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Extrinsic and Intrinsic Curvatures in Thermodynamic Geometry Physics Letters B 759 (2016) 298–305 Contents lists available at ScienceDirect 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 geom- etry 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 .
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