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Extrinsic and Intrinsic Curvatures in Thermodynamic Geometry

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Extrinsic and Intrinsic Curvatures in Thermodynamic Geometry Extrinsic and intrinsic curvatures in thermodynamic geometry Seyed Ali Hosseini Mansoori1;2,∗ Behrouz Mirza2,y and Elham Sharifian2z 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 ther- modynamic 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-Nordstr¨om(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-Nordstr¨omanti 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 hyper- surface 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 y [email protected] magnetic field. Section VI contains a discussion of our z 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 fS; Qg 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) fT;QgS;Q 0 T It is obvious that the phase transitions of CQ are the gij = Jik gkl Jlj (8) zeros of fT;QgS;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) fT; Φg 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 fS; Φg @ (S; Φ) @ (T; −Q) @S S;Q g¯0 = det = CΦ = T = (4) @ (S; Q) @ (S; Q) @T Φ fT; ΦgS;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 fT;QgS;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; jTCSj (14) 3 It is easy to show that the above vector is a normalized µ vector (¯nµn¯ = −1).
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