Ruppeiner Geometry of (2 + 1)-Dimensional Spinning Dilaton Black Hole

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1 Commun. Theor. Phys. 55 ( Vol. 55, No. 5, May 15, 2011 Ruppeiner Geometry of (2 + 1-Dimensional Spinning Dilaton Black Hole CHEN Xiu-Wu (í 1,2, WEI Shao-Wen (ï, 2, and LIU Yu-Xiao ( 2, 1 School of Electronics and Information Engineering, Gansu Lianhe University, Lanzhou , China 2 Institute of Theoretical Physics, Lanzhou University, Lanzhou , China (Received September 1, 2010; revised manuscript received November 12, 2010 Abstract In this paper, we study the geometrothermodynamics of (2 + 1-dimensional spinning dilaton black hole. We show that the Ruppeiner curvature vanishes, which implies that there exist no phase transitions and thermodynamic interactions. However when the thermodynamics fluctuation is included, the geometry structure is reconsidered. The non-vanishing Ruppeiner curvature is obtained, which means the phase space is non-flat. We also study the phase transitions and show that it can indeed take place at some points. PACS numbers: Bw, Dy, Lf Key words: black hole, geometrothermodynamics 1 Introduction Since the pioneer work presented by Hawking and Bekenstein, [1 2] the black hole is generally believed to be a thermodynamics system. It is considered to satisfy the four laws of the elementary thermodynamics regarding the surface gravity and the outer horizon area as its temperature and entropy, respectively. [3] Although, it is widely believed that a black hole is a thermodynamic system, the statistical origin of the black hole entropy is still unclear. The investigation of thermodynamic properties of a black hole is an interesting subject. It is Weinhold who first introduced the geometrical concept into a thermodynamics. He suggested that a Riemannian metric can be defined as the second derivatives of internal energy U respect to the entropy and other extensive quantities of a thermodynamic system. [4] However, the Weinhold geometry is generally considered to be physical meaningless. A few years later, Ruppeiner introduced another geometry into the thermodynamics. [5] Different from the Weinhold one, the thermodynamic potential of the Ruppeiner geometry is the entropy S rather than the internal energy U. These two geometries are proved to be conformally related to each other as ds 2 R = 1 T ds2 W, (1 where the conformal factor 1/T is the inverse of the temperature of the thermodynamic system. The Ruppeiner geometry was first used to study the ideal gas and the van der Waals gas. It was found that the curvature vanishes for the ideal gas whereas, for the van der Waals gas, the curvature is non-zero and diverges only at those points, where the phase transitions take place. [6] As a thermodynamics system, the black hole has been extensively investigated and the Weinhold geometry and Ruppeiner geometry were obtained for various black holes and black branes. [4 27] In particular, the Ruppeiner geometry is generally considered to carry the information of phase structure of a thermodynamic system, i.e. the Ruppeiner curvature is singular at the points, where the phase transitions take place. This property is held for many black hole except the Banados Teitelboim Zanelli (BTZ and Reissner Nordström (RN black holes. The Ruppeiner curvatures for them are all zero, which imply that the phase space is flat and no thermodynamics interactions exist. For this contradiction, many researches had been carried out to explain it. It is generally thought that the thermodynamic potential should be chosen different for the different purposes. Queved et al. presented their explanation on the vanishing Ruppeiner curvature few years ago. [28 29] They attributed the failure of the Ruppeiner curvature depicted the phase transition to the Ruppeiner metric is not Legendre invariant. A new Legendre invariant metric was introduced by them, which could reproduce the corrected behavior of the thermodynamic interactions and phase transitions for the BTZ and RN black holes. [30 31] Other black hole configurations and models are also investigated. [32 35] In this paper, we will investigate the Ruppeiner geometry of (2 + 1-dimensional spinning dilaton black hole. The results show that when thermodynamics fluctuation is included, the non-vanishing Ruppeiner curvature can be Supported by the National Natural Science Foundation of China under Grant No Corresponding author, cxw660715@sina.com weishaow06@lzu.cn liuyx@lzu.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 818 Communications in Theoretical Physics Vol. 55 reproduced and which implies that the thermodynamic interactions exist. The phase transitions can also take place at some points. The paper is organized as follows. In Sec. 2, we first review some thermodynamic quantities of (2 + 1-dimensional spinning dilaton black hole. In Sec. 3, both the Ruppeiner and Weinhold geometry structures are obtained. When the thermodynamics fluctuation is included, the geometry structure is considered in Sec. 4 and the non-vanishing geometry curvature is obtained. Finally, the paper ends a brief summary. 2 Thermodynamic Quantity of (2 + 1-Dimensional Spinning Dilaton Black Hole In this section, we will give a brief review of the spinning black hole. The action which leads to spinning black hole solutions is S = d 3 x g ( R 4( ϕ 2 + 2e bϕ Λ, (2 where Λ is the cosmological constant and ϕ is the dilaton field and R are scalar curvature. The spinning black hole solutions were presented in Refs. [36 38] ( ds 2 8Λr N = (3N 2N + ηr1 N/2 dt 2 + [ 8Λr N (3N 2N + (η T = 1 dr2 2Λγ2 (3N 2Nηr1 N/2] γr 1 N/2 dtdθ + (r N γ2 4η r1 N/2 dθ 2, (3 where the mass M, angular momentum J, and parameter η are given by M = N [ 2Λγ 2 ( 4 ] 2 (3N 2Nη N 3 η, (4 J = 3N 2 γ, (5 4 η = M N M 2 ( 4 N 2 + N 3 2Λγ 2 (3N 2N. (6 The parameters η and γ are integration constants. Another parameter N [2/3, 2] and cosmological constant Λ is restricted to be positive. One may find that the metric (3 is not asymptotically flat. So, the magnitude of the timelike killing vector field of an asymptotically non-flat spacetime diverges as it approaches spatial infinity and there exist some distinctions between the mass M and the energy E. Generally, the temperature is defined as T = M/ S for an asymptotically flat spacetime. While, for an asymptotically non-flat spacetime, the thermodynamic temperature T(r at a fix value of radial coordinate r is defined as T(r = E(r S, (7 where S is the entropy of the black hole. The temperature of (2+1-dimensional spinning dilaton black hole was obtained in Ref. [37] N ( 3N 2 [ M ( 2 4πr H 4 3N N N 1 4M + 2 N 2 + 8Λγ2 (4 3N ( 1 N ] (3N 2N 2. (8 N For the dilaton coupling parameters N = 1, the quasi-normal modes and the area spectrum have been studied in detail in Ref. [39]. In this paper, we will focus on the case of N = 4/3 and the metric for this spinning black hole is reduced to ( ds 2 = r 1/3 (3rΛ + η dt 2 + 4ηr 1/3 (12rηΛ 3γ 2 Λ+ 4η 2 dr2 r 1/3 γdtdθ + r 1/3( r γ2 dθ 2. (9 4η The radius of the event horizon satisfies 3rh 2 Λ = 3M 2 γ2 Λ 2M. (10 The parameter γ corresponds to the angular momentum J as γ = 2J. The entropy of the black hole is S = 2πr h. With Eq. (10, the entropy S of the black hole can be expressed as the function of the mass M and angular momentum J 2M S = π Λ 8J2 3M. (11 Note that when 3M 2 = 4JΛ, the entropy S will vanish and the radius of the horizon r h also vanishes. 3 Ruppeiner Geometry and Weinhold Geometry In this section, we will study the Ruppeiner geometry and Weinhold geometry for the 3-dimensional spinning dilaton black hole, respectively. The Ruppeiner geometry is defined as the second derivatives of entropy S respect to the entropy and other extensive quantities, which is defined as ds 2 R = g ij dx i dx j = 2 S(M, J x i x j dx i dx j, (i, j = 1, 2, (12 x 1 = M, x 2 = J. With the help of Eq. (11, the form of the metric is obtained g R MM = 9M2 S K 2 3S 4M 2, g R MJ = g R JM = 2JSΛ(4J2 Λ 9M 2 K 2, M g R = 12M2 SΛ JJ K 2, (13

3 No. 5 Communications in Theoretical Physics 819 where K = 3M 2 4J 2 Λ. Note that the metric diverges at K = 0 (3M 2 = 4J 2 Λ, which corresponds to a black hole vanishing entropy. So, the Ruppeiner metric for the black hole at that point is ill-defined. Thus the Ruppeiner geometry could not describe the spinning dilaton black hole at K = 0. With the metric (13, we can obtain the Ruppeiner curvature for the 3-dimensional spinning dilaton black hole. General, the non-vanishing Ruppeiner curvature R R implies the existence of the thermodynamic interactions and phase transitions take place at some cases. Here, we want to known whether the Ruppeiner curvature for the black hole is non-zero. The Christoffel symbols is calculated Γ λ µν = 1 2 gλτ (g ντ,µ + g µτ,ν g µν,τ. (14 The Riemannian curvature tensor, Ricci curvature and scalar curvature are given, respectively R µ σντ = Γµ σν,τ Γµ στ,ν + Γµ λ,τ Γλ σ,ν Γµ λ,ν Γλ σ,τ, R µν = R λ µλν, R = gµν R µν. (15 Using the above equations, we get the Ruppeiner curvature R R for the 3-dimensional spinning dilaton black hole: R R = 0. (16 The curvature R R vanishes for arbitrary values of mass M and angular momentum J. This case is the same like the BTZ black hole and the vanishing curvature implies no thermodynamic interactions and phase transitions. Here, we show a vanishing Ruppeiner curvature. Now, we would like to study the Weihold curvature for the spinning dilaton black hole. Taking the mass M as the thermodynamic potential, the Weinhold is defined as ds 2 W = g µνdy µ dy ν = 2 M(S, J y µ y ν dy µ dy ν, (µ, ν = 1, 2, (17 where y 1 = S and y 2 = J. Solving Eq. (11, we get a smarr-like formula: 3H + 3S 2 Λ M = 12π 2, (18 H = 64π 4 J 2 Λ + 3S 4 Λ 2. Substituting Eq. (18 into Eq. (17, we can obtain the form of the Weinhold metric, which will produce a non-vanishing curvature and it is given by α R W = HS 2 ( 3H + 3S 2 Λ, (19 3 α = 24π 2 Λ(48π 4 J 2 S 2 (H 3S 2 Λ + 9S 6 Λ(H 3S 2 Λ π 8 J 4. (20 It is clear that the Weinhold curvature R W is nonvanishing. So the non-vanishing Ruppeiner curvature R R does not mean the non-vanishing Weinhold curvature. The difference between the two geometries is the different choice of the thermodynamic potential. For the Ruppeiner geometry, the thermodynamic potential is the entropy S of the black hole. While for the Weinhold geometry, the thermodynamic potential is the mass M. Both the geometries have different descriptions on the black hole thermodynamics. 4 Thermodynamics Fluctuation and Geometrothermodynamics For a canonical ensemble, the entropy of a thermodynamical system is known to have a logarithmic correction. [40 42] So, we would like to discuss the thermodynamics fluctuation and its effect on the geometrothermodynamics. The corrected-entropy formula has the form of S c = S 1 2 ln(ct 2, (21 S c denotes the corrected entropy. C and T are related to the heat capacity and temperature of the black hole, respectively. The heat capacity and temperature of (2 + 1-dimensional spinning dilaton black hole are given in Sec. 2. We have approximate forms which leads to C Q S, T 2 S 2, (22 S c = S 3 lns. (23 2 This corrected entropy is the same like the ( dimensional BTZ black hole. [42] Then, we can express the corrected entropy S c as a function of mass M and angular momentum J 2M S c = π Λ 8J2 3M 3 ( 2M 2 ln π Λ 8J2. (24 3M So, it is natural to ask how the geometry curvature behaves when the thermodynamics fluctuation is included. With this question, we start the new calculation. With Eq. (24, the Ruppeiner geometry when the thermodynamics fluctuation is included is ( ds 2 2 S c (M, J ( R = M 2 dm 2 2 S c (M, J + 2 dm dj M J ( 2 S c (M, J + J 2 dj 2. (25 The corrected Ruppeiner metric reads MM = 8J2 K(2 6πL 9Λ + ( 6πL 9(4J 2 Λ + 3M K 2 M 2, MJ = 64 6πJ 5 Λ πJ 3 M 2 Λ 104JM 3 ( 6πM 2LΛ 3K 2 LM 2,

4 820 Communications in Theoretical Physics Vol. 55 JJ = 2Λ(12J2 Λ + (9 2 6πLM 2 K 2, (26 L = K/MΛ. It is clear that the corrected Ruppeiner metric is ill-defined when K = 0. This result is the same for the uncorrected Ruppeiner metric (13. After some calculations, we obtain the corrected Ruppeiner curvature R R β = ((9 6 4πL(3π 2 M 2 4π 2 J 2 Λ 27LMΛ, (27 3 β = 54π(MΛ 2 L 4 ((13 6 4πLπ 2 K + 27( 6 3πLMΛ. (28 Note that the corrected Ruppeiner curvature R R does not vanish, which behaves very difference from the uncorrected Ruppeiner curvature R R. So the thermodynamics fluctuation indeed has influence on the thermodynamics geometry. This corrected Ruppeiner geometry implies that this spacetime has thermodynamic interactions and phase transitions. From Eq. (27, we can see that the corrected Ruppeiner curvature R R has a divergence point at L = 9 6K 16π 3 J 2 Λ + 12π 3 M MΛ, (29 which is thought to be a point for the phase transition takes place. Also, the corrected Ruppeiner curvature R R vanishes at or L 1 = 0, (30 L 2 = 52 6π 2 J 2 Λ π 2 M MΛ π( 16π 2 J 2 Λ + 12π 2 M MΛ. (31 On the other hand, the heat capacity for fixed angular momentum J is C J = 3L(Kǫ 1/2 + M(64J 4 Λ 2 /3M (2π 4 1J 2 Λ + K 3/2 4 2MΛ(144π 4 J 2 KΛ + K 3 /M 2 + Mǫ 3/2, (32 ǫ = 16J4 Λ 2 M (2π 4 1J 2 Λ + K. (33 Note that the heat capacity C J vanishes at L 1 = 0. So, the vanished heat capacity C J corresponds to the vanished Ruppeiner curvature. However, this case just describes the case for an extremal black hole. At the other point L 2, the curvature R R vanishes. While the heat capacity C J is non-zero for fixed M, J, and Λ. We also need to note that the vanishing curvature R R at the point L 2 is a local property and which does not imply the spacetime is flat. It is worth noting that the singular points of the C J are not consistent the divergence points of the heat capacity, which is mainly because the Ruppeiner metric is not Legendre invariant as suggested by Quevedo. [28] However, the Ruppeiner curvature is still an effective method as a measure of the thermodynamic interaction. From Sec. 3, we can see that the uncorrected Ruppeiner curvature R R always vanishes. However, when the thermodynamics fluctuation is included, the thermodynamics geometry gives a non-vanishing thermodynamics curvature, which implies the existence of the thermodynamic interactions. It is also found that the phase transition may take place at some points. So, the structure of the phase space becomes more richer when the thermodynamics fluctuation is considered. 5 Summary We have studied the Ruppeiner and Weinhold geometry of (2 + 1-dimensional spinning dilaton black hole. Like the rotating BTZ black hole, the geometry structure of (2 + 1-dimensional spinning dilaton black hole is found to be flat, which means there exist no thermodynamic interactions and implies the phase transitions will not take place at some special points. We also calculate the corrected entropy of the black hole from the thermodynamics fluctuation. The corrected term turns to be a logarithmic term and its coefficient is 3/2. With this corrected entropy we restudy the geometry structure for the black hole. We show that when the thermodynamics fluctuation is considered, richer structures are appeared. The corrected Ruppeiner curvature is non-zero, which implies the existence of the thermodynamic interactions. It is also found that the phase transition may take place at L = 9 6K/( 16π 3 J 2 Λ + 12π 3 M MΛ. References [1] J.D. Bekenstein, Phys. Rev. D 7 ( [2] S.W. Hawking, Commun. Math. Phys. 43 ( [3] J.M. Bardeen, B. Carter, and S.W. Hawking, Commun. Math. Phys. 31 ( [4] F. Weinhold, J. Chem. Phys. 63 ( [5] G. Ruppeiner, Phys. Rev. A 20 (

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