Sound Modes in Holographic Hydrodynamics for Charged AdS Black Hole
Yoshinori Matsuo, Sang-Jin Sin, Shingo Takeuchi, Takuya Tsukioka, Chul-Moon Yoo
aa r X i v : . [ h e p - t h ] J a n APCTP-Pre2009-001arXiv:0901.0610[hep-th]
January 2009
Sound Modes in Holographic Hydrodynamicsfor Charged AdS Black Hole
Yoshinori Matsuo ∗ , Sang-Jin Sin †∗ , Shingo Takeuchi ∗ , Takuya Tsukioka ∗ and Chul-Moon Yoo ∗ ∗ Asia Pacific Center for Theoretical Physics,Pohang, Gyeongbuk 790-784, Korea ymatsuo, shingo, tsukioka, c m [email protected] † Department of Physics, Hanyang University, Seoul 133-791, Korea [email protected]
Abstract
In the previous paper we studied the transport coefficients of Quark-Gluon-Plasma in finite temperature and finite density in vector and tensor modes. In thispaper, we extend it to the scalar modes. We work out the decoupling problem andhydrodynamic analysis for the sound mode in charged AdS black hole and calcu-late the sound velocity, the charge susceptibility and the electrical conductivity. Wefind that Einstein relation among the conductivity, the diffusion constant and thesusceptibility holds exactly.
Introduction
The discovery of low viscosity in the theory with gravity dual [1] and its possible relationto the RHIC (Relativistic Heavy Ion Collider) experiment induced a great deal of efforts toestablish the relevant calculational scheme that may be provided by AdS/CFT correspon-dence [2–4]. An attempt has been made to map the entire process of RHIC experiment interms of the gravity dual [5]. The way to include a chemical potential in the theory wasfigured out in the context of probe brane embedding [6–13]. Phases of these theories werediscussed and new phases were reported where instability due to the strong attraction is afeature [8–10].In spite of the difference between QCD and N = 4 SYM, it is expected that someof the properties are shared by the two theories based on the universality of low energyphysics. In this respect, the hydrodynamic limit is particularly interesting since such limitcan be shared by many theories. The calculation scheme for transport coefficients is touse Kubo formula, which gives a relation to the low energy limit of Wightman Greenfunctions. In AdS/CFT correspondence, one calculate the retarded Green function whichis related to the Wightman function by fluctuation-dissipation theorem. Such scheme hasbeen developed in a series of papers [14–18].For the hydrodynamic analysis, one may need to have master equations for the decou-pled modes in vector and scalar at hands. Although the analysis for the decoupling problemwere analyzed in [19], it was based on SO (3) decomposition while more useful work forhydrodynamics should be based on SO (2) decomposition, where longitudinal direction ofthe spatial direction is distinguished. For this purpose, Kovtun and Starinets worked outthe decoupling problem based on SO (2) for the AdS black hole case [18] before doing thehydrodynamic analysis. For the charged cases, there are extra difficulties: vector modesof gravity and those of the gauge fields couple. Furthermore there are extra couplings inscalar modes which are not present in the chargeless cases.In the previous paper [20], some of us extended this work to charged case using theReissner-Nordstr¨om-Anti-deSitter (RN-AdS) black hole, which corresponds to the diagonal(1 , ,
1) R-charged STU black hole ∗ . However, analysis for the scalar mode of charged case ∗ In fact much works had been done for charged case by various groups [21–25]. In [21, 22], ther-modynamics for STU black hole [26, 27] and the hydrodynamic calculations for the (1 , ,
0) charge wereperformed. In [23, 24], charged AdS and AdS black hole backgrounds were considered, respectively, andit was shown numerically that the ratio ( η/s ) was 1 / (4 π ) with very good accuracy. Later, it was also U (1) chargeand the electrical conductivity in this section. Conclusions and discussions are given in thefinal section. Three appendices are provided. In Appendix A, we summarize the resultsin the vector and the tensor type perturbations in our previous work [20]. The details ofcalculations to solve equations of motion are given in Appendix B and C. Before introducing RN-AdS black hole, we briefly summarize to obtain Minkowskian cor-relators in AdS/CFT correspondence. We follow the prescription proposed in [14]. Wework on the five-dimensional background,d s = g µν d x µ d x ν + g uu (d u ) , (2.1)where x µ and u are the four-dimensional and the radial coordinates, respectively. We referthe boundary as u = 0 and the horizon as u = 1. A solution of the equation of motionmay be given, φ ( u, x ) = Z d k (2 π ) e ikx f k ( u ) φ ( k ) , (2.2) proven that the ratio might be universal in more general setup [25]. f k ( u ) is normalized such that f k (0) = 1 at the boundary. An on-shell action mightbe reduced to surface terms by using the equation of motion, S [ φ ] = Z d k (2 π ) φ ( − k ) G ( k, u ) φ ( k ) (cid:12)(cid:12)(cid:12) u =1 u =0 . (2.3)Here, the function G ( k, u ) can be written in terms of f ± k ( u ) and ∂ u f ± k ( u ). Accommo-dating Gubser-Klebanov-Polyakov/Witten relation [3, 4] to Minkowski spacetime, Son andStarinets proposed the formula to get retarded Green functions, G R ( k ) = 2 G ( k, u ) (cid:12)(cid:12)(cid:12) u =0 , (2.4)where the incoming boundary condition at the horizon is imposed. In general, there areseveral fields in the model. We write the Green function as G ij ( k ), where indices i and j distinguish corresponding fields.In this paper, we work in RN-AdS background and consider its perturbations so thatessential ingredients are perturbed metric field and U (1) gauge field. Here we define theprecise form of the retarded Green functions which we discuss later: G µν ρσ ( k ) = − i Z d x e − ikx θ ( t ) h [ T µν ( x ) , T ρσ (0)] i ,G µν ρ ( k ) = − i Z d x e − ikx θ ( t ) h [ T µν ( x ) , J ρ (0)] i ,G µ ν ( k ) = − i Z d x e − ikx θ ( t ) h [ J µ ( x ) , J ν (0)] i , (2.5)where the operators T µν ( x ) and J µ ( x ) are energy-momentum tensor and U (1) current whichcouple to the metric and the gauge field, respectively. The charge in RN-AdS black hole is usually regarded as R -charge of SUSY [28]. We hereconsider an another interpretation in the following way: One can introduce quarks andmesons by considering the bulk-filling branes in AdS space. The overall U (1) of the flavorbranes is identified as the baryon charge. The U (1) charge in this model [29] minimallycouples to the bulk gravity since the bulk and the world volume of brane are identified.Then, the baryon charge and the R -charge have the same description in terms of the U (1)gauge field living in the AdS space. A charged black hole (RN-AdS black hole) is then3nduced by its back reaction. Therefore the U (1) charge in RN-AdS can be identical to thebaryon charge. As a result, we can interpret our result as a calculation of the transportcoefficients in the presence of the baryon density.The effective action of this gauge field is given the quadratic piece of Dirac-Born-Infeldaction ∗ S DBI = − e Z d x √− g Tr ( F mn F mn ) , (2.6)where the gauge coupling constant e is given by [29] le = N c N f π , (2.7)with l the radius of the AdS space. We pick up an overall U (1) part of this gauge field inorder to consider the baryon current at the boundary. Together with the gravitation part,we arrive at the following action which is our starting point: S = 12 κ Z d x √− g (cid:16) R − (cid:17) − e Z d x √− g F mn F mn , (2.8)where we denote the gravitation constant and the cosmological constant as κ = 8 πG andΛ, respectively. The U (1) gauge field strength is given by F mn ( x ) = ∂ m A n ( x ) − ∂ n A m ( x ).The gravitation constant is related to the gauge theory quantities by l κ = N c π . (2.9)Suppose we have baryon charge Q . This should be identified to the source of U (1) chargeon the brane hence on the bulk. Then we can relate it to the parameter in RN black holesolution by considering the full solution to the equation of motion, R mn − g mn R + g mn Λ = κ T mn , (2.10)where energy-momentum tensor T mn ( x ) is given by T mn = 1 e (cid:16) F mk F nl g kl − g mn F kl F kl (cid:17) . (2.11)An equation of motion for the gauge field A m ( x ) gives Maxwell equation, ∇ m F mn = 1 √− g ∂ m (cid:16) √− gg mk g nl ( ∂ k A l − ∂ l A k ) (cid:17) = 0 . (2.12) ∗ The indices m and n run through five-dimensional spacetime while µ and ν would be reserved forfour-dimensional Minkowski spacetime. Their spatial coordinates are labeled by i and j . s = r l (cid:16) − f ( r )(d t ) + X i =1 (d x i ) (cid:17) + l r f ( r ) (d r ) , (2.13a) A t = − Qr + µ, (2.13b)with f ( r ) = 1 − ml r + q l r , Λ = − l , if and only if q is related to the Q by e = 2 Q q κ . (2.14)It should be noted that a ratio of the gauge coupling constant e to the gravitation constant κ is e κ = N c N f l − . (2.15)Since the gauge potential A t ( x ) must vanish at the horizon, the charge Q and the chemicalpotential µ are related † . The parameters m and q are the mass and charge of AdS space,respectively. This is nothing but Reissner-Nordstr¨om-Anti-deSitter (RN-AdS) backgroundin which we are interested throughout this paper.The horizons of RN-AdS black hole are located at the zero for f ( r ) ‡ , f ( r ) = 1 − ml r + q l r = 1 r (cid:16) r − r (cid:17)(cid:16) r − r − (cid:17)(cid:16) r − r (cid:17) , (2.16)where their explicit forms of the horizon radiuses are given by r = (cid:18) m q (cid:18) (cid:16) θ π (cid:17)(cid:19)(cid:19) − , (2.17a) r − = (cid:18) m q (cid:18) (cid:16) θ (cid:17)(cid:19)(cid:19) − , (2.17b) r = (cid:18) m q (cid:18) (cid:16) θ π (cid:17)(cid:19)(cid:19) − , (2.17c) † The chemical potential µ can be expressed by using gauge invariant quantity as µ = Z ∞ r + d r F rt = A t ( ∞ ) , where r + and ∞ represent the horizon and the boundary, respectively. This definition gives thermodynamicrelations consistently. ‡ In order to define the horizon, the charge q must satisfy a relation q ≤ m l / θ = arctan (cid:18) √ q p m l − q m l − q (cid:19) , and satisfy a relation r + r − = − r . The positions expressed by r + and r − correspond tothe outer and the inner horizon, respectively. It is useful to notice that the charge q canbe expressed in terms of θ and m by q = 4 m l
27 sin (cid:18) θ (cid:19) . The outer horizon takes a value in r m l ≤ r ≤ √ ml, where the upper bound and the lower bound correspond to the case for q = 0 and theextremal case, respectively.We shall give various thermodynamic quantities of RN-AdS black hole [28, 29]. Thetemperature is defined from the conical singularity free condition around the horizon r + , T = r f ′ ( r + )4 πl = r + πl (cid:16) − q l r (cid:17) ≡ πb (cid:16) − a (cid:17) , ( > , (2.18)where we defined the parameters a and b by a ≡ q l r , b ≡ l r + . (2.19)In the limit q →
0, these parameters go to a → , b → l / m / , and the temperature becomes T → T = m / πl / . The entropy density s , the energy density ǫ , the pressure p , the chemical potential µ andthe density of physical charge ρ can be also computed as s = 2 πr κ l = πl b κ , (2.20) ǫ = 3 m κ l = 3 l b κ (cid:16) a (cid:17) , (2.21) p = ǫ , (2.22) µ = Qr = 4 b Ql , (2.23) ρ = 2 Qe l = le µ b . (2.24)6n order to obtain a well-defined boundary term from the gravitational part, we haveto add the Gibbons-Hawking term into the action, which is given by S GH = 1 κ Z d x p − g (4) K, (2.25)where integration is taken on the boundary of the AdS space. The four-dimensional metric g (4) µν ( x ) is the induced metric on the boundary and K ( x ) is the extrinsic curvature. We alsoneed to add counter terms to regularize the action [30], S ct = 1 κ Z d x p − g (4) (cid:18) l − l R (4) (cid:19) . (2.26) In RN-AdS background, we study small perturbations of the metric g mn ( x ) and the gaugefield A m ( x ), g mn ≡ g (0) mn + h mn , A m ≡ A (0) m + A m , (3.1)where the background metric g (0) mn ( x ) and the background gauge field A (0) m ( x ) are given in(2.13a) and (2.13b), respectively. In the metric perturbation, one can define an inversemetric as g mn = g (0) mn − h mn + h ml h ln + O ( h ) , and raise and lower indices by using the background metric g (0) mn ( x ) and g (0) mn ( x ).Let us now consider a linearized theory of the symmetric tensor field h mn ( x ) and thevector field A m ( x ) propagating in RN-AdS background. We shall work in the h rm ( x ) = 0and A r ( x ) = 0 gauges and use the Fourier decomposition h µν ( t, z, r ) = Z d k (2 π ) e − iωt + ikz h µν ( k, r ) ,A µ ( t, z, r ) = Z d k (2 π ) e − iωt + ikz A µ ( k, r ) , where we choose the momenta which are along the z -direction. In this case, one cancategorize the metric perturbations to the following three types by using the spin underthe SO (2) rotation in ( x, y )-plane [15]: • vector type: h tx = 0, h zx = 0, (others) = 0 (cid:16) equivalently, h ty = 0, h zy = 0, (others) = 0 (cid:17) tensor type: h xy = 0, (others) = 0 (cid:16) equivalently, h xx = − h yy = 0, (others) = 0 (cid:17) • scalar type: h tt = 0, h tz = 0, h xx = h yy = 0, and h zz = 0, (others) = 0First two types of the perturbations were studied in [20]. We list the result in AppendixA. In this paper we consider the scalar type perturbation. From explicit calculation, one can show that t and z -components of the gauge field A µ ( x )could participate in the linearized perturbative equations of motion. Thus independentvariables are h tt ( x ) , h tz ( x ) , h xx ( x ) = h yy ( x ) , h zz ( x ) ,A t ( x ) , A z ( x ) . In the hydrodynamic regime, it is standard to introduce new dimensionless coordinate u = r /r which is normalized by the outer horizon. In this coordinate system, the horizonand the boundary are located at u = 1 and u = 0, respectively. We also define new fieldvariables h tt = g (0) tt h tt = − l ur f h tt ,h zt = g (0) zz h zt = l ur h zt ,h xx = g (0) xx h xx = l ur h xx ,h zz = g (0) zz h zz = l ur h zz ,B µ ≡ A µ µ = l Qb A µ , where µ is the chemical potential given by (2.23). Nontrivial equations in the Einsteinequation (2.8) appear from ( t, t ), ( t, u ), ( t, z ), ( u, u ), ( u, z ), ( x, x ) and ( z, z ) components,respectively:0 = h tt ′′ + ( u − f ) ′ u − f (cid:16) h tt ′ + h xx ′ + 12 h zz ′ (cid:17) − b k uf h tt + 2 b uf (cid:16) ω h xx + 12 ω h zz + ωkh zt (cid:17) + 2 a uf h tt + 4 a uf B ′ t , (3.2a)8 = ω (cid:18) h xx ′ + h zz ′ − f ′ f (cid:16) h xx + 12 h zz (cid:17)(cid:19) + k (cid:16) h zt ′ − f ′ f h zt (cid:17) , (3.2b)0 = h zt ′′ − u h zt ′ + 2 b ωkuf h xx − auB ′ z , (3.2c)0 = h tt ′′ + 2 h xx ′′ + h zz ′′ + f ′ f (cid:16) h tt ′ + h xx ′ + 12 h zz ′ (cid:17) + 2 a uf h tt + 4 a uf B ′ t , (3.2d)0 = kh tt ′ + 2 kh xx ′ − ωf h zt ′ + kf ′ f h tt + 3 a uf (cid:16) kB t + ωB z (cid:17) , (3.2e)0 = h xx ′′ + ( u − f ) ′ u − f h xx ′ − u (cid:16) h tt ′ + h zz ′ (cid:17) + b uf (cid:16) ω − k f (cid:17) h xx − a uf h tt − a uf B ′ t , (3.2f)0 = h zz ′′ + ( u − f ) ′ u − f h zz ′ − u (cid:16) h tt ′ + h xx ′ (cid:17) + b uf (cid:16) ω h zz + 2 ωkh zt − k f h tt − k f h xx (cid:17) − a uf h tt − a uf B ′ t , (3.2g)with f ( u ) = (1 − u )(1 + u − au ) , where the prime implies the derivative with respect to u . On the other hand, in the Maxwellequation (2.12), t , u and z -components give nontrivial contributions0 = B ′′ t − b uf (cid:16) k B t + kωB z (cid:17) + 12 (cid:16) h tt ′ − h xx ′ − h zz ′ (cid:17) , (3.3a)0 = ωB ′ t + kf B ′ z + ω (cid:16) h tt − h xx − h zz (cid:17) − kh zt , (3.3b)0 = B ′′ z + f ′ f B ′ z + b uf (cid:16) ω B z + ωkB t (cid:17) − f h zt ′ . (3.3c)The equations (3.3a) and (3.3b) imply (3.3c). In the set of equations for the metric per-turbation (3.2a)-(3.2g), together with (3.3a) and (3.3b), the following four independentrelations are obtained: h xx ′ = 3( ω − k f ) + k uf ′ k (3 f − uf ′ ) h tt ′ + 2 b ω f (3 f − uf ′ ) h xx − f ′ (3 f − uf ′ ) − b ω f (3 f − uf ′ ) h tt + ω (cid:16) f ′ (3 f − uf ′ ) − b ω (cid:17) k f (3 f − uf ′ ) (cid:16) ωh xx + ω h zz + kh zt (cid:17) + 3 aω u k f (3 f − uf ′ ) (cid:16) h tt + 2 B ′ t (cid:17) − au kf (cid:16) kB t + ωB z (cid:17) , (3.4a) h zz ′ = − ω + k uf ′ k (3 f − uf ′ ) h tt ′ − b k f − uf ′ (cid:16) h tt + 2 h xx (cid:17) + 2 b f (3 f − uf ′ ) (cid:16) ω (2 h xx + h zz ) + 2 ωkh zt (cid:17) f (3 f − uf ′ ) (cid:18)(cid:16) f ′ (3 f − uf ′ ) − b ω (cid:17) h tt − b ω h xx (cid:19) − ω (cid:16) f ′ (3 f − uf ′ ) − b ω (cid:17) k f (3 f − uf ′ ) (cid:16) ωh xx + ω h zz + kh zt (cid:17) − au ( ω + k f ) k f (3 f − uf ′ ) (cid:16) h tt + 2 B ′ t (cid:17) + 3 aukf (cid:16) kB t + ωB z (cid:17) , (3.4b) h zt ′ = 3 ωfk (3 f − uf ′ ) h tt ′ + 2 b ωk f − uf ′ (cid:16) h tt + 2 h xx (cid:17) + f ′ (3 f − uf ′ ) − b ω kf (3 f − uf ′ ) (cid:16) ωh xx + ω h zz + kh zt (cid:17) + 3 aωu k (3 f − uf ′ ) (cid:16) h tt + 2 B ′ t (cid:17) , (3.4c)0 = h tt ′′ + 12 uf (3 f − uf ′ ) (cid:26) − f − uf ′ )(2 f − uf ′ ) h tt ′ +4 b (cid:18) − k f h tt + (cid:16) ω + ( f − uf ′ ) k (cid:17) h xx + ω h zz + 2 ωkh zt (cid:19) + au (15 f − uf ′ ) (cid:16) h tt + 2 B ′ t (cid:17)(cid:27) . (3.4d)The equations of motion (3.2a)-(3.2g) can be derived by using the above relations. Takingthe limit q →
0, the relations (3.4a)-(3.4d) coincide with the result in [18].
Before solving the equations of motion, we shall give a surface action in oder to obtainGreen functions. By using the equations of motion, bilinear parts of on-shell action (2.8)reduce to surface terms: S = l κ b Z d k (2 π ) (cid:26) + fu h tt h tt ′ + fu h xx h xx ′ + fu h zz h zz ′ − u h zt h zt ′ − fu h xx h tt ′ − f u h zz h tt ′ − fu h tt h xx ′ − fu h zz h xx ′ − f u h tt h zz ′ − fu h xx h zz ′ + uf ′ − f u (cid:0) h tt (cid:1) − f u ( h zz ) + uf ′ + fu f ( h zt ) − uf ′ − f u h tt h xx − uf ′ − f u h tt h zz + fu h xx h zz +3 a (cid:16) B t B ′ t − f B z B ′ z + 12 B t h tt + B z h zt − B t h xx − B t h zz (cid:17)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) u =0 . (3.5)10elevant terms of the Gibbons-Hawking term (2.25) are explicitly given by S GH = l κ b Z d k (2 π ) (cid:26) − fu h tt h tt ′ − fu h zz h zz ′ + 4 u h zt h zt ′ + 2 fu h xx h tt ′ + fu h zz h tt ′ + 2 fu h tt h xx ′ + 2 fu h zz h xx ′ + fu h tt h zz ′ + 2 fu h xx h zz ′ − uf ′ − f u (cid:0) h tt (cid:1) − uf ′ − f u ( h zz ) − uf ′ + 4 fu f ( h zt ) + uf ′ − fu h tt h xx + uf ′ − f u h tt h zz + uf ′ − fu h xx h zz (cid:27) . (3.6)The counter term (2.26) also can be evaluated as S ct = 3 l κ b Z d k (2 π ) p f (cid:26) − u (cid:0) h tt (cid:1) − u ( h zz ) + 1 u f ( h zt ) + 1 u h tt h xx + 12 u h tt h zz + 1 u h xx h zz (cid:27) , (3.7)up to O ( ω , k , ωk ). We now look for solutions of our set of equations (3.4a)-(3.4d), and (3.3a) and (3.3b).We will consider these equations of motion in low frequency limit so-called hydrodynamicregime. In this regime we could obtain the sound velocity, the diffusion constant for U (1)charge and the electrical conductivity from retarded Green functions. By using master variables derived by Kodama and Ishibashi in [19], the following field Φ( u )is introduced: Φ ≡ u / (4 b k − f ′ ) (cid:18)(cid:16) b k − f ′ (cid:17) h xx + 2 f (cid:16) h xx ′ + h zz ′ (cid:17)(cid:19) . (4.1)For the gauge field, the corresponding variable is given by A ≡ a (cid:16) − h tt + 3 h xx − B t ′ (cid:17) . (4.2)11n terms of these new variables Φ( u ) and A ( u ), Einstein equations (3.4a)-(3.4d) andMaxwell equations (3.3a) and (3.3b) can be combined as follows:0 = ( u / f Φ ′ ) ′ − u / f (4 b k − f ′ ) (cid:26) − b ω u (4 b k − f ′ ) + f (cid:16) − b k + 108 ab k u + 162 a u )+27 f ′ (8 b k + 16 au + 5 f ′ ) (cid:17) +4 uf (4 b k − f ′ ) (cid:16) b k ( b k + 3 au )+ f ′ (8 b k + 36 au + 9 f ′ ) (cid:17)(cid:27) Φ+ 18 u / (4 b k − f ′ ) (cid:26) f (cid:16) b k + 3 f ′ + 18 au (cid:17) + u (cid:16) b k − f ′ ) (cid:17)(cid:27) A , (4.3a)0 = ( uf A ′ ) ′ + 1 f (4 b k − f ′ ) (cid:26) b (cid:16) b k − f ′ (cid:17)(cid:16) ω − k f (cid:17) − auf (cid:27) A− au / f Φ ′ + 4 au / (4 b k − f ′ ) (cid:26) b k u + 12 f (cid:16) b k + 9 au (cid:17) + 3 f ′ (cid:16) − b k u + 9 f (cid:17)(cid:27) Φ . (4.3b)Next we would like to try to obtain decoupled equations from the equations (4.3a) and(4.3b). This will be done by introducing the following linear combinations of the variables:Φ ± ≡ α ± Φ + β A , (4.4)where the coefficients α ± and β are α ± = C ± − au,β = u / , with the constants C ± = (1 + a ) ± p (1 + a ) + 4 ab k . As a result, we can obtain second order ordinary differential equations in term of thesenew variables, 0 = Φ ′′± + ( u / f ) ′ u / f Φ ′± + V ± Φ ± , (4.5)12here potentials V ± ( u ) are given by V ± = 116 u f (4 b k − f ′ ) (cid:26) b ω u (4 b k − f ′ ) − uf (4 b k − f ′ ) (cid:16) b k ( b k − C ± u + 3 au ) − f ′ (2 b k + 3 C ± u − au ) − f ′ ) (cid:17) + f n (cid:16) b k + 12 C ± b k u − ab k u + 54 C ± au − a u (cid:17) − f ′ (cid:16) b k − C ± u − au (cid:17) + 9( f ′ ) o(cid:27) . (4.6)Considering the perturbative expansion with respect to small ω and k , it might beconvenient to introduce new variables e Φ ± ( u ),Φ ± = H ± e Φ ± , (4.7)where the factors F ± ( u ) are H ± = u − / (for Φ + ) u / (1 + a ) − au (for Φ − ) , so that the second order differential equations (4.5) are reduced to be much simpler formsto solve, 0 = e Φ ′′± + ( H ± u / f ) ′ H ± u / f e Φ ′ + e V ± e Φ ± , (4.8)where the potential e V ± ( u, ω, k ) is newly defined from the original potential V ± ( u, ω, k ), e V ± ( u, ω, k ) ≡ V ± ( u, ω, k ) − V ± ( u, , . (4.9) Let us proceed to solve the differential equations. First we consider the equation for e Φ + ( u ). Following the usual way to solve differential equations, we impose a solution as e Φ + ( u ) = (1 − u ) ν F + ( u ) where F + ( u ) is a regular function at the horizon u = 1. Pluggingthis form into the equation of motion, one can fix the parameter ν as ν = ± iω/ (4 πT )where T is the temperature defined by eq. (2.18). We here choose ν = − i ω πT (4.10)13s the incoming wave condition. We are now in the position to solve the equation of motionin the hydrodynamic regime. We start by introducing the following series expansion withrespect to small ω and k : F + ( u ) = F +0 ( u ) + ωF +1 ( u ) + k G +1 ( u ) + ω F +2 ( u ) + O ( ω , ωk ) , (4.11)where F +0 ( u ), F +1 ( u ) and G +1 ( u ) are determined by imposing suitable boundary condi-tions. The solution can be obtained recursively ∗ . The result is as follows: F +0 ( u ) = C, (const.) (4.12a) F +1 ( u ) = iCb − a ) (cid:26) log (cid:0) u − au (cid:1) − K ( u ) √ a (cid:27) , (4.12b) G +1 ( u ) = 23 Cb (cid:26) K ( u ) √ a − a ) u (cid:27) , (4.12c) F +2 ( u ) = Z u d u Cb (1 − u )(1 + u − au ) × (cid:26) − u + (1 − u )(1 + au ) log (1 + u − au )2(2 − a ) − − u )(1 + au ) K (0)2(2 − a ) √ a − (1 + a ) K (1) u √ a + (cid:16) a − a + 2 a ) u − au (cid:17) K ( u )2(2 − a ) √ a (cid:27) , (4.12d)where K ( u ) = 12 log(1 + u − au ) − log (cid:18) − au √ a (cid:19) ,K ( u ) = log (cid:18) √ a − au − √ a + 2 au (cid:19) . Next, we shall study the equation of motion for e Φ − ( u ). Assuming again e Φ − ( u ) =(1 − u ) ν F − ( u ) where F − ( u ) is a regular function at u = 1, the singularity might be extracted.We fix the constant as ν = − iω/ (4 πT ) to use the incoming wave condition. We now imposea perturbative solution as F − ( u ) = F − ( u ) + ωF − ( u ) + k G − ( u ) + ω F − ( u ) + O ( ω , ωk ) , (4.13)and then we obtain the following result † : F − ( u ) = e C, (const.) (4.14a) ∗ The derivation of the solution is given in Appendix B. † The detail is given in Appendix C. − ( u ) = i e Cb − a ) n a ) log( u ) − (2 + a )(1 + 4 a ) log (cid:16) u − au (cid:17) − √ a (2 + 5 a ) K ( u ) o , (4.14b) G − ( u ) = e Cb (cid:26) − a u a ) (cid:16) a − au (cid:17) − a )(2 + a ) log( u )(2 − a ) + (1 + a )(2 + a ) log (cid:16) u − au (cid:17) (2 − a ) + 2(2 + 5 a + 6 a ) K ( u )(2 − a ) √ a (cid:27) , (4.14c) F − ( u ) = Z u d u e Cb − a ) (1 + 4 a ) / (1 − u ) u (1 + u − au ) × (cid:26) − a )(1 + a ) (1 + 4 a ) / u (1 + u − au ) log( u ) − (2 − a )(1 + 4 a ) / (cid:16) a ) + (2 − a − a ) u + (2 + 9 a + 13 a ) u − a (2 + a )(1 + 4 a ) u (cid:17) log (cid:16) u − au (cid:17) +(1 + 4 a ) (2 + 5 a ) K (0)(1 − u ) (cid:16) a ) + (2 − a − a ) u − (2 − a ) au (cid:17) − a )(2 − a + 41 a ) K (1) (cid:16) a ) − au (cid:17) − (2 − a ) (cid:16) a (1 + 4 a ) (2 + 5 a ) u − (2 − a )(1 + 11 a + 46 a + 18 a ) u − (cid:0) a + 180 a + 224 a + 24 a (cid:1) u − a ) (1 − a − a ) (cid:17) K ( u ) (cid:27) . (4.14d)Using these, we can get behaviors for the solutions of Φ + ( u ) and Φ − ( u ) around theboundary u = 0,Φ + = Cu / (cid:26) − k b a ) u + 13 (cid:0) k + 3 ω (cid:1) b u + · · · (cid:27) , (4.15a)Φ − = e Cu / (cid:26)
11 + a + (cid:18) i (1 + a ) bω (2 − a ) − a ) b k (2 − a ) + b D − ω (cid:19) log( u ) + · · · (cid:27) , (4.15b)where the constant D − is D − = 2(2 − a ) (1 + 4 a ) / (cid:26) − − a ) a √ a +4(1 + 4 a ) / (1 + a ) log(2 − a )154(2 − a + 41 a )(1 + a ) K (1) (cid:27) . Let us now consider the integration constants C and ˜ C . These could be estimated interms of boundary values of the fieldslim u → h tt ( u ) = ( h tt ) , lim u → B t ( u ) = ( B t ) , etc . Using equations of motion, the integration constants C and e C are determined as C = 12 ( k − ω ) (cid:26) ak ( B t ) + 3 akω ( B z ) +(1 + a ) (cid:16) − k ( h tt ) + 2 kω ( h zt ) + (cid:0) k − ω (cid:1) ( h xx ) + ω ( h zz ) (cid:17)(cid:27) , (4.16a) e C = (2 − a ) abk (cid:16) k ( B t ) + ω ( B z ) (cid:17) D p ( ω, k ) , (4.16b)where D p ( ω, k ) = 2(2 + a ) bk − i (1 + a ) ω − (2 − a ) bD − ω . In the equations (4.16a) and (4.16b), one can observe the existence of the sound anddiffusion poles in the complex ω -plane. Let us evaluate the Minkowskian correlators. The relevant action is given by the sum ofthree parts (3.5), (3.6) and (3.7), S = S + S GH + S ct = l κ b Z d k (2 π ) (cid:26) u h zt h zt ′ + fu h xx h xx ′ + fu h tt h xx ′ + f u h tt h zz ′ + fu h xx h tt ′ + fu h xx h zz ′ + f u h zz h tt ′ + fu h zz h xx ′ + 34 u (cid:16) f − p f (cid:17) ( h tt ) + 14 u (cid:16) f − uf ′ − p f (cid:17) ( h zz ) − u f (cid:16) f − p f (cid:17) ( h zt ) − u (cid:16) f − uf ′ − p f (cid:17) h tt h xx − u (cid:16) f − uf ′ − p f (cid:17) h tt h zz − u (cid:16) f − uf ′ − p f (cid:17) h xx h zz +3 a (cid:16) B t B ′ t − f B z B ′ z + 12 B t h tt + B z h zt − B t h xx − B t h zz (cid:17)(cid:27)(cid:12)(cid:12)(cid:12)(cid:12) u =0 . (4.17)16sing equations of motion and solutions of Φ + ( u ) and Φ − ( u ), derivatives of h ’s and B ’scan be expressed in terms of their boundary values:1 u h tt ′ → k − ω (cid:26) − ak (cid:16) k ( B t ) + ω ( B z ) (cid:17) +(1 + a ) (cid:16) k ( h tt ) − ωk ( h zt ) − ω (2( h xx ) + ( h zz ) ) (cid:17)(cid:27) , (4.18a)1 u h xx ′ → k − ω (cid:26) ak (cid:16) k ( B t ) + ω ( B z ) (cid:17) +(1 + a ) (cid:16) − k ( h tt ) + 2 kω ( h zt ) + ω (2( h xx ) + ( h zz ) ) (cid:17)(cid:27) , (4.18b)1 u h zz ′ → k − ω (cid:26) ak (cid:16) k ( B t ) + ω ( B z ) (cid:17) +(1 + a ) (cid:16) − k ( h tt ) + 2 kω ( h zt ) + ω (2( h xx ) + ( h zz ) ) (cid:17)(cid:27) , (4.18c)1 u h zt ′ → k − ω (cid:26) − aω (cid:16) k ( B t ) + ω ( B z ) (cid:17) − (1 + a ) k (cid:16) k ( h zt ) + ω ( − h tt ) + 2( h xx ) + ( h zz ) ) (cid:17)(cid:27) , (4.18d) B ′ t → a )( k − ω ) (cid:26) − ak (cid:16) k ( B t ) + ω ( B z ) (cid:17) +(1 + a ) (cid:16) (2 k + 3 ω )( h tt ) − kω ( h zt ) − ω (2( h xx ) + ( h zz ) ) (cid:17)(cid:27) − (2 − a ) bk (cid:16) k ( B t ) + ω ( B z ) (cid:17) (1 + a ) D p ( ω, k ) , (4.18e) B ′ z → a )( k − ω ) (cid:26) aω (cid:16) k ( B t ) + ω ( B z ) (cid:17) +(1 + a ) (cid:16) − kω ( h tt ) + 2 k ( h zt ) + kω (2( h xx ) + ( h zz ) ) (cid:17)(cid:27) + (2 − a ) bω (cid:16) k ( B t ) + ω ( B z ) (cid:17) (1 + a ) D p ( ω, k ) . (4.18f)Substituting these expressions to the surface term (4.17), we can read off the Green func-tions defined by (2.5). Through the counter terms, the singularities around the boundaryvanish completely. The results are listed below in Table 1, 2 and 3.In the final expression we rescaled the gauge field ( B µ ) to the original one ( A µ ) =4 Qb l ( B µ ) and raised and lowered the indices by using the flat Minkowski metric η µν =17 able 1: G ∗∗ ∗∗ / (cid:18) − (1 + a ) l κ b ( k − ω ) (cid:19) . tt xx zz tztt k − ω ) 6 ( k + ω ) 3 ( k + ω ) 24( k + ω ) xx — 16 ω k + ω ) 16 kωzz — — − k + 7 ω kωtz — — — 4( k + 9 ω ) Table 2: G ∗∗ ∗ / (cid:18) − lµ e b ( k − ω ) (cid:19) . tt xx zz tzt k k k kωz kω kω kω ω Table 3: G ∗ ∗ / (cid:18) − l e (1 + a ) b (cid:18) ak − ω + 2(2 − a ) bD p ( ω, k ) (cid:19)(cid:19) . t zt k kωz — ω − , + , + , +) in four-dimensional boundary theory. Taking the limit which the chargegoes to zero, the correlators for energy-momentum tensors coincide with the known onesin [15]. In this limit, the correlators for the energy-momentum tensor and the U (1) currentvanish, while ones for the U (1) currents have no sound poles, as we could see in the caseof vector type perturbation for (1 , ,
0) R-charged [22] and RN-AdS black holes [20].Remember l /κ = N c / (4 π ) and it should be noticed that the factor l/e = N c / (16 π )for the R-charge since e = 2 κ/l in that case, while l/e = N c N f / (4 π ) for the branecharge [29]. From the obtained Green functions, we can observe the value of the speed of sound withoutthe medium effect, v s = 1 √ . (4.19)One should notice that there is no effect of the charge on the sound velocity.We can also find the diffusion pole in the current-current correlators. The diffusionconstant can be read off D A = (2 + a ) b a ) . (4.20)It should be compared with the diffusion constant for gravitation fields D H obtained in theprevious work [20], D H = b a ) , so that the relation between them is D A = (2 + a ) D H . (4.21)In the chargeless case, we can reproduce the result in [15].The electrical conductivity σ of the medium could be also determined by the current-current correlators via Kubo formula, σ ≡ − lim ω → e ω Im (cid:16) δ ij G ij ( ω, k = 0) (cid:17) , where e E is a four-dimensional gauge coupling. Together with the result for vector typeperturbation [20], we can obtain σ = le (2 − a ) e (1 + a ) b × (cid:18) le (cid:19) πe E2 (2 − a )2(1 + a ) T. (4.22)We can also access to the charge susceptibility Ξ defined byΞ ≡ T h Q i (volume) . (4.23)Using Green function G t t ( k ) which might give an expectation value of Q , one can obtainthe following relation in thermal equilibrium, h Q i (volume) = Z d ω π (cid:16) − Im (cid:16) G t t ( ω, k → (cid:17)(cid:17) n b ( ω ) , (4.24)where n b ( ω ) = e ω/T − is Bose-Einstein distribution function [31]. From Table 3, we cansee − Im (cid:16) G t t ( ω, k ) (cid:17) = 4 π lT e (1 + a )(2 + a ) (cid:18) ωD A k ω + ( D A k ) (cid:19) . It should be noted that a quantity D A k / (cid:16) ω + ( D A k ) (cid:17) approaches to 2 πδ ( ω ) for k → (cid:18) le (cid:19) π (1 + a )(2 + a ) T . (4.25)We can then observe that Einstein relation σ/ ( e E2 Ξ) = D A (4.26)holds exactly. (See also [32].) ‡ In R-charge case, taking the charge-free limit, the electric conductivity and the chargesusceptibility coincide with the results in [31].It is interesting to express physical constants in terms of the boundary variables: thetemperature and the chemical potential. In fact, it is easy to verify that a = 2 −
41 + p µ/T ) , b = (cid:18) πT (cid:19)
11 + p µ/T ) , (4.27)where ˜ µ ≡ κ √ πel µ . Notice that for the R-charge, ˜ µ = 12 √ π µ , while for the branecharge, ˜ µ = r N f N c π µ . The behaviors of the diffusion constants D A and D H , the electrical ‡ It is interesting to notice that ( ∂ρ/∂µ ) T gives a different value for Ξ given in eq.(4.25). The authorsthank J. Mas and J. Shock for pointing this out. σ and the charge susceptibility Ξ are drawn as functions of the ˜ µ/T in Figure 1,2, 3 and 4 respectively. Notice that for the fixed temperature, all of them are decreasingfunctions of the chemical potential. One should notice that there is no upper bound of µ/T for any of these quantities unlike (1,0,0) charged black hole studied in [22]. ˜ µ/TTD A Figure 1:
T D A vs. ˜ µ/T ˜ µ/TTD H Figure 2:
T D H vs. ˜ µ/T ˜ µ/Tσ/T Figure 3: σ/ ( T le /e ) vs. ˜ µ/T . ˜ µ/T Ξ /T Figure 4: Ξ / ( T l/e ) vs. ˜ µ/T : Notice therapid change between two finite values as T runs from 0 to ∞ It is particularly interesting to observe that the charge susceptibility modulo T factor,which is an indicator of the degree of freedom, shows rapid change between low and hightemperature (density) indicating a mild phase transition. Such behavior does not exist forchargeless case. See also [33]. 21 Conclusions and Discussions
In this paper, we worked out the decoupling of scalar modes of the charged AdS blackhole background in SO (2) basis for the mode classifications. We also perform the hydro-dynamic analysis for the holographic Quark-Gluon Plasma system. Master equations forthe decoupled modes are worked out explicitly. The sound velocity is not modified by thepresence of the charge. We calculated the diffusion constants, the charge susceptibility andthe conductivity as a consequence and observed that Einstein relation holds between them.These transport coefficients are modified due to the charge effect. Interestingly, the sus-ceptibility modulo T factor, which is an indicator of the degree of freedom, shows rapidchange between low and high temperature (density) indicating a mild phase transition.Such behavior does not exist for chargeless case.One can give an explanation of hydrodynamic mode in meson physics. In our interpre-tation, the Maxwell fields are the fluctuations of bulk-filling branes, therefore they shouldbe interpreted as master fields of the mesons. Then hydrodynamic modes are lowest lyingmassless meson spectrum. In terms of brane embedding picture, this massless-ness is dueto the touching of the brane on the black hole horizon. Near the horizon, the tension ofthe brane is zero due to the metric factor and it can lead to the massless fluctuation. Thenthe massless spectrum can not go far from the horizon in radial direction. In this picture,hydrodynamic nature is closely related to the near horizon behavior of the branes. Wewill discuss the spectrum of meson mode by considering the quasinormal mode [34] of thevector modes in elsewhere. Acknowledgments
We would like to thank X.-H. Ge and F.-W. Shu for useful discussion at the early stageof this work and S. Nakamura for stimulating discussions. We especially want to thank J.Mas and J. Shock for pointing out interesting points after first version of the paper wasuploaded. The work of SJS was supported by KOSEF Grant R01-2007-000-10214-0. Thiswork is also supported by Korea Research Foundation Grant KRF-2007-314-C00052 andSRC Program of the KOSEF through the CQUeST with grant number R11-2005-021.22 ppendix A Results for the vector and the tensortype perturbations
Appendix A.1 Vector type perturbation
In the vector type perturbation, independent variables are h tx ( x ) , h zx ( x ) , A x ( x ) . One can observe the diffusion constant for the metric perturbation D H as D H = b a ) . (A.1)We list the retarded Green functions in Table 4, 5 and 6. Notice that l /κ = N c / (4 π ) Table 4: G ∗∗ ∗∗ / (cid:18) l κ b ( iω − D H k ) (cid:19) . tx zxtx k − ωkzx — ω Table 5: G ∗∗ ∗ / (cid:18) lµ e b ( iω − D H k ) (cid:19) . tx zxx − iω b (1 + a ) ωk Table 6: G ∗ ∗ / (cid:18) l e (1 + a ) b (cid:18) aiω − D H k − (2 − a ) b a ) (cid:19)(cid:19) . xx iω and l/e = N c / (16 π ) for the R-charge, l/e = N c N f / (4 π ) for the brane charge. From23he Green function for U (1) currents G x x ( ω, k ), one can read off the thermal conductivity κ T via Kubo formula, κ T ≡ − ( ǫ + p ) ρ T lim ω → ω Im (cid:16) G x x ( ω, k = 0) (cid:17) = 2 π N c N f ηTµ . (A.2) Appendix A.2 Tensor type perturbation
In the tensor type perturbation, an independent variable is just h xy ( x ) . By using Kubo formula, one can obtain the shear viscosity η as η ≡ − lim ω → ω Im (cid:16) G xy xy ( ω, k = 0) (cid:17) = l κ b , (A.3)where the retarded Green function is given by Table 7: G ∗∗ ∗∗ / (cid:18) − l κ b (cid:19) . xyxy iω + bk One can confirm the universal (within Einstein gravity ∗ ) ratio that is the ratio of the shearviscosity to the entropy density s , ηs = 14 π . (A.4) Appendix B Perturbative Solutions for e Φ +Substituting the equation (4.11) into the equation (4.8), one can read off one for F +0 ( u ),0 = (cid:16) u − (1 − u )(1 + u − au ) F ′ +0 (cid:17) ′ . (B.1)A general solution is given by F +0 ( u ) = C + D (cid:26) (cid:16) − u (cid:17) − log (cid:16) u − au (cid:17) − K ( u ) √ a (cid:27) . (B.2) ∗ If one consider higher derivative corrections to Einstein gravity, the viscosity bound could be modi-fied [35–39]. C and D should be determined to be a regular function at thehorizon. So we here choose D = 0 and set F +0 ( u ) = C = C. (const.) (B.3)By using this solution, one can get an equation for F +1 ( u ) from the equation (4.8),0 = (cid:16) u − (1 − u )(1 + u − au ) F ′ +1 (cid:17) ′ − i Cb (1 + au )(2 − a ) u . (B.4)A general solution is F +1 ( u ) = C + i (cid:16) C (1 − a ) b − (2 − a ) D (cid:17) log (cid:16) − u (cid:17) (2 − a ) + i (cid:16) Cb + (2 − a ) D (cid:17) log (cid:16) u − au (cid:17) − a ) + 3 i (cid:16) Cb + (2 − a ) D (cid:17) K ( u )2(2 − a ) √ a . (B.5)Removing the singularity at the horizon, the constant D should be D = Cb − a − a . We also impose a boundary condition F +1 ( u = 0) = 0 so as to fix the constant C .Therefore the final form is F +1 ( u ) = iCb − a ) (cid:26) log (cid:16) u − au (cid:17) − K ( u ) √ a (cid:27) . (B.6)A differential equation for G +1 ( u ) is0 = (cid:16) u − (1 − u )(1 + u − au ) G ′ +1 (cid:17) ′ + Cb (cid:16) − (1 + a ) u (cid:17) a ) u . (B.7)A general solution is G +1 ( u ) = e C + 16(2 − a )(1 + a ) (cid:26) − C (2 − a ) b u + (cid:16) C (1 + 5 a − a ) b − a ) e D (cid:17) K ( u ) √ a + (cid:16) C (1 − a ) b + 3(1 + a ) e D (cid:17) × (cid:18) (cid:16) − u (cid:17) − log (cid:16) u − au (cid:17)(cid:19)(cid:27) (B.8)25nd the constant e D can be fixed as e D = − C (1 − a ) b a ) . From the condition ( uG +1 ) ′ | u =0 = 0, we can fix the constant e C . The final form of thesolution becomes G +1 ( u ) = 23 Cb (cid:26) K ( u ) √ a − a ) u (cid:27) . (B.9)A differential equation for F +2 ( u ) is0 = (cid:20) u − (1 − u ) (cid:0) u − au (cid:1) × (cid:18) Cb log (cid:16) − u (cid:17) (2 − a ) (1 − u ) + ibF +1 ( u )(2 − a )(1 − u ) − ib log(1 − u ) F ′ +1 ( u )2 − a + F ′ +2 ( u ) (cid:19)(cid:21) ′ − Cb u (1 − u )(1 + u − au ) . (B.10)Integrating over u , we have F ′ +2 ( u ) = 12(2 − a ) √ a (1 − u )(1 + u − au ) × (cid:26) √ a (2 − a ) (cid:0) Cb − D u (cid:1) − Cb K (0) (cid:0) u − au (cid:1) + Cb (cid:16) a − a + 2 a ) u − au (cid:17) K ( u )+ C √ ab (1 − u )(1 + au ) log (cid:16) u − au (cid:17)(cid:27) (B.11)and the constant D can be fixed as D = Cb n − K (0)2(2 − a ) √ a + (1 + a ) K (1) √ a o , so that (cid:0) (1 − u ) F ′ +2 ( u ) (cid:1)(cid:12)(cid:12) u =1 = 0. Then, we find F ′ +2 ( u ) = Cb + O ( u ) . (B.12)Hence, we can write the solution of F +2 ( u ) as F +2 ( u ) = Z u d u Cb (1 − u )(1 + u − au )26 (cid:26) − u + (1 − u )(1 + au ) log (cid:16) u − au (cid:17) − a ) − K (0)(1 − u )(1 + au )2(2 − a ) √ a − (1 + a ) K (1) u √ a + (cid:16) a − a + 2 a ) u − au (cid:17) K ( u )2(2 − a ) √ a (cid:27) , (B.13)which satisfies a boundary condition F +2 (0) = 0. Appendix C Perturbative Solutions for e Φ − For F − ( u ), one can get an equation0 = (cid:18) u (1 − u )(1 + u − au ) (cid:16) a − au (cid:17) − F ′− (cid:19) ′ . (C.1)A general solution is given by F − ( u ) = C + D (cid:26) (2 − a ) log (cid:16) − u (cid:17) − a ) log ( u )+ 12 (2 + a )(1 + 4 a ) log (cid:16) u − au (cid:17) − √ a (2 + 5 a ) K ( u )2 (cid:27) . (C.2)Since the function F − ( u ) should be regular at the horizon, we choose D = 0 and get F − ( u ) = C = e C. (const.) (C.3)Substituting the solution to the equation (4.8), we get an equation for F − ( u ),0 = (cid:18) u (1 − u )(1 + u − au ) (cid:16) a − au (cid:17) − F ′− (cid:19) ′ + i e Cb (cid:16) a ) + (4 + 7 a ) u − a (1 + a ) u + 3 a u (cid:17) − a ) (cid:16) a − au (cid:17) . (C.4)A general solution is given as F − ( u ) = C i − a ) a × (cid:26) a ) (cid:16) e C (1 + 4 a ) b − D (2 − a ) a (cid:17) log( u ) − (cid:16) e C (2 + 7 a + 23 a ) b − D (2 − a ) a (cid:17) log (cid:16) − u (cid:17) − (2 + a )(1 + 4 a ) (cid:16) e C (1 + 4 a ) b − D (2 − a ) a (cid:17) log (cid:16) u − au (cid:17) + √ a (2 + 5 a ) (cid:16) e C (1 + 4 a ) b − D (2 − a ) a (cid:17) K ( u ) (cid:27) . (C.5)The constant of integration D should be D = e Cb (2 + 7 a + 23 a )27(2 − a ) a , so that the singularity at the horizon could be removed. In addition, we require thecondition (cid:20) F − ( u ) − log( u ) lim u → (cid:18) F − ( u )log( u ) (cid:19)(cid:21) u =0 = 0to fix the constant C . Then we get the final form of the solution F − ( u ) = i e Cb − a ) n a ) log( u ) − (2 + a )(1 + 4 a ) log (cid:16) u − au (cid:17) − √ a (2 + 5 a ) K ( u ) o . (C.6)Similarly we have a differential equation for G − ( u ),0 = (cid:18) u (1 − u )(1 + u − au ) (cid:16) a − au (cid:17) − G ′− (cid:19) ′ + e Cb a ) (cid:16) a − au (cid:17) (cid:26) − a )(2 + 6 a + 3 a + 2 a )+2 a (10 + 30 a + 57 a + 10 a ) u − a (1 + a ) u + 21 a (1 + a ) u (cid:27) . (C.7)A general solution of this equation can be obtained G − ( u ) = e C − e Ca b a (1 + a ) (cid:16) a − ua (cid:17) + 154 a (1 + a )(2 − a ) 28 (cid:26) a ) (2 − a ) (cid:16) e Cb + 54 e D a (1 + a ) (cid:17) log( u ) − (cid:16) e C (14 + 13 a + 17 a ) b + 54 e D (2 − a ) a (1 + a ) (cid:17) log (cid:16) − u (cid:17) − (2 + a ) (cid:16) e C (7 + 11 a − a ) b + 27 e D a (2 − a )(1 + a )(1 + 4 a ) (cid:17) × log (cid:16) u − au (cid:17) + 1 √ a (cid:16) e C (14 + 57 a + 81 a − a ) b +27 e D a (2 − a )(1 + a )(1 + 4 a )(2 + 5 a ) (cid:17) K ( u ) (cid:27) . (C.8)The constant of integration e D might be fixed to remove the singularity at u = 1, e D = − e Cb (14 + 13 a + 17 a )27 a (1 + a )(2 − a ) . Another constant of integration e C is fixed to satisfy the condition (cid:20) G − ( u ) − log( u ) lim u → (cid:18) G − ( u )log( u ) (cid:19)(cid:21) u =0 = 0 . The final result of the solution is G − ( u ) = 13 e Cb (cid:26) − a u a ) (cid:16) a − au (cid:17) − a )(2 + a )(2 − a ) log( u )+ 3(1 + a )(2 + a )(2 − a ) log (cid:16) u − au (cid:17) + 6(2 + 5 a + 6 a )(2 − a ) √ a K ( u ) (cid:27) . (C.9)A differential equation for F − ( u ) is0 = (cid:20) u (1 + u − au )(2 − a ) (cid:16) a − au (cid:17) × (cid:18) (2 − a ) (1 − u ) F ′− ( u ) + i (2 − a ) bF − ( u ) − i (2 − a ) b (1 − u ) log (cid:16) − u (cid:17) F ′− ( u ) + e Cb log (cid:16) − u (cid:17)(cid:19)(cid:21) ′ + e Cb (cid:16) a − au (cid:17) (1 − u ) (cid:0) u − au (cid:1) . (C.10)Integrating over u , we have F ′− ( u ) = (cid:16) a − au (cid:17) (1 − u ) u (1 + u − au ) 29 (cid:26) D + 18 e Cab (2 − a ) (1 + 4 a ) (cid:16) a − au (cid:17) + 4 e C (1 + a ) b u (1 + u − au ) log( u )(2 − a ) (cid:16) a − au (cid:17) − e Cb (2 − a ) a )(1 + 4 a ) u (1 + u − au )4 (cid:16) a − au (cid:17) log (cid:16) u − au (cid:17) − e C √ a (2 + 5 a ) b K (0) u (1 + u − au )2(2 − a ) (cid:16) a − au (cid:17) + 2 e Cb (2 − a ) (1 + 4 a ) / (cid:18) − a (5 + a )+ (2 + 5 a )(1 + 4 a ) u (1 + u − au )4 (cid:16) a − au (cid:17) (cid:19) K ( u ) (cid:27) , (C.11)and the constant D can be fixed as D = 4 e Cb (2 − a ) (1 + 4 a ) / (cid:26) √ a (cid:16) a )(1 + a ) log(2 − a ) − a (2 − a ) (cid:17) + 12 (2 + 5 a )(1 + 4 a ) K (0) − (1 + a )(2 − a + 41 a ) K (1) (cid:27) , so that (cid:0) (1 − u ) F ′− ( u ) (cid:1)(cid:12)(cid:12) u =1 = 0. 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