Hessian Matrix, Specific Heats, Nambu Brackets, and Thermodynamic

JHEP04(2015)115 Springer April 1, 2015 April 21, 2015 : : b , November 11, 2014 cal Methods, 2D : Accepted Published 10.1007/JHEP04(2015)115 ng our bracket approach, Received atrix with any number of doi: les with three spins. formal transformations that this approach to investigate counterparts in thermodynamic Published for SISSA by [email protected] and Mohamadreza Fazel ], we employ the Nambu brackets to prove , a 1 Behrouz Mirza a . 3 1411.2582 The Authors. Black Holes, Diﬀerential and Algebraic Geometry, Statisti c

As an extension to our earlier work [ , [email protected] [email protected] Department of Physics, IsfahanIsfahan University 84156-83111, of Iran Technology, Department of Physics, UniversityAlbuquerque, of New New Mexico, Mexico, 87131-0001, U.S.A. E-mail: b a Gravity ArXiv ePrint: we obtain interestingparameters exact and relations specific between heatsome the capacities. thermodynamic Hessian properties Finally, of m we the employ Meyers-Perry black ho Keywords: that the divergences of heatgeometry. capacities correspond We also to their obtainconnect a simple different representation thermodynamics for metrics the con to each other. Usi Hessian matrix, specific heats,thermodynamic Nambu geometry brackets, and Open Access Article funded by SCOAP Abstract: Seyed Ali Hosseini Mansoori, JHEP04(2015)115 ]. 1 2 8 5 14 17 19 20 21 , 4 [ T ]. Prior 8 t nota- he Myers- nstein proposed ] that this outcome 3 rtional to the fluctuation se of Reimanian geometry he known first and second , Hawking later discovered acterized as an equilibrium brium manifold and, hence, . The Ruppeiner’s metric is e. y considering the propagation ic thermodynamics [ modynamics, namely that the ody with temperature hysical systems. In fluctuation ate of the system. The physical Bekenstein [ ]. However, the statistical origin of 6 , of the event horizon of a black hole never A – 1 – ] proposed a novel method to study the thermodynamics of 7 ] showed that the area, 2 , of an isolated system never decreases. Consequently, Beke S In 1979, Ruppeniner [ tion Perry black hole with three spins theory, every thermodynamic equilibriummanifold system and each can point on be thissystem char manifold is could an equilibrium fluctuate st betweenthe different line points element on ofprobability. the the equili This stable line points element is on the the basic manifold idea is for propo geometr laws of thermodynamics wasblack further hole investigated thermodynamics in is [ still an open problem tophysical realiz systems, such asto black explore holes. the This thermodynamicalbased on approach properties the makes of fluctuation u theory these of equilibrium systems states of the p total entropy, Correspondence between thermodynamics of black holes and t is similar to the statement of the ordinary second law of ther 1 Introduction Stephen Hawking [ C Properties of heat capacities 5 Conclusion A Partial derivative and braket notation B Maxwell’s equations and Nambu brackets 3 Correspondence between transitions4 and singularities The relation between phase transitions and singularities for t Contents 1 Introduction 2 Conformal transformation between different metrics and bracke that the entropy of aof black hole fields is in proportional a tothat curved its black background area. holes and radiate B with close the to spectrum the of an event ideal horizon black b decreases in physical processes. It was later pointed out by JHEP04(2015)115 , 19 (2.3) (2.1) (2.2) ]. The 9 acket no- ass and the ]. Subsequently, 9 eometry of black holes nhold metrics through u brackets, and employ he curvature of the equi- eters are also worked out. ntation [ o been employed to study re conformally related [ eats correspond exactly to nd Nambu brackets. Using ibrium space of thermody- f thermodynamics [ ets notation is also utilized s, respectively, by Weinhold ins. ular momentum (J-metric). al conformal transformation etric as follows: ed the phase thermodynamic s follows: s. Finally, we investigate the ps us with the powerful tools f thermodynamics and statisti- odynamic geometry and phase ct to the entropy and the other , we introduce Q-metric and J- ies are exactly at the same points j . 2 . j j dX i dX dX i i , we show that for Meyers-Perry black 4 dX , we discuss our results. dX j 5 dX j j M ∂X i S 2 M ∂X ∂X i 2 2 i ∂ – 2 – ∂ ]. ∂ ∂X , we study the relationship between singularities ∂X ∂X 18 3 1 T – = = = 10 2 R 2 W 2 R ds ds ds ] in which the second derivatives of entropy with respect to m 7 In this paper, we explore new metrics for the thermodynamic g The outline of this paper is as follows. In section tation ]: other extensive variables of the system give the Ruppeiner m second derivatives of the internalextensive energy variables (mass) of with the respe system give the Weinhold metric a transitions of different heathole capacities. with three In spins, section the phaseof transitions the of curvature heat singularities. capacit Finally in section 2 Conformal transformation between different metrics and br metric and show thatconformal they transformations. are In relatedof section to the the scalar Ruppenier curvature in and various Wei representations of therm needed for dealing withthe thermodynamic thermodynamics problems of , black holes as [ they als the metric was also extendedand to Ruppeiner mass and and they entropylibrium were, description manifold. consequently, This able formulation to of work thermodynamics out equi t Weinhold was the first to introduce a geometric formulation o 20 space and had developednamic a states. geometric The description Weinhold of metric the is equil based on energy represe to Ruppeiner, it was Weinhold who had additionally introduc The line elements of the Weinhold and Ruppeiner geometries a Ruppeiner later developed another geometriccal formulation mechanics o [ thermodynamics of the Meyers-Perry black hole with three sp between any two metrics arethis also approach, obtained the using Hessian theMoreover, with Poisson we an a prove arbitrary that numberthe the of singularities divergent param points of of theto the thermodynamic work specific out geometry. h heat The capacities, brack metric elements, and curvature in the representationWe of write electric the charge partialthem (Q-metric) to derivatives and represent in the ang terms conformal of transformations. Poisson The and gener Namb JHEP04(2015)115 (2.6) (2.7) (2.9) (2.4) (2.5) spec- (2.10) (2.11) ormation about this proof for the In the case of the eraction. It also plays a lation of thermodynamics . This argument has been stem. As an example, the nces of the Ricci scalar of sed in the third section of Φ lows: dS dQ C ransitions associated with the ions of the system, which are e Ricci scalar of this geometry j hermodynamics is: ) (2.8) Φ T Φ T dX dQ i Φ S, Q d d . What is more, the first law can be ( d dX dS S dQ, − − dQ j 2 W + Φ T Φ T ds Q − ∂X and + Φ dM dM 1 − 2 i T ∂ Q − dT dS 1 dM ∂X 1 Φ T dM – 3 – T dS 1 Φ 1 T ) and we dubbed it Q-metric. This argument will ) = ) = ]. Investigation of different types of black holes, = d d T ) = 1 = = C S, Q ) = dM ) = M,Q ( dQ dS ( M,S ) gives: 2 W ( 2 R 2 Q ds 2.7 ds M,S M,Q ( ds ( 2 Q 2 R ds ds ) and ( ) are the intensive and extensive variables of the system, re 2.5 S, Q are the explicit functions of ), one might infer that the Q-metric is: φ ) represents a Riemannian geometry, which contains some inf ] for black holes with three parameters. A generalization of 2.9 are entropy and the other extensive variables of the system. Φ) and ( 1 2.7 and i T, X T The first law of thermodynamics might be recast as follows: Equation ( proved in [ be demonstrated in the third section. which implies that the Ruppeiner metric can be written as fol black holes with anthis arbitrary paper. number In ofin this parameters which section, the is we singularities discus developheat of capacity the a at Ricci new a scalar geometric fixed are temperature formu the ( phase t key role in figuringdivergences out of the this thermodynamic variablethe stability are singularities of linked of the to the sy such the specific as phase heats the transit [ chargedthe black Ruppenier hole, geometry corresponds has to revealed the that singularities the of diverge the thermodynamics of the physicalcontains system. a Specifically, certain th amount of knowledge and is a measure of int Therefore its line element is: The combination of ( Regarding ( charged black hole the (Reissner-Nordström), first law of t where, where, ( tively. The Weinhold metric can be rewritten as: where, transformed into: JHEP04(2015)115 (2.13) (2.14) dT dS ) (2.12) 1 Φ ase transitions of − S, Q ( W dS 2 i scalar of the metrics Φ nt metrics. e related to each other ) (2.15) ds transformations between T d 1 rmal transformations, we Φ 2 1 law as follows: − B,C Φ certain functions with respect he proofs will not be easy to = + = ( duce a new convenient notation dC i B dM X dT dS Φ d ; 1 Φ j ∂A ∂C 2 1 -metric and the Ruppeiner metric is: Φ − − Q dX i + = dS Φ dX dB dS j – 4 – C T d 2 A ∂X 1 Φ T 2 i Φ ∂ ). Besides, Q-metrics are conformally related to the ∂A ∂B + d ∂X ) 2.8 . Moreover, one can define the A-metric as follows: − C = dM ) = Φ M,Q dM d ( dA ) below: and 2 1 R B,C 2 Φ − ( 1 B Φ 2 A 2.12 ds . The conformal transformations between various metrics. Φ T ds d ) = = ) to ( ) = M,S ( 2.10 2 Q Figure 1 M,S ds ( , respectively, coincide with the singularities of the Ricc Q S s are mass and entropy. Furthermore, we will prove that the ph is a function of 2 i depicts the conformal transformations that connect differe C A 1 X ds It is interesting to have a general formula for the conformal As already mentioned, the Ruppeiner and Weinhold metrics ar and Q The conformal transformation that connects the different geometries. To get such a formula, we write the first Ruppenier and Weinhold metrics.transform eq. In ( order to obtain the confo where, C where, by the conformal transformation ( Figure to extensive/intensive variables.present It in should the be conventional algebra;by noted however, using we that poisson will t and intro Nambu brackets. whose elements can be derived from the second derivatives of JHEP04(2015)115 ): A (2.24) (2.19) (2.23) (2.20) (2.21) (2.22) a,b } a,b a,b ) (2.17) ) (2.16) b } g, h } and Poisson bracket, we { ct results for the Hessian : rms of Poisson and Nambu f, g h, k A, C A, B a,b l help us not only to gain a ) (2.18) paces as we will see in some ∂a ∂h as: { { tions between A-metric and } = ( = ( = a , c,d c,d i i f, k ity (for a proof, see appendix A, C ut it is also useful for generalizing } } ( a,b X X h,k B ). The following useful relation can } } ) can be rewritten as: ∂b − { 2 ∂f g, h f, h ; ; { { a,b a,b j j ds a, b c, d a,b } } f, g { C } 2.19 { − = dX dX a c,d i i g, h f, h = } g, k { a,b { a,b

{ } ∂A } ∂B dX dX h h ∂b = ∂h f, g j j a,b – 5 – { h } g, h f, h − b ∂g ∂k ∂k ∂f { { C B ∂X ∂X = 2 2 i i f, h ∂ ∂ = ) = ∂g ∂f k k { a,b ∂a ∂f h ∂X ∂X } = ∂g ∂h ∂h ∂f B,C f, g ). Therefore, eq. ( ( = a,b ∂g ∂f ) = ) = ), we reach the relation below, which connects the A-metric {

} A 2 c, d a,b = ( } as explicit functions of ( ds 2.16 b h, k A, C A, B ) ) ( ( { h = 2 B 2 C f, h a,b f, g { b h, k ( ds ds } ( and ∂ ) and ( ∂ g f, g , { ) and 2.15 f c, d ( a = a In the following, we will write the partial derivatives in te In order to obtain the relation between partial derivatives where, The Jacobian transformation in this notation can be written to the B-metric via a conformal transformation. is the Poisson bracket. Consider the following useful ident Contemplating ( C-metric and also between B-metric and C-metric. It is also straightforward to gain the conformal transforma It is also simple to demonstrate that: The B-metric and C-metric are also defined as in the following where, parts of this paper.metric and Moreover, to it calculate allows heat capacities us very to easily. obtain some exa brackets. The bracket methodbetter insight has into invaluable the different benefits. thermodynamicthe relations It b thermodynamic wil formula to arbitrary dimensional phase s should consider then be obtained by a simple calculation: JHEP04(2015)115 ). 3 , q 2 (2.29) (2.26) (2.27) (2.28) (2.25) (2.32) (2.30) , q 1 are fixed. q ( k k y operator of = (2.31) ), and following and }} ) and 5 h 3 , f , q a, b, c bu one also has some 2 ( 2 , f 2 D , q 1 when f 1 f bu bracket. For a better B q }

{ g 2 ( 4 , k h ns, we get: 4 (3) ds he conformal transformation, ) and , f p ∂k 3 ), ∂q 3 , f 3 ,g a,b a,b 3 , f j 2 , f } } B f 3 3 , q ease the task of generalizing it to 1 ,g 2 ∂h { (2) 2 ∂q ,q ,q a, b, c 1 f p 2 2 g ( i { ds , q + ,q } ,q , f C 1 A, C B,C 1 1 3 ∂f ∂q }} } + q q q { { 5 5 ( ), } } (1) , f } a,b,c a,b,c g p ijk − 2 4 } } , f , f ε f descriptions, as follows: 4 ), } , f with respect to , f = 3 4 1 3 ) =1 , f a, b, c B 3 f p f B g, h, k f, h, k 3 ( , q ( , f X , f { 2 { { f 2 ijk π 2 2 B { – 6 – f ds A, C, D = , q B,C,D , 1) , f = = and { 1 2 { { C 1 ) ) as: ) q − 1 3 ]. f 3 ( 3 A − , f f ,q { h,k 1 f 2 22 , , f , g f = ( = ,q 3 = ). As we will see below, writing the conformal trans- 2 , 2.18 2 ∂A { 1 ∂B f } q ), where } A { , f 21 3 , g ∂g ∂f } 4 2 a, b 1 1 ( f − + g , f , f ds ( ( 2 3 C } ∂ = ∂ 5 , f , f f, h, k = 1 A 2 B,C,D , f { f 2 ( f 4 { 1 C A ds f , f { } = 3 ) to write eq. ( A , f ) and 2 2.23 are the permutation operator of the subscripts and the parit , f a, b 1 ( p f B {{ = and is the Levi-Civita symbol. Like the Poisson bracket, the Nam B π ijk ε Contemplating The Leibniz rule is expressed by: three or more dimensional spaces. formation in the bracket representation will significantly an approach like the two dimensionalwhich case, connects we also the show two that metrics t in the The Jacobian transformation will be as follows: Now, we want to work out the derivative of useful properties such as the Skew-symmetry: where, the Nambu bracket is defined as follows: in which we haveunderstanding, utilized consider another the functions new notation called the Nam where, these permutations, respectively [ where, and Following the above method and doing some simple calculatio an the fundamental identity by: Now, we utilize ( JHEP04(2015)115 , - n 1 1 h q X (2.33) (2.34) (2.35) (2.36) and g , f l n ∂q ∂h 3 3 ), we show that the ,q ,q n 2 2 ... 1 ,q ,q 2 cs for Kerr Newman black k 1 1 X q q s in two and three dimen- 2 +1 +1 } } ∂q ∂h es and generalize the partial n n 3 3 n the A-representation , . . . , x ds 1 j 1 , f n , g ), is depicted schematically in ackets. This method could also ,...,q x ,...,q 2 2 ∂q ∂h ions of the thermodynamic met- 2 2 ( i n i ,q ,q , g , f ,...,x 1 1 al, respectively. 1 1 1 X q q ∂f + 1 variables, we have: ∂q g f x } } ,...,x { { } n n 1 S, Q, J 3 determinant whose elements are the = n ( x = } i × ijk...l n M 3 ε + 1 variables. Contemplating X , . . . , n ,g 2 n , . . . , h , . . . , h =1 ,g 1 1 ,...,X 1 +1 = 1 3 g X n } – 7 – ,...,X g, h f, h , i ijk...l 3 ) and 2 i ,X { { n q 2 , f = 2 = ,X +1 , f A, X n 1 n 1 { f X { . Considering all the functions as explicit functions of ,...,X ,...,h { ,...,q i 2 1 2 g h = − ,q 1 ) ,X ) q = 3 1 3 } ∂g n ∂f A , f X , g 2 ( 2 2 A ds , f , g 1 1 f g = , . . . , h ( ( 1 ∂ ∂ A ) which are functions of with respect to f, h i { f ,... 3 , we have: . Representation of conformal factor between various metri , 3 2 . q , 2 Considering Thus, in this section, we showed that the partial derivative = 1 and 2 n ( figure representation is given by: q conformal transformation that connects the line elements i rics of Kerr Newman black hole with the mass, The conformal transformation between various representat where, In the above equation, we usederivative a for new a notation function as which in is previous a cas function of Figure 2 hole. Here, Ω and Φ are angular velocity and electricIn potenti the above equation, thederivative Nambu of bracket is the 3 sions, respectively, are related to the Poisson and Nambu br JHEP04(2015)115 (3.4) (3.3) (3.1) (3.2) (3.5) he metrics ation for black holes pecific heats. Taking ific heat at fixed electric herefore, the Ricci scalar making use of the bracket or’s indices, one could show rder type. The most impor- to any arbitrary dimensional Besides, it has a lot of merits . ns out that the divergences of effortlessly and the thermody- ature, and entropy. Moreover, ties, which are one of the most 2 ociated with the divergences of rbitrary number of dimensions. various thermodynamic geome- 2 ,q 1 q S,Q 2 ) below: S,Q } } ,q k } 1 Φ q 3.2 Φ , a } i j 2 S, T, a { dq , b { i { i j T i 1 b b 1212 a g ∂b ∂q { = 2 R =1 i ijk X 2 Φ 4 ε – 8 – = = − ]. Using the bracket method, we generalize that = ] that show the variation of an extensive variable ijk P 1 . In general, the phase transitions of the physical ij R g ∂S ∂T 23 12 dM g ) = 2 T = , q 3 = 1 a q Φ ( C R and are intensive and extensive variables, respectively, and t 22 i g q = 2 and a , i 11 ∂q ∂M g = = is the determinant of the metric. Consider the following equ 1 i a g b We start with a brief review of the Ricci scalar and different s important thermodynamics variables [ systems are attributed to the divergences of the heat capaci where, The Ricci scalar may be written as follows: with a two-dimensional thermodynamic space as in ( due to the changepotential in as its following equation: conjugate parameter. By defining spec that it has onlycould one be independent written element as: in two dimensions. T where, work to an arbitrarynotation, number we of particularly parameters generalize in our former this work section. to an By a into account the permutation symmetries of the Reimann tens elements in the energy (mass) representation are defined by: where, tries, for which we provided a proof in [ In most black holes,tant the phase phase transitions transitions in are blackspecific hole of thermodynamics heat the are second with ass o aby fixed studying charge, various electric charged potential, andspecific rotating temper heats black correspond holes, to it the tur curvature singularities of and assists us to performnamics very formula difficult in algebraic this operations phase notation spaces. could be simply generalized 3 Correspondence between transitions and singularities be extended to cover every higher dimensional phase spaces. JHEP04(2015)115 ). as S 1 1 (3.9) (3.7) (3.8) S, Q (3.10) TC ( S C . So, we R = Φ Qα S,Q C Q S,Q o } Φ − S C Φ 1 S = T QkC TC C S,Q S,Q T, , Q } } Φ 1 − { α S occur accurately in kC C T TC Φ C Φ S, Q S, Q == C = { { err-Newman black hole C = n S S,Q S,Q r-Newman black hole, one S,Q S,Q . Considering the scalar Φ Φ } } S,Q S,Q C } } ightforward that the phase 1 eat S } } Φ Φ heats and Ruppenier metric Φ S, S, Qα 1 ) (3.6) } C } C Φ Φ . T, T, T,Q T,Q { { { { ,S − S, S, { { S, Q Φ Q, S S,Q S,Q S,Q S,Q S,Q } } { S,Q { Φ } } S,Q } } T ) are the roots of Φ Φ C S,Q = ( o are the same by following a procedure = T,Q T,Q } T, T, 2 i S ,T { { { { Ω , Φ C T T 2 S 2 Φ X Q, T S, Q Φ Φ i C 1 ; C ( T { ) using the metrics elements in table S,Q { S,Q T S,Q S,Q T, C S,Q S,Q S C Φ C } } } } R } } { , C C = Φ j ,S = 1 S Φ ,S S, Q Φ Q,S = C T Q,S Q, = ( { { Φ Q, S ! { { Φ TC Qα . Moreover, we know that the Ruppenier metric – 9 – M ∂X { { ) and R 1 T C Q i h 1 2 T 1 S,Q Φ 2 C ∂ S,Q S,Q − } = S,Q ∂ = ∂Q correspond precisely to the singularities of } } − = } ∂X n Φ = S Φ Φ S Φ 1 S,Q S, Q, J S,Q T, C S,Q S,Q = S, S, Q } ( } TC 1 { } T, S,Q S,Q } T Φ { { } } { 2 ,S R T ∂ ∂Q 1 − Φ Q,S = T T,S Q,S C { { { { S,Q T,Q { is defined as follows: { R ! ij 1 T = 1 T = S,Q g 1 T S

o ), which is the Ricci scalar of the Ruppenier metric. Here, we S = S,Q S,Q = C S = } } C S SQ QQ C S T T Φ Φ Φ Qα Q M M S, Q . The elements of the two dimensional Ruppenier metric. C ( Φ S, T, ∂ ∂Q ∂T ∂Q R − { { ∂S ∂T SS SQ , T T are the zeros of

1 S T M M = 1 and 1 T 1 T 1 2

Φ 1 TC Table 1 = T = C = S,Q n } S = QQ SQ T SS T is given by: C T T ), it is easy to show that the denominator is given by: M M Φ C M T S, Q C 3.4 C { = = = = In the case of black holes with three parameters, like the Ker QQ SQ SS R g g g The elements of the Ruppenier metric are listed in table as the above. In orderand to we prove show this, finally we first the study correspondence the between specific them. For the K where is related to Weinhold metric by a conformal transformation use the new notationtransitions of of this paper to show this fact. It is stra where Moreover, we can rewrite the Ricci scalar can show that the singularities of the following equation. Ricci ( the singularities of Thus, it is easyassert to that see the that phase the transitions singularities of of it can be demostrated that the phase transitions of specific h JHEP04(2015)115 . In J,Q J,Q Ω T,Q α (3.14) (3.12) (3.13) (3.11) Φ . C S,Q,J S,Q,J S,Q,J S,Q,J Q, kC } } } } Q, C C S,Q,J C J,Q ), we can − ,Q ,Q JηT C } Ω Ω B ,T,Q TC Ω = J,T,Q J, = T, Ω rmodynamic , { { { { = Φ 1 J 1 Q S,J S,Q − T, = { S,Q +1) parameters in Ω ∂T ∂Q = ∂J ∂ following equation: n ,Q Φ and compressibilities. ∂J ∂ Ω 1 1 T T ) (3.15) h three parameters is: T,Q n th ( 1 T = = ∂J ∂T Ω and compressibilities (See ∂J ∂ . = 1 J ) S,Q 2 se, the first law of thermody- M 1 J M C 2 2 ,...,Q = ∂J ∂ Ω − ∂ 2 M een shown in appendix ∂S∂Q 2 β S, ∂ i = ∂J∂Q 1 ,Q 1 C T S, Q, J at a constant electric potential and 1 1 T η Ω , ) are the zeros of dQ Ω 1 = T i Φ , = ( = Φ Φ C i S, Q = 2 JJ ,Q C SQ g X n Φ T =1 ; g i X = ( S, Q, J JQ C ( i g = + j R = X ; – 10 – ,J M ∂X . Using Maxwell’s relation (appendix Ω i S,Q,J T dS Q,J 2 2 j ,J } C Q,J ∂ β Φ = Ω C ∂X ηC , M ∂X 1 J,Q i 2 QkT C Φ − C ∂ dM 1 T = S,Q,J S,Q,J = T, ∂X } = } { = 3 S,Q,J S,Q,J J,S 1 S,Q } } ,T,J T 1 J,Q R ij T Φ Q,T,J ,J g ,J Φ { { Φ = Φ ∂ = ∂Q ∂J ∂T ∂S ∂T 1 Q

T, Q, R ij { { 1 1 − T T g 1 T JJ SJ QJ 1 Q T T T = M M M = = = = s are the intensive variables. The metric elements of the the i T,J 2 ,J JQ SQ QQ 2 T T T M M M Φ 2 2 M M 2 M ∂Q ∂ ∂S Φ ∂ ∂ ∂S∂J ∂ ∂Q ∂T ∂Q JS SS QS and Φ 1 . The elements of the Ruppenier metric with three parameters T T T T 1 1 T T 1 Q M M M T 1 Q ). =

− = = 2 = One might also generalize these outcomes to a general case wi For the KN black hole, the denominator of the Ricci scalar wit The elements of the Ruppenier metric could be deﬁned as in the = SS SJ QQ Compressibility parameters α k g g g Ruppenier metric elements Table 2 geometry can be also deﬁned as: where, which the intensive variables arenamics more can than expressed three. by the In following this equation: ca angular velocity. addition, these zeros are the phase transitions of Hence, the singularities of the Ricci scalar Other interesting relations between heat capacities have b show that: the compressibilities are given in table These metrics cantable be written as functions of heat capacities JHEP04(2015)115 (3.16) (3.22) (3.18) (3.19) (3.17) (3.20) = ! ! n n n n 2 n n . . . (3.21) n n n n n n n n − ,...,Q ,...,Q ,...,Q ,...,Q n n 2 2 2 2 n ,...,Q ,...,Q ,Q 2 ,Q 2 ,...,Q ,...,Q ,...,Q ,...,Q ,...,Q ,...,Q ,...,Q ,...,Q ,Q ,Q 1 1 1 1 1 1 1 1 1 1 1 1 ,...,Q ,...,Q ,Q ,Q 2 2 2 ,...,Q 1 1 ) between these heat 2 S,Q S,Q S,Q S,Q S,Q S,Q − S,Q S,Q S,Q S,Q S,Q S,Q ,Q ,Q

} } } } } } } } } } 1 1 } } ,Q S,Q S,Q n n n n n n n n n n n 1 n n n } } n Q Φ Q Φ Φ Φ Φ Φ Φ Φ Φ C.13 n n S,Q S,Q = (3.23) = Φ 1 n , } } ,Q S,Q SQ Φ Φ n Q Q 1 n n 1 n n } − n − Φ Φ n n ,..., ,..., ,..., ,..., ,..., ,..., ,..., ,..., ,..., ,...,Q Φ 2 ,...,Q 2 2 2 2 2 2 2 2 2 ,...,Q ,...,Q 2 . . . g . . . g . . . g ,..., ,..., 2 2 Φ Φ Φ Φ Φ Φ Φ 2 2 , 1 , , , ,Q 2 , , , ,Q ,Q ,Q ,Q ,Q 2 ,..., ,..., Φ Φ 1 1 1 1 1 1 1 1 1 1 1 1 1 Q Q 2 2 ,..., , , ) as: 1 n Φ Φ Φ ) to a general case with an arbitrary Φ 2 1 1 ,...,Q ,...,Q SQ Q Q S,Q 2 2 S,Q S,Q g ,Q ,Q Φ Φ Φ g g } S, S, S, Q S, T, S, Q } } S, Q S, Q 1 1 , S, Q S, Q 3.16 n { { { { { { n n { 1 . . . { { { ,Q 3.13 1 ,Q S, S, S S 1 Φ 1 1 n { { Φ T :::: = = = SQ Q Q n S, Q S, Q g g ! g T, = { { n n n

{ n S, Q n S, Q Φ Φ n n n ,..., { { – 11 – ,...,Q ,...,Q ,...,Q Φ ) is proportional to the inverse of the square de- 2 ) and ( ∝ 2 2 2 ,..., ,..., +1) n ,...,Q

2 Φ 1 ) ,...,Q 1 ,...,Q n ,Q ,..., ,...,Q ,...,Q , ( 2 Φ 2 ) to the following form: n 3.7 1 2 ,Q ,Q 2 2 1 − 1 1 S, ... Φ S,Q ,Q ,Q S,Q , ,Q ,Q Φ 1 1 T 1 1 1 S,Q ! 3.17 Φ } 1 1 2 2 T, ,...,Q n n n n n S, Q S, Q S,Q S,Q 2 { S,Q S,Q Φ Φ ∝ { { } } Φ ,...,Q Φ } } ∂ ∂Q ∂ ∂Q n 2 n ) ∂ ∂Q n n numbers of relations similar to ( +1 ,Q ,...,Q ∂S ,...,Q ∂T n 1 Φ Φ n 2 1 2 ... Φ Φ ,Q ! ), we can rewrite eq. ( T ,Q 1 ,Q = = ,..., 1 = T 1 2 n n S, Q 1 ,..., ,..., Φ Φ ( = ,..., ,..., Φ 3.15 S,Q 2 S,Q 2 , S, Q + 1) 2 2 ,...,Q − } R } 1 ( n ,...,S 2 Φ Φ ,..., Φ Φ 2 n n n Φ , 2 , Φ n R , , ,S,..., 1 1 Φ Φ Φ ,Q 1 1 2( ,Q 1 1 ,..., Φ 1 Φ T, S, 2 Q Q {

) and ( C Φ C , S, T, C S, Q S, Q 1 { ,..., { ,..., +1 { { S, Q Φ 1 2 2 n ( 3.14

C Φ T R , ,Q 1 1 +1 1 n T S, Q S, Q { {

terminant of the metric. Using eqs. ( capacities. To extend ournumber of results parameters, in we ( recast ( One might get On the other hand, the heat capacities can be deﬁned by: The scalar curvature JHEP04(2015)115 . Φ ty ase S, } (3.29) (3.24) (3.25) (3.28) repre- = s, which ,S M T,Q 1 { ); however, − n n ,S 1 ) without the − ,...,Q S, Q n 2 ,...,Q ( 2 . R ,Q 3.25 1 ,Q 1 ,S ,...,Q 1 1 2 − Q S,Q n ,Q ck hole and verify the } 1 Φ n Q . It is important to note Φ denominator of the next ) and ( S, n Φ ...C ,...,Q C S, } t the Hessian (in 2 n } Q Φ ,Q Φ) (3.27) S 3.23 1 ,...,Q 2 C Φ) (3.26) ,..., S, Q Q ... S, ngularities of TC T,Q 2 { ( ,Q ,S,..., { 1 S, Φ 1 − T T Q n , Q 1 Φ ...C Φ = S,Q = = ( C = 1. The singularities of the Ricci Φ n } n − i Φ Φ n Φ 1 n S, S, ,S,..., X Φ S,Q 1 { } ; S,Q } ,..., Φ)) Q ), and the nominator of the last fraction } 2 ,S,..., = ,...,Q 1 Φ C Φ) is: S, 2 j 1 Q 1 ( T,Q S, ,..., S, Q ,Q − 3.22 2 { C T,Q S, { C 1 ( 2 Φ). The line elements of the metric related to n M ∂X { Φ n – 12 – i 1 T 2 Φ , ,S )–( n T R S, Q Φ 1 1 S,Q ∂ S, ( Φ ( − } = − ,..., Φ ∂X ,..., n 2 n R 2 M ,..., 3.18 Q Φ = 1 2 Φ T, ,

Φ S, { 1 1 ,...,Q T S, Φ C 2 n ∂S ∂T C C Φ) = Φ 2 ¯ ¯ = ,Q ,...,Q Φ M M 1 Φ 2 2 2 are the roots of the denominator or the roots of S, ij = ∂ Q ∂ ∂ ∂S∂ ( ,..., T ), we can rewrite heat capacity as: g

2 Q ,Q n ) occur only at the phase transition points of the heat capaci Φ n 1 n M = , C n 1 T 1 Φ n T 1 Q Q ...C 2.23 . The interesting point that arises in this case is that the ph 1 Q n Φ SQ n ,..., Q Q Q C C S, Q 2 Φ T ), we multiplied, in the ﬁrst step, the early term by fraction n C { 2 ¯ ¯ Φ ∂S , M M T 1 2 2 Φ ), the denominator of ∂S ,...,Q ∂ ∂ Φ ∂ ,S,..., 2 3.23 1 ...M ...M ...M C 3.4 Q 1 1 2 T 1 ,Q T 2 C 1 Q Q

n 1 n which are the roots of SQ +1 Φ Q Q 1 n n M S, Q Φ M ,..., M T ( 2 1 Φ S R S ,..., 1 2 n S, :::: Φ Q SQ Q , C 1 M M M Φ To accomplish (

they are equal to the divergences of Making use of ( C this Ricci scalar are: that the multi-term transitions of this heat capacity do not correspond to the si has no divergences. What is more, one might also conclude tha bracket notation used here.phase transitions Now, of let us to return to the RN bla one, which are theequals heat one, capacities namely ( are identity, and then the nominator of every fraction by the scalar Clearly, it would be quite impossible to obtain equations ( The phase transitions of sentation) is of the following form: Additionally, using eq. ( where, the enthalpy is defined by: JHEP04(2015)115 i Q i Φ (3.30) (3.33) (3.34) ponds =1 n i 2 P − are the roots − n T Φ M C ,..., = 1 Φ . So, we assert that S, Φ M ) or the one with fixed nd phase transitions. S, T } Φ). Moreover, we should . ...C Φ ) (3.32) n n the answer is the following is given by the singularities S, ences of the Ricci scalar is ∂ . ∂Q n n d the singularities of other ) (3.31) ( Φ ) T,Q Ricci scalar: Φ Q R { ,...,Q ,..., C Q, T 2 ,...,Q 1 ,..., ≡ ( Q, T 3 1 S, Q Φ ,Q T Φ ) are the divergences of this heat } 1 ,Q } n TS 1 n C = ( Q = ( n Φ Φ i C i − S, ) X X C ; ; n ...Q ,Q ,..., ,...,Q ! 1 ) 2 1 j j Φ ,...,Q , 2 1 S, Q ∂X T,Q ∂X ) and phase transition points of M T,Q i { ( i – 13 – S,Q { ∂ +1) parameters, one can define the heat capacity ∂M (Φ S Φ) are the roots of C ( n T R ∂X ∂X S, Q n ( S, M =

( Q 1 n R 1 1 Φ Φ R T − ) = − − ,...,Q 2 = = Q, T ,Q are equal to the singularities of ij ( ∝ Q ij 1 g g ) ), corresponding to the following metric: Q Q M ) and considering the conjugate potential n C C Φ ). One might ask the singularities of which curvature corres T,Q S 3.26 ( for the RN black hole. R ,..., Φ 3 2 ∂ ∂Q Φ ) is the free Helmholtz energy and is defined as , 1 ≡ Φ . One could show that the phase transitions of . The connection between the singularities of Ricci scalar a S Q, T S, ( C S,Q ( } M R Φ Furthermore, for a black hole with ( T, Figure 3 { -metric of the Ricci scalar, of one constructs the elements of the metric with the following capacity. Using eq. ( It is clear that both singularities of as follows, check the correspondancesheat between capacities the such as phase the transitions one at an a fixed temperature ( Q to the phase transitions of this heat capacity? We argue that depicted in figure The connection between the phase transitions and the diverg We will show that the singularities of the phase transitions of entropy ( where, It is plain that the divergences of JHEP04(2015)115 (4.3) (4.2) (4.4) (4.1) (4.5) k hole more than 2) 1 − d ( for the Myers- 2 2 2 3 S 1 − J d capacities divergences e bracket notation, we en by: ith two rotating planes, 2 th rotating plane. The i ), respectively: 2 al spins. This is perhaps ave been extensively ex- iven by: i e number of the rotating J 1 + 4 S 4 4.4 lack hole with three spinning 2 the conventional methods for ic geometries. The correct co- 2) 1 icci scalar and the phase tran- − 1 d − 1 + d ( ularities of the metric correspond 2 2 i 1 ) and ( − 2 i N d 2 J 2 Y S 2 4 4.3 S # J 2 i 2 2 k J J S 1 1 + 4 is the dimensions of the space time. The ]. However, specific heats and phase tran- 2 k 2 J − d 1 + 4 34 + 4 8 i d N 2 1 + Y is attributed to the S 3 2 2) – 14 – − − 1 i − ≤ d d d ] has been of the particular interest due to its rich J N =1 =1 ( N i Y k X S N 26 2 2 P − 2 2 J 1 4 − 3 S P J d J − + 4 8 2 d = " S 1 + 4 3 2 2 M 1 − − − − 4 4 d d d 3 2 − − S S ]. This black hole is a direct generalization of the Kerr blac d d S 33 = = – p 2) T 27 Ω 4 − ].The Hawking temperature for a Myers-Perry black hole with d 34 ( – = 32 is the entropy and the spin ) are confined by , M N S 27 The mass of the Myers-Perry black hole with three spins is giv Although the thermodynamics of the Myers-Perry black hole w In this section, we proved a correspondence between the heat Perry black hole with three spins temperature and angular momentums are given by ( plored [ equal spins, and a particular number of nonzero equal spins h where, where, the bracket gives the integer part and mass of this black hole in both the odd and even dimensions is g exploit it to derive theplanes, thermodynamics whereby of we the gain Myers-Perry its b phase transitions. two and unequal spins havesitions been have obtianed not in yet [ due been to reported the for complicated more algebracases than that with two is more and often than unequ associated two with spins. In order to show the power of th in higher dimensions withplanes rotation ( in more than one plane. Th thermodynamics [ sition is clear. Thus, the correspondence between the singularities of the R 4 The relation between phase transitions and singularities and the singularities of theordinates Ricci were also scalar identified of in the whichto the thermodynam Ricci the scalar divergences sing of heat capacity. Recently, the Myers-Perry black hole [ JHEP04(2015)115 (4.6) (4.8) (4.7) 3 J 3 3 6= J J 2 J 6= 6= 2 2 6= J J 3 3 1 ,J 3 ,J 3 J 6= 6= 3 3 2 2 ,J ,J in an arbitrary number 1 1 ,J ,J ,J 2 ,J 2 2 3 2 1 1 J J with an arbitrary number ,J ,J tead, we have listed the Ω ,J ,J 1 1 , 2 1 1 2 S,J S,J Ω Ω 3 3 } } , S,J S,J , 1 S,J 3 3 S,J J J 1 ady derived; however, they } } Ω } } Ω 3 3 Ω Ω ry black hole in this work: 3 2 2 = = , , T, C J 2 2 Ω ,J ,J Ω C 2 2 2 , 2 , Ω Ω J J = 1 1 , , ,J ,J 1 2 1 Ω Ω = = 1 1 J Ω . For dimensions above 31, there is no Ω 1 1 3 J J ,T, = ,T, ,J S, T, S, J T,J 3 2 3 and some relevant graphs are included, { { 1 { { J ,J Ω J 5 3 3 1 T { { T J J J C = = 3 = 6= 6= 3 J 3 2 2 2 Ω , and Ω , ,J – 15 – , J J 6= 2 2 4 1 Ω ,J 2 , Ω , = = 1 1 J 3 J T, 1 1 Ω J J = 3 3 1 Ω J ∂S ∂T ∂S ∂T ∂J ∂ = 0 = 0 T T 3 3 = J J = 0 = = 2 3 3 3 Ω J , ,J Ω 1 2 , and and 2 Ω ,J 2 2 Ω 1 , T, J J and J 1 C Ω C 2 = = J C 1 1 = 1 101112 Onepoint Onepoint Twopoints Onepoint Twopoints Onepoint Twopoints Onepoint Threepoints Onepoint Threepoints Threepoints 101112 Two points Two points Two points Two points Twod points One Two point7 points J Two points8 Two Two points points Two9 points Two points No point One point One point One point One point One point No point No point No Two point point Two point Two point d7 J 89 Onepoint Twopoints Twopoints Onepoint Onepoint Onepoint Onepoint Onepoint Onepoint Onepoint Onepoint Onepoint . The number of phase transitions for . The number of phase transitions for the heat capacity . The number of phase transitions for the heat capacity 89101112 No point No point No point One point No point One point Two points No One point point Two points Two One points point Two points Two points Two two or points three points Two Three point points Two point Three points d7 J No point One point One point Two point The exact statement forappeared these to heat be very capacitiesnumber long have of and been we phase alre would transitions not in repeat tables them here. Ins Table 5 Table 4 Table 3 of dimensions. of dimensions. We examine the following three heat capacities of Myers-Per phase transition. JHEP04(2015)115 , , ) 3 2 3 Ω Ω are ) , , Ω 6 2 1 , 2 3 5 Ω Ω 2 − − , S dS d d 1 ) . This Ω T, − 3 S Ω , 6 6) C 1 J S C − 3 d Ω S ( = 0. From ! +3 ) = . Moreover, 2 4 2 2 3 2 and S, 3 1 J − ( J 2 J S d dS J 3 2 ) J 2 R 1 28 2 ,J J J 6= 2 2 − J ) 4 ) 4 J 2 2 = 2 ,J +4 2 2 S 2 J and 36 S . We observe that 1 dJ 4 S +4 dJ +16 S J J ) J 3 1 6 − 2 ) S 2 ) 2 2 2 J 2 − 2 S +4 2 C S J J 2 2 S +4 J ) J 6= J +80 2 +112 d 4 S 3 4 28

4 dJ S ! 1 36 − ) )( − = +4 dimensions lower than 31 − able 4+ 2 2 2 2 2 J 2 dS − 2 )( dS 1 +4 − 2 1 S − 2 S + d + 2 ( dS dS dS dJ of higher order types. 1 S 2 J )( 3 2 4 ) J − − 2 J 3 2 J J dS − 2 2 2 2 1 +4 2 − 2 2 S S 2 J dS S +4 48 2 )( S J 2 2 2 + − dJ − dS S 24 +3 S 2 2 dJ S 4 +4 +3 24 ions in the right side of ﬁgure 2 2 2 2 = 10) when S − J )( − )( +4 J 2 S 2 2 2 2 +4 +2 2 ( dJ 2 dJ − J 7) S d 2 2 J +3 4 d 2 2 dS 2 − dS +4 − dS dS d ( − +4 J − 2 4 = 0 in dimensions higher than thirteen dS ! ( ( 2 dS 2 − J 2 2) 2 S 8 +144 J 2 2 36 S 2 2) 3 d 1 − − 16 2 − 36 2) J 6 J − d − J d 4 28 S 2 dJ − 2 ( − − J d +2 1 J 4 ( ) − d 2 )( J 2 ( ( ) 2 J dJ +4 J + 4 1 S 2 J 2 J )( 560 – 16 – J (4 2 and 2 + +192 J 32 S − +16 6 7) J +4 +4 S 1008 − 4 2 2 2 J +4 24 2 − 2 S S − S 2 J )(2 d S − 2 4 ( 3 S ( S 5) ( dJ 2 +3 − S ( 2 − 1728 2

4 d − d = − 3 − ( J d J − S − ( 2 − 1 ( 3 2 ) 1 − J 2 − J − − 2(2 − 1 d d J S + ) = +16 has no phase transitions in space time dimensions above 31 2 has not phase transitions when ) = ) = 2 . The heat capacity at fixed angular momentums, 2 = 0) = J 2 3 S = 0) = = 0) = J J J 3 2 3 J ( 5) = 3 3 Ω ,J J , 4 = = 2 − J J ; 1 3 − 6= d ; ; ,J 3 3 . The representation of heat capacities in different states. Ω J J ( 4 1 ) J J J J = 0) = 2 S J T, 2 = 0) = ) exactly correspond to the phase transitions of = = J 1 3 3) C C = = = = 3 J 3 2 2 J − 2 2 + 2 2 J ; d 6= J J ,J , it is obvious that the singularities of the Ricci scalar ( 2 ; 2 J J J J 2 ( 2 7 2 1 J Table 6 J = = J ; J ( = = = = ,J ; 1 1 1 1 1 1 1 1 1 J J J +4 ( ( ( J J J J J 4 ( ( ( ( ( 3 3 3 S , and 3 3 2 2 2 S, J Ω Ω Ω , , , 4 ( Ω Ω Ω 3) ,J ,J 2 2 2 , , , 2 2 J 1 1 1 − Ω Ω Ω , , , d ,J ,J Ω Ω Ω R ( 1 1 1 1 1 ( = 31), neither has the case = 31) for T, T, T, Ω Ω J J Ω C − C C C C C C C d d figures the heat capacity has no phase transitions in dimensions below 10 ( heat capacity has either two or three phase transitions when (d=13). There are( either one or two transitions points in the ( not the second order phase transitions, rather they might be the heat capacity and too. Some special cases of heat capacities are presented in t respectively. Interestingly, we see that the phase transit JHEP04(2015)115 = 9 = 1, = 1, = 10 d 1 1 d J J espond to = 7, = 8, , for , for d d S S ) with respect 3 Ω , 2 Ω , espond to 1 espond to with respect to entropy, Ω 3 S, ,J etry of black holes in the ( 2 mple representation of the R ,J 1 J namic metrics to each other. C with respect to entropy, with respect to entropy, 3 3 ,J ,J 2 2 ,J ,J 1 1 J J C C – 17 – = 3. The dot red, dot-dashed blue, and solid green curves 3 J = 11, respectively. d = 2, and 2 = 0. The dot red, dot-dashed blue, and solid green curves corr J 3 J = 9 and = 15, respectively. d = 1, d 1 J = 7, d = 2, and = 3. The dot red, dot-dashed blue, and solid green curves corr , for = 3. The dot red, dot-dashed blue, and solid green curves corr 2 3 3 J S J J = 14 and . Left: Graph of the the heat capacity . Left: Graph of the the heat capacity = d 1 = 11, respectively; Right: Graph of the the scalar curvature J = 12, respectively; Right: Graph of the the heat capacity d d = 2 and = 2 and = 13, , for 2 2 correspond to Figure 5 Figure 4 to entropy, J J and and 5 Conclusion In this work, theconformal Nambu brackets transformations were that exploited to connectWe explored obtain different two a thermody new si formulations for the thermodynamic geom S d JHEP04(2015)115 = 1 J , for = 7, = 7, S d d espond , for S with respect 2 with respect to Ω , espond to 3 espond to 1 Ω Ω , 2 T, Ω , C 1 Ω C ) with respect to entropy, 3 he singularities of the thermo- arities of the scalar curvature in ,J he Nambu bracket approach, we with respect to entropy, ons yield the correct relation be- 2 he Hessian matrix in an arbitrary 3 ,J Ω , 1 2 Ω , 1 S, J Ω ( Q C R – 18 – = 3. The dot red, dot-dashed blue, and solid green curves 3 J = 11, respectively. d = 1. The dot red, dot-dashed blue, and solid green curves corr 3 = 2, and 2 J J = = 12, respectively. = 9 and 2 d d J =1, = 3. The dot red, dot-dashed blue, and solid green curves corr 1 3 J = J = 3. The dot red, dot-dashed blue, and solid green curves corr 1 = 7, 3 J J d = 11, respectively; Right: Graph of the the scalar curvature , for = 11, respectively; Right: Graph of the the scalar curvature S d = 10 and d , for . Left: Graph of the the scalar curvature . Left: Graph of the the scalar curvature d = 2, and S 2 J = 2, and = 8, 2 d J =1, = 9 and = 9 and 1 correspond to electric charge and spin representations.tween heat These capacities formulati and curvaturealso singularities. obtained some Using interesting t exactdimensions results to that specific relates t heats.dynamic Thus, scalar one curvature. can The easily relationship identify between t singul to entropy, to d J entropy, 1, Figure 7 Figure 6 d JHEP04(2015)115 = 1, 2 (A.2) (A.3) (A.1) (A.4) J = = 10 and 1 d J = 8, , for d S a,b a,b } } n this area in the near , we rewrite the above k g, h g, h espond to { { h a,b a,b s-Perry black hole with three } se transitions of different heat a,b } } of the Meyers-Perry black holes nding of different aspects of the ) and the Jacobian transforma- ∂g ∂f f, k dg f, k h g, h { 2.21 − { − − { a,b k ∂g ∂f } a,b with respect to entropy, a,b } } 2 Ω g, k , ∂g ∂f + 1 { g, k g, k Ω { { dh T, g = a,b a,b C – 19 – } k } ∂h ∂f f, h f, h { ∂g ∂h { = = = g ): df a,b a,b a,b . By considering a new function } } } 2.21 h ∂h ∂f g, k h, k h, k ), we have: { { { and a,b a,b a,b g 2.19 } } } f, g f, g g, h { { { ). We could write the following equation: . Graph of the the scalar curvature = 0. The dot red, dot-dashed blue, and solid green curves corr 2.24 3 is a function of J f = 12, respectively. Taking into account ( future. Our exact results willthermodynamic be geometry useful of for physical a systems. clear understa A Partial derivative and braketIn notation the following, we provide the proofs for identity ( various representations of thermodynamic geometrycapacities and pha were also studied.spins. We It also will investigated be interesting thewith to Myer more explore the than thermodynamics three spins. We hope to report on our progress i d tion ( and At last, we get the identity ( if Figure 8 equation as: JHEP04(2015)115 (B.1) (B.2) (B.4) (B.5) (B.6) (B.3) (A.5) (A.7) (A.6) n ,...,q 2 S,Q,J S,Q,J ,q } } 1 q Φ , one could get the } n ! dQ T, S, J a,b = T, S, a,b n n } { } k ole as follows: −{ + Φ , . . . , f = tation in the form below: ct and the chain rule: ,...,q ,...,q 2 g, k h, k on. 2 2 = ns: { { ∂g ∂h ,q ,q , f 1 1 1 q q T dS a,b a,b f S,Q,J } } h S,Q,J S,Q,J } } { } ! n n S,Q,J } } = } Φ S,Q = Q a,b a,b ,J f, g } ,J h, k } } ∂k ∂f n { { Ω Ω , J, ,Q dM Φ , . . . , f Ω , . . . , f , Q, J = S, T, ,...,q f, h k, h ∂S ∂ 2 Φ 2 { 2 − { { Φ ,q , f { , f h −{ −{ a,b ; 1 −{

1 1 2 q } = f h } ; = = − = { { n S – 20 – ∂g ∂k a,b h, k ! = } { S,Q,J S,Q k n } a,b } a,b S,Q,J S,Q,J a,b S,Q,J ∂T ∂Q } } } } Ω h, k } } , . . . , h { 2 ,...,f 3 T,S ,Q ,Q ∂h ∂f g, h k, h , yields f, g { ,f , h Φ Φ { { 2 n { 1 T, S, f h h S, T, = T, S, Q { = { −{ { { 1 = n ! 1 ) ) = n = ∂f a,b ∂h a,b ,...,h } f, g } h, k 2 ( ( ,h ∂ ∂ 1 S,Q,J f, k h, k S,Q,J h , . . . , f } { { } } 2 Ω n h

= , J, Q , Q, 2 Ω Φ f , . . . , f { { 2 , f 1 f { Based on the first law of thermodynamics, We can also obtain the following equation using matrix produ One can also derive the Maxwell’s relations for the KN black h which is equivalent to: The Jacobian transformation can be written in the bracket no The Maxwell’s equations can be recast in the brackets notati B Maxwell’s equations and Nambu brackets Maxwell’s relation as follows: Furthermore, one could extract some more Maxwell’s relatio Considering JHEP04(2015)115 # =1 (C.3) (C.4) (C.1) (C.9) (C.5) (C.2) (C.7) (C.6) 1 (C.12) al po- (C.11) (C.10) S,Q S,Q − } } !# ,Q T,Q Φ S,Q S,Q { { } } ermal compress- S,Q S,Q Q ,S } } Φ Q, S T Φ { { Φ ∂ ∂T T, S, T # Φ { { ! ∂S ∂ " α S,Q S,Q Φ Φ T tion. It is suitable to prove } } S,Q S,Q = 1 S, S, Q } } } } = Φ Φ = # 2 S, Q T,Q ,S S,Q S,Q S, { T, { 2 ¨mblackröm hole, respectively. } } Φ Q, S T { { k { Φ { S,Q S,Q S,Q − C k T 2 } } } they can be rewritten as follows: 2 T Qα k = Q, Q, T Φ Φ T Qα Qk S,Q S,Q − { { S,Q " S,Q S,Q S,Q } } T α − − } T, T,S T, } } } α × = S,Q Φ S,Q { { S,Q { S,Q Φ Φ Φ } } = = = = 1 (C.8) } } = # 1 S, Q S, Q !# 1 S,Q T,Q S, T, T, { T { Q Q ,T ,S } { { − { { S,Q S,Q S,Q C T = C C T ) Φ Q, T S,Q Φ S,Q } Q, S S,Q S,Q " } } S { { } { } { } } – 21 – − − = Φ Φ = T C S, Q S,Q S,Q = Φ Φ ,T = Φ ,T Q Φ T,Q } T, T, } = C C Φ { Q, T C T S Φ { { Q, T Φ − { Φ ( { { # { { T Q S,Q 1 T, Q Q, ∂S ∂S ∂T ∂T S,Q Φ Φ C S,Q } S,Q { { } } ∂ ∂ ∂Q ∂Q ! Q − } Φ ), has been used. We can also obtain some well-known 1 Q C T T ∂S ∂T ,Q = T, S,Q S,Q Φ { T,Q = } = } = = = T,Q T 2.23 { { { Φ − S T Φ Q ), respectively, by: C Φ S,Q C C S,Q C Φ S,Q S, Q T,Q } } ∂ ∂Q } { { Φ C.8 ∂T ∂Q Φ ), we can define the specific heats with a fixed charge, electric T ,Q 1 ∂S ∂T Q T, S, Φ " " { { { 1 Q − 2.19 = " " " are the electric potential coefficient of expansion and isoth 1 = = ) and ( T T T − k ) k α S = = = C.5 C and Q Φ C α C ( − T Φ C C Q In the last case, Relation ( In which Maxwell equations have beenRelations used ( to get the first equa C relations between different heat capacities as follows: tential, temperature, and entropy for the Reissner-Nordst ibility, respectively. Using the bracket representation, Using equation ( C Properties of heat capacities where, JHEP04(2015)115 , , es (1979) ommons , , A 23 (1975) 63 A 20 ]. (1975) 199. , 43 = 1 (C.13) = 1 = 1 SPIRE 1 ]. 1 1 IN − ]. − − , ) ) ) , Phys. Rev. ][ Φ Mod. Phys. Lett. Ω Ω , , , , , Ω Φ (2014) 2681 Q, SPIRE J. Chem. Phys. redited. C SPIRE C C , IN [ ,S IN ,S n the heat capacities with three Ω ]. Φ Q,S ][ ]. C 74 C C C ( ( ( ,T ,T Ω SPIRE Φ T,Q SPIRE C gr-qc/0512035 IN (1974) 30 C ]. Commun. Math.Phys. [ C IN , ,Q ][ ,J Ω ][ Φ ]. Q,J Thermodynamic geometry and critical behavior 248 C C The Four laws of black hole mechanics C SPIRE Geometry of black hole thermodynamics Eur. Phys. J. Black holes and critical points in moduli space IN , ]. ]. – 22 – SPIRE ][ (2007) 11 arXiv:1210.2011 [ IN Nature [ = 1 ; = 1 ; = 1 ; , ]. 1 1 1 SPIRE − − SPIRE − A 22 ) ) Ruppeiner Geometry of RN Black Holes: Flat or Curved? IN ) gr-qc/0304015 [ IN [ Φ Correspondence of phase transition points and singulariti [ Φ T,J hep-th/9702103 S, J, SPIRE ), which permits any use, distribution and reproduction in [ ]. C C C IN [ Φ (1973) 161 S,J T, J,S (2013) 012138 C C C 31 ( ( ( SPIRE arXiv:0706.3450 [ Ω Ω IN (1995) 605 (1971) 1344 410 (2003) 1733 (1997) 75 S, J,T T, ][ C CC-BY 4.0 C C 67 Extraction of energy and charge from a black hole 26 35 (1973) 949 Gravitational radiation from colliding black holes Black hole explosions Particle Creation By Black Holes Riemannian geometry in thermodynamic fluctuation theory Thermodynamics: A Riemannian geometric model Thermodynamic curvature: pure fluids to black holes Int. J. Mod. Phys. This article is distributed under the terms of the Creative C J,Q S,Q T,Q Metric geometry of equilibrium thermodynamics , C C C A commentary on Ruppeiner metrics for black holes B 500 D 7 (2007) 059 06 arXiv:1308.1543 1608. 2479. of thermodynamic geometry of black holes JHEP J. Phys. Conf. Ser. Gen. Rel. Grav. of black holes Rev. Mod. Phys. Nucl. Phys. Commun. Math. Phys. Phys. Rev. Lett. Phys. Rev. [ (2008) 2149. [8] G. Ruppeiner, [1] S.A.H. Mansoori and B. Mirza, [9] F. Weinhold, [7] G. Ruppeiner, [3] J.D. Bekenstein, [4] S.W. Hawking, [5] S.W. Hawking, [2] S.W. Hawking, [6] J.M. Bardeen, B. Carter and S.W. Hawking, [10] S. Ferrara, G.W. Gibbons and R. Kallosh, [14] A. Medved, [11] J.E. Aman, I. Bengtsson and N. Pidokrajt, [12] J.-y. Shen, R.-G. Cai, B. Wang and R.-K. Su, [13] B. Mirza and M. Zamani-Nasab, [15] G. Ruppeiner, any medium, provided the original author(s) and source areReferences c parameters as follows: Moreover, one could exploit the important relations betwee Open Access. Attribution License ( JHEP04(2015)115 ] , ] ]. ]. (2008) 6 , , SPIRE 11 IN SPIRE , , ][ t IN , [ (2010) 165004 ]. 27 arXiv:1002.1550 , [ arXiv:1303.2077 [ SPIRE IN Living Rev. Rel. (1973) 2405 (2013) 024005 , ][ ]. ]. ]. Geometrothermodynamics of Quasilocal formalism and ]. ]. ]. arXiv:1004.5550 D 7 [ a, ]. a, (2012) 20004 D 88 SPIRE SPIRE (2013) 549808 SPIRE 99 IN SPIRE IN SPIRE SPIRE IN IN ][ SPIRE Class. Quant. Grav. ][ IN IN ][ ][ , IN 2013 Phase transition, critical behavior and ][ ][ ][ Thermodynamic Geometry of Black Hole in Thermodynamic instability of doubly Phys. Rev. (2013) 2601 , (1984) 436. Phys. Rev. arXiv:1003.2421 , [ , Cambridge University Press, Cambridge U.K. Geometry of black hole thermodynamics 80 C 73 – 23 – On the relation between entropy and energy versions Thermodynamical Metrics and Black Hole Phase ]. Europhys. Lett. ]. , hep-th/0110260 arXiv:1008.4482 [ [ arXiv:1311.6963 (2010) 046 gr-qc/0304015 hep-th/0510139 arXiv:1208.0399 hep-th/0212267 SPIRE [ [ SPIRE [ [ [ Classical and quantum Nambu mechanics IN IN [ On Explicit Thermodynamic Functions and Extremal Limits of Geometry of higher-dimensional black hole thermodynamics 08 [ Thermodynamic Geometry: Evolution, Correlation and Phase A Rotating black ring solution in five-dimensions Black Holes in Higher Dimensions ]. ]. Black Holes in Higher Dimensional Space-Times J. Chem. Phys. Thermodynamic curvature and ensemble nonequivalence Eur. Phys. J. , Adv. High Energy Phys. , , (2010) 054 JHEP SPIRE SPIRE , IN IN (2014) 2930 (2002) 101101 (2003) 1733 12 (1986) 304 ][ ][ (2006) 024017 (2014) 044064 (2003) 085001 Geometrothermodynamics: comments, criticisms and suppor 88 35 C 74 172 Calculus Concepts and Methods JHEP Generalized Hamiltonian dynamics arXiv:1010.5148 D 73 D 90 D 68 , , ]. ]. ]. SPIRE SPIRE SPIRE IN IN arXiv:1306.4516 arXiv:0801.3471 IN [ Transitions Myers-Perry Black Holes critical exponents of Myers-Perry black holes thermodynamics of asymptotically flat black objects (2007) pg. 190. [ [ [ spinning black objects Annals Phys. Phys. Rev. Phys. Rev. Lett. Gen. Rel. Grav. Phys. Rev. the Deformed Gravity Hoˇrava-Lifshitz [ of thermodynamics length Eur. Phys. J. Transition Phys. Rev. Myers-Perry black holes [20] H. Liu, H. M. Lü, Luo and K.-N. Shao, [34] J.E. Aman and N. Pidokrajt, [32] M.B.J. Poshteh, B. Mirza and Z. Sherkatghanad, [24] D. Binmore, [25] J.E. Aman, I. Bengtsson and N. Pidokrajt, [33] A. Bravetti, D. Momeni, R. Myrzakulov and A. Altaibayev [29] R. Emparan and H.S. Reall, [30] D. Astefanesei, M.J. Rodriguez and S. Theisen, [31] D. Astefanesei, R. B. Mann, M. J. Rodriguez, and C. Stele [27] J.E. Aman and N. Pidokrajt, [28] R. Emparan and H.S. Reall, [26] R.C. Myers and M.J. Perry, [22] T. Curtright and C.K. Zachos, [23] S.-W. Wei, Y.-X. Liu, Y.-Q. Wang and H. Guo, [19] P. Salamon, J.D. Nulton and E. Ihring, [21] Y. Nambu, [17] S. Bellucci and B.N. Tiwari, [18] A. Bravetti and F. Nettel, [16] M. Azreg-Ainou,