Extrinsic and Intrinsic Curvatures in Thermodynamic Geometry

Physics Letters B 759 (2016) 298–305

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Physics Letters B

www.elsevier.com/locate/physletb

Extrinsic and intrinsic curvatures in thermodynamic geometry

Seyed Ali Hosseini Mansoori a,b, Behrouz Mirza b, Elham Shariﬁan b a Department of Physics, Boston University, 590 Commonwealth Ave., Boston, MA 02215, USA b Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran a r t i c l e i n f o a b s t r a c t

Article history: We investigate the intrinsic and extrinsic curvatures of a certain hypersurface in thermodynamic geom- Received 19 February 2016 etry of a physical system and show that they contain useful thermodynamic information. For an anti- Received in revised form 29 May 2016 Reissner–Nordström-(A)de Sitter black hole (Phantom), the extrinsic curvature of a constant Q hypersur- Accepted 31 May 2016 face has the same sign as the heat capacity around the phase transition points. The intrinsic curvature of Available online 2 June 2016 the hypersurface can also be divergent at the critical points but has no information about the sign of the Editor: N. Lambert 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 hypersurface) ones [1]. This approach can easily be generalized to an arbitrary thermodynamic system. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction identify a phase transition, several works [18–21] have exploited it to explain the black hole phase transitions. Bekenstein and Hawking showed that a black hole has a behav- The Ruppeiner geometry has also been analyzed for several ior similar to a common thermodynamic system [2,3]. They drew black holes to find out the thermodynamic properties [22,23]. As a a parallel relationship between the four laws of thermodynamics result, the Ruppeiner curvature is flat for the BTZ and Reissner– and the physical properties of black holes by considering the sur- Nordström (RN) black holes, while curvature singularities occur face gravity and the horizon area as the temperature and entropy, for the Reissner–Nordström anti de Sitter (RN-AdS) and Kerr black respectively [4]. An interesting topic is to study phase transitions holes. Moreover, it has been argued in [1] that all possible physical in black hole thermodynamics where the heat capacity diverges fluctuations could be considered for calculating curvature because [5,6]. These divergence points of heat capacity are usually asso- neglecting one parameter may lead to inadequate information ciated with a second order phase transition for some fixed black about it. Therefore, the thermodynamic curvature of RN should hole parameters [7]. be reproduced from the Kerr–Newmann anti-de Sitter (KN-AdS) Geometric concepts can also be used to study the properties black hole when the angular momentum J → 0and cosmologi- of an equilibrium space of thermodynamic systems. Riemannian cal constant → 0. This approach leads to a non-zero value for geometry in the space of equilibrium states was introduced by the Ruppeiner scalar, which is in contrast to the reports on RN in Weinhold [8] and Ruppeiner [9,10] who defined metric elements previous works [22,23]. as the Hessian matrix of the internal energy and entropy. These ge- The present letter seeks to explain this contrast by obtaining ometric structures are used to find the significance of the distance intrinsic and extrinsic curvatures of the related submanifolds. The between equilibrium states. Consequently, various thermodynamic induced metric (intrinsic curvature) and the extrinsic curvature properties of the system can be derived from the properties of of a constant J hypersurface contain the necessary information these metrics, especially critical behaviors, and stability of various about the properties of this hypersurface. The zero limit of an types of black hole families [11–13]. For the second order phase angular momentum for a KN-AdS black hole is equivalent to the transitions, Ruppeiner curvature scalar (R) is expected to diverge two-dimensional constant J hypersurface embedded in a three- at critical points [7,14–17]. Due to the success of this geometry to dimensional complete thermodynamic space. The curvature scalar of KN-AdS black hole on this hypersurface can be decomposed into an intrinsic curvature (Ruppeiner curvature of RN black hole), E-mail addresses: [email protected], [email protected] (S.A. Hosseini Mansoori), [email protected] (B. Mirza), e.sharifi[email protected] which is zero, and an extrinsic part that give the curvature singu- (E. Sharifian). larities. http://dx.doi.org/10.1016/j.physletb.2016.05.096 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305 299 ∂(T , −Q ) 1 C S We also prove that there is a one-to-one correspondence be- g = det(g) = det =− {T , Q } =− (6) 2 S, tween divergence points of the heat capacities and those of the T ∂(S,) T TCQ extrinsic curvature for thermodynamic descriptions where poten- It is straightforward to show that the phase transitions of C Q are tials are related to the mass (rather than the entropy) by Legendre equal to the singularities of R(S, ). We now examine a relation- transformations. In spite of this correspondence, we can get other ship between the divergences of the extrinsic curvature and the information about thermodynamics like stability and non-stability phase transition points. As already mentioned, the extrinsic curva- regions around phase transitions from singularities of extrinsic cur- ture can be constructed by living on a certain hypersurface with a vature and certain elements of the Ricci tensor. normal vector (see Appendix B). Since the heat capacity, C Q , is de- The organization of the letter is as follows. In Section 2 and 3, fined at a constant electric charge, we should set on a constant Q we analyze the nature of the phase transition through the dia- hypersurface. To do this, we change the coordinate from (S, ) to grams of the Riemann tensor elements and extrinsic curvature. (S, Q ) by using the following Jacobian matrix. In Section 4, we try to provide an answer to the question aris- ing from the article [1], ” Ruppeiner geometry of RN black holes: flat ∂ (S,) J ≡ (7) or curved?” using the concept of thermodynamic hypersurface in ∂ (S, Q ) lower dimensions. In Section 5, we consider a Pauli paramagnetic gas and investigate a hypersurface in its thermodynamic geome- The metric elements of M(S, ) in the new coordinate (S, Q ) can try that corresponds to a zero magnetic field. Section 6 contains a also be changed as follows: discussion of our results. = T gij J ik gkl Jlj (8) 2. Thermodynamic extrinsic curvature where, J T is the transpose of J . One can also rewrite Eq. (5) as a Jacobian matrix by: We begin with a review of our previous results on the corre- ∂ (T , −Q ) spondence between second order phase transitions and singular- g = (9) ities of the thermodynamic geometry [20,21]. We also introduce T ∂ (S,) extrinsic curvature as a new concept of the thermodynamic geom- Thus, the new metric takes the following form: etry. We will use this quantity in determining some information about stability and non-stability regions around phase transitions. T − ¯ ∂ (S,) ∂ (T , Q ) ∂ (S,) For charged black holes, a specific heat at a fixed electric charge is g = ∂ (S, Q ) T ∂ (S,) ∂ (S, Q ) defined as follows: ∂ (S,) T ∂ (T , −Q ) T {S, Q } = = ∂ S = ∂(S, Q ) = S,Q (10) C Q T Tdet { } (1) ∂ (S, Q ) T ∂ (S, Q ) ∂ T Q ∂(T , Q ) T , Q S,Q Furthermore by regarding, given the property of the determinant, It is obvious that the phase transitions of C Q are the zeros of i.e., det( J T ) = det( J), the determinant of the above relation can be { } T , Q S,Q (Appendix A may be consulted for a brief introduction calculated as follows: to the bracket notation). Moreover, the Ruppeiner metric in the mass representation can be expressed as: ∂ (S,) T ∂ (T , −Q ) g¯ = det = ∂ (S, Q ) ∂ (S, Q ) Hi, j M gR = (2) −1 T ∂ (S,) ∂ (T , −Q ) C S det det =− (11) 2 i j where Hi, j M = ∂ M/∂ X ∂ X is called the Hessian matrix and ∂ (S, Q ) ∂ (S, Q ) TCQ X i = (S, Q ) are extensive parameters. Therefore, according to the On the other hand, when we restrict ourselves to live on the con- first law of thermodynamics, dM = TdS + dQ , one could define =− QQ the denominator of the scalar curvature R(S, Q ) by: stant Q hypersurface with a normal vector nQ 1/ g , the { } extrinsic curvature will be given by: ∂(T ,) T , S,Q 1 g = det(gR ) = det = = (3) 2 T ∂(S, Q ) T TCC S = 1 μ ¯ + μ K (S, Q ) ¯ n ∂μ g ∂μn (12) ≡ 2g where, C S (∂ Q /∂)S and, { } μ = S Q = SQ QQ ∂ S T S, S,Q where, n n ,n g , g nQ (see Appendix B). From C = T = (4) { } Eq. (10), the metric tensor in the new coordinate can be calculated ∂ T T , S,Q as follows: As a result, the scalar curvature R(S, Q ) is not able to explain the −1 properties of the phase transitions of C . From Eq. (3), it is ob- − C Q g¯ = diag(C 1, − S ) (13) vious that the phase transitions of C correspond precisely to the Q T singularities of R(S, Q ). Now, one is able to prove an exact corre- ¯ Therefore, on the constant Q hypersurface, we have: spondence between singularities of this new metric (R(S, )) and phase transitions of C Q [20] by redefining the Ruppeiner metric as μ Q n = 0,n = 0, |TCS | (14) follows: It is easy to show that the above vector is a normalized vector Hi, j M μ g = (5) (n¯μn¯ =−1). Therefore, the extrinsic curvature can be rewritten T as follows: = − where, M(S, ) M(S, Q ) Q is the enthalpy potential for −1 i = = − TCQ C M(S, Q ) and X (S, ). From the first law, dM(S, ) TdS K¯ = |TC |∂ S + ∂ |TC | (15) −1 S Q Q S dQ , the denominator of R(S, ) is obtained by: C TCQ S Q =cte 300 S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305

Table 1 Thermodynamic variables and extrinsic curvature functions for Kerr, RN, BT Z and EMGB black holes.

kerr RN √ 2 −1 Q J2π Sπ π + S +4 S = π S = M(S, J) 2 M(S, Q ) 2 √ 2 2 2 =−4 J √π −S 1 =− Sπ Q 2 T (S, J) 2 T (S, Q ) 2 4 π S 4 J2π2+S2 2S S √ √ 2 Jπ 48 J 4π 4+8 J 2π 2 S2 +3 S4 2 S K (S, J) = K (S, Q ) = 0 48 J 4π 4+24 J 2π 2 S2−S4 16 J 4π 4−S4 2S 16 J 4π 4−S4 C (S, J) =− C (S, Q ) =−S/2 J 48 J 4π 4+24 J 2π 2 S2−S4 Q BT Z EMGB √ √ 2 2 2 2 3 = S + π J = + π√Q + 2 3 2 − π S4 M(S, J) 2 2 4 2 M(S, Q ) πα 3 π S 16π l S 6 S2 12 √ 4 3 2 2 2/3 4 2/3 2 4 2/3 2 2 π S S −6 π S S +Q S = S − π J =− √ T (S, J) 2 2 8 3 T (S, Q ) 3 2/3 8π l S 9S S2 S4 √ 7/2 2 2QS3/2 S2/3−4 π 3 K (S, J) = 32S Jπ l K (S, Q ) = 192 J 2π 4l2+S4 64 J 2π 4l2−S4 S8/3−5 Q 2 S2/3+6 π S2 S2 −6 π S4/3+Q 2 64 J 2π 4l2−S4 S 3S S8/3+Q 2 S2/3−6 π S2 C (S, J) =− C (S, Q ) = J 192 J 2π 4l2+S4 Q S8/3−5 Q 2 S2/3+6 π S2

Indeed this relation tells us that the singularities of this curvature occur exactly at phase transitions of the C Q . 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 do not always imply non-differentiable metric elements. In Table 1, we compare the heat capacity and the extrinsic curvature function for Kerr, RN, BT Z, 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 at the phase transition points which is exactly a similar behavior to heat capacity. Generally, for thermodynamic systems with (n + 1) variables, one could consider the following metric,

Hi, j M ∂ (T , −Q 1, −Q 2, ..., Q n) g = = (16) T T ∂ (S,1,2, ..., n) = − n i = where M M i i Q i and X (S, Q 1, ..., Q n). Furthermore, (T ,1, ..., n) are called extensive parameters. Then utilizing the Jacobian matrix as follows: Fig. 1. Graph of the extrinsic curvature K¯ (solid blue curve) and the heat capacity C J,Q (dashed red curve) as a function of entropy, S, for an electric charge Q = 1 ∂ (S,1,2, ..., n) = J ≡ (17) and an angular momentum J 1. (For interpretation of the references to color in ∂ (S, Q 1, Q 2, ..., Q n) this figure legend, the reader is referred to the web version of this article.) in a similar way, the metric tensor can be represented by below where the non-zero term places in ith column. The extrinsic cur- block-diagonal matrix. ¯ vature, K (S, Q 1, Q 2, ..., Q n), can be calculated using Eq (B7). It is − g¯ = diag(C 1 , −G) (18) interesting that the extrinsic curvature has the same behavior as Q 1,Q 2,...,Q n the specific heat, C Q 1,Q 2,...,Q n . where G is a square matrix of order n defined by the following For a Kerr–Newman (KN) black hole with the following mass, relation, S(4 J 2 + S2 + 2Q 2 S + Q 4) M = (22) Hi, j M i G = ; X = (Q 1, Q 2, ..., Q n) (19) 2S T metric elements are defined as follows: Thus the metric determinant in the new coordinates can be writ- Hi, j M ∂ (T , −Q , − J) ten as: g = = (23) T T ∂ (S,,) −1 C S,Q ,Q ,...,Q C S, ,Q ,...,Q ...C S, , ,..., g¯ = 1 2 n 1 2 n 1 2 n (20) where T is the Hawking temperature, is the angular velocity, − n ( T ) C Q 1,Q 2,...,Q n and is the potential deference [20,21]. When somebody restricts himself to live on the constant J hypersurface which has the or- where above functions were defined in [20]. Now, a constant Q i =− hypersurface has the following unit normal vector, thogonal normal vector, n J 1/ TCS, , the extrinsic curvature diverges at the phase transition point and exhibit a similar sign −1 ¯ = − 2 nμ (0, 0, ..., (TCS,1,...,i−1,i+1,...,n ) , .., 0) (21) behavior around the transition points. In Fig. 1, the graph of the S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305 301

extrinsic curvature, K , and the heat capacity, C J,Q , for the Kerr– Newman black hole (KN) shows an exact correspondence between singularities and phase transitions. (Note that the ﬁrst divergence point is related to T = 0.) It is surprising that the same result ob- tains by considering a constant Q hypersurface with unit normal vector, nQ =−1/ TCS, . Moreover, it will be easy to show that the signs of such Ricci tensor elements as R SS, R SQ and R SJ are similar to that of the C J,Q around the transition points.

3. Extrinsic curvature of the phantom RN-AdS black hole

The action for the Einstein–Hilbert theory with phantom Maxwell field reads: √ μν 4 S = −g R + 2 + 2ηFμν F d x , (24) where, is the cosmological constant, and η =±1. RN-AdS black hole corresponds to η = 1, while phantom couplings of the Maxwell field (Phantom RN-AdS black hole) are obtained for η =−1. The metrics of these solutions, derived in [24], take the below form. Fig. 2. Graph of the scalar of curvature R (solid blue curve) and the heat capacity C (dashed red curve) as a function of entropy, S, in the RN-AdS case, for an 2 2 1 2 2 2 2 2 Q ds = f (r)dt − dr − r (dθ + sin θ dφ ), (25) electric charge Q = 0.25 and a cosmological constant =−1. (For interpretation f (r) of the references to color in this figure legend, the reader is referred to the web where f (r) is given by: version of this article.)

2M Q 2 specific heat (30) are depicted in Fig. 2 as a function of entropy f (r) = 1 − − r2 + η (26) r 3 r2 and for a fixed value of electric charge Q = 0.25. On the other hand, the extrinsic curvature opens an interesting and impressive The event horizon, r+, of this solution can be determined by calcu- avenue to the investigation of how phase transitions behave. In lating the roots for the equation f (r+) = 0. The mass of this black this case, we need to sit on a hypersurface with the normal vector hole is expressed as a function of the thermodynamic variables. QQ nQ =−1/ g¯ in which the extrinsic curvature associated with 1 π ηπ2 Q 2 M = (S/π)3/2( − + ) (27) this hypersurface is given by: 2 2 S 3 S √ √ 2 − | | where S = πr+ is the Bekenstein–Hawking entropy. According to ¯ (π 2 S) η π Q S K =− the first law of thermodynamics, one can calculate the Hawking −π S − S2 + 3 ηπ2 Q 2 −π S + S2 + ηπ2 Q 2 temperature, T , the electric potential, , and the specific heat capacity, C Q , as follows: (33) ∂ M −π S + S2 + ηπ2 Q 2 T = ( ) = (28) The first term of the denominator shows the phase transition 3/2 ∂ S Q −4(π S) points, while the second is only zero at T = 0. Interestingly, we see in Fig. 3 that the extrinsic curvature has the same sign as heat ca- ∂ M (S/π)3/2ηπ2 Q = ( ) = (29) pacity does, while in Fig. 2, the scalar curvature does not have the 2 ∂ Q S S same sign as the heat capacity around the phase transition points. ∂ S −2S(−π S + S2 + ηπ2 Q 2) In other words, extrinsic curvature reveals more information such C = T ( ) = (30) Q 2 2 2 as the stability/non-stability of heat capacities than the Ricci scalar ∂ T Q (−π S − S + 3ηπ Q ) does. Fig. 4 indicates that the special Ricci tensor elements R SS Using Eq. (5), it is easy to obtain the metric elements of the en- also exhibit a similar behavior around phase transition points such thalpy potential in the coordinate (S, ). Then by applying Eqs. (7) as heat capacity. Therefore, the extrinsic curvature K (S, Q ) and the and (8), the scalar curvature, R, takes the following form: R SS component of the Ricci tensor describe the phase transitions and the sign of the heat capacity, C . Our work is a new method ¯ F (S, Q ) Q R = (31) to identify stable regimes in the parameter space of black holes by − + 2 + 2 2 − − 2 + 2 2 2 ( Sπ S π ηQ )( Sπ S 3π ηQ ) studying the extrinsic curvature in their thermodynamic geometry. where, 4. Ruppeiner curvature of RN black hole as an intrinsic curvature 2 4 5 2 2 4 4 2 4 F =−η Q π + 2 η Q + 10 η Q Sπ + 2 S π on a constant J hypersurface + 2 − 2 − 3 + − 2 2 3 2 S 9 η Q 1 π 3 6 η Q S π (32) In article [1], the authors proposed a new measure of mi- croscopic interactions and its effects on Ruppeiner curvature by The first part of the denominator is zero only at T = 0 and the considering a complete phase space of extensive variables. They roots of the second part of the curvature denominator gives us the obtained a new non-zero Ruppeiner curvature for RN black holes phase transition points. Therefore, the curvature diverges exactly at →∞ = −3 → these points where heat capacity diverges with no other additional by setting l , l , where J 0 limits in the scalar cur- roots. For the RN-AdS black hole, the scalar curvature (31) and the vature for the Kerr–Newman-AdS (KN-AdS) black hole as follows: 302 S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305

Ruppeiner curvature of the RN black hole, and an extrinsic part, which measures the bending of the constant J hypersurface; that is:

(3) (2) 2 ab α β α β R − = R − (K − K K ) − 2(n n − n n ) KN AdS in ab ;β ;β ;α (35)

(2) where, Rin = R RN = 0. Here the extrinsic part is expected to be exactly the same as Eq. (34). Now, let us investigate the accuracy of our claim about the KN-AdS black hole in the limits l →∞ and J → 0. The mass relation of the KN-AdS black hole [25] as a function of thermodynamic variables can be written as: S π 4 J 2 + Q 2 Q 2 J 2 M =[ + + + (36) 2 4π 4S 2 l 2 S 2 S S 1 + Q + + ] 2 πl2 π 2π 2l2 The Hawking temperature T is also deﬁned by: Fig. 3. Graph of the scalar of curvature K (solid blue curve) and the heat capac- ∂ M S2π 2l4 − 4π 4l4 J 2 − π 4l4 Q 2 ity C (dashed red curve) as a function of entropy, S, in the RN-AdS case, for an T = = (37) Q S 3 2 4 electric charge Q = 0.25 and a cosmological constant =−1. (For interpretation ∂ Q , J 8π MS l of the references to color in this ﬁgure legend, the reader is referred to the web 2S2 Q 2π 2l2 + 4S3πl2 + 3S4 version of this article.) + 8π 3 MS2l4 It should be noted that, within the hypersurface framework, setting J to zero is tantamount to living on the constant J hypersurface ( J -zero hypersurface) which has the following normal vector, 1 ∂2 M JJ n J =−1/ g ; g JJ = . (38) T ∂ J 2 Therefore, by making use of Eqs. (B7) and (B10), we have:

2 αβ K = Kαβ K = 0 (39) On the other hand, the metric elements induced on the constant J hypersurface can be given by: 1 ∂2 M 3Q 2 − S gSS = lim = (40) l→∞; J→0 T ∂ S2 2S S − Q 2 1 ∂2 M −2Q gSQ = g QS = lim = (41) l→∞; J→0 T ∂ S∂ Q S − Q 2 1 ∂2 M 4S g QQ = lim = (42) l→∞; J→0 T ∂ Q 2 S − Q 2

Fig. 4. Graph of the scalar of curvature R SS (solid blue curve) and the heat capac- The above elements are the same as those of the Ruppeiner metric ity C Q (dashed red curve) as a function of entropy, S, in the RN-AdS case, for an (2) for the RN black hole [23]. Therefore, the intrinsic curvature Rin electric charge Q = 0.25 and a cosmological constant =−1. (For interpretation of the references to color in this figure legend, the reader is referred to the web of the constant J hypersurface equals the Ruppeiner curvature of version of this article.) the RN black hole; i.e., (2) = = S2 + Q 2 S + 2Q 4 Rin R RN 0 (43) lim R − = (34) KN AdS 2 l→∞; J→0 S + Q 2 Q 2 − S Finally, based on Eq. (38), the last statement of Eq. (35) is given by: This non-vanishing scalar curvature is the result of another dimen- S2 + Q 2 S + 4 sion specified by J which fluctuates even if we set it to zero. It β Q lim (nα nβ − nαn ) =− 2 2 (44) is surprising that this result is in contrast to a direct calculation ;β ;β ; 2 l→∞; J→0 α S + Q 2 Q 2 − S on the Ruppeiner metric of the RN which is zero [23]. One of our objectives in this work is to explain the difference between these We can, therefore, conclude that the non-vanishing Ruppeiner cur- two results by applying the basic concepts of the extrinsic/ intrin- vature in the limits J → 0and l →∞ is extracted from the cur- sic geometry for a particular hypersurface (see Appendix B). In this vature of the KN-AdS black hole when one lives on the constant J framework, the Ruppeiner curvature of the KN-AdS black hole can hypersurface. In addition, the Ruppeiner curvature of the RN black be broken down into a purely intrinsic part, which yields a zero hole can be interpreted as the intrinsic curvature produced by the S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305 303 induced metric on the two dimensional hypersurface ( J -zero hy- thermodynamic geometry is flat [27]. Now we consider thermody- persurface). namic geometry of a Pauli paramagnetic gas with identical spin 1/2 We may also check Eq. (B9) for the Kerr black hole at Q → 0 fermions in the presence of an external magnetic field [30]. From and l →∞ limits of the KN-AdS black hole to show that the in- the grand canonical distribution through the Fermi–Dirac statistics, trinsic part of Eq. (B9) is the Ruppeiner curvature of the Kerr black the thermodynamic potential can be obtained as follows: hole. Using the definition of the Ruppeiner metric for the complete − 3 −y− Jz −y+ Jz phase space of the parameters (KN-AdS black hole), and assuming φ(x, y, z) = Ix 2 [ f 5 (e ) + f 5 (e )], (53) 2 2 the limits Q → 0, l →∞, we obtain the following equation for the 3 Ricci scalar for a non-charged KN-AdS black hole. 2 where I = (2πm) and x = 1/T , y =−μ/T , z =−H/T are thermo- h3 2 + 2 dynamic coordinates. T , μ, and H are also temperature, chemical (3) 36 J S S lim R − = (45) potential, and external magnetic field, respectively. The fn(η) func- →∞; → KN AdS 4 − 4 l Q 0 16 J S tion is defined by: Also, the induced metric for the Q -zero hypersurface is calculated ∞ − by: 1 Xn 1dX fn(η) = (54) e X 2 − 2 2 − 4 + 4 (n) + 1 1 ∂ M 24 J S 48 J S 0 η gSS = lim = (46) l→∞;Q →0 T ∂ S2 2S 4 J 2 + S2 4 J 2 − S2 μ where η = exp( ) is called fugacity. The Ruppenier metric in K B T 1 ∂2 M 4 J 4 J 2 + 3 S2 thermodynamic geometry is given by: gSJ = g JS = lim = →∞; → 2 + 2 2 − 2 l Q 0 T ∂ S∂ J 4 J S 4 J S ∂2φ g = ; Y i = (x, y, z) (55) (47) ij i j ∂Y ∂Y 2 − 3 1 ∂ M 8S The Ricci scalar, (3) R was already obtained as a symmetric function g JJ = lim = (48) l→∞;Q →0 T ∂ J 2 4 J 2 + S2 4 J 2 − S2 of z [30]. It means that the scalar curvature doesn’t depend on the orientation of external magnetic field. In the classical limit in the One can obtain the following relation for the intrinsic curvature. ± lack of the external magnetic field (H → 0), when η → η and ± → (3) 12 J 2 + S2 S fn (η) η; the R is rewritten as follows: Rin = Rkerr = (49) 16 J 4 − S4 9 lim (3) R =− (56) → → Moreover, for the Q -zero hypersurface with the normal vector, H 0(z 0) 2ρ 2 1 2 = = 2 1 ∂ M where in the classical regime, ρ 3 η and λ h/(2πm0 K B T ) . QQ λ nQ =−1/ g ; g QQ = (50) Eq. (56) shows that in the classical limit, the curvature of a Pauli T ∂ Q 2 paramagnetic gas depends on the volume occupied by a single par- 2 αβ the extrinsic curvature is zero and K = Kαβ K = 0. Now, we can ticle. The scalar curvature is also similar to the curvature that was successfully examine the validity of the following equation; obtained for a two-component ideal gas by Ruppeiner [31]. In this letter we explore the physical properties of a Pauli para- = (2) − α β − α β magnetic gas by studying the intrinsic curvature of a hypersurface lim R KN−AdS R Kerr 2lim(n;βn n n;β ); l→∞;Q →0 l→∞;Q →0 α corresponding to a zero magnetic field (z → 0). The intrinsic cur- (51) vature of H-zero hypersurface can be calculated as: where, (2) 5 Rin =− (57) 2 2 ρ α β − α β =− SJ lim (n;βn n n;β ); 12 (52) Because the intrinsic curvature is negative, so the statistical inter- l→∞;Q →0 α 16 J 4 − S4 actions of a Pauli paramagnetic gas can be repulsive which indi- In summary, the Ruppeiner curvature of the Kerr black hole is sim- cates a more stable paramagnetic gas. It should be noted that in ilar to the intrinsic curvature produced by the induced metric on the classical limit the extrinsic curvature vanish (K = 0) and the the two dimensional hypersurface (i.e., constant Q hypersurface). Gauss–Codazzi relation is held in this case. On the other hand, Our study indicates that in thermodynamic geometry, properties of the Ruppenier curvature for a non-interacting classical paramag- the intrinsic and the extrinsic curvatures are important to obtain a netic gas through Maxwell–Boltzmann statistics with the thermo- − 3 − sinh( Jz) complete geometric representation of thermodynamics in physical = 2 y dynamic potential, φc 4π Ix e Jz ( J is magnetic momen- systems. The intrinsic curvature also help us to identify attrac- tum) [30], takes the following form: tive, repulsive and non-interacting statistical interaction between 3 the constituent parts of a thermodynamic system as we discuss in (3) λ Jz Rc = (58) the next Section. 8πηsinh ( Jz) = 5. Hypersurfaces and their intrinsic curvature in thermodynamic From the equation of state (PV NKB T ), one can see the cur- geometry of the Pauli paramagnetic gas vature is proportional to the volume occupied by single particle (3) ( Rc = 1/2ρ) [30]. It is surprising that in the absence of an (3) = λ3 Thermodynamic curvature may explain the statistical interac- external magnetic field, the curvature is not zero ( Rc 8πη), tion between particles in a thermodynamic system [27–29]. The while we would expect the curvature to be zero because of non- thermodynamic curvature is positive for attractive interaction be- interaction particles [27]. To obtain a correct result we have to tween particles, and negative for a repulsive interaction [29]. In calculate the induced metric on a zero magnetic field hypersur- case that particles have not any interaction with each other, the face in the thermodynamic geometry. It can easily be shown that 304 S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305

H-zero hypersurface is flat which indicates a non-interacting gas Appendix B. The concepts of extrinsic and intrinsic curvatures as expected. Our analysis indicates that the intrinsic geometry of for a hypersurface hypersurfaces in thermodynamic geometry has important physical information. In order to have correct information we must study In this section, we briefly review the concept of extrinsic curva- hypersurfaces in thermodynamic geometry. ture. For an n-dimensional manifold M, a special hypersurface can be defined as follows: 6. Conclusion (xα) = 0(B1) This work analyzed the thermodynamic geometry of a black where, xαs are the coordinates of the manifold M. The induced hole from the perspective of an extrinsic curvature. It was found metric on can be written as: that the extrinsic scalar curvature represents the critical behavior of a second order phase transition in a thermodynamic system. α β 2 α β ∂x a ∂x b a b ds = gαβdx dx = gαβ ( dy )( dy ) = habdy dy (B2) Some particular Ricci tensor elements were found to have the ∂ ya ∂ yb same sign behavior as heat capacities. Another part of the article explained the relationship between the intrinsic, the extrinsic, where, and the total curvatures of thermodynamic geometry of a system α β ∂x by sitting on a certain hypersurface. For the KN-AdS black hole on h = g Eα E ; Eα = (B3) ab αβ a b a a a constant J hypersurface, the curvature scalar was broken down ∂ y into two parts. One was a zero intrinsic curvature (the Ruppeiner hab defines the induced metric on the hypersurface. A unit normal curvature of the RN black hole) while the other was an extrin- α nα can be introduced if nαn = where ε = 1 when is timelike sic part whose divergence points were the singularities of a non- and ε =−1 when is spacelike [7]. We select that nα points in rotating KN-AdS black hole. We also used the intrinsic curvature α the direction of increasing : n ,α0. We can also easily show of the relevant hypersurface to investigate some thermodynamic that: properties such as stability and the statistical interaction. As a result, the critical behavior of a thermodynamic system ,α nα = (B4) on an explicit hypersurface can be explained consistently by using 1 gαβ 2 intrinsic and extrinsic curvatures of this hypersurface. ,α ,β The inverse of the induced metric is obtained as follows: Appendix A. Partial derivative and bracket notation ab α β = αβ − α β h Ea Eb g εn n (B5) When f , g, and h are explicit functions of (a, b), we can obtain the following relation for the partial derivative. One can also introduce the extrinsic curvature tensor on the hypersurface using the following relation: { } ∂ f f , h a,b = (A1) { } α β 1 α β ∂ g h g, h a,b Kab = nα;β E a E b = Ln gαβ E a E b (B6) 2 where, ; where, the symbol and Ln are the covariant and Lie derivatives of g along nα, respectively. Therefore, the extrinsic curvature is { } = ∂ f ∂h − ∂ f ∂h αβ f , h a,b (A2) defined by: ∂a b ∂b a ∂b a ∂a b √ Moreover, if one considers a = a(c, d) and b = b(c, d), Eq. (A1) can ab α 1 α K = h Kab = n; = √ ∂α gn (B7) be rewritten as: α g ∂ f { f , h} { f , h} = a,b = c,d where, g = det(gαβ ). Suppose that a two-dimensional manifold is { } { } (A3) ∂ g h g, h a,b g, h c,d embedded in a three-dimensional space. The induced metric hab and the extrinsic curvature K contain the necessary informa- And, the determinant of the Jacobian transformation can be writ- ab tion about the properties of the hypersurface . The full Riemann ten in the bracket notation as in the form below: ⎛ ⎞ curvature tensor of the 3-dimensional space and the curvature ∂ f ∂ f tensor of the 2-dimensional hypersurface are related by the Gauss– { } ∂(f , g) ∂h ∂k f , g a,b det = det ⎝ k h ⎠ = (A4) Codazzi equation: ∂(h,k) ∂ g ∂ g {h,k} a,b β γ ∂h k ∂k h (3) α σ = (2) + − RαβγσEa Eb Ec Ed Rabcd ε(Kac Kbd Kad Kbc) (B8) Generally, the partial derivative for functions with n + 1 variables This indicates that the three-dimensional Riemann tensor can be can be calculated as follows: expressed in terms of the intrinsic and extrinsic curvature tensors { } ∂ f f , h1, ..., hn q ,q ,...,q + of the hypersurface [32]. Writing the Gauss–Codazzi equation in = 1 2 n 1 (A5) ∂ g {g, h , ..., h } the contracted form, we obtain: h1,.....,hn 1 n q1,q2,...,qn+1 = = (3) = (2) + 2 − ab + α β − α β where, the f , g, and hn (n 1, 2, 3, ...) are functions of qi , i R Rin (K Kab K ) 2(n;βn n n;β );α (B9) 1, ..., n + 1 variables [20] and, This relation is the three-dimensional Ricci scalar evaluated on the (2) { f h h } = (A6) hypersurface . The R is intrinsic Ricci scalar of the hypersur- , 1, ..., n q1,q2,...,qn+1 face. The third term, on the right hand side of the Equation, can n+1 ∂ f ∂h1 ∂h2 ∂hn be expressed as: εijk...l ... . ∂qi ∂q j ∂qk ∂ql ijk....l=1 ab = α β Kab K n;βn;α (B10) S.A. Hosseini Mansoori et al. / Physics Letters B 759 (2016) 298–305 305

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