Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 865354, 5 pages http://dx.doi.org/10.1155/2013/865354

Research Article Legendre Invariance and Geometrothermodynamics Description of the 3D Charged-Dilaton Black Hole

Yiwen Han1 and XiaoXiong Zeng2

1 College of Computer Science, Chongqing Technology and Business University, Chongqing 400067, China 2 School of Science, Chongqing Jiaotong University, Chongqing 400074, China

Correspondence should be addressed to XiaoXiong Zeng; [email protected]

Received 13 February 2013; Revised 9 June 2013; Accepted 15 June 2013

Academic Editor: Hernando Quevedo

Copyright © 2013 Y. Han and X. Zeng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We first review Weinhold information geometry and Ruppeiner information geometry of 3D charged-dilaton black hole. Then, we use the Legendre invariant to introduce a 2-dimensional thermodynamic metric in the space of equilibrium states, which becomes singular at those points. According to the analysis of the heat capacities, these points are the places where phase transitions occur. This result is valid for the black hole, therefore, provides a geometrothermodynamics description of black hole phase transitions in terms of curvature singularities.

1. Introduction geometry can overcome the covariant and self-consistent problem of general thermodynamics. Based on the Ruppeiner Since Ferrara et al. [1] investigated the critical points of mod- and Weinhold metrics, consideration of different black hole uli space by using Weinhold metric and Ruppeiner metric; the families under various assumptions has led to numerous black hole thermodynamic in geometry framework becomes puzzling results for both metrics [4–13]. So it is then natural a hot spot of theoretical physics. to try to describe the phase transitions of black holes in Itiswellknownthatanequilibriumthermodynamic termsofcurvaturesingularitiesinthespaceofequilibrium system poses interesting geometric features. An interesting states. Unfortunately, the obtained results, at least some, are inner product on the equilibrium thermodynamic state space contradictory. For instance, for Reissner-Nordstrom¨ black in the energy representation was provided by Weinhold as hole, the Ruppeiner metric is flat [14], whereas the Wein- the Hessian matrix of the internal energy 𝑈 with respect to hold metric presents a curvature singularity. Similarly, the 𝑎 𝑊 the extensive thermodynamic variables 𝑁 ,namely,[2] 𝑔𝑖𝑗 = 3D charged-dilaton black hole also showed similar result 𝑎 [15]. Nevertheless, a simple change of the thermodynamic 𝜕𝑖𝜕𝑗𝑀(𝑈, 𝑁 ). However, there was no physical interpretation associated with this metric structure. As a modification, potential [16] affects Ruppeiner’s geometry in such a way Ruppeiner introduced Riemannian metric into thermody- that the resulting curvature singularity now corresponds to namic system once more and defended it as the second a phase transition. A dimensional reduction of Ruppeiner’s derivative of entropy 𝑆 (here, entropy is a function of curvature seems to affect its properties too17 [ ]. However, it is 𝑎 𝑅 well known that ordinary thermodynamics does not depend internal energy 𝑈 and its extensive variables 𝑁 )[3] 𝑔𝑖𝑗 = 𝑎 on the thermodynamic potential. −𝜕𝑖𝜕𝑗𝑆(𝑈, 𝑁 ). An interesting phenomenon is that these two 𝑊 𝑅 Recently, Quevedo [18]proposedaformalismofgeom- metrics are conformally related, that is, 𝑔𝑖𝑗 =𝑇𝑔𝑖𝑗 ,andthe etrothermodynamics (GTDs) as a geometric approach that conformal factor is the temperature, 𝑇=𝜕𝑀/𝜕𝑆.Ithas incorporates Legendre invariance in a natural way and allows beenappliedtoallkindsofthermodynamicmodels,for us to derive Legendre invariant metrics in the space of example, the ideal gas, the van der Waals gas, and the two- equilibrium states. Since Weinhold and Ruppeiner metrics dimensional Fermi gas et al. Studies showed that Ruppeiner are not Legendre invariant, one of the first results in the 2 Advances in High Energy Physics context of GTD was the derivation of simple Legendre invari- The temperature is ant generalizations of these metrics and their application to 𝜕𝑀 2Λ black hole thermodynamics [19–26]. These results end the 𝑇=( ) = (𝑟 −𝑟 ). 𝜕𝑆 𝑟2 + − (6) controversy regarding the application of geometric structures 𝑄 + in black hole thermodynamics. The phase transition struc- Byusingthearealaw,theentropyoftheblackholeisgivenby ture contained in the heat capacity of black holes becomes completely integrated in the scalar curvature of the Legendre 𝑘 2 2 𝑆= 𝐵 𝐴=𝑘 𝜋𝑟 =𝑟 , (7) invariant metric so that a curvature singularity corresponds 4 𝐵 + + to a phase transition. 𝑘 𝑘 =1/𝜋 In this paper, we first review Weinhold information ge- with 𝐵 Boltzmann’s constant, and 𝐵 . In its natural ometry and Ruppeiner information geometry of 3D charged- coordinates, the Weinhold metric can be obtained as follows: dilaton black hole. Then, based on our previous work, 1 3𝑄2 𝑑𝑠2 = ( −Λ)𝑑𝑆2 −8𝑄𝑑𝑆𝑑𝑄+𝑟2 𝑑𝑄2, the thermodynamics scalar curvature of the black hole is 𝑊 𝑟3 𝑟2 + (8) described. We explored the geometrothermodynamics of 3D + + charged-dilaton black hole. where the index 𝑊 denotes the Weinhold information The organization of this paper is outlined as follows. In geometry. Here, we have made the choice that the mass 𝑀 Section 2, we review Weinhold information geometry and corresponds to the thermodynamic potential; entropy 𝑆 and Ruppeiner information geometry of the 3D charged-dilaton charge 𝑄 correspond to the extensive variables, from which black hole. Then, in Section 3,geometrothermodynamicsof it can be shown that the Weinhold scalar curvature in the thechargedblackholeisedescribed.Finally,somediscus- entropy representation becomes sions and conclusions are given in Section 4.Throughoutthe 𝑟 paper, the units (𝐺=𝑐=ℎ=1)areused. R =− + . 𝑊 2 (9) 4Λ(𝑟+ −𝑟−) 2. Information Geometry Description of We see that the curvature R𝑊 naively diverges at the extreme 3D Charged-Dilaton Black Hole limit of the black hole, where 𝑟+ =𝑟−,whichisofless interest physically since at the extreme limit, the Hawking Let us first review Weinhold information geometry and temperature vanishes, and the thermodynamics description Ruppeiner information geometry of the 3D charged-dilaton breaks down as mentioned above. We interpret this result black hole. as an indication of the limit of applicability of geometric The starting action with the dilaton field 𝜙 is given by [27] thermodynamics as a geometric model for equilibrium ther- 3 4𝜙 2 −4𝜙 𝜇] modynamics. 𝑆=∫ 𝑑 𝑥√−𝑔 [𝑅 +2𝑒 Λ+4(∇𝜙) −𝑒 𝐹𝜇]𝐹 ]. (1) By using the coordinate transformation 𝑢=𝑄/𝑟+ [14, 15], we obtain the diagonalized Ruppeiner metric for the 3D Λ>0 The cosmology constant for anti-de Sitter space-time. charged-dilaton black hole as follows: Thisactionisconformablyrelatedtothelow-energystring 1 1 4𝑆 action black hole that is given by 𝑑𝑠2 = 𝑑𝑠2 =− 𝑑𝑆2 + 𝑑𝑢2. 𝑅 𝑇 𝑊 2𝑆 Λ−𝑢 (10) 4𝑟2 𝑑𝑠2 =−𝑓(𝑟) 𝑑𝑡2 + 𝑑𝑟2 +𝑟2𝑑2𝜑, 𝛾4𝑓 (𝑟) Letusdoanewtransformationasfollows: (2) 𝜎 1 𝑟 𝑄 𝜏=√2𝑆, √Λ sin ( )=𝑢. (11) 𝜙= [ ]; 𝐹 = , √2 4 ln 𝛾2 𝑟𝑡 𝑟2

2 2 Then, the Ruppeiner metric can be written in the above where the metric function 𝑓(𝑟) = −2𝑀𝑟 +8Λ𝑟 +8𝑄 and Rindler coordinates as 1/2 an integration constant 𝛾 with dimension [𝐿] is necessary 𝑑𝑠2 =−𝑑𝜏+𝜏2𝑑𝜎2. to have correct dimensions. 𝑅 (12) In our previous work, the mass and electric charge of the Obviously, this is a flat metric; its curvature is zero. The 3D charged-dilaton black hole have been expressed in terms vanished thermodynamic curvature implies that no phase of the inner and outer horizons as [15] transition points exist and no thermodynamic interactions 2 𝑀=4Λ(𝑟+ +𝑟−), 𝑄 =Λ𝑟+𝑟−, (3) appear. This result implies that the Ruppeiner curvature cannot describe the phase transitions of the black hole either. and, the electric potential is given by 𝜕𝑀 8𝑄 3. Geometrothermodynamics Description of Φ=( ) = . (4) 𝜕𝑄 𝑆 𝑟+ 3D Charged-Dilaton Black Hole According to the energy conservation law, In this section, we turn to use the recent geometric formula- tion of extended thermodynamic behavior of the 3D charged- 𝑑𝑀 = 𝑇𝑑𝑆 +Φ𝑑𝑄. (5) dilaton black hole. Advances in High Energy Physics 3

TheformulationofGTDisbasedontheuseofcontact This metric determines all the geometric properties of the geometry as a framework for thermodynamics [18]. Consider equilibrium space 𝜀. We see that in order to obtain the the (2𝑛 + 1)-dimensional thermodynamic phase space I explicit form of the metric, it is only necessary to specify the coordinated by the thermodynamic potential Φ, extensive thermodynamic potential 𝑀 as a function of 𝑆 and 𝑄.In 𝑎 𝑎 variables 𝐸 ,andintensivevariables𝐼 (𝑎 = 1,...,𝑛). ordinary thermodynamics, this function is usually referred 𝐴 Consider on I a nondegenerate metric 𝐺=𝐺(𝑍),with to as the fundamental equation from which all the equations 𝐴 𝑎 𝑎 𝑎 𝑏 𝑍 ={Φ,𝐸,𝐼 },andtheGibbs1-formΘ=𝑑Φ−𝛿𝑎𝑏𝐼 𝑑𝐸 , of state can be derived. From (3), the fundamental equation 𝑀=𝑀(𝑆,𝑄) with 𝛿𝑎𝑏 = diag(1,1,...,1).Theset(I,Θ,𝐺) defines a is given by 𝑛 contact Riemannian manifold if the condition Θ ∧ (𝑑Θ) =0̸ 𝑄2 is satisfied. Moreover, the metric 𝐺 is Legendre invariant 𝑀 (𝑆, 𝑄) =4Λ√𝑆(1+ ). 𝐴 (17) if its functional dependence on 𝑍 does not change under Λ𝑆 a Legendre transformation. The Gibbs 1g-form Θ is also invariant with respect to Legendre transformations. Legendre The first-order and the second-order partial differentials can invariance guarantees that the geometric properties of 𝐺 be expressed, respectively, as do not depend on the thermodynamic potential used in its 𝜕𝑀 2 𝑄2 𝜕𝑀 8𝑄 construction. = (Λ − ), = , 𝜕𝑆 𝑟 𝑟2 𝜕𝑄 𝑟 The thermodynamic phase space I which in the case + + + (18) of the 3D charged-dilaton black hole can be defined as a 4- 2 2 2 𝐴 𝜕 𝑀 1 3𝑄 𝜕 𝑀 8 dimensional space with coordinates 𝑍 ={𝑀,𝑆,𝑇,𝑄}, 𝐴= =− (Λ − ), = . 𝜕𝑆2 𝑟3 𝑟2 𝜕𝑄2 𝑟 0,...,4.Equation(3) represents the fundamental relationship + + + 𝑀 = (𝑆, 𝑄) from which all the thermodynamic information Substituting (18)into(16), the lines of GTD for the 3D can be obtained; therefore, we would like to consider a 5- charged-dilaton black hole are written as dimensional phase space I with coordinates (𝑀,𝑆,𝑇,𝑄,Φ), a contact one-form 2 16 𝑑𝑆2 = (Λ𝑟4 −9𝑄4)𝑑𝑆2 + (Λ𝑟2 +3𝑄2)𝑑𝑄2, 𝐺 6 + 2 + (19) Θ = 𝑑𝑀 − 𝑇𝑑𝑆 −Φ𝑑𝑄, (13) 𝑟+ 𝑟+ and an invariant metric where the index 𝐺 denotes the geometrothermodynamics. 𝐺=(𝑑𝑀 − 𝑇𝑑𝑆 −Φ𝑑𝑄)2 + (𝑇𝑆 + Φ𝑄)(−𝑑𝑇 𝑑𝑆 +) 𝑑Φ𝑑𝑄 . Thus,thecurvaturescalarcanbeobtainedby (14) 9𝑟+𝑟− (𝑟+ −𝑟−) (I,Θ,𝐺) R𝐺 = . (20) The triplet defines a contact Riemannian mani- 2Λ4(3𝑟 −𝑟 )2(𝑟 +3𝑟 )3 fold that plays an auxiliary role in GTD. It is used to properly − + + − handle the invariance with respect to Legendre transfor- We see in our setup that the scalar curvature R𝐺 vanishes mations. In fact, for the charged black hole, a Legendre only at the extremal limit to where 𝑟+ =𝑟−.Inageneralcase, transformation involves in general all the thermodynamic the scalar curvature R𝐺 does not vanish and it goes positive variables 𝑀, 𝑆, 𝑄,𝑇,andΦ so that they must be independent infinity when 𝑟+ =3𝑟−,whichstandsforakindofphase from each other as they are in the phase space. We introduce transition or long rang correlation of the system according also the geometric structure of the space of equilibrium states to the Ruppeiner theory [28]. It is interesting to note that the 𝜀 in the following manner: 𝜀 is a 2-dimensional submanifold I 𝜑:𝜀󳨃→ I divergence point of the scalar curvature is just the transition of that is defined by the smooth embedding map , point of Davies [29]. In the fact, it is easy to check this by satisfying the condition that the “projection” of the contact ∗ ∗ calculating the heat capacity with a fixed charge as follows: form Θ on 𝜀 vanishes, namely, 𝜑 (Θ) = 0,where𝜑 is the pullback of 𝜑,andthat𝐺 induces a Legendre invariant metric 𝜕𝑆 2𝑟2 (𝑟 −𝑟 ) 𝑔 𝜀 𝜀 𝐶 =𝑇( ) = + + − , on by means of .Inprinciple,any2-dimensionalsubset 𝑄 𝜕𝑇 3𝑟 −𝑟 (21) of the set of coordinates of I can be used to label 𝜀.Forthe 𝑄 − + sake of simplicity, we will use the set of extensive variables 𝑟 =3𝑟 𝑀2 = 266𝑄2/3 𝑆 and 𝑄 which in ordinary thermodynamics corresponds to which is singular at + − corresponding to the energy representation. Then, the embedding map for this and indicates that the black hole has a second-order phase specific choice is transition. Moreover, we see that all thermodynamic variables are well behaved, except perhaps in the extremal limit 𝑟+ =𝑟−, 𝜕𝑀 𝜕𝑀 𝜑:{𝑆, 𝑄} 󳨃󳨀→ { 𝑀 (𝑆, 𝑄) ,𝑆,𝑄, , }. (15) at this point, changes sign and the scalar curvature diverge. 𝜕𝑆 𝜕𝑄 Therefore, there will be a phase transition R𝐺. ∗ The condition 𝜑 (Θ) = 0 is equivalent to (5)(the first law of thermodynamics) and4 ( ), (6) (the conditions of 4. Discussion and Conclusions thermodynamic equilibrium); the induced metric is obtained as follows: In this work, we investigated the Weinhold metric and the 𝜕𝑀 𝜕𝑀 𝜕2𝑀 𝜕2𝑀 Ruppeiner metric as well as the geometrothermodynamics 𝑔=(𝑆 +𝑄 )(− 𝑑𝑆2 + 𝑑𝑄2). of a 3D Charged-Dilaton Black Hole. In all these cases, 𝜕𝑆 𝜕𝑄 𝜕𝑆2 𝜕𝑄2 (16) our results showed that the thermodynamic curvature is in 4 Advances in High Energy Physics general different, indicating the presence of thermodynamic [9] A. J. 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