ordinary systems, chemical interactions, black holescongurations, and cosmological solutions (see []). Inparticular, the correspondence between the divergences othe scalar curvature o the equilibrium maniold o GD andthe phase transition points signaled by the divergences o the

heat capacity (i.e., phase transitions a la Davies) seems to bea general act, according to the variety o systems analyzed soar and to the general expressions given in []. In addition,a recent study [] o the thermodynamics o the Reissner-Nordstrom and Kerr black holes in any dimensions suggestedthat the GD approach can detect not only the points ophase transitions due to singularities o the heat capacitiesbut also divergences o the ull spectrum o the generalizedsusceptibilities.

On the other side, the thermodynamic properties othe Myers-Perry black holes in ve dimensions have beenextensively studied in the literature rom completely differentpoints o view (see, e.g., [, , ]). In this work, we

give special emphasis on the relation between divergenceso the generalized susceptibilities and curvature singularitieso the metric rom GD. For example, we do not considerhere possible phase transitions related to change in thetopology o the event horizon, an intriguing question whichwas addressed, or example, in []. We nd out that the

GD thermodynamic geometry is always curved or theconsidered cases, showing the presence o thermodynamicinteraction and that its singularities always correspond todivergences o the susceptibilities or to points where there is achangeroma stable to anunstable phase. Tis will allow ustoiner new results on the physical meaning o the equilibriummaniold o GD.

Te work is organized as ollows. In Section , wepresent the basic aspects o GD and introduce all themathematical concepts that are needed. In Section , weperorm the parallel between the thermodynamic quantitiesand the Geometrothermodynamic description o the ve-dimensional Myers-Perry black holesor threedifferent cases,depending on the values o the angular momenta. Finally, inSection , we comment on the results and discuss possibledevelopments.

2. Basics of Geometrothermodynamics

Geometrothermodynamics (GD) is a geometric theoryo thermodynamics recently proposed by Quevedo []. Itmakes use o contact geometry in order to describe thephase space T o thermodynamic systems and express therst law o thermodynamics in a coordinate-ree ashion.Furthermore, GD adds a Riemannian structure to thephase space and requests to be invariant under Legendretransormations, in order to give it the same properties whichone expects or ordinary thermodynamics. Moreover, GDintroduces the maniold o the equilibrium space E as themaximum integral submaniold o the contact structure oT, characterized by the validity o the rst law o thermo-dynamics []. At the same time, GD prescribes also to pullback the Riemannian structure to the equilibrium space.Tis results in a naturally induced Riemannian structurein E, which is supposed to be the geometric counterparto the thermodynamic system. Such a description has beenproposed in order to give thermodynamic geometry a newsymmetry which was notpresent in previousapproaches, thatis, the Legendre invariance.

Let us see now the mathematical denitions o the GD

objects that we shall use in this work. I we are given a systemwith thermodynamic degrees o reedom, we introducerst a(2 + 1)-dimensional spaceT with coordinates ={,, }, with = 0, . . . ,2 and = 1, . . . , , which isknown asthe thermodynamic phase space[]. We make useo the phase space T in order to correctly handle both theLegendre transormations and the rst law o thermodynam-ics. In act, in classical thermodynamics, we can change thethermodynamic potential using a Legendre transformation,which is dened in T as the change o coordinates given by[]:

,

,

,

,

,


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= , = , =, =, =,

()

where

can be any disjoint decomposition o the set o

indices{1, . . . , }and, = 1, . . . , . We remark that Legen-dre transormations are change o coordinates in T and thatthey are not dened in the equilibrium space. Moreover, thephasespace T is equipped with a canonical contact structurecalledthe Gibbs1-formdened as

= = , ()which extremely resembles the rst law o thermodynamicsand hence will be the starting point to dene the equilibriumspace.

Furthermore, the equilibrium spaceE isthe-dimension-al submaniold oT dened by the embedding : E Tunder the condition

()= 0, that is, = , = , ()where is the pullback o. It ollows immediately rom() that () represents both the rst law and the equilibriumconditions or the thermodynamic system under analysis, sothat E results to be (by denition) the submaniold o pointswhere the rst law and the equilibrium conditions hold, thatis, the geometric counterpart o the thermodynamic system.

It also ollows that the coordinates{}oT assume a phys-ical meaning in E. In act, the set{}, with = 1, . . . ,, canbe identied with the extensive thermodynamic variables,

while = (

) with the undamental equation or thethermodynamic potential, and nally the coordinates{} ={()} {}, = 1, . . . , represent the intensivequantities corresponding to the extensive set{}(see, e.g.,[] or these denitions).

Now, let us add the Riemannian structure. Since we wantthe Riemannian structure to share the same properties o therst law and since the rst law is invariant under Legendretransormations, we introduce in the phase space T a metric which is invariant under Legendre transormations. InGD, there are several amilies o metrics which have thisproperty (or a recent work on this topic see []). Amongthem, one has been proven particularly successul to describesystems with second-order phase transitions, as black holesare supposed to be. Tus, the candidate metric we shall use inthis work is

= 2 + , ()where and are diagonal constant tensors and isan arbitrary Legendre invariant unction o the coordinates{}. In particular, we choose to x = 1,= diag(1 , . . . , 1 )and= diag(1,1, . . . ,1)in order toget the exact expression or the metric describing black holesphase transitions (see also []).

On the other side, we are not interested in the geometricrepresentation o the phase space, while we care about the

geometric properties o the thermodynamic system, which isparalleled by the equilibrium space E. Tus, we pull back themetric onto E and obtain a Riemannian structure or theequilibrium space which reads

()=

2

, ()

where is the pullback o as in () and =diag(1,1, . . . ,1). We remark that is (by denition)invariant under (total) Legendre transormations (see, e.g.,

[]). Moreover, we also note that can be calculatedexplicitly once the undamental equation = () isknown.

Te main thesis o GD is that the thermodynamicproperties o a system described by a undamental equation() can be translated into geometrical eatures o theequilibrium maniold E, which in our case is described by

the metric

. For example, the scalar curvature oE should

give inormation about the thermodynamic interaction. Tismeans that systems without interaction shall correspondto at geometries and systems showing interaction andphase transitions should correspond to curved equilibriummaniolds having curvature singularities. It has been testedin a number o works (see, e.g. []) that indeed suchcorrespondence works. Furthermore, a previous work []studying the thermodynamics and GD o the Reissner-Nordstrom and o the Kerr black holes in any dimensions,

highlighted that curvature singularities o are exactlyat the same points where the generalized susceptibilitiesdiverge.

In this work, we extend the work in [] to the case o

Myers-Perry black holes in ve dimensions, with the aim oboth to analyze their thermodynamic geometry rom a newperspective andto ocus on theidea o checking what happenswith a change o the potential rom = to = in the GD analysis and when the equilibrium maniold is dimensional. Te investigation o the phase structure oMyers-Perry black holes in ve dimensions is thus a matterwhich is interesting by itsel and that will provide us withthe necessary ground or a new test o the correspondence

between the thermodynamic geometry o GD and blackholes thermodynamics.

3. Myers-Perry Black Holes

Te Kerr black hole can be generalized to the case o arbitrarydimensions and arbitrary number o spins. It turns outthat, provided, is the number o spacetime dimensions,that the maximum number o possible independent spinsis( 1)/2 i is odd and( 2)/2 i is even [].Such general congurations are called Myers-Perry blackholes. Tey deserve a special interest because they are thenatural generalization o the well-known Kerr black holeto higher number o spins and because they are shown tocoexist with the Emparan-Reall black ring solution or some

values o the parameters, thus providing the rst explicitexample o a violation in a dimension higher than our o


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the uniqueness theorem (see, e.g., [] or more details).Te line element o the Myers-Perry black hole with anarbitrary number o independent angular momenta in Boyer-Lindquist coordinates or = 2 + 1(i.e., odd)reads []

2

= 2

+2

+

=12

2

+ 2 2

+ =1

2 + 2 2+ 22 ,()

with

1 =1

222 + 2 , =1

2 + 2 , 16( 2)(2) ,

( 2)2 ,()

where(2) = 2

/(), is the mass o the blackhole,= 1, . . . , are the( 1)/2independent angularmomenta, and the constraint=12= 1holds. Solving theequation = 1/= 0, one nds the radius o the eventhorizon (in any dimensions) and thus derives the area andthe corresponding entropy, using Bekenstein area law [].

In particular, in this work, we are interested in the vedimensional case, that is, when = 5. Myers-Perry blackholes in ve-dimensions can have up to independentangular momenta, and the general equation or the area reads[]

=22+

2+

+ 21

2+

+ 22

, ()

where +is the radius o the event horizon. From the previousexpression the entropy can be calculated, being

=4=1+ 2++ 21 2++ 22 , ()

where we choose and such that simplies as in thesecond equality in ().

Since it is rather complicated to calculate explicitly theprevious expression or the entropy, we will use therepresentation throughout the paper. Tis is possible sincethemass can be written in terms o

,

1, and

2as []

,1, 2 =342/31 + 4212

1/31 + 4 2221/3. ()

Equation () thus represents the undamental equationor the Myers-Perry black hole in ve dimensions as athermodynamic system. Starting rom (), we can analyzeboth the thermodynamic properties and their geometrother-modynamic counterparts. We will split the work in order toconsider the three most interesting cases, that is, when oneo the two angular momenta is zero, when they are bothnonzero but equal, and nally when they are both nonzeroand different among each other.

.. Te Case2= 0. I either1 = 0 or2 = 0, weobtain the Kerr black hole in dimensions, which has beenanalyzed in []. We briey review here some o the resultspresented there and improve the analysis, including theinvestigation o the response unctions dened in the totalLegendre transormation o the mass, which we will callthe Gibbsian response unctions, in analogy with standardthermodynamics []. Tereore, let us suppose that2=0. According to our previous results [], we know that theresponse unctions dened in the mass representation read

1 = 32 + 4212 1221 , =

32 + 4215/32132 421 ,

= 382 + 4215/321 ,

()

where we make use o the notation / and

2

/

, or

= ,1. It ollows that does not show any singularity (apart rom the extremallimit = 0), while1 diverges at the Davies point2 =1221 and shows an additional possible phase transition at32 = 421 . As it was pointed out in [], both singularitieso the heat capacity and o the compressibility are in theblack hole region and hence are physically relevant. It wasalso shown that the GD geometry () with undamentalequation () (with2= 0) is curved, indicating the presenceo thermodynamic interaction, and that the singularities othe scalar curvature are situated exactly at the same pointswherethe response unctions1 , and diverge,both in themass and in the entropy representations. Furthermore, it was

also commented that Weinhold geometry is at in this caseand Ruppeiner thermodynamic metric diverges only in theextremal limit = 0(see, e.g., [] or a complete analysisusing these metrics).

Moreover, since the thermodynamics o black holes candepend on the chosen ensemble (see, e.g., [,]), we nowproceed to calculate the Gibbsian response unctions, whichcan possibly give new inormation about the phase structure.Using the relations between thermodynamic derivatives (see,[]), we nd out that the expressions or such responseunctions in the coordinates(, 1)used here are

1 = 32 4212 + 421 , =

2

1221

212 + 4211/3 ,1 = 82 + 4211/3 .

()

It is immediate to see that1 never diverges and itvanishes exactly at the same points where diverges. On theother side,is never divergent and it vanishes exactly where1 diverges, while1 is always nite. It ollows that theGibbsian response unctions do not add any inormation tothe knowledge o the phase structure o this conguration, asthey change the sign exactly atthe points that we have alreadyanalyzed; thereore, we conclude that the divergences o the


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scalar curvature o the metric () match exactly the points osecond-order phase transitions.

Let us now add a second spin parameter and show thatthere is still a concrete correlation between the geometric

description perormed with and the thermodynamicproperties. o do so, we rst ocus on the special case o ()

in which1= 2= , and aferwards we will consider thecompletely general case, that is, with1and2both differentrom zero and rom each other. In particular, in the lattercase, we will get a -dimensional thermodynamic maniold

labelled by(1 = , 2 = 1, 3 = 2), and hence we willconsider the -dimensional version o the metric ().

.. Te Case 1= 2 . Another special case in () whichis o interest is the case in which the two angular momentaare xed to be equal, that is,1= 2 . Tis is interestingrom the mathematical and physical point o view since it istheonly case in which the angular momenta are both differentrom zero, andat the same time the thermodynamic maniold

is -dimensional. In act, the mass undamental equation ()in this case is given by

(, )=342/31 + 4 2

22/3, ()

and the response unctions can then be accordingly calcu-lated to give

= 34 164

4 3222 804 , =32/32 + 424/3

432 + 42 ,

= 3165/3

2

+ 42

4/3

2 2 + 22 .()

From (), it ollows that in this case and do notshow any singularity, while diverges at the roots o thedenominatorD= 4 3222 804. We also observe thatthe temperature o this black hole is given by

=12

2 425/32 + 421/3 , ()

thereore, the extremal limit = 0is reached when2

/2

=1/4.SolvingD= 0, we nd that the singularities o the heat

capacity are situated at a value criticalor the entropy such that22=critical =

21 420 , ()which is less than the extremal limit. Tereore Davies pointo phase transition belongs to the black hole region and weshall investigate it.

It is convenientalsoin this case to write the ull set other-modynamic response unctions, including the Gibbsian ones.

Again, making use o the relations between thermodynamicderivatives, we nd out that they read

= 2 4232 + 42

2

+ 42

2

,

= 2/34 D2 + 425/3 ,

=85/3 2 + 222 + 425/3.

()

In this case, we observe that the only divergence o theresponse unctions in (), that is, the divergence o, isagain controlled by the vanishing o. Furthermore, both and vanish at the extremal limit, but this does notcorrespond to any divergence o

, and hence we expect the

curvature o the thermodynamic metric to diverge only at thepoints whereD= 0.

From the point o view o Geometrothermodynamics,given the undamental equation () and the general metric(), we can calculate the particular metric and the scalarcurvature or the equilibrium maniold o the MP black holewith two equal angular momenta, both in the mass and in theentropy representations.

Te metric in therepresentation reads= 14/32 + 422/3

112D22 +2 32 + 423 2 .

()

Tereore, its scalar curvature is

= 2410/32 + 422/3 56 + 4824 36842 8966 D232 + 4221.

()

Te numerator is a not very illuminating unction that nevervanishes when the denominator is zero, and D is exactlythe denominator o the heat capacity. Tereore, thesingularities o correspond exactly to those o (resp.,to the vanishing o). We remark that the actor32 +42 inthe denominator o, despite being always different rom0 (thus not indicating any phase transition in this case), isexactly the denominator o the compressibility (resp., aactor in the numerator o).

o continue with the analysis, in [], a general relation

was presented (see () therein) to expresswith = (i.e., in the

representation) in the coordinates o the


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representation (i.e.,{} = (, )). Such relation in the presentcase reads

= 3

2

+ 2

+2 2 ,()

where / is the temperature, / is theangular velocity at the horizon and 2/,or = , . Using () and () or the mass in termso and, we can calculate the expression or metric inthe coordinates(, ), which reads= 13 2 + 42 ( + 2)2( 2)2

32

42

2 D

2

++8 32 42( + 2) ( 2)D 42

96 + 15642 + 11224 4486( + 2) ( 2) 2 .

()

Consequently, the scalar curvature is

= N32 4232 + 422D2 , ()where N is again a unction which never vanishes at thepoints where the denominator is zero. From (), we see thatthe denominator o is present in the denominator o.Furthermore, the actor2 + 42 is never zero, and hence itdoes not give any additional singularity. On the other hand,

the actor32 42 is clearly vanishing when2/2 = 3/4,which is readily greater than the extremal limit2/2 = 1/4,and hence it has no physical relevance in our analysis.

We thus conclude that also in this case the GDgeometry exactly reproduces the phase transition struc-ture o the Myers-Perry black holes both in the mass and inthe entropy representation. We comment that in the entropyrepresentation there is an additional singularity which does

not correspond to any singularity o the response unctions.However, such singularity is situated out o the black holeregion, and thus it is not to be considered here. We alsoremark that Ruppeiner curvature in this case reads =(2 + 122)/(4 164), and hence it diverges only in theextremal limit, while Weinhold metric is at.

In the next subsection, we will analyze the general case othe Myers-Perry black hole in ve dimensions, that is, whenthe two angular momenta are allowed to vary reely.

.. Te General Case in Which1= 2= 0. Perhaps the mostinteresting case is the most general one, in which the twoangular momenta are allowed to vary reely. In this case,

the thermodynamic maniold is dimensional and the massundamental equation is given by ().

Te generalized susceptibilities can then be accordinglycalculated. Te heat capacity at constant angular momenta 1and2reads

1,2 == 3 2 + 421 2 + 422 4 162122

D

,()

where

D= 8 12 21+ 22 6 32021224 576212221+ 22 2 12804142 .

()

Furthermore, one can dene the analogues o the adiabatic

compressibility as

11 11= 32/32 + 421

5/3

22 + 4221/3 32 421 ,

22 22

= 32/32 + 4225/3

22 + 4211/3

32 422 ,12 12=316

2/32 + 4212/32 + 4222/312 .

()

Finally, the analogues o the expansion are given by

,2 1 = 38

5/32 + 4215/32 + 4222/314 + 6222+ 82122 ,,1 1

= 385/32 + 4225/32 + 4212/324 + 6221+ 82122 .

()

In this case, neither()12 nor the expansions show anysingularity, while1,2 diverges when D= 0 and thecompressibilities

(

)11and

(

)22diverge when

32

421

= 0


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0.250

0.245

0.240

0.235

0.230

0.225

0 0

2 2

4 4

6 6

8 8

10

JaJb

F : Te difference between the extremal limit (12/2 = 1/4)and the value o12/2 at the critical point o the heat capacity,plotted or values o1and2in the interval[0,10].

and 32422= 0, respectively. Furthermore, the temperaturereads

= 125/34 1621222 + 421 2/32 + 4222/3 . ()

Hence; the extremal limit is reached or12/2 = 1/4. Teheat capacity diverges when D= 0, which is an algebraicequation o degree in. We can solve numerically suchequation and obtain the critical value = Scriticalin terms o1 and2. aking only the roots which are real and positive,we can compare them with the extremal limit by doing

122=Sextremal

122=Scritical =

14122=Scritical. ()

Te plot o the result is given in Figure or some valueso1 and2. As we can see romFigure , the difference in() is alwayspositive, and hence the point o phase transitionsignaled by the divergence o the heat capacity is always in the

black hole region.In the same way, we can solve32 421= 0 and see

whether the divergence o()11lies in the black hole regionor not. It turns out that the denominator o()11vanishes or

values o such that21 /2 = 3/4, which means that12/2 =(3/4)(2/1). Tereore, we have that(1/4) (3/4)(2/1) ispositive provided that1> 32 or1> 0 or1< 32or1< 0 . Summing up, the divergences o()11 can bein the black hole region or appropriate values o1 and2.Analogously, the divergences o()22can also be in the blackhole region.

As in the preceding sections, we will now ocus on theGibbsian response unctions, in order to make the analysis

complete. Te heat capacity at constant angular velocitiesread

1,2 1 ,2

= 3 4

162

12

232

42

1 32 422 2 + 421 2 + 422 D ,1, 21,

()

where the denominator is given by

D ,1, 2 = 912 + 72 21+ 22 10+ 16941+ 952122+ 942 8+ 5376212221+ 22 6

2562122941 1012122+ 942 4 6144414221+ 22 2 532486162 .

()

Furthermore, one can dene three generalized susceptibili-ties, analogous to the isothermal compressibility, as

11 11, 12 12,

22 22. ()

For the Myers-Perry black hole it can be written as

11= 2/32D 2 + 421

1/32 + 4221/3

6 12224 + 4821222 + 1922242 1 ,22

= 2/32D 2 + 4211/32 + 4221/3

6 12214 + 4821222 + 1922241 1 ,()

while()12 has a more cumbersome expression and wewill not write it here, since it has the same propertieso()11 and()22 as regards to our analysis; that is, it isproportional to the denominator o1,2 dened in ()andit has a nontrivial denominator (one can also introduce thetwo analogues o the thermal expansion, but or the sake osimplicity, we are not going to write them here, since they donot show any singularities, and hence they do not play anyrole in our analysis).


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Tereore, rom the thermodynamic point o view, weremark that the divergences o1,2 are matched by the

vanishing o the three quantities(), while the diver-gences o()11 and()22 are reproduced as zeroes othe heat capacity1,2 . Tis behavior is in agreementwith the analysis o the preceding sections. Furthermore,

in this case, the heat capacity1 ,2 and the generalizedcompressibilities()possibly show additional phase tran-sitions, which is a urther indication o the act that blackholes exhibit different thermodynamic behavior in differentpotentials.

Now, let us turn to the GD analysis. Given the un-damental equation () and the general metric (), we cancalculate the particular metric and the scalar curvature orthe MP black hole with two ree angular momenta, both inthe mass and in the entropy representations. Te metric inthe representation reads

= 1

34/32 + 421 1/3

2 + 4221/3

14 D2 2 + 421 2 + 4222

+32 421 2 + 4222 + 421 21

+32 422 2 + 4212 + 422 22

+161

2

1

2

.

()

Hence, its scalar curvature is

=N D2 34 42 21+ 22 1621222

2 + 4212/32 + 4222/31,()

where D is as usual the denominator o1 ,2 dened in(). Since there is no term in the numerator N which

cancels out the divergences that happen when D= 0, wecan conclude that every phase transition related to the heatcapacity1,2 is properly reproduced by the scalar curvature. In addition, in this case, the actor34 42(21+22 ) 162122 can also vanish, possibly giving an additionalsingularity which does not correspond to the ones shown by

theresponseunctions. It is easy to calculate that 3442(21+22 ) 162122= 0or values osuch that

12

2=1

8

21 22+41+ 142122+ 42

1

2

. ()

We can thus calculate the difference between the extremallimit12/2 = 1/4and the critical value (). Te result is

1

41

8

21 22+41+ 142122+ 42

1

2

= 14821+ 22 612+41+ 142122+ 42 21+ 2241+ 142122+ 42

21221 ,()

which can be positive or appropriate values o1 and2.Tereore, such points o divergence o are in the blackhole region or some values o the parameters. Hence, we

conclude that the behavior o perectly matches thebehavior o1,2 , but in this case, it does not reproducethe additional possible phase transitions indicated by thesingularities o the compressibilities()11 and()22 andpossibly shows some additional unexpected singularities.

However, we can give a precise physical meaning to suchadditionalsingularities.In act, i we evaluate the determinanto the Hessian o the mass with respect to the angularmomenta1and2, we get

det Hess()12 1122 212

=4334

42

2

1+ 2

2 162

12

24/32 + 4214/32 + 422 4/3 , ()

rom which we can see that the numerator is exactly theactor in the denominator o whose roots give theadditional singularities. Since the Hessians o the energyin thermodynamics are related to the stability conditions,we suggest that the physical meaning o such additionaldivergences o is to be ound in a change o stability othe system, or example, rom a stable phase to an unstableone.

On the other side, using the relation () or betweentheand the representations, naturally extended to the-dimensional case with coordinates(, 1, 2), that is,= 11 223

2 + 21 1+ 22 2+ 211 11 21+ 222 22 22212 12 21 12 ,

()


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wecannowcalculatethemetricinthe representation, whichreads

= 34 + 42 21+ 22 162122D

3 2 + 421 2 + 422 2 41222 + 4122

122 + 4 12

+ 4222 412 2 + 4121

+ 4 22 + 4212 412 2 + 4122 222 + 4222

36 + 26421+ 14422122+ 3204122 D 2 412 2 + 4121 21

222 + 4212 36 + 26422+ 14422122+ 3204221 D 2 412 2 + 4121 22

32 2122 + 421 2 + 422 54 + 122 21+ 22 + 162122 D 2 412

2 + 412112 .()

Te scalar curvature can thus be calculated to obtain

=N D234 + 42 21+ 22 1621223

2 2 + 421 2 + 422 1. ()

In this case, we see again that the denominator o theheat capacity D is present in the denominator o

.

Furthermore, the second actor, which is slightly differentrom the actor in the denominator o, vanishes or valueso such that

122=1821+ 22+41+ 142122+ 4212 . ()

Te earlier discussion or the additional singularity odoes not apply in this case, since one can easily show thatthe points described by () do not belong to the black holeregion or any values o1 and2. However, we commentin passing that such additional singularities are still relatedto the vanishing o the determinant o the Hessian o the

entropy with respect to the angular momenta1 and2.Tereore, they still indicate the points where the Hessian

vanishes, although they are not situated in the black holeregion in this case. We iner rom these results that thephysical meaning o the divergences o the scalar curvature o

the metric

or such a -dimensional equilibrium maniold

is related to the divergences o the heat capacity at constantangular momenta and to the zeroes o the Hessian o thepotential with respect to those momenta, both in the massand in the entropy representations. On the other side, romthe ull analysis o the divergences o the generalized responseunctions, we see that there are other possible points o phasetransitions related to divergences o the compressibilities,which appear to be not enclosed by the analysis given with. We also comment that we could have used the potential = 11 22 in writing the metric() to study the GD analysis in the representation, butsuch investigation would have led to exactly the same results,as it has to be, since the metric () is invariant under total

Legendre transormations.o conclude, we observe that in [] the case o theull Myers-Perry black hole thermodynamics has beeninvestigated using Weinhold and Ruppeiner thermodynamicgeometries. Te authors proved that both Weinhold andRuppeiner scalar curvatures only diverge in the extremallimit.

4. Conclusions

In this work, we have analyzed the thermodynamics andthermodynamic geometry o different Myers-Perry blackholes congurations in ve dimensions, classiying them

according to the values o the two possible independentangular momenta.

o this end, we ollowed the approach o Davies orthe standard analysis o the thermodynamic properties indifferent potentials and used the approach o GD or thethermodynamic geometric investigation. Te present workhas been carried out with the twoold aim o understand-ing the phase structure o Myers-Perry black holes in vedimensions and inerring new conclusions on the physical

meaning o the metric, both in the massand in the entropyrepresentations.

Our results indicate that the Myers-Perry black holesin ve dimensions have a nontrivial phase structure in the

sense o Davies. In particular, the analysis o the responseunctions indicates that both the heat capacities and thecompressibilities dened in thepotential diverge at somepoints, which is usually interpreted as the hallmark o a phasetransition. Interestingly, such a behavior is matched by the

vanishing o the corresponding Gibbsian response unctionsin all the cases studied here. Moreover, in the most generalcase when the two angular momenta vary reely, we haveshown that the Gibbsian response unctions provide someadditional singularities, indicating that the analysis in thepotential is different rom that perormed in thepotential.

In all the cases studied in this work, the phase transi-tions are well reproduced by the GD analysis, while they


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are not reproduced by the thermodynamic geometries oWeinhold and Ruppeiner, whose analysis has been observedto correspond to other approaches (see e.g., []). We have

also ound that the scalar curvature o the metric showsa very similar behavior in the representation to that othe

representation. In particular, or the cases in which we

have only two degrees o reedom we argue that no physicaldifference has been detected and we have shown that notonly the phase transitions indicated by are reproduced,but also the ones indicated by divergences o. Moreover, adetailed analysis o the Gibbsian response unctions showedthat such divergences correspond to points where andvanish and change their character. We thereore concludethat or such cases the divergences o the scalar curvature o reproduce the ull set o second order phase transitionsconsidered here.

On the other side, it seems that analyzing the generalcase in which both angular momenta are switched on, thatis, a thermodynamic system with three degrees o reedom,

some differences might appear. In act, the phase transitionssignaled by1 ,2 are still obtained as curvature singularitiesin both representations. Nevertheless, the scalar curvaturehas some additional divergences, which or the case othe representation can be in the black hole region orappropriate values o the angular momenta and that appar-ently are not directly related to the response unctions o thesystem. However, we claim that such additional divergencesare linked to the vanishing o the Hessian determinant othe potential with respect to the two angular momenta,and thereore they mark the transition rom a stable phaseto an unstable one. In our opinion, this means that thephysical meaning o the scalar curvature o the metric

or

thermodynamic systems with three degrees o reedom goesbeyond the well-established correspondence with the gener-alized susceptibilities, that is, second-order derivatives o thepotential, encompassing also questions o stability related totheir mutual relation, that is, determinants o the Hessian othe potential. Tis is also supported by the analysis o thescalar curvature in the representation, which again showssingularities exactly at those points where the Hessian o theentropy with respect to the two angular momenta vanishes,so rom the mathematical point o view the situation isbasically the same. It is interesting, however, to note that inthe representation such points are not in the black holeregion, a direct evidence o the act that black hole ther-modynamics strictly depends on the potential being used.Moreover, in the completely general case, some additionaldivergences appear when considering the Gibbsian responseunctions, which are not present in the thermodynamicanalysis in the potential nor are indicated as curvaturesingularities o. Te study o such additional singularitiesgoes beyond the scope o this work and may be the mat-ter o urther investigation. We also expect to extend thiswork in the nearest uture and nd a number o urtherexamples which support (or discard) the interpretation othe thermodynamic metric or thermodynamic systemswith

3degrees o reedom given here.

Acknowledgments

Te authors want to thank Proessor H. Quevedo or insight-ul suggestions. Alessandro Bravetti wants to thank ICRA ornancial support.

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