Transport Coefficients for Holographic Hydrodynamics at Finite Energy Scale
We investigate the relations between black hole thermodynamics and holographic transport coefficients in this paper. The formulae for DC conductivity and diffusion coefficient are verified for electrically single-charged black holes. We examine the correctness of the proposed expressions by taking charged dilatonic and single-charged STU black holes as two concrete examples, and compute the flows of conductivity and diffusion coefficient by solving the linear order perturbation equations. We then check the consistence by evaluating the Brown-York tensor at a finite radial position. Finally, we find that the retarded Green functions for the shear modes can be expressed easily in terms of black hole thermodynamic quantities and transport coefficients.
The experiments at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC) show that the quark-gluon plasma (QGP) does not behave as a weakly coupled gas of quarks and gluons, but rather as a strongly coupled fluid. This places limitations on the applicability of perturbative methods. The AdS/CFT correspondence provides a powerful tool for studying the dynamics of strongly coupled quantum field theories [1–5]. Moreover, the result of RHIC experiment on the viscosity/entropy ratio turns out to be in favor of the prediction of AdS/CFT [6–8] and some attempt has been made to map the entire process of RHIC experiment in terms of gravity dual .
Aimed to develop a model independent theory of the hydrodynamics, the membrane paradigm and a holographic version of Wilsonian Renormalization Group (hWRG) have been proposed to describe strongly coupled field theories with a finite cutoff [10–19]. The radial flow in the bulk geometry can be regarded as the renormalization group flow of the boundary and the radial direction marks the energy scale of the boundary field theory [5, 20–29] and it was found in  that the holographic renormalization group may also lead to instable AdS background. For neutral black hole duals, some universal transport coefficients of the generic boundary theory can be expressed in terms of geometric quantities evaluated at the horizon . It was later proved that the expressions given by the membrane paradigm are universal for the various neutral black holes.
On the other side, we know that the calculation of the linear response functions (i.e., the retarded Green functions) of a strongly coupled system is very important but obviously a tough task. Even for the transport coefficients of translational-invariant charged black holes, the complete solutions of the shear modes and sound modes are coupled together and hard to be solved [11, 15]. At finite momentum, it has been found that the equation of motion for the shear modes of the charged black hole in higher derivative gravity turned out to be impossible to decouple , let alone the sound modes. However, compared with the tedious calculation of the linear response, the black hole thermodynamics seems simple and concise. The black hole no-hair theorem asserts that all black hole solutions in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum.
A natural question is that why the linear response functions of a two parameters (mass and charge) system is so complicated to deal with and do we have a more general and powerful method to deal with the linear type perturbation of charged black holes? We are going to present a positive answer.
In [11, 15], the membrane paradigm and the hWRG approach were utilized to investigate the transport coefficients of the charged AdS black hole at an arbitrary cutoff surface between the horizon and the boundary. In particular, it was found in  that the diffusion coefficient for Reissner- Nordstrom-Anti de Sitter (RN-AdS) black holes  can be computed in the scaling limit without explicit decoupling procedure. In , they applied the hWRG approach to the Einstein-Maxwell-Gauss-Bonnet theory and utilized a formula for DC conductivity at an arbitrary cutoff surface for charged black holes where denotes the gauge coupling, and are the determinant and metric components, and , , , and are the entropy density, temperature, energy density, and pressure evaluated at the cutoff surface, respectively. Formula (1) reflects the fact the conductivity varies with the sliding membrane, and there is a RG flow between the horizon and the boundary. For neutral black holes, conductivity (1) recovers the result given by Iqbal and Liu  because of the Euler relation .
Our logic of this paper is as follows: when a system is perturbed slightly, its response will be linear in the perturbation and this regime is called the linear response regime, although the system is in a nonequilibrium state of which all characteristics can be inferred from the properties of its equilibrium state. Because all the scalar, shear, and sound modes are linear response to small perturbations to the black hole thermodynamic equilibrium state, all the transport coefficients can be determined by the black hole thermodynamic variables in its equilibrium state.
The purpose of this paper is to verify the unified form of DC conductivity and diffusion constant for translational-invariant hydrodynamics with a chemical potential, as a first step towards general formulae of transport coefficients of anisotropic and inhomogeneous hydrodynamics. Formula (1) holds for most single charged black holes (For multicharge black holes, the conductivity should be a matrix and has been investigated in dual rotating D3, M2, and M5 brane by Jain in .) and we can rewrite (1) in terms of the metric components. So, if somebody knows the black hole solution, the DC conductivity can be computed directly by using where is the bulk spacetime dimensions and the prime “” hereafter denotes the derivative with respect to .
As to the diffusion coefficient, we also conjecture the following formula for holographic hydrodynamics with a chemical potential: where is determinant of the metric.
After that, we will prove that actually is the solution of the transverse mode perturbation equation of the gauge field in the zero frequency and zero momentum limit. As it was noticed that the charge makes the hydrodynamics analysis complicated [15, 16] because vector modes of gravitational fluctuation mix with transverse Maxwell modes.
In fact, formula (1) has been proposed in the previous paper . In that paper, the author investigated the electrical conductivity and thermal conductivity by using the proposed formulae for RN black hole and charged Lifshitz black hole. However, the author did not mention how to calculate the diffusion coefficient. In this work, we are going to calculate the transport coefficients for charged dilatonic black hole and R-charged black hole. In addition, the formulae of diffusion coefficient and retarded Green functions will be proposed in a generalized way.
Before going on, let us summarize the new features and the main result of this paper.(i)The unified form of the diffusion coefficient and the retarded Green functions are proposed for the first time.(ii)By evaluating the black hole thermodynamic quantities as a function of radial coordinate, we can write down the transport coefficients and the retarded Green functions in terms of black hole metric line-element and black hole thermodynamic variables. This result implies that there exists deep connection between black hole thermodynamics and the linear response functions.(iii)Our work can be regarded as a first step towards easy computation of the transport coefficients with respect to the sound modes and holographic lattice in which partial differential equations are involved.
The contents of this paper are organized as follows. In Section 2, we will consider the shear modes of charged dilatonic black holes. At first part of this section, we will review the black hole geometry and thermodynamics. By using the RG flow approach developed in , we will compute the conductivity and diffusion coefficients at a cutoff surface. Then we will provide a consistent check on the result by using the black hole thermodynamic relation and the Brown-York tensor. The unified form of the retarded Green functions evaluated on the boundary related to the shear modes will be presented in the Appendix. In Section 3, we will work on the single-charged STU black holes. The conclusion will be presented in the last section.
2. Charged Dilatonic Black Hole
In this section, we study the transport coefficients and the RG flow of holographic hydrodynamics for the charge dilatonic black holes. We will first review the black hole geometry and thermodynamics. Then, we will compute the transport coefficients.
2.1. Backgrounds and Thermodynamics
We start by introducing the following action for charged dilatonic black hole in with mixing dilatonic field and gauge field : where is the radius of the space and plays the role of a dilaton with respect to the radial coordinate . We denote the gravitational constant as and the gauge field strength is given by .
In this paper, we will introduce a dimensionless coordinate for simplicity. The spatially uniform, electrically charged solution for this action  can be obtained as follows: where
The horizon of the black hole locates at where is a constant related to the mass with the form
The black hole is extremal if , which implies .
The equation of motion for the gauge field is given by
The Einstein equation is written as where
The equation of motion for the scalar field is
The Hawking temperature yields
The volume density of Bekenstein-Hawking entropy is given by
According to thermodynamic relation and the first law of thermodynamics we can obtain the energy density and the pressure, which can be expressed as
The charge density and the chemical potential are 
Furthermore, there are several relations satisfied by the thermodynamic variables
The temperature and chemical potential can also be obtained as
The susceptibility can be calculated which is given by
According to the definition of the special heat
In particular, when the temperature of the charged dilatonic black hole becomes very low, there is a linear special heat which can be expressed as
2.2. Perturbations and Transport Coefficients
Considering the metric perturbations  to the background and , one can use background metric and inverse metric lower and raise tensor indices. The inverse metric can be expressed as .
We choose the momentum along the -direction and as the radial direction which describes the energy scale in field theory. In the gauge and by using Fourier decomposition we will consider the scalar mode and shear mode , and in the following.
2.2.1. Scalar Mode and Shear Viscosity
From symmetry analysis, one can find that off-diagonal perturbation decouples from all other perturbations. We obtain an equation of motion for the scalar mode as
Following the sliding membrane argument , we define a cutoff dependent tensor response function
We define the shear viscosity as
At zero momentum limit, the flow equation is given by
The shear viscosity is requested to be because of the horizon regularity. Because the entropy density was given in (14), we can easily check the shear viscosity to entropy density ratio
There are some debate on whether the shear viscosity flow depends on the position of the cutoff surface [11, 15]. From the fluid-gravity computation , both the shear viscosity and the entropy density depend on the cutoff position, but their ratio does not.
Actually, if we do not consider the quantum corrections to the geometry, the radial evolution of the total entropy remains a constant in nature. Therefore, we can see that the entropy density must depend on the radial coordinate. Isentropic evolution equation  where and is the -dependent volume. We can derive -dependent entropy density from the thermodynamical relation (15). Multiplying (15) with the volume and considering the derivative of the total entropy along the radial direction, we obtain
The right hand side of the above equation is exactly a component of Einstein equation. Here is the bulk matter stress tensor and is any null vector tangent to the cutoff, implies the null energy condition. So, we can turn around to state that the radial Einstein equation implies the isentropic character, the total entropy keeps a constant in everywhere (32). In this sense, the entropy density is evaluated as
We will follow  and assume that (obeys the KSS bound ) at an arbitrary cutoff surface (The KSS bound is violated by higher derivative gravity (see [39–42] and references there in). The holographic RG flow in such gravity was done in ). Under such an assumption, we find the shear viscosity
2.2.2. Shear Modes: , , and
The linear perturbative Einstein equation can be read off from the , , and components, respectively, where the prime denotes the derivative with respect to . Among these three equations for vector modes, there are only two independent equations, because (37) is a constraint equation. From the Maxwell equation (9), the -component gives
One shall notice that in the limit of zero momentum, the equations for the metric, and the gauge perturbations are completely decoupled.
For convenience of calculation, we would like to define “current” and “strength” for the vector field and as follows
We also introduce the effective coupling and
The vector part off-shell action in charged dilatonic black hole can be written as follows
Note that is the strength of the Maxwell fields and should not be confused with the effective strength of the shear modes of gravity .
One the other hand, we must introduce a new current related to
Then we can define the shifted current
The Bianchi identity holds as
For the same reason, we can define and as where
The equation of motion for can be written as
In the zero momentum limit, the equation of motion for decouples from
The relevant on-shell action for at boundary can be written as
2.2.3. DC Electric Conductivity
By defining we have
So the flow equation (51) for electric conductivity can be rewritten as
The regularity condition at the horizon requires
On the other hand, the DC conductivity can be evaluated by using the Kubo formula
From (56), we know the regularity at the horizon corresponds to
At the horizon , the above formula becomes which agrees with (56). In the boundary , the DC conductivity is reduced to
We can see from Figure 1 that the DC conductivity on the boundary depends on the charge . The slope of lines is proportional to the charge .
As a side note, we will check whether the transport coefficients calculated satisfies the Einstein relation . By using the relation which the value of susceptibility is defined by (21), we find the expression for can be written as
We will see later that the right hand of equation is not the diffusion constant and, thus, the Einstein relation is not satisfied. In the absence of the chemical potential (i.e., ), the above equation becomes with .
As a byproduct, we calculate the thermal conductivity by using the relation
The ratio can be computed as
2.2.4. Diffusion Coefficient
Now we are going to calculate the diffusion coefficient. Let us evaluate the “conductivity” introduced by the metric perturbation at any momentum. The conductivity is defined as
In the absence of the momentum, the decoupled flow equation for yields
Again, the regularity condition at the event horizon gives
For the fields and equations of motion, we treat the vector modes in a long wave-length expansion. We will find that the diffusion constant depends on , charge, and dilaton field. The equation of motion of is coupled with other modes when momentum is nonzero. To proceed, we take the scaling limit for temporal and spatial derivatives as
The in-falling boundary condition at the horizon implies is linearly related to . So, we have
Through the charge conservation equation (45), we have
To zeroth order, (46) can be reduced to
The zeroth order of the Bianchi identity becomes
Integrating the above equation from to the horizon , we obtain
For the gauge perturbation , the Maxwell equation in the scaling limit becomes
Following the sliding membrane paradigm , we defined the momentum conductivity by the current and electric fields as
Note that the dimensional can be set to any value by a coordinate transformation. So it is necessary to obtain a dimensionless diffusion constant in the following sections.
We can define the proper frequency and the proper momentum on the hypersurface which is conjugate to the proper time and the proper distance, respectively. By using the Tolman relation, we can obtain the Hawking temperature at the cutoff surface which is .
We define a dimensionless diffusion constant which is coordinate-invariant as
Thus, in terms of the normalized momentum and the diffusion constant, the conductivity can be expressed as where the dimensionless diffusion coefficient is given by (see Figure 2)
Figure 2 shows the dimensionless diffusion coefficient of charged dilatonic black hole runs for different charges on cutoff surface. The dash line shows the dimensionless diffusion coefficient is a constant when . On the boundary, different charge gives different diffusion constants while at the horizon , all these lines end at the same value . Note that the slope of these lines is also proportional to the charge .
2.3. Transport Coefficients from Brown-York Stress Tensor
In this section, we will provide a consistent check by using the black hole thermodynamics. We will verify that by using the formula one can easily obtain the DC conductivity. For -dimensional charged black holes, the dimensionless diffusion coefficient can be written in a simple form where
On an arbitrary cutoff surface outside the horizon, there is a thermodynamic description of the fluid dual to the background configuration (5). The Brown-York tensor is defined as  with the extrinsic curvature
The induced metric can be expressed by bulk metric and the normal vector in direction, so where is a unit normal vector and the trace of extrinsic curvature is given by . One may note that the indices of the bulk and the indices live on the cutoff surface. By calculating with the metric and induced metric, we obtain
The stress tensor at an arbitrary cutoff surface can be written as
On the other hand, the stress-energy tensor of a relativistic fluid in equilibrium is where denotes the energy density, is the pressure, and is the normalized fluid four-velocity. For simplicity, we can redefine the energy density and pressure which leaves the combination invariant. In this sense, we can omit the constant in the Brown-York tensor (Note that is the counterterm of the action and essential for the regularity of as ). Comparing (100) with (101), we find
Notice that the entropy density is given by
The local Hawking temperature is given by
We can express thermodynamic variables in terms of metric components
By detailed calculation, we can obtain the value
Apparently, one can verify the following identity by introducing the quantities
Finally, we verify that
In concrete, it can be written as
The above result agrees with (65).
Moreover, we can verify the diffusion constant by using where is to be obtained from (35). Note that throughout the paper, we consider the case in which the KSS bound is satisfied. In terms of the metric component, the diffusion coefficient can be expressed as
For our case, the diffusion coefficient is given by
This is consistent with (92). As , the diffusion constant becomes .
3. R-charged Black Holes
The hydrodynamics of R-charged black holes was studied by several authors [43, 45–48]. The retarded Green functions were evaluated on the boundary and the boundary theory transport coefficients were computed up to first and second orders. In order to prove that the formula conjectured for transport coefficients has their universal applications, we will compute the RG flow of the single-charged black holes in the following.
We know the effective Lagrangian of a single charged black hole can be written as  where the potential for the scalar field and is
The single-charged metric and gauge fields are given by where with denoting the R-charge while denotes the Hawking temperature of neutral black hole. The energy density, pressure, and entropy density are given by
The charge density and conjugated chemical potential are
The Newton constant is given by . The system of the gauged supergravity equations of motion for the fields , , and reads
The vector type perturbation takes the form where and are dimensionless frequency and momentum. We are interested in gravitational fluctuations of the shear type, where the only nonzero components of are , , . One can show that fluctuations of all other fields except , , can be consistently set to zero. Introduce the new variables
The above four equations are not independent. Combining (124a) with (124b), one obtains (124c). Thus it is sufficient to consider (124a), (124b), and (124d). In the zero momentum limit, (124a) becomes
Now we introduce and to denote the current and strength for the vector modes
By further define we can recast the equations of motion as
The Bianchi identity holds as
Similarly we can define and as where
The equation of motion for can be written as
One can see that decouples from in the limit
3.1. DC Conductivity
By defining we rearrange (134) as the flow equation for electric conductivity
We can immediately write down the regularity condition at the horizon
It is convenient to define the radial momentum as
The equation of motion for then takes the form  where
The regularity at the horizon corresponds to
Finally we can obtain the DC conductivity at the cutoff surface (see Figure 3)
Figure 3 shows the DC conductivity of single charged black hole runs on a cutoff surface. We can see that for different charges, the DC conductivity on the boundary and at the horizon is different.
3.2. Shear Viscosity
We can write the same equation of motion according to minimally coupled massless scalar as
The flow equation for shear viscosity is the same as in the previous section
On the horizon, the regularity gives
The entropy density is
So the shear viscosity to entropy density ratio . Following , we assume that this ratio does not change with cutoff surface. Under this assumption, we obtain the shear viscosity of any cutoff surface as
On any cutoff surface, the entropy density can be expressed as