Extrinsic and Intrinsic Curvatures in Thermodynamic Geometry

Extrinsic and intrinsic curvatures in thermodynamic geometry

Seyed Ali Hosseini Mansoori1,2,∗ Behrouz Mirza2,† and Elham Shariﬁan2‡ 1Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA 2Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran (Dated: October 4, 2018) We investigate the intrinsic and extrinsic curvature of a certain hypersurface in thermodynamic geometry of a physical system and show that they contain useful thermodynamic information. For an anti-Reissner-Nordstr¨om-(A)deSitter black hole (Phantom), the extrinsic curvature of a constant Q hypersurface has the same sign as the heat capacity around the phase transition points. The intrinsic curvature of the hypersurface can also be divergent at the critical points but has no information about the sign of the heat capacity. Our study explains the consistent relationship holding between the thermodynamic geometry of the KN-AdS black holes and those of the RN (J-zero hypersurface) and Kerr black holes (Q-zero hapersurface) ones [1]. This approach can easily be generalized to an arbitrary thermodynamic system.

I. INTRODUCTION lead to inadequate information about it. Therefore, the thermodynamic curvature of RN should be reproduced Bekenstein and Hawking showed that a black hole has from the Kerr-Newmann anti-de Sitter (KN-AdS) black a behavior similar to a common thermodynamic system hole when the angular momentum J → 0 and cosmolog- [2, 3]. They drew a parallel relationship between the ical constant Λ → 0. This approach leads to a non-zero four laws of thermodynamics and the physical properties value for the Ruppeiner scalar, which is in contrast to of black holes by considering the surface gravity and the the reports on RN in pervious works [22, 23]. horizon area as the temperature and entropy, respectively The present letter seeks to explain this contrast by ob- [4]. An interesting topic is to study phase transitions in taining intrinsic and extrinsic curvatures of the related black hole thermodynamics where the heat capacity di- submanifolds. The induced metric (intrinsic curvature) verges [5, 6]. These divergence points of heat capacity are and the extrinsic curvature of a constant J hypersurface usually associated with a second order phase transition contain the necessary information about the properties for some fixed black hole parameters [7]. of this hypersurface. The zero limit of an angular mo- Geometric concepts can also be used to study the prop- mentum for a KN-AdS black hole is equivalent to the erties of an equilibrium space of thermodynamic systems. two-dimensional constant J hypersurface embedded in a Riemannian geometry in the space of equilibrium states three-dimensional complete thermodynamic space. The was introduced by Weinhold [8] and Ruppeiner [9, 10] curvature scalar of KN-AdS black hole on this hypersur- who defined metric elements as the Hessian matrix of face can be decomposed into an intrinsic curvature (Rup- the internal energy and entropy. These geometric struc- peiner curvature of RN black hole), which is zero, and an tures are used to find the significance of the distance extrinsic part that give the curvature singularities. between equilibrium states. Consequently, various thermodynamic properties of the system can be derived from We also prove that there is a one-to-one correspon- the properties of these metrics, especially critical behav- dence between divergence points of the heat capacities iors, and stability of various types of black hole families and those of the extrinsic curvature for thermodynamic [11–13]. For the second order phase transitions, Rup- descriptions where potentials are related to the mass peiner curvature scalar (R) is expected to diverge at crit- (rather than the entropy) by Legendre transformations. ical points [7, 14–17]. Due to the success of this geometry In spite of this correspondence, we can get other infor- arXiv:1602.03066v2 [gr-qc] 11 Jun 2016 to identify a phase transition, several works [18–21] have mation about thermodynamics like stability and non- exploited it to explain the black hole phase transitions. stability regions around phase transitions from singular- The Ruppeiner geometry has also been analyzed for ities of extrinsic curvature and certain elements of the several black holes to find out the thermodynamic prop- Ricci tensor. erties [22, 23]. As a result, the Ruppeiner curva- The organization of the letter is as follows. In Section ture is flat for the BTZ and Reissner-Nordström(RN) II and III, we analyze the nature of the phase transi- black holes, while curvature singularities occur for the tion through the diagrams of the Riemann tensor ele- Reissner-Nordströmanti de Sitter (RN-AdS) and Kerr ments and extrinsic curvature. In Section IV, we try to black holes. Moreover, it has been argued in [1] that all provide an answer to the question arising from the ar- possible physical fluctuations could be considered for cal- ticle [1], ” Ruppeiner geometry of RN black holes: flat culating curvature because neglecting one parameter may or curved?” using the concept of thermodynamic hypersurface in lower dimensions. In Section V, we consider a Pauli paramagnetic gas and investigate a hypersurface in ∗ [email protected] & [email protected] its thermodynamic geometry that corresponds to a zero † [email protected] magnetic field. Section VI contains a discussion of our ‡ e.sharifi[email protected] results. 2

II. THERMODYNAMIC EXTRINSIC examine a relationship between the divergences of the CURVATURE extrinsic curvature and the phase transition points. As already mentioned, the extrinsic curvature can be con- We begin with a review of our previous results on the structed by living on a certain hypersurface with a nor- correspondence between second order phase transitions mal vector (See Appx. B). Since the heat capacity, CQ, and singularities of the thermodynamic geometry [20, 21]. is defined at a constant electric charge, we should set on We also introduce extrinsic curvature as a new concept of a constant Q hypersurface. To do this, we change the the thermodynamic geometry. We will use this quantity coordinate from (S, Φ) to (S, Q) by using the following in determining some information about stability and non- Jacobian matrix. stability regions around phase transitions. For charged ∂ (S, Φ) black holes, a specific heat at a fixed electric charge is J ≡ (7) defined as follows: ∂ (S, Q) ∂S ∂(S, Q) T {S, Q} The metric elements of M(S, Φ) in the new coordinate C = T = T det = S,Q (1) Q (S, Q) can also be changed as follows: ∂T Q ∂(T,Q) {T,Q}S,Q 0 T It is obvious that the phase transitions of CQ are the gij = Jik gkl Jlj (8) zeros of {T,Q}S,Q (Appx. A may be consulted for a brief introduction to the bracket notation). Moreover, where, J T is the transpose of J. One can also rewrite the Ruppeiner metric in the mass representation can be Eq. (5) as a Jacobian matrix by: expressed as: ∂ (T, −Q) H M g = (9) gR = i,j (2) T ∂ (S, Φ) T

2 i j Thus, the new metric takes the following form: where Hi,jM = ∂ M/∂X ∂X is called the Hessian i matrix and X = (S, Q) are extensive parameters. There- ∂ (S, Φ) T ∂ (T, −Q) ∂ (S, Φ) fore, according to the first law of thermodynamics, dM = g¯0 = ∂ (S, Q) T ∂ (S, Φ) ∂ (S, Q) T dS + ΦdQ, one could define the denominator of the T scalar curvature R(S, Q) by: ∂ (S, Φ) ∂ (T, −Q) = (10) ∂ (S, Q) T ∂ (S, Q) ∂(T, Φ) {T, Φ} 1 g = det(gR) = det = S,Q = T ∂(S, Q) T 2 TC C Furthermore by regarding, given the property of the de- Φ S T (3) terminant, i.e., det(J ) = det(J), the determinant of the above relation can be calculated as follows: where, CS ≡ (∂Q/∂Φ)S and, T ! T {S, Φ} ∂ (S, Φ) ∂ (T, −Q) ∂S S,Q g¯0 = det = CΦ = T = (4) ∂ (S, Q) ∂ (S, Q) ∂T Φ {T, Φ}S,Q ∂ (S, Φ) ∂ (T, −Q) C −1 As a result, the scalar curvature R(S, Q) is not able to ex- det det = − S (11) ∂ (S, Q) ∂ (S, Q) TCQ plain the properties of the phase transitions of CQ. From Eq. (3), it is obvious that the phase transitions of CΦ cor- On the other hand, when we restrict ourselves to live respond precisely to the singularities of R(S, Q). Now, on the constant Q hypersurface with a normal vector one is able to prove an exact correspondence between q 0QQ singularities of this new metric (R¯(S, Φ)) and phase tran- nQ = −1/ g , the extrinsic curvature will be given sitions of CQ [20] by redefining the Ruppeiner metric as by: follows: 1 K(S, Q) = (nµ∂ ) g¯0 + (∂ nµ) (12) H M ¯0 µ µ g = i,j (5) 2g T SQ QQ where, nµ = nS, nQ = g0 , g0 n (See Appx. B). where, M(S, Φ) = M(S, Q) − ΦQ is the enthalpy poten- Q tial for M(S, Q) and Xi = (S, Φ). From the first law, From Eq. (10), the metric tensor in the new coordinate dM(S, Φ) = T dS − ΦdQ, the denominator of R(S, Φ) is can be calculated as follows: obtained by: C−1 ¯0 −1 S g = diag(CQ , − ) (13) ∂(T, −Q) 1 CS T g = det(g) = det = − 2 {T,Q}S,Φ = − T ∂(S, Φ) T TCQ (6) Therefore, on the constant Q hypersurface, we have: It is straightforward to show that the phase transitions µ Q p of CQ are equal to the singularities of R(S, Φ). We now n = 0, n = 0, |TCS| (14) 3

It is easy to show that the above vector is a normalized µ vector (¯nµn¯ = −1). Therefore, the extrinsic curvature can be rewritten as follows: −1 ¯ TCQ p CS p K = |TCS|∂Q + ∂Q |TCS| −1 TC CS Q Q=cte (15) Indeed this relation tells us that the singularities of this curvature occur exactly at phase transitions of the CQ. It should be noted that in this case when the extrinsic curvature diverges, the metric components are differentiable. However, the metric elements are non-differentiable for extremal black holes in the Ruppeiner geometry. Our study indicates that in thermodynamic geometry divergences of the extrinsic curvature does not always implies non-differentiavble metric elements. In table I, we com- pare the heat capacity and the extrinsic curvature function for Kerr, RN, BTZ, and Einstein-Maxwell-Gauss- Bonnet (EMGB) [26] black holes. In all cases, the roots of the extrinsic curvature denominator show phase transition points. The extrinsic curvature also changes its sign FIG. 1. Graph of the extrinsic curvature K¯ (solid blue curve) at the phase transition points which is exactly a similar and the heat capacity CJ,Q (dashed red curve) as a function behavior to heat capacity. Generally, for thermodynamic of entropy, S, for an electric charge Q = 1 and an angular systems with (n + 1) variables, one could consider the momentum J = 1. following metric, H M ∂ (T, −Q , −Q , ..., Q ) g = i,j = 1 2 n (16) T T ∂ (S, Φ , Φ , ..., Φ ) 1 2 n For a Kerr-Newman (KN) black hole with the follow- Pn i ing mass, where M = M − i ΦiQi and X = (S, Q1, ..., Qn). Fur- thermore, (T, Φ1, ..., Φn) are called extensive parameters. Then utilizing the Jacobian matrix as follows: pS(4J 2 + S2 + 2Q2S + Q4) ∂ (S, Φ1, Φ2, ..., Φn) M = (22) J ≡ (17) 2S ∂ (S, Q1,Q2, ..., Qn) in a similar way, the metric tensor can be represented by below block-diagonal matrix. metric elements are defined as follows: g¯0 = diag(C−1 , −G) (18) Q1,Q2,...,Qn H M ∂ (T, −Q, −J) where G is a square matrix of order n defined by the g = i,j = (23) following relation, T T ∂ (S, Φ, Ω) H M G = i,j ; Xi = (Q ,Q , ..., Q ) (19) T 1 2 n where T is the Hawking temperature, Ω is the angu- Thus the metric determinant in the new coordinates can lar velocity, and Φ is the potential deference [20, 21]. be written as: When somebody restricts himself to live on the constant −1 J hypersurface which has the orthogonal normal vector, [CS,Q ,Q ,...,Q CS,Φ ,Q ,...,Q ...CS,Φ ,Φ ,...,Φ ] g¯0 = 1 2 n 1 2 n 1 2 n n = −1/p|TC |, the extrinsic curvature diverges at (−T )nC J S,Φ Q1,Q2,...,Qn the phase transition point and exhibit a similar sign be- (20) havior around the transition points. In Figure 1, the where above functions were defined in [20]. Now, a con- graph of the extrinsic curvature, K, and the heat capac- stant Q hypersurface has the following unit normal vec- i ity, C , for the Kerr-Newman black hole (KN) shows tor, J,Q an exact correspondence between singularities and phase −1 transitions (Note that the first divergence point is related n¯µ = (0, 0, ..., −(TCS,Φ ,...,Φ ,Φ ,...,Φ ) 2 , .., 0) (21) 1 i−1 i+1 n to T = 0.). It is surprising that the same result obtains where the non-zero term places in ith column. The by considering a constant Q hypersurface with unit nor- p extrinsic curvature, K¯ (S, Q1,Q2, ..., Qn), can be calcu- mal vector, nQ = −1/ |TCS,Ω|. Moreover, it will be lated using Eq (B7). It is interesting that the extrin- easy to show that the signs of such Ricci tensor elements sic curvature has the same behavior as the specific heat, as RSS, RSQ and RSJ are similar to that of the CJ,Q

CQ1,Q2,...,Qn . around the transition points. 4 kerr RN

√ 2 q 2 −1 Q S J π Sπ π + S π +4 S M(S, J) = 2 M(S, Q) = 2 √ 2 2 2 2 T (S, J) = − 4 J √π −S 1 T (S, Q) = − SπQ 4 πS2 q 4 J2π2+S2 2S2 S √ √ 2Jπ (48 J4π4+8 J2π2S2+3 S4) 2 S K(S, J) = √ K(S, Q) = 0 (48 J4π4+24 J2π2S2−S4) 16 J4π4−S4

2S(16 J4π4−S4) CJ (S, J) = − 48 J4π4+24 J2π2S2−S4 CQ(S, Q) = −S/2

BTZ EMGB

√ 2 2 2 2 √ 3 S π J π Q 2 3 2 π Λ S4 M(S, J) = 2 2 + 4 2 M(S, Q) = πα + √3 + π S − 16π l S 6 S2 12

√3 2/3 2/3 2/3 π Λ S4 S2−6 π S2 S4 +Q2 S4 S π2J2 ( ) ( ) ( ) T (S, J) = 2 2 − 8 3 T (S, Q) = − √3 2/3 8π l S 9S S2(S4)

√ 7/2 2 2QS3/2(Λ S2/3−4 π) 3 K(S, J) = 32S √Jπ l K(S, Q) = √ (192 J2π4l2+S4) 64 J2π4l2−S4 (Λ S8/3−5 Q2S2/3+6 π S2) Λ S2−6 π S4/3+Q2

(64 J2π4l2−S4)S 3S(Λ S8/3+Q2S2/3−6 π S2) C (S, J) = − C (S, Q) = J 192 J2π4l2+S4 Q Λ S8/3−5 Q2S2/3+6 π S2

TABLE I. Thermodynamic variables and extrinsic curvature functions for Kerr, RN, BTZ and EMGB black holes.

2 III. EXTRINSIC CURVATURE OF THE where S = πr+ is the Bekenstein-Hawking entropy. Ac- PHANTOM RN-ADS BLACK HOLE cording to the first law of thermodynamics, one can calculate the Hawking temperature, T , the electric potential, The action for the Einstein-Hilbert theory with phan- Φ, and the specific heat capacity, CQ, as follows: tom Maxwell field reads: ∂M −πS + ΛS2 + ηπ2Q2 Z T = ( ) = (28) √ µν 4 3/2 S = −g (R + 2Λ + 2ηFµν F ) d x , (24) ∂S Q −4(πS) where, Λ is the cosmological constant, and η = ±1. RN- 3/2 2 AdS black hole corresponds to η = 1, while phantom ∂M (S/π) ηπ Q Φ = ( ) = 2 (29) couplings of the Maxwell field ( Phantom RN-AdS black ∂Q S S hole) are obtained for η = −1. The metrics of these solutions, derived in [24], take the below form. ∂S −2S(−πS + ΛS2 + ηπ2Q2) CQ = T ( ) = (30) 1 ∂T Q (−πS − ΛS2 + 3ηπ2Q2) ds2 = f(r)dt2 − dr2 − r2(dθ2 + sin2 θ dφ2) , (25) f(r) Using Eq. (5), it is easy to obtain the metric elements of where f(r) is given by: the enthalpy potential in the coordinate (S, Φ). Then by applying Eqs. (7) and (8), the scalar curvature, R, takes 2M Λ Q2 f(r) = 1 − − r2 + η (26) the following form: r 3 r2 F (S, Q) R¯ = The event horizon, r+, of this solution can be determined 2 2 2 2 2 2 2 by calculating the roots for the equation f(r ) = 0. The (−Sπ + ΛS + π ηQ )(−Sπ − ΛS + 3π ηQ ) + (31) mass of this black hole is expressed as a function of the where, thermodynamic variables. 2 4 5 2 2 4 4 2 4 2 2 F = −η Q π + 2 η Q + 10 Λ η Q Sπ + 2 Λ S π 1 3/2 π Λ ηπ Q M = (S/π) ( − + ) (27) 2 2 3 2 2 3 2 2 S 3 S2 + S −9 η Q Λ − 1 π + 3 Λ − 6 Λ η Q S π (32) 5

The first part of the denominator is zero only at T = 0 and the roots of the second part of the curvature denominator gives us the phase transition points. Therefore, the curvature diverges exactly at these points where heat capacity diverges with no other additional roots. For the RN-AdS black hole, the scalar curvature (31) and the specific heat (30) are depicted in Figure 2 as a function of entropy and for a fixed value of electric charge Q = 0.25. On the other hand, the extrinsic curvature opens an interesting and impressive avenue to the inves- tigation of how phase transitions behave. In this case, we need to sit on a hypersurface with the normal vec- p QQ tor nQ = −1/ |g¯ | in which the extrinsic curvature associated with this hypersurface is given by: √ (π − 2 Λ S) p|η|π Q S K¯ = − (−π S − Λ S2 + 3 η π2Q2) p|−π S + Λ S2 + η π2Q2| (33) The first term of the denominator shows the phase transition points, while the second is only zero at T = 0. Inter- estingly, we see in Figure 3 that the extrinsic curvature FIG. 2. Graph of the scalar of curvature R (solid blue curve) has the same sign as heat capacity does, while in Figure and the heat capacity CQ (dashed red curve) as a function 2, the scalar curvature does not have the same sign as of entropy, S, in the RN-AdS case, for an electric charge the heat capacity around the phase transition points. In Q = 0.25 and a cosmological constant Λ=- 1. other words, extrinsic curvature reveals more information such as the stability/non-stability of heat capacities than the Ricci scalar does. Figure 4 indicates that the special Ricci tensor elements RSS also exhibit a similar behavior around phase transition points such as heat capacity. Therefore, the extrinsic curvature K(S, Q) and the RSS component of the Ricci tensor describe the phase transitions and the sign of the heat capacity, CQ. Our work is a new method to identify stable regimes in the parameter space of black holes by studying the extrinsic curvature in their thermodynamic geometry.

IV. RUPPEINER CURVATURE OF RN BLACK HOLE AS AN INTRINSIC CURVATURE ON A CONSTANT J HYPERSURFACE

In article [1], the authors proposed a new measure of microscopic interactions and its effects on Ruppeiner curvature by considering a complete phase space of extensive variables. They obtained a new non-zero Ruppeiner cur- FIG. 3. Graph of the scalar of curvature K (solid blue curve) vature for RN black holes by setting l → ∞, l = −3 , Λ and the heat capacity CQ (dashed red curve) as a function where J → 0 limits in the scalar curvature for the Kerr- of entropy, S, in the RN-AdS case, for an electric charge Newman-AdS (KN-AdS) black hole as follows: Q = 0.25 and a cosmological constant Λ =- 1. S2 + Q2S + 2Q4 lim RKN−AdS = (34) l→∞;J→0 (S + Q2)2 (Q2 − S) Appx. B). In this framework, the Ruppeiner curvature of This non-vanishing scalar curvature is the result of an- the KN-AdS black hole can be broken down into a purely other dimension specified by J which fluctuates even if intrinsic part, which yields a zero Ruppeiner curvature of we set it to zero. It is surprising that this result is in the RN black hole, and an extrinsic part, which measures contrast to a direct calculation on the Ruppeiner met- the bending of the constant J hypersurface; that is: ric of the RN which is zero [23]. One of our objectives in this work is to explain the difference between these (3)R = (2)R −(K2−K Kab)−2(nα nβ − nαnβ ) two results by applying the basic concepts of the extrin- KN−AdS in ab ;β ;β ;α sic/ intrinsic geometry for a particular hypersurface (See (35) 6

have:

2 αβ K = KαβK = 0 (39)

On the other hand, the metric elements induced on the constant J hypersurface can be given by:

1 ∂2M 3Q2 − S gSS = lim = (40) l→∞;J→0 T ∂S2 2S (S − Q2) 1 ∂2M −2Q gSQ = gQS = lim = (41) l→∞;J→0 T ∂S∂Q S − Q2 1 ∂2M 4S gQQ = lim = (42) l→∞;J→0 T ∂Q2 S − Q2

The above elements are the same as those of the Rup- peiner metric for the RN black hole [23]. Therefore, the (2) intrinsic curvature Rin of the constant J hypersurface equals the Ruppeiner curvature of the RN black hole; i.e.,

FIG. 4. Graph of the scalar of curvature RSS (solid blue (2) curve) and the heat capacity CQ (dashed red curve) as a func- Rin = RRN = 0 (43) tion of entropy, S, in the RN-AdS case, for an electric charge Q = 0.25 and a cosmological constant Λ=- 1. Finally, based on Eq. (38), the last statement of Eq. (35) is given by:

(2) where, Rin = RRN = 0. Here the extrinsic part is 2 2 S Q S 4 expected to be exactly the same as Eq. (34). Now, let us α β α β 2 + 2 + Q lim (n;βn − n n;β) = − investigate the accuracy of our claim about the KN-AdS l→∞;J→0 ;α (S + Q2)2 (Q2 − S) black hole in the limits l → ∞ and J → 0. (44) The mass relation of the KN-AdS black hole [25] as a We can, therefore, conclude that the non-vanishing Rup- function of thermodynamic variables can be written as: peiner curvature in the limits J → 0 and l → ∞ is ex- tracted from the curvature of the KN-AdS black hole S π 4J 2 + Q2 Q2 J 2 M = [ + + + (36) when one lives on the constant J haypersurface. In addi- 4π 4S 2 l2 tion, the Ruppeiner curvature of the RN black hole can 2 S 2 S S 1 be interpreted as the intrinsic curvature produced by the + Q + + ] 2 πl2 π 2π2l2 induced metric on the two dimensional hypersurface (J- zero hypersurface). The Hawking temperature T is also deﬁned by: We may also check Eq. (B9) for the Kerr black hole ∂M S2π2l4 − 4π4l4J 2 − π4l4Q2 at Q → 0 and l → ∞ limits of the KN-AdS black hole to T = = (37) ∂S 8π3MS2l4 show that the intrinsic part of Eq. (B9) is the Ruppeiner Q,J curvature of the Kerr black hole. Using the deﬁnition 2S2Q2π2l2 + 4S3πl2 + 3S4 + of the Ruppeiner metric for the complete phase space of 8π3MS2l4 the parameters (KN-AdS black hole), and assuming the It should be noted that, within the hypersurface frame- limits Q → 0, l → ∞, we obtain the following equation work, setting J to zero is tantamount to living on the for the Ricci scalar for a non- charged KN-AdS black constant J hypersurface (J- zero hapersurface) which has hole. the following normal vector, 2 2 (3) 36J + S S lim RKN−AdS = (45) q 1 ∂2M l→∞;Q→0 16J 4 − S4 n = −1/ |gJJ | ; g = . (38) J JJ T ∂J 2 Also, the induced metric for the Q-zero hypersurface is Therefore, by making use of Eqs. (B7) and (B10), we calculated by:

1 ∂2M −24J 2S2 − 48 J 4 + S4 gSS = lim = (46) l→∞;Q→0 T ∂S2 2S (4 J 2 + S2) (4 J 2 − S2) 7

1 ∂2M 4J 4 J 2 + 3 S2 gSJ = gJS = lim = (47) l→∞;Q→0 T ∂S∂J (4 J 2 + S2) (4 J 2 − S2) 1 ∂2M −8S3 gJJ = lim = (48) l→∞;Q→0 T ∂J 2 (4 J 2 + S2) (4 J 2 − S2)

3 One can obtain the following relation for the intrinsic (2πm) 2 where I = h3 and x = 1/T, y = −µ/T, z = −H/T curvature. are thermodynamic coordinates. T , µ, and H are also 12 J 2 + S2 S temperature, chemical potential, and external magnetic R = R = (49) in kerr 16 J 4 − S4 field, respectively. The fn(η) function is defined by: Moreover, for the Q-zero hypersurface with the normal 1 Z ∞ Xn−1dX f (η) = (54) vector, n Γ(n) eX 0 η + 1 2 q 1 ∂ M µ n = −1/ |gQQ| ; g = (50) where η = exp( ) is called fugacity. The Ruppenier Q QQ 2 KB T T ∂Q metric in thermodynamic geometry is given by: 2 αβ the extrinsic curvature is zero and K = KαβK = ∂2φ g = ; Y i = (x, y, z) (55) 0. Now, we can successfully examine the validity of the ij ∂Y i∂Y j following equation; The Ricci scalar, (3)R was already obtained as a symmet- (2) α β α β lim RKN−AdS = RKerr−2 lim (n;βn − n n;β) l→∞;Q→0 l→∞;Q→0 ;ricα function of z [30]. It means that the scalar curvature (51) doesn’t depend on the orientation of external magnetic where, field. In the classical limit in the lack of the external ± ± magnetic field (H → 0), when η → η and fn (η) → η; 2 (3) α β α β SJ the R is rewritten as follows: lim (n;βn − n n;β) = −12 (52) l→∞;Q→0 ;α 16 J 4 − S4 9 lim (3)R = − (56) In summary, the Ruppeiner curvature of the Kerr black H→0(z→0) 2ρ hole is similar to the intrinsic curvature produced by the 2 induced metric on the two dimensional hypersurface ( where in the classical regime, ρ = λ3 η and λ = 1 i.e., constant Q hypersurface). Our study indicates that h/(2πm0KBT ) 2 . Eq. (56) shows that in the classical in thermodynamic geometry, properties of the intrinsic limit, the curvature of a Pauli paramagnetic gas depends and the extrinsic curvatures are important to obtain a on the volume occupied by a single particle. The scalar complete geometric representation of thermodynamics in curvature is also similar to the curvature that was ob- physical systems. The intrinsic curvature also help us to tained for a two-component ideal gas by Ruppeiner [31]. identify attractive, repulsive and non-interacting statis- In this letter we explore the physical properties of a tical interaction between the constituent parts of a ther- Pauli paramagnetic gas by studying the intrinsic curva- modynamic system as we discuss in the next Section. ture of a hypersrface corresponding to a zero magnetic field ( z → 0). The intrinsic curvature of H-zero hypersurface can be calculated as: V. HYPERSURFACES AND THEIR INTRINSIC (2) 5 CURVATURE IN THERMODYNAMIC Rin = − (57) GEOMETRY OF THE PAULI PARAMAGNETIC 2ρ GAS Because the intrinsic curvature is negative, so the statistical interactions of a Pauli paramagnetic gas can be Thermodynamic curvature may explain the statistical repulsive which indicates a more stable paramagnetic gas. interaction between particles in a thermodynamic system It should be noted that in the classical limit the extrinsic [27–29]. The thermodynamic curvature is positive for at- curvature vanish (K = 0) and the Gauss-Codazzi rela- tractive interaction between particles, and negative for a tion is held in this case. On the other hand, the Rup- repulsive interaction [29]. In case that particles have not penier curvature for a non-interacting classical paramag- any interaction with each other, the thermodynamic ge- netic gas through Maxwell-Boltzman statistics with the 3 sinh(Jz) ometry is flat [27]. Now we consider thermodynamic ge- − 2 −y thermodynamic potential, φc = 4πIx e Jz (J is ometry of a Pauli paramagnetic gas with indentical spin magnetic momentum) [30], takes the following form: 1/2 fermions in the presence of an external magnetic field [30]. From the grand canonical distribution through the λ3Jz (3)R = (58) Fermi-Dirac statistics, the thermodynamic potential can c 8π η sinh (Jz) be obtained as follows: From the equation of state (PV = NKBT ), one can see − 3 −y−Jz −y+Jz φ(x, y, z) = Ix 2 [f 5 (e ) + f 5 (e )]. (53) the curvature is proportional to the volume occupied by 2 2 8

(3) single particle ( Rc = 1/2ρ) [30]. It is surprising that And, the determinant of the Jacobian transformation can in the absence of an external magnetic field, the cur- be written in the bracket notation as in the form below: (3) λ3 vature is not zero ( Rc = 8π η ), while we would expect   the curvature to be zero because of non-interaction parti- ∂f ∂f ∂(f, g)  ∂h ∂k  {f, g}a,b cles [27]. To obtain a correct result we have to calculate det = det  k h  = the induced metric on a zero magnetic field hypersur- ∂(h, k)  ∂g ∂g  {h, k}a,b ∂h ∂k face in the thermodynamic geometry. It can easily be k h shown that H-zero hypersurface is flat which indicates a (A4) non-interacting gas as expected. Our analysis indicates Generally, the partial derivative for functions with n + 1 that the intrinsic geometry of hypersurfaces in thermo- variables can be calculated as follows: dynamic geometry have important physical information. ∂f {f, h1, ..., hn} In order to have correct information we must study hy- = q1,q2,...,qn+1 (A5) ∂g {g, h , ..., h } persurfaces in thermodynamic geometry. h1,.....,hn 1 n q1,q2,...,qn+1

where, the f, g, and hn (n = 1, 2, 3, ...) are functions of VI. CONCLUSION qi, i = 1, ..., n + 1 variables [20] and,

{f, h1, ..., hn} = (A6) This work analyzed the thermodynamic geometry of a q1,q2,...,qn+1 n+1 black hole from the perspective of an extrinsic curvature. X ∂f ∂h1 ∂h2 ∂hn It was found that the extrinsic scalar curvature represents εijk...l ... ∂qi ∂qj ∂qk ∂ql the critical behavior of a second order phase transition in ijk....l=1 a thermodynamic system. Some particular Ricci tensor elements were found to have the same sign behavior as heat capacities. Another part of the article explained the Appendix B: The concepts of extrinsic and intrinsic curvatures for a hypersurface relationship between the intrinsic, the extrinsic, and the total curvatures of thermodynamic geometry of a system by sitting on a certain hypersurface. For the KN-AdS In this section, we brieﬂy review the concept of ex- black hole on a constant J hypersurface, the curvature trinsic curvature. For an n-dimensional manifold M, a scalar was broken down into two parts. One was a zero special hypersurface Σ can be deﬁned as follows: intrinsic curvature (the Ruppeiner curvature of the RN α black hole) while the other was an extrinsic part whose Φ(x ) = 0 (B1) divergence points were the singularities of a non-rotating where, xαs are the coordinates of the manifold M. The KN-AdS black hole. We also used the intrinsic curvature induced metric on Σ can be written as: of the relevant hypersurface to investigate some thermodynamic properties such as stability and the statistical α β 2 α β ∂x a ∂x b a b interaction. ds = gαβdx dx = gαβ( dy )( dy ) = habdy dy Σ ∂ya ∂yb As a result, the critical behavior of a thermodynamic (B2) system on an explicit hypersurface can be explained con- where, sistently by using intrinsic and extrinsic curvatures of this hypersurface. ∂xα h = g EαEβ; Eα = (B3) ab αβ a b a ∂ya

Appendix A: Partial derivative and bracket notation hab deﬁnes the induced metric on the hypersurface. A α unit normal nα can be introduced if nαn = where When f, g, and h are explicit functions of (a, b), we can ε = 1 when Σ is timelike and ε = −1 when Σ is spacelike obtain the following relation for the partial derivative. [7]. We select that nα points in the direction of increasing α Φ: n Φ,αi0. We can also easily show that: ∂f {f, h} = a,b (A1) Φ,α ∂g h {g, h}a,b nα = 1 (B4) |gαβΦ Φ | 2 where, ,α ,β ∂f ∂h ∂f ∂h The inverse of the induced metric is obtained as follows: {f, h} = − (A2) a,b β ∂a b ∂b a ∂b a ∂a b ab α αβ α β h Ea Eb = g − εn n (B5) Moreover, if one considers a = a(c, d) and b = b(c, d), Eq. One can also introduce the extrinsic curvature tensor on (A1) can be rewritten as: the hypersurface Σ using the following relation: ∂f {f, h}a,b {f, h}c,d = = (A3) α β 1 α β Kab = nα;βE aE b = LngαβE aE b (B6) ∂g h {g, h}a,b {g, h}c,d 2 9 where, the symbol ; and Ln are the covariant and Lie the Gauss-Codazzi equation in the contracted form, we α derivatives of gαβ along n , respectively. Therefore, the obtain: extrinsic curvature is deﬁned by: (3) (2) 2 ab α β α β R = Rin + (K − KabK ) + 2(n;βn − n n;β);α 1 √ (B9) K = habK = nα = √ ∂ ( gnα) (B7) ab ;α g α This relation is the three-dimensional Ricci scalar evalu- ated on the hypersurface Σ. The (2)R is intrinsic Ricci scalar of the hypersurface. The third term, on the right where, g = det(gαβ). Suppose that a two-dimensional hand side of the Equation, can be expressed as: manifold is embedded in a three-dimensional space. The ab α β induced metric hab and the extrinsic curvature Kab con- KabK = n;βn;α (B10) tain the necessary information about the properties of the hypersurface Σ. The full Riemann curvature tensor For the two-dimensional space, the hypersurface Ξ is of the 3-dimensional space and the curvature tensor of a one-dimensional space with the normal vector n = a the 2-dimensional hypersurface are related by the Gauss- p ab b 1/ |g | δa. For this case, we have: Codazzi equation: 2 ab K = KabK (B11) (3) α β γ σ (2) RαβγσEa Eb Ec Ed = Rabcd + ε(KacKbd − KadKbc) (B8) Thus, the Ricci scalar (2)R is determined as follows: This indicates that the three-dimensional Riemann ten- (2) a b a b sor can be expressed in terms of the intrinsic and extrin- R = 2(n;bn − n n;b);a (B12) sic curvature tensors of the hypersurface [32]. Writing

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