Sound Modes in Holographic Hydrodynamics for Charged AdS Black Hole

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 this paper, we extend it to the scalar modes. We work out the decoupling problem and hydrodynamic analysis for the sound mode in charged AdS black hole and calculate the sound velocity, the charge susceptibility and the electrical conductivity. We find that Einstein relation among the conductivity, the diffusion constant and the susceptibility holds exactly.


Introduction
The discovery of low viscosity in the theory with gravity dual [1] and its possible relation to the RHIC (Relativistic Heavy Ion Collider) experiment induced a great deal of efforts to establish the relevant calculational scheme that may be provided by AdS/CFT correspondence [2][3][4]. An attempt has been made to map the entire process of RHIC experiment in terms of the gravity dual [5]. The way to include a chemical potential in the theory was figured out in the context of probe brane embedding [6][7][8][9][10][11][12][13]. Phases of these theories were discussed and new phases were reported where instability due to the strong attraction is a feature [8][9][10].
In spite of the difference between QCD and N = 4 SYM, it is expected that some of the properties are shared by the two theories based on the universality of low energy physics. In this respect, the hydrodynamic limit is particularly interesting since such limit can be shared by many theories. The calculation scheme for transport coefficients is to use Kubo formula, which gives a relation to the low energy limit of Wightman Green functions. In AdS/CFT correspondence, one calculate the retarded Green function which is related to the Wightman function by fluctuation-dissipation theorem. Such scheme has been developed in a series of papers [14][15][16][17][18].
For the hydrodynamic analysis, one may need to have master equations for the decoupled modes in vector and scalar at hands. Although the analysis for the decoupling problem were analyzed in [19], it was based on SO(3) decomposition while more useful work for hydrodynamics should be based on SO(2) decomposition, where longitudinal direction of the spatial direction is distinguished. For this purpose, Kovtun and Starinets worked out the decoupling problem based on SO(2) for the AdS black hole case [18]  In the previous paper [20], some of us extended this work to charged case using the Reissner-Nordström-Anti-deSitter (RN-AdS) black hole, which corresponds to the diagonal (1, 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][22][23][24][25]. In [21,22], thermodynamics for STU black hole [26,27] and the hydrodynamic calculations for the (1, 0, 0) charge were performed. In [23,24], charged AdS 5 and AdS 4 black hole backgrounds were considered, respectively, and it was shown numerically that the ratio (η/s) was 1/(4π) with very good accuracy. Later, it was also was not done due to difficulties caused by extra mixing between various scalar modes in charged AdS black hole. In this paper, we work out the decoupling problem and hydrodynamics for the sound (scalar) mode in such case. Green functions are explicitly obtained.
Our results show that the behavior of the transport coefficients in RN-black hole are very different from those in the (1,0,0) charged black hole: the formers are much more smoother than the latters. We find that Einstein relation among the conductivity, the diffusion constant and the susceptibility holds exactly. This paper is organized as follows: In Section 2, we introduce RN-AdS black hole and review correlation function calculation at finite temperature in AdS/CFT correspondence. In Section 3, a formulation on the metric and the gauge perturbations in RN-AdS background is considered. We then solve linearized perturbative equations of motion in hydrodynamic regime and obtain retarded Green functions in Section 4. We also observe the transport coefficients including the speed of sound, the diffusion constant for U(1) charge and the electrical conductivity in this section. Conclusions and discussions are given in the final section. Three appendices are provided. In Appendix A, we summarize the results in the vector and the tensor type perturbations in our previous work [20]. The details of calculations to solve equations of motion are given in Appendix B and C.
2 Basic Setup 2.1 Minkowskian correlators in AdS/CFT correspondence Before introducing RN-AdS black hole, we briefly summarize to obtain Minkowskian correlators in AdS/CFT correspondence. We follow the prescription proposed in [14]. We work on the five-dimensional background, where x µ and u are the four-dimensional and the radial coordinates, respectively. We refer the boundary as u = 0 and the horizon as u = 1. A solution of the equation of motion may be given,

2)
proven that the ratio might be universal in more general setup [25].
where f k (u) is normalized such that f k (0) = 1 at the boundary. An on-shell action might be reduced to surface terms by using the equation of motion, Here, the function G(k, u) can be written in terms of f ±k (u) and ∂ u f ±k (u). Accommodating Gubser-Klebanov-Polyakov/Witten relation [3,4] to Minkowski spacetime, Son and Starinets proposed the formula to get retarded Green functions, where the incoming boundary condition at the horizon is imposed. In general, there are several 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 that essential ingredients are perturbed metric field and U(1) gauge field. Here we define the precise form of the retarded Green functions which we discuss later: where the operators T µν (x) and J µ (x) are energy-momentum tensor and U(1) current which couple to the metric and the gauge field, respectively.

Reissner-Nordström-AdS background
The charge in RN-AdS black hole is usually regarded as R-charge of SUSY [28]. We here consider an another interpretation in the following way: One can introduce quarks and mesons by considering the bulk-filling branes in AdS 5 space. The overall U(1) of the flavor branes is identified as the baryon charge. The U(1) charge in this model [29] minimally couples 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 5 space. A charged black hole (RN-AdS black hole) is then induced by its back reaction. Therefore the U(1) charge in RN-AdS can be identical to the baryon charge. As a result, we can interpret our result as a calculation of the transport coefficients in the presence of the baryon density.
The effective action of this gauge field is given the quadratic piece of Dirac-Born-Infeld where the gauge coupling constant e is given by [29] l with l the radius of the AdS space. We pick up an overall U(1) part of this gauge field in order to consider the baryon current at the boundary. Together with the gravitation part, we arrive at the following action which is our starting point: where we denote the gravitation constant and the cosmological constant as κ 2 = 8πG 5 and Λ, respectively. The U(1) gauge field strength is given by The gravitation constant is related to the gauge theory quantities by where energy-momentum tensor T mn (x) is given by An equation of motion for the gauge field A m (x) gives Maxwell equation, Here we assumed that there is no electromagnetic source outside the black hole. One can confirm that the following metric and gauge potential satisfy the equations of motion (2.10) and (2.12), (2.14) It should be noted that a ratio of the gauge coupling constant e 2 to the gravitation constant Since the gauge potential A t (x) must vanish at the horizon, the charge Q and the chemical potential µ are related † . The parameters m and q are the mass and charge of AdS space, respectively. This is nothing but Reissner-Nordström-Anti-deSitter (RN-AdS) background in which we are interested throughout this paper.
The horizons of RN-AdS black hole are located at the zero for f (r) ‡ , where their explicit forms of the horizon radiuses are given by (2.17c) † The chemical potential µ can be expressed by using gauge invariant quantity as where r + and ∞ represent the horizon and the boundary, respectively. This definition gives thermodynamic relations consistently. ‡ In order to define the horizon, the charge q must satisfy a relation q 4 ≤ 4m 3 l 2 /27.
and satisfy a relation r 2 + + r 2 − = −r 2 0 . The positions expressed by r + and r − correspond to the outer and the inner horizon, respectively. It is useful to notice that the charge q can be expressed in terms of θ and m by The outer horizon takes a value in where the upper bound and the lower bound correspond to the case for q = 0 and the extremal case, respectively.
We shall give various thermodynamic quantities of RN-AdS black hole [28,29]. The temperature is defined from the conical singularity free condition around the horizon r + , where we defined the parameters a and b by a ≡ q 2 l 2 r 6 In the limit q → 0, these parameters go to a → 0, b → l 3/2 2m 1/4 , and the temperature becomes The entropy density s, the energy density ǫ, the pressure p, the chemical potential µ and the density of physical charge ρ can be also computed as In order to obtain a well-defined boundary term from the gravitational part, we have to add the Gibbons-Hawking term into the action, which is given by 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 also need to add counter terms to regularize the action [30],

Perturbations in RN-AdS Background
In RN-AdS background, we study small perturbations of the metric g mn (x) and the gauge field A m (x), where the background metric g m (x) are given in (2.13a) and (2.13b), respectively. In the metric perturbation, one can define an inverse metric as and raise and lower indices by using the background metric g mn (x) and g (0)mn (x). Let us now consider a linearized theory of the symmetric tensor field h mn (x) and the vector field A m (x) propagating in RN-AdS background. We shall work in the h rm (x) = 0 and A r (x) = 0 gauges and use the Fourier decomposition where we choose the momenta which are along the z-direction. In this case, one can categorize the metric perturbations to the following three types by using the spin under the SO(2) rotation in (x, y)-plane [15]: • vector type: h tx = 0, h zx = 0, (others) = 0 equivalently, h ty = 0, h zy = 0, (others) = 0 • tensor type: h xy = 0, (others) = 0 equivalently, h xx = −h yy = 0, (others) = 0 • scalar type: h tt = 0, h tz = 0, h xx = h yy = 0, and h zz = 0, (others) = 0 First two types of the perturbations were studied in [20]. We list the result in Appendix A. In this paper we consider the scalar type perturbation.

Linearized equations of motion
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 independent variables are In the hydrodynamic regime, it is standard to introduce new dimensionless coordinate u = r 2 + /r 2 which is normalized by the outer horizon. In this coordinate system, the horizon and the boundary are located at u = 1 and u = 0, respectively. We also define new field variables where µ is the chemical potential given by (2.23). Nontrivial equations in the Einstein where the prime implies the derivative with respect to u. On the other hand, in the Maxwell equation (2.12), t, u and z-components give nontrivial contributions The equations (3.3a) and (3.3b) imply (3.3c). In the set of equations for the metric perturbation (3.2a)-(3.2g), together with (3.3a) and (3.3b), the following four independent relations are obtained: The equations of motion (3.2a)-(3.2g) can be derived by using the above relations. Taking the limit q → 0, the relations (3.4a)-(3.4d) coincide with the result in [18].

Surface terms
Before solving the equations of motion, we shall give a surface action in oder to obtain Green functions. By using the equations of motion, bilinear parts of on-shell action (2.8) reduce to surface terms: . (3.5) Relevant terms of the Gibbons-Hawking term (2.25) are explicitly given by The counter term (2.26) also can be evaluated as up to O(ω 2 , k 2 , ωk).

Pole Structures and Transport Coefficients from Hydrodynamics
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 hydrodynamic regime. In this regime we could obtain the sound velocity, the diffusion constant for U (1) charge and the electrical conductivity from retarded Green functions.

Master variables
By using master variables derived by Kodama and Ishibashi in [19], the following field Φ(u) is introduced: For the gauge field, the corresponding variable is given by In terms of these new variables Φ(u) and A(u), Einstein equations (3.4a)-(3.4d) and Maxwell equations (3.3a) and (3.3b) can be combined as follows: (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: where the coefficients α ± and β are with the constants As a result, we can obtain second order ordinary differential equations in term of these new variables, where potentials V ± (u) are given by (4.6) Considering the perturbative expansion with respect to small ω and k, it might be convenient to introduce new variables Φ ± (u), where the factors F ± (u) are so that the second order differential equations (4.5) are reduced to be much simpler forms to solve, where the potential V ± (u, ω, k) is newly defined from the original potential V ± (u, ω, k),

Perturbative solutions
Let us proceed to solve the differential equations. First we consider the equation for Φ + (u). Following the usual way to solve differential equations, we impose a solution as where T is the temperature defined by eq. (2.18). We here choose (4.10) as the incoming wave condition. We are now in the position to solve the equation of motion in the hydrodynamic regime. We start by introducing the following series expansion with respect to small ω and k: where F +0 (u), F +1 (u) and G +1 (u) are determined by imposing suitable boundary conditions. The solution can be obtained recursively * . The result is as follows: Next, we shall study the equation of motion for Φ − (u). Assuming again Φ − (u) = is a regular function at u = 1, the singularity might be extracted.
We fix the constant as ν = −iω/(4πT ) to use the incoming wave condition. We now impose a perturbative solution as and then we obtain the following result † : (4.14d) Using these, we can get behaviors for the solutions of Φ + (u) and Φ − (u) around the boundary u = 0, where the constant D − is Let us now consider the integration constants C andC. These could be estimated in terms of boundary values of the fields Using equations of motion, the integration constants C and C are determined as In the equations (4.16a) and (4.16b), one can observe the existence of the sound and diffusion poles in the complex ω-plane.

Retarded Green functions
Let us evaluate the Minkowskian correlators. The relevant action is given by the sum of three parts (3.5), (3.6) and (3.7), . (4.17) Using equations of motion and solutions of Φ + (u) and Φ − (u), derivatives of h's and B's can be expressed in terms of their boundary values: Substituting these expressions to the surface term (4.17), we can read off the Green functions defined by (2.5). Through the counter terms, the singularities around the boundary vanish completely. The results are listed below in Table 1, 2 and 3.
In the final expression we rescaled the gauge field (B µ ) 0 to the original one (A µ ) 0 = 4Qb 2 l 4 (B µ ) 0 and raised and lowered the indices by using the flat Minkowski metric η µν = Table 1: Table 2: Table 3: diag(−, +, +, +) in four-dimensional boundary theory. Taking the limit which the charge goes to zero, the correlators for energy-momentum tensors coincide with the known ones in [15]. In this limit, the correlators for the energy-momentum tensor and the U(1) current vanish, while ones for the U(1) currents have no sound poles, as we could see in the case of vector type perturbation for (1, 0, 0) R-charged [22] and RN-AdS black holes [20].
Remember l 3 /κ 2 = N 2 c /(4π 2 ) and it should be noticed that the factor l/e 2 = N 2 c /(16π 2 ) for the R-charge since e = 2κ/l in that case, while l/e 2 = N c N f /(4π 2 ) for the brane charge [29].

Transport coefficients
From the obtained Green functions, we can observe the value of the speed of sound without the medium effect, 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 diffusion constant can be read off (4.20) It should be compared with the diffusion constant for gravitation fields D H obtained in the previous work [20], so that the relation between them is In the chargeless case, we can reproduce the result in [15].
The electrical conductivity σ of the medium could be also determined by the currentcurrent correlators via Kubo formula, where e E is a four-dimensional gauge coupling. Together with the result for vector type perturbation [20], we can obtain We can also access to the charge susceptibility Ξ defined by . (4.23) Using Green function G t t (k) which might give an expectation value of Q 2 , one can obtain the following relation in thermal equilibrium, where n b (ω) = 1 e ω/T −1 is Bose-Einstein distribution function [31]. From Table 3, we can see It should be noted that a quantity D A k 2 / ω 2 + (D A k 2 ) 2 approaches to 2πδ(ω) for k → 0 limit. Taking the relation (4.23) into account, we can read off the charge susceptibility Ξ as Ξ = l e 2 We can then observe that Einstein relation holds exactly. (See also [32].) ‡ In R-charge case, taking the charge-free limit, the electric conductivity and the charge susceptibility coincide with the results in [31].
It is interesting to express physical constants in terms of the boundary variables: the temperature and the chemical potential. In fact, it is easy to verify that whereμ ≡ κ √ 3πel µ. Notice that for the R-charge,μ = 1 2 √ 3π µ, while for the brane charge,μ = N f 3N c π 2 µ. The behaviors of the diffusion constants D A and D H , the electrical conductivity σ and the charge susceptibility Ξ are drawn as functions of theμ/T in Figure 1, 2, 3 and 4 respectively. Notice that for the fixed temperature, all of them are decreasing functions 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]. It is particularly interesting to observe that the charge susceptibility modulo T 2 factor, which is an indicator of the degree of freedom, shows rapid change between low and high temperature (density) indicating a mild phase transition. Such behavior does not exist for chargeless case. See also [33].

Conclusions and Discussions
In this paper, we worked out the decoupling of scalar modes of the charged AdS black hole background in SO(2) basis for the mode classifications. We also perform the hydrodynamic analysis for the holographic Quark-Gluon Plasma system. Master equations for the decoupled modes are worked out explicitly. The sound velocity is not modified by the presence of the charge. We calculated the diffusion constants, the charge susceptibility and the conductivity as a consequence and observed that Einstein relation holds between them.
These transport coefficients are modified due to the charge effect. Interestingly, the susceptibility modulo T 2 factor, which is an indicator of the degree of freedom, shows rapid change 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 interpretation, the Maxwell fields are the fluctuations of bulk-filling branes, therefore they should be interpreted as master fields of the mesons. Then hydrodynamic modes are lowest lying massless meson spectrum. In terms of brane embedding picture, this massless-ness is due to the touching of the brane on the black hole horizon. Near the horizon, the tension of the brane is zero due to the metric factor and it can lead to the massless fluctuation. Then the 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. We will discuss the spectrum of meson mode by considering the quasinormal mode [34] of the vector modes in elsewhere.

Appendix A Results for the vector and the tensor type perturbations Appendix A.1 Vector type perturbation
In the vector type perturbation, independent variables are One can observe the diffusion constant for the metric perturbation D H as .

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 where the retarded Green function is given by A general solution is given by 2) * If one consider higher derivative corrections to Einstein gravity, the viscosity bound could be modified [35][36][37][38][39].
Constants of integration C 0 and D 0 should be determined to be a regular function at the horizon. So we here choose D 0 = 0 and set By using this solution, one can get an equation for F +1 (u) from the equation (4.8), Removing the singularity at the horizon, the constant D 1 should be We also impose a boundary condition F +1 (u = 0) = 0 so as to fix the constant C 1 .
Therefore the final form is A general solution is and the constant D 1 can be fixed as From the condition (uG +1 ) ′ | u=0 = 0, we can fix the constant C 1 . The final form of the solution becomes . (B.10) Integrating over u, we have and the constant D 2 can be fixed as Hence, we can write the solution of F +2 (u) as which satisfies a boundary condition F +2 (0) = 0.
Appendix C Perturbative Solutions for Φ − where C 2 is an integration constant.