# Transport Coefficients for Holographic Hydrodynamics at Finite Energy Scale

Authors: Xian-Hui Ge; Hong-Qiang Leng; Li Qing Fang; Guo-Hong Yang
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

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 [9].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 [30] 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 [14]. 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 [16], 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 [15] that the diffusion coefficient for Reissner- Nordstrom-Anti de Sitter (RN-AdS) black holes [31] can be computed in the scaling limit without explicit decoupling procedure. In [16], 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
(1)σx(uc)=1q2(u)-gguugttgxx|u=1(Tcscε+P)2,
where q2(u) denotes the gauge coupling, g and gμν are the determinant and metric components, and s, Tc, ϵ, and P 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 [14] because of the Euler relation ε+P=Ts.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 [32].) 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
(2)σDC(uc)=Ω(1)Ϝ2(uc),Ω(1)=(gxx(5d-2)/4gtt′q(u)-g)2|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc,
where (d+2) is the bulk spacetime dimensions and the prime “′” hereafter denotes the derivative with respect to u.As to the diffusion coefficient, we also conjecture the following formula for holographic hydrodynamics with a chemical potential:(3)D-(uc)=14πΞ(1)Ϝ(uc),Ξ(1)=-gxxdgtt′-g|u=1,
where  -g is determinant of the metric.After that, we will prove thatϜ(u)∝(ϵ+P)/Ts 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 U(1) 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 [33]. 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 Section2, 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 [11], 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 inAdS5 with mixing dilatonic field and U(1) gauge field [34]:
(4)L=12κ2[R-14e4αFμν2-12(∂μα)2+1L2(8e2α+4e-4α)],
where L is the radius of the AdS space and α plays the role of a dilaton with respect to the radial coordinate r. We denote the gravitational constant as κ2=8πG5 and the U(1) gauge field strength is given by Fμν(x)=∂μAν(x)-∂νAμ(x).In this paper, we will introduce a dimensionless coordinateu=r0/r for simplicity. The spatially uniform, electrically charged solution for this action [34] can be obtained as follows:
(5)ds2=e2A(-hdt2+dx→2)+e2Br02hu4du2,Aμdxμ=ϕdt,
where
(6)A=log⁡⁡r0uL+13log⁡⁡(1+Q2u2r02),B=-log⁡⁡r0uL-23log⁡⁡(1+Q2u2r02),h=1-u4mL2(r02+Q2u2)2,ϕ=Qu22mr02+Q2u2-Q2mr02+Q2,α=16log⁡⁡(1+Q2u2r02).The horizon of the black hole locates at(7)r0=Lm-Q2,
where m is a constant related to the mass with the form
(8)m=(Q2+r02)2L2.The black hole is extremal ifr0=0, which implies m=Q4/L2.The equation of motion for the gauge fieldAμ(x) is given by
(9)∂ν(-ge4αFμν)=0.The Einstein equation is written as(10)Rμν-12gμνR=Gμν+Tμν,
where
(11)Gμν=12e4αFμρFνρ-18e4αgμνF2,Tμν=12(∂μα∂να)-6gμν(∂α)2+12L2gμν(8e2α+4e-4α).The equation of motion for the scalar field is(12)-34∇2α+1L2(e2α-e-4α)=116e4αF2.The Hawking temperature yields(13)T=h′u24πr0eA-B|u=1=r0πL2.The volume density of Bekenstein-Hawking entropy is given by(14)s=e3A4G5|u=1=2r0πκ2L3(Q2+r02).According to thermodynamic relation(15)ε+P-sT-μρ=0,
and the first law of thermodynamics
(16)dε=Tds+μdρ,
we can obtain the energy density and the pressure, which can be expressed as
(17)ε=3(Q2+r02)22κ2L5,P=(Q2+r02)22κ2L5.The charge density and the chemical potential are [34]
(18)ρ=Q2κ2L3(Q2+r02),μ=2QL2.Furthermore, there are several relations satisfied by the thermodynamic variables(19)s=2π2LκT23Lε,sμ=2π2ρT,μ22+T2π2=mL3,ε=3κ2/327/3π4/3L(s2+2π2ρ2)2/3.The temperature and chemical potential can also be obtained as(20)T=(∂ε∂s)ρ,μ=(∂ε∂ρ)s.The susceptibility can be calculated which is given by(21)χ=(∂ρ∂μ)T=(Q2+r02)κ2L+2Q2κ2L.According to the definition of the special heat(22)Cμ=T(∂s∂T)|μ.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(23)Cμ=2π2Q2κ2LT.


### 2.2. Perturbations and Transport Coefficients

Considering the metric perturbations [35]
(24)gμν⟶gμν+hμν,Aμ⟶Aμ+δAμ,
to the background gμν and Aμ, one can use background metric gμν and inverse metric gμν lower and raise tensor indices. The inverse metric can be expressed as gμν=g(0)μν-hμν+O(h2).We choose the momentum along thez-direction and u as the radial direction which describes the energy scale in field theory. In the gauge Au(x)=0 and by using Fourier decomposition
(25)hμν(t,z,u)=∫d5k(2π)5e-iωt+ikzhμν(k,u),Aμ(t,z,u)=∫d5k(2π)5e-iωt+ikzAμ(k,u),
we will consider the scalar mode hxy(t,z,u) and shear mode htx(t,z,u), hzx(t,z,u) and Ax(t,z,u) in the following.


#### 2.2.1. Scalar Mode and Shear Viscosity

From symmetry analysis, one can find that off-diagonal perturbationhxy decouples from all other perturbations. We obtain an equation of motion for the scalar mode as
(26)∂μ(-g∂μhxy)=0.Following the sliding membrane argument [14], we define a cutoff dependent tensor response function
(27)Gxy(uc,k)=---gguu∂uhxy2κ2hxy(uc,k).We define the shear viscosity as(28)η(uc,k):=Gxy(uc,k)iω.At zero momentum limit, the flow equation is given by(29)∂ucη(uc,ω)=iω(2κ2η2(uc,ω)-gguu--ggtt2κ2).The shear viscosity is requested to be(30)η(r0)=r032κ2L3(1+Q2r02),
because of the horizon regularity. Because the entropy density was given in (14), we can easily check the shear viscosity to entropy density ratio
(31)ηs=14π.This result agrees with [35–37] and obeys the KSS bound [38].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 [11], 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 [11]
(32)∂uS=0,
where S=sV3 and V3=e3A is the u-dependent volume. We can derive u-dependent entropy density from the thermodynamical relation (15). Multiplying (15) with the volume V3 and considering the derivative of the total entropy along the radial direction, we obtain
(33)∂uS=Vp16πGr0TH(h′′+(4A′-B′+2u)h′)eA-Bu2-VpTHTμνζμζν.The right hand side of the above equation is exactly a component of Einstein equation. HereTμν is the bulk matter stress tensor and ζμ is any null vector tangent to the cutoff, Tμνζμζν≥0 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
(34)s(uc)=uc34G5(Q2+r02)(Q2uc2+r02).We will follow [11] and assume that η/s=1/4π (obeys the KSS bound [38]) 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 [16]). Under such an assumption, we find the shear viscosity
(35)η(uc)=uc32κ2(Q2+r02)(Q2uc2+r02).


#### 2.2.2. Shear Modes:htx(t,z,u), hzx(t,z,u), and Ax(t,z,u)

The linear perturbative Einstein equation can be read off from the(t,x), (u,x), and (x,z) components, respectively,
(36)0=htx′′+(2u+4A′-B′)htx′-r02u4he2B-2A(ωkhzx+k2htx)+ϕ′e4α-2AAx′,(37)0=khhzx′+ωhtx′+ϕ′e4α-2AωAx,(38)0=hzx′′+(2u+4A′-B′+h′h)hzx′+r02u4h2e2B-2A(ω2hzx+ωkhtx),
where the prime denotes the derivative with respect to u. 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 x-component gives
(39)0=Ax′′+(2u+2A′-B′+4α′+h′h)Ax′+r02u4h2e2B-2A(ω2-k2h)Ax+ϕ′hhtx′.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 fieldhtx and hzx as follows
(40)jμ=-1geff2(u)-gfuμ,fμν=∂μhνx-∂νhμx.We also introduce the effective couplingq2(u) and geff2(u)(41)G=-ggeff2(u),1q2(u)=e4α2κ2,1geff2(u)=gxx2κ2.The vector part off-shell action inAdS5 charged dilatonic black hole can be written as follows
(42)S=∫d5x-g(-14q2(u)FμνFμν-14geff2(u)fμνfμν+1q2(u)htxAx′Fut).Note thatFμν is the strength of the Maxwell fields Aμ and should not be confused with the effective strength of the shear modes of gravity fμν.One the other hand, we must introduce a new current related toAx(43)jλ=-1q2(u)-gFutAx.Then we can define the shifted current(44)j~t≡jt+jλ.In terms of the defined “current” and “strength,” the equations of motion (36)-(38) can be recast as
(45)∂tj~t+∂zjz=0,(46)∂uj~t+Ggttgzz∂zfzt=0,∂ujz-Ggttgzz∂tfzt=0.The Bianchi identity holds as(47)-guugzzG∂tjz+guugttG∂zjt+∂ufzt=0.For the same reason, we can definejx and j~x as
(48)jx≡-1q2(u)-gFux,j~x≡jx+jρ,
where
(49)jρ=1q2(u)-gFuthtx.The equation of motion forAx can be written as
(50)-∂uj~x+1q2(u)-ggxx(gtt∂tFtx+gzz∂zFzx)=0.In the zero momentum limit, the equation of motion forAx decouples from htx(51)∂u(1q2(u)-gguugxx∂uAx)-1q4(u)-ggeff2(u)guugtt(ϕ′)2Ax-1q2(u)-ggxxgttω2Ax=0.The relevant on-shell action forAx at boundary u=uc can be written as
(52)Son-shell=∫u=ucj~xAx.


#### 2.2.3. DC Electric Conductivity

By defining(53)σx=jxEx=jxFxt=jxiωAx,
we have
(54)∂uσx=∂ujxFxt-jx∂u(Fxt)Fxt2,σx2=(jx)2(Fxt)2.So the flow equation (51) for electric conductivity can be rewritten as
(55)∂uσxiω-q2(u)σx2-gguugxx+geff2(u)-gguugttϕ′2ω2q4(u)-1q2(u)-ggxxgtt=0.The regularity condition at the horizon requires(56)σx(u=1)=1q2(u)-gguugttgxx|u=1=r02κ2L(1+Q2r02).On the other hand, the DC conductivity can be evaluated by using the Kubo formula(57)σDC=-lim⁡ω→0Im⁡Gx,xR(ω,k=0)ω.The retarded Green’s function is given by [35, 37]
(58)Gx,xR(ω,k=0)=-i∫dtdxeiωtθ(t)〈[Jx(x),Jx(0)]〉,
where Jμ is the conformal field theory (CFT) current dual to the bulk gauge field Aμ. It is convenient to define the radial momentum as [16]
(59)Jkx=1κ2K(u)Ax′(u,k).The equation of motion (51) for Ax can be rewritten as [43]
(60)∂uJkx=1κ2L(u)Ax(u,k),
where
(61)K(u)=--gq2(u)gxxguu,L(u)=ω2-gq2(u)gxxgtt-geff2-ggttguuϕ′2q4(u).From (56), we know the regularity at the horizon corresponds to
(62)Jkx(1)=-iωK(u)guu-gttAx(1).The DC conductivity can be calculated by using the following relation [43]:
(63)σDC=-K(u)guu-gtt|u=1Ax(1)Ax(1)Ax(uc)Ax(uc),
where Ax(u) is the solution of (51). We can solve Ax(u) by imposing boundary condition at u=0 and set ω to zero, which leads to
(64)Ax(u)=Ax(0)(r02+Q2)2(r02+Q2uc2).Inserting (64) into (63), finally we can obtain the DC conductivity at the cutoff surface (see Figure 1)
(65)σDC(uc)=r02κ2L(1+Q2r02)-1(1+Q2uc2r02)2.Figure 1
The DC conductivity as a function of the radial coordinateu with different charges, where we set L=r0=1=κ2=1. The dashed line corresponds to Q=0.At the horizonuc=1, the above formula becomes
(66)σ(u=1)=r02κ2L(1+Q2r02),
which agrees with (56). In the boundary uc→0, the DC conductivity is reduced to
(67)σ(u=0)=r02κ2L(1+Q2r02)-1.Without the chemical potential, (65) reduces to the formula for DC conductivity given in [14].We can see from Figure1 that the DC conductivity on the boundary depends on the charge Q. The slope of lines is proportional to the charge Q.As a side note, we will check whether the transport coefficients calculated satisfies the Einstein relationσDC/χ|ω=0=Dc. By using the relation which the value of susceptibility is defined by (21), we find the expression for σDC/χ can be written as
(68)σDC(0)χ=r032(Q2+r02)(3Q2+r02).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 potentialμ=0 (i.e., Q=0), the above equation becomes σDC/χ|ω=0=Dc with Dc=1/2πT.As a byproduct, we calculate the thermal conductivity by using the relation(69)κT=(ϵ+PTHρ)2THσDC(0)=πr04κ2L3Q2(1+Q2r02).The ratioκTμ2/ηTH can be computed as
(70)κTμ2ηTH=4π2.


#### 2.2.4. Diffusion Coefficient

Now we are going to calculate the diffusion coefficient. Let us evaluate the “conductivity” introduced by the metric perturbationhzx at any momentum. The conductivity is defined as
(71)σh:=jzfzt.In the absence of the momentum, the decoupled flow equation forσh yields
(72)∂ucσh-iω+σh2guugzzG+Ggttgzz=0.Again, the regularity condition at the event horizon gives(73)σh(1)=r032κ2L3(1+Q2r02).Actually, the value of this conductivity is equivalent to shear viscosity. Noting that from (71) and (73), we can recast the the conductivity at the horizon as
(74)jz(1)=σh(1)fzt(1).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 onuc, charge, and dilaton field. The equation of motion of σh is coupled with other modes when momentum kz is nonzero. To proceed, we take the scaling limit for temporal and spatial derivatives as
(75)∂t~ϵ2,∂z~ϵ,fzt~ϵ3(fzt(0)+ϵfzt(1)+⋯).The in-falling boundary condition at the horizon impliesjz is linearly related to fzt. So, we have
(76)jz~ϵ3(jz(0)+ϵjz(1)+⋯).Through the charge conservation equation (45), we have
(77)j~t~ϵ2(j~t(0)+ϵj~t(1)+⋯),jt~ϵ2(jt(0)+ϵjt(1)+⋯).To zeroth order, (46) can be reduced to
(78)∂uj~t(0)=0,∂ujz=0.The first equation of (78) figures that
(79)j~t(0)=jt(0)-1q2-gguugtt∂uAtAx(0)=C0,
where C0 is a constant. The equation (45) implies
(80)jz(0)=j~t(0)ωk.The zeroth order of the Bianchi identity becomes(81)∂ufzt(0)+guugttG∂zjt(0)=0.Integrating the above equation fromu to the horizon u=1, we obtain
(82)fzt(0)(u)=fzt(0)(u)|u=1+2κ2∫u1du∂zjt(0)guugtt-ggxx.For the gauge perturbationAx(0), the Maxwell equation in the scaling limit becomes
(83)∂u(1q2(u)-gguugxx∂uAx(0))=1q2(u)-gguugtt∂uAthtx′(0).Following [11], imposing the boundary condition Ax(0)(uc)=0, and solving equation (79) and (83), we obtain
(84)Ax(0)(u)=Qr0(2Q2uc2+r02/u2+r02uc2)42(Q2+r02)2(u2uc2-1),
and substitute it into (79). Consequently, we have
(85)fzt(u)=f(0)(1)+ik(1-uc2)(2Q2uc2+(1+uc2)r02)r0L24(Q2uc2+r02)(Q2+r02).Following the sliding membrane paradigm [14], we defined the momentum conductivity by the current and electric fields as
(86)σh(uc)=jz(0)(uc)fzt(0)(uc)=ωkj~t(0)(uc)fzt(0)(uc).By using (85) and the boundary condition given in (74), we find the expression of the conductivity σh satisfying
(87)1σh(uc)=1σh(1)-k2iωD(uc)σh(1),
where the diffusion constant is given by
(88)D(uc)=r0L2(2Q2uc2+(1+uc2)r02)4(Q2uc2+r02)(Q2+r02)(1-uc2).Note that the dimensionalD(uc) 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 frequencyωc and the proper momentum kc on the hypersurface u=uc(89)ωc≡ω-gtt,kc≡kgii,
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 u=uc which is Tc(uc)=T/-gtt.We define a dimensionless diffusion constantD-(uc) which is coordinate-invariant as
(90)D-(uc)=D(uc)Tcgzz-gtt.Thus, in terms of the normalized momentum and the diffusion constant, the conductivity can be expressed as(91)σh(1)σh(uc)=1-kc2iωcD-(uc)Tc,
where the dimensionless diffusion coefficient is given by (see Figure 2)
(92)D-(uc)=14π(Q2uc2+r02)(Q2+r02).Figure 2
The dimensionless diffusion coefficient as a function of radial coordinateu with L=r0=1=κ2=1.Figure2 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 1/4π when Q=0. On the boundary, different charge gives different diffusion constants while at the horizon u=1, all these lines end at the same value D-=1/4π. Note that the slope of these lines is also proportional to the charge Q.


### 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(93)σDC(uc)=Ω(1)Ϝ2(uc),
one can easily obtain the DC conductivity. For d+2-dimensional charged black holes, the dimensionless diffusion coefficient can be written in a simple form
(94)D-(uc)=14πΞ(1)Ϝ(uc),
where
(95)Ξ(1)=-gxx3gtt′-g|u=1,Ω(1)=(gxx(5d-2)/4gtt′q(u)-g)2|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.On an arbitrary cutoff surfaceu=uc outside the horizon, there is a thermodynamic description of the fluid dual to the background configuration (5). The Brown-York tensor is defined as [44]
(96)Tij=18πG5(Θγij-Θij-Cγij),
with the extrinsic curvature
(97)Θij=-12(∇inj+∇jni),i,j=t,xi.The induced metricγμν can be expressed by bulk metric gμν and the normal vector in u direction, so
(98)γμν=gμν-nμnν,
where nu=u2h/r0eB is a unit normal vector and the trace of extrinsic curvature is given by Θ≡γijΘij. One may note that the indices μ,ν of the bulk and the indices i,j live on the cutoff surface. By calculating with the metric and induced metric, we obtain
(99)Θtt=-hu2r0eB(h′2h+A′)γtt,Θxx=-hu2A′r0eBγxx,Θ=-hu2r0eB(h′2h+4A′).The stress tensor at an arbitrary cutoff surface can be written as(100)Tijdxidxj=18πG5[hhu2e2Ar0eB3A′dt2-hu2e2Ar0eB(3A′+h′2h)dxidxi-Cdsc2hhu2e2Ar0eB].On the other hand, the stress-energy tensor of a relativistic fluid in equilibrium is(101)Tij=(ε+P)uiuj+Pγij,
where ε denotes the energy density, P is the pressure, and ui={e-A/h,0,0,0} is the normalized fluid four-velocity. For simplicity, we can redefine the energy density and pressure
(102)P⟶P-C8πG5,ϵ⟶ε+C8πG5,
which leaves the combination ε+P invariant. In this sense, we can omit the constant C in the Brown-York tensor (Note that C is the counterterm of the action and essential for the regularity of Tij as uc→0). Comparing (100) with (101), we find
(103)ε+P=116πG5nu(γtt′γtt-γxx′γxx)=-uc2h′2κ2hr0eB|u=uc.Notice that the entropy density is given by(104)sc=AH4G5V3=2πr0mκ2L2e3A|u=uc.The local Hawking temperature is given by(105)Tc=TH-gtt=r0πL2heA|u=uc,TH=14π-gtt′-gttguu|u=1.We can express thermodynamic variables in terms of metric components(106)scTcε+P=-4r03muc2L4h′e4A-B|u=uc.By detailed calculation, we can obtain the value(107)scTcε+P=Q2uc2+r02Q2+r02.Apparently, one can verify the following identity(108)scTcε+P=Ξ(1)Ϝ(uc),
by introducing the quantities
(109)Ξ(1)=-gxx3gtt′-g|u=1,Ω(1)=(gxx(5d-2)/4gtt′q(u)-g)2|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.Finally, we verify that(110)σDC(uc)=Ω(1)Ϝ2(uc).In concrete, it can be written as(111)σDC(uc)=r02κ2L(1+Q2r02)-1(1+Q2uc2r02)2.The above result agrees with (65).Moreover, we can verify the diffusion constant by using(112)D-(uc)=ηε+PTc=14πscTcε+P,
where η(uc) 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
(113)D-(uc)=Ξ(1)4πϜ(uc).For our case, the diffusion coefficient is given by(114)D-(uc)=14π(Q2uc2+r02)(Q2+r02).This is consistent with (92). As uc→1, the diffusion constant becomes D-(uc)=1/4π.


## 2.1. Backgrounds and Thermodynamics

We start by introducing the following action for charged dilatonic black hole inAdS5 with mixing dilatonic field and U(1) gauge field [34]:
(4)L=12κ2[R-14e4αFμν2-12(∂μα)2+1L2(8e2α+4e-4α)],
where L is the radius of the AdS space and α plays the role of a dilaton with respect to the radial coordinate r. We denote the gravitational constant as κ2=8πG5 and the U(1) gauge field strength is given by Fμν(x)=∂μAν(x)-∂νAμ(x).In this paper, we will introduce a dimensionless coordinateu=r0/r for simplicity. The spatially uniform, electrically charged solution for this action [34] can be obtained as follows:
(5)ds2=e2A(-hdt2+dx→2)+e2Br02hu4du2,Aμdxμ=ϕdt,
where
(6)A=log⁡⁡r0uL+13log⁡⁡(1+Q2u2r02),B=-log⁡⁡r0uL-23log⁡⁡(1+Q2u2r02),h=1-u4mL2(r02+Q2u2)2,ϕ=Qu22mr02+Q2u2-Q2mr02+Q2,α=16log⁡⁡(1+Q2u2r02).The horizon of the black hole locates at(7)r0=Lm-Q2,
where m is a constant related to the mass with the form
(8)m=(Q2+r02)2L2.The black hole is extremal ifr0=0, which implies m=Q4/L2.The equation of motion for the gauge fieldAμ(x) is given by
(9)∂ν(-ge4αFμν)=0.The Einstein equation is written as(10)Rμν-12gμνR=Gμν+Tμν,
where
(11)Gμν=12e4αFμρFνρ-18e4αgμνF2,Tμν=12(∂μα∂να)-6gμν(∂α)2+12L2gμν(8e2α+4e-4α).The equation of motion for the scalar field is(12)-34∇2α+1L2(e2α-e-4α)=116e4αF2.The Hawking temperature yields(13)T=h′u24πr0eA-B|u=1=r0πL2.The volume density of Bekenstein-Hawking entropy is given by(14)s=e3A4G5|u=1=2r0πκ2L3(Q2+r02).According to thermodynamic relation(15)ε+P-sT-μρ=0,
and the first law of thermodynamics
(16)dε=Tds+μdρ,
we can obtain the energy density and the pressure, which can be expressed as
(17)ε=3(Q2+r02)22κ2L5,P=(Q2+r02)22κ2L5.The charge density and the chemical potential are [34]
(18)ρ=Q2κ2L3(Q2+r02),μ=2QL2.Furthermore, there are several relations satisfied by the thermodynamic variables(19)s=2π2LκT23Lε,sμ=2π2ρT,μ22+T2π2=mL3,ε=3κ2/327/3π4/3L(s2+2π2ρ2)2/3.The temperature and chemical potential can also be obtained as(20)T=(∂ε∂s)ρ,μ=(∂ε∂ρ)s.The susceptibility can be calculated which is given by(21)χ=(∂ρ∂μ)T=(Q2+r02)κ2L+2Q2κ2L.According to the definition of the special heat(22)Cμ=T(∂s∂T)|μ.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(23)Cμ=2π2Q2κ2LT.


## 2.2. Perturbations and Transport Coefficients

Considering the metric perturbations [35]
(24)gμν⟶gμν+hμν,Aμ⟶Aμ+δAμ,
to the background gμν and Aμ, one can use background metric gμν and inverse metric gμν lower and raise tensor indices. The inverse metric can be expressed as gμν=g(0)μν-hμν+O(h2).We choose the momentum along thez-direction and u as the radial direction which describes the energy scale in field theory. In the gauge Au(x)=0 and by using Fourier decomposition
(25)hμν(t,z,u)=∫d5k(2π)5e-iωt+ikzhμν(k,u),Aμ(t,z,u)=∫d5k(2π)5e-iωt+ikzAμ(k,u),
we will consider the scalar mode hxy(t,z,u) and shear mode htx(t,z,u), hzx(t,z,u) and Ax(t,z,u) in the following.


### 2.2.1. Scalar Mode and Shear Viscosity

From symmetry analysis, one can find that off-diagonal perturbationhxy decouples from all other perturbations. We obtain an equation of motion for the scalar mode as
(26)∂μ(-g∂μhxy)=0.Following the sliding membrane argument [14], we define a cutoff dependent tensor response function
(27)Gxy(uc,k)=---gguu∂uhxy2κ2hxy(uc,k).We define the shear viscosity as(28)η(uc,k):=Gxy(uc,k)iω.At zero momentum limit, the flow equation is given by(29)∂ucη(uc,ω)=iω(2κ2η2(uc,ω)-gguu--ggtt2κ2).The shear viscosity is requested to be(30)η(r0)=r032κ2L3(1+Q2r02),
because of the horizon regularity. Because the entropy density was given in (14), we can easily check the shear viscosity to entropy density ratio
(31)ηs=14π.This result agrees with [35–37] and obeys the KSS bound [38].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 [11], 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 [11]
(32)∂uS=0,
where S=sV3 and V3=e3A is the u-dependent volume. We can derive u-dependent entropy density from the thermodynamical relation (15). Multiplying (15) with the volume V3 and considering the derivative of the total entropy along the radial direction, we obtain
(33)∂uS=Vp16πGr0TH(h′′+(4A′-B′+2u)h′)eA-Bu2-VpTHTμνζμζν.The right hand side of the above equation is exactly a component of Einstein equation. HereTμν is the bulk matter stress tensor and ζμ is any null vector tangent to the cutoff, Tμνζμζν≥0 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
(34)s(uc)=uc34G5(Q2+r02)(Q2uc2+r02).We will follow [11] and assume that η/s=1/4π (obeys the KSS bound [38]) 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 [16]). Under such an assumption, we find the shear viscosity
(35)η(uc)=uc32κ2(Q2+r02)(Q2uc2+r02).


### 2.2.2. Shear Modes:htx(t,z,u), hzx(t,z,u), and Ax(t,z,u)

The linear perturbative Einstein equation can be read off from the(t,x), (u,x), and (x,z) components, respectively,
(36)0=htx′′+(2u+4A′-B′)htx′-r02u4he2B-2A(ωkhzx+k2htx)+ϕ′e4α-2AAx′,(37)0=khhzx′+ωhtx′+ϕ′e4α-2AωAx,(38)0=hzx′′+(2u+4A′-B′+h′h)hzx′+r02u4h2e2B-2A(ω2hzx+ωkhtx),
where the prime denotes the derivative with respect to u. 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 x-component gives
(39)0=Ax′′+(2u+2A′-B′+4α′+h′h)Ax′+r02u4h2e2B-2A(ω2-k2h)Ax+ϕ′hhtx′.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 fieldhtx and hzx as follows
(40)jμ=-1geff2(u)-gfuμ,fμν=∂μhνx-∂νhμx.We also introduce the effective couplingq2(u) and geff2(u)(41)G=-ggeff2(u),1q2(u)=e4α2κ2,1geff2(u)=gxx2κ2.The vector part off-shell action inAdS5 charged dilatonic black hole can be written as follows
(42)S=∫d5x-g(-14q2(u)FμνFμν-14geff2(u)fμνfμν+1q2(u)htxAx′Fut).Note thatFμν is the strength of the Maxwell fields Aμ and should not be confused with the effective strength of the shear modes of gravity fμν.One the other hand, we must introduce a new current related toAx(43)jλ=-1q2(u)-gFutAx.Then we can define the shifted current(44)j~t≡jt+jλ.In terms of the defined “current” and “strength,” the equations of motion (36)-(38) can be recast as
(45)∂tj~t+∂zjz=0,(46)∂uj~t+Ggttgzz∂zfzt=0,∂ujz-Ggttgzz∂tfzt=0.The Bianchi identity holds as(47)-guugzzG∂tjz+guugttG∂zjt+∂ufzt=0.For the same reason, we can definejx and j~x as
(48)jx≡-1q2(u)-gFux,j~x≡jx+jρ,
where
(49)jρ=1q2(u)-gFuthtx.The equation of motion forAx can be written as
(50)-∂uj~x+1q2(u)-ggxx(gtt∂tFtx+gzz∂zFzx)=0.In the zero momentum limit, the equation of motion forAx decouples from htx(51)∂u(1q2(u)-gguugxx∂uAx)-1q4(u)-ggeff2(u)guugtt(ϕ′)2Ax-1q2(u)-ggxxgttω2Ax=0.The relevant on-shell action forAx at boundary u=uc can be written as
(52)Son-shell=∫u=ucj~xAx.


### 2.2.3. DC Electric Conductivity

By defining(53)σx=jxEx=jxFxt=jxiωAx,
we have
(54)∂uσx=∂ujxFxt-jx∂u(Fxt)Fxt2,σx2=(jx)2(Fxt)2.So the flow equation (51) for electric conductivity can be rewritten as
(55)∂uσxiω-q2(u)σx2-gguugxx+geff2(u)-gguugttϕ′2ω2q4(u)-1q2(u)-ggxxgtt=0.The regularity condition at the horizon requires(56)σx(u=1)=1q2(u)-gguugttgxx|u=1=r02κ2L(1+Q2r02).On the other hand, the DC conductivity can be evaluated by using the Kubo formula(57)σDC=-lim⁡ω→0Im⁡Gx,xR(ω,k=0)ω.The retarded Green’s function is given by [35, 37]
(58)Gx,xR(ω,k=0)=-i∫dtdxeiωtθ(t)〈[Jx(x),Jx(0)]〉,
where Jμ is the conformal field theory (CFT) current dual to the bulk gauge field Aμ. It is convenient to define the radial momentum as [16]
(59)Jkx=1κ2K(u)Ax′(u,k).The equation of motion (51) for Ax can be rewritten as [43]
(60)∂uJkx=1κ2L(u)Ax(u,k),
where
(61)K(u)=--gq2(u)gxxguu,L(u)=ω2-gq2(u)gxxgtt-geff2-ggttguuϕ′2q4(u).From (56), we know the regularity at the horizon corresponds to
(62)Jkx(1)=-iωK(u)guu-gttAx(1).The DC conductivity can be calculated by using the following relation [43]:
(63)σDC=-K(u)guu-gtt|u=1Ax(1)Ax(1)Ax(uc)Ax(uc),
where Ax(u) is the solution of (51). We can solve Ax(u) by imposing boundary condition at u=0 and set ω to zero, which leads to
(64)Ax(u)=Ax(0)(r02+Q2)2(r02+Q2uc2).Inserting (64) into (63), finally we can obtain the DC conductivity at the cutoff surface (see Figure 1)
(65)σDC(uc)=r02κ2L(1+Q2r02)-1(1+Q2uc2r02)2.Figure 1
The DC conductivity as a function of the radial coordinateu with different charges, where we set L=r0=1=κ2=1. The dashed line corresponds to Q=0.At the horizonuc=1, the above formula becomes
(66)σ(u=1)=r02κ2L(1+Q2r02),
which agrees with (56). In the boundary uc→0, the DC conductivity is reduced to
(67)σ(u=0)=r02κ2L(1+Q2r02)-1.Without the chemical potential, (65) reduces to the formula for DC conductivity given in [14].We can see from Figure1 that the DC conductivity on the boundary depends on the charge Q. The slope of lines is proportional to the charge Q.As a side note, we will check whether the transport coefficients calculated satisfies the Einstein relationσDC/χ|ω=0=Dc. By using the relation which the value of susceptibility is defined by (21), we find the expression for σDC/χ can be written as
(68)σDC(0)χ=r032(Q2+r02)(3Q2+r02).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 potentialμ=0 (i.e., Q=0), the above equation becomes σDC/χ|ω=0=Dc with Dc=1/2πT.As a byproduct, we calculate the thermal conductivity by using the relation(69)κT=(ϵ+PTHρ)2THσDC(0)=πr04κ2L3Q2(1+Q2r02).The ratioκTμ2/ηTH can be computed as
(70)κTμ2ηTH=4π2.


### 2.2.4. Diffusion Coefficient

Now we are going to calculate the diffusion coefficient. Let us evaluate the “conductivity” introduced by the metric perturbationhzx at any momentum. The conductivity is defined as
(71)σh:=jzfzt.In the absence of the momentum, the decoupled flow equation forσh yields
(72)∂ucσh-iω+σh2guugzzG+Ggttgzz=0.Again, the regularity condition at the event horizon gives(73)σh(1)=r032κ2L3(1+Q2r02).Actually, the value of this conductivity is equivalent to shear viscosity. Noting that from (71) and (73), we can recast the the conductivity at the horizon as
(74)jz(1)=σh(1)fzt(1).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 onuc, charge, and dilaton field. The equation of motion of σh is coupled with other modes when momentum kz is nonzero. To proceed, we take the scaling limit for temporal and spatial derivatives as
(75)∂t~ϵ2,∂z~ϵ,fzt~ϵ3(fzt(0)+ϵfzt(1)+⋯).The in-falling boundary condition at the horizon impliesjz is linearly related to fzt. So, we have
(76)jz~ϵ3(jz(0)+ϵjz(1)+⋯).Through the charge conservation equation (45), we have
(77)j~t~ϵ2(j~t(0)+ϵj~t(1)+⋯),jt~ϵ2(jt(0)+ϵjt(1)+⋯).To zeroth order, (46) can be reduced to
(78)∂uj~t(0)=0,∂ujz=0.The first equation of (78) figures that
(79)j~t(0)=jt(0)-1q2-gguugtt∂uAtAx(0)=C0,
where C0 is a constant. The equation (45) implies
(80)jz(0)=j~t(0)ωk.The zeroth order of the Bianchi identity becomes(81)∂ufzt(0)+guugttG∂zjt(0)=0.Integrating the above equation fromu to the horizon u=1, we obtain
(82)fzt(0)(u)=fzt(0)(u)|u=1+2κ2∫u1du∂zjt(0)guugtt-ggxx.For the gauge perturbationAx(0), the Maxwell equation in the scaling limit becomes
(83)∂u(1q2(u)-gguugxx∂uAx(0))=1q2(u)-gguugtt∂uAthtx′(0).Following [11], imposing the boundary condition Ax(0)(uc)=0, and solving equation (79) and (83), we obtain
(84)Ax(0)(u)=Qr0(2Q2uc2+r02/u2+r02uc2)42(Q2+r02)2(u2uc2-1),
and substitute it into (79). Consequently, we have
(85)fzt(u)=f(0)(1)+ik(1-uc2)(2Q2uc2+(1+uc2)r02)r0L24(Q2uc2+r02)(Q2+r02).Following the sliding membrane paradigm [14], we defined the momentum conductivity by the current and electric fields as
(86)σh(uc)=jz(0)(uc)fzt(0)(uc)=ωkj~t(0)(uc)fzt(0)(uc).By using (85) and the boundary condition given in (74), we find the expression of the conductivity σh satisfying
(87)1σh(uc)=1σh(1)-k2iωD(uc)σh(1),
where the diffusion constant is given by
(88)D(uc)=r0L2(2Q2uc2+(1+uc2)r02)4(Q2uc2+r02)(Q2+r02)(1-uc2).Note that the dimensionalD(uc) 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 frequencyωc and the proper momentum kc on the hypersurface u=uc(89)ωc≡ω-gtt,kc≡kgii,
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 u=uc which is Tc(uc)=T/-gtt.We define a dimensionless diffusion constantD-(uc) which is coordinate-invariant as
(90)D-(uc)=D(uc)Tcgzz-gtt.Thus, in terms of the normalized momentum and the diffusion constant, the conductivity can be expressed as(91)σh(1)σh(uc)=1-kc2iωcD-(uc)Tc,
where the dimensionless diffusion coefficient is given by (see Figure 2)
(92)D-(uc)=14π(Q2uc2+r02)(Q2+r02).Figure 2
The dimensionless diffusion coefficient as a function of radial coordinateu with L=r0=1=κ2=1.Figure2 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 1/4π when Q=0. On the boundary, different charge gives different diffusion constants while at the horizon u=1, all these lines end at the same value D-=1/4π. Note that the slope of these lines is also proportional to the charge Q.


## 2.2.1. Scalar Mode and Shear Viscosity

From symmetry analysis, one can find that off-diagonal perturbationhxy decouples from all other perturbations. We obtain an equation of motion for the scalar mode as
(26)∂μ(-g∂μhxy)=0.Following the sliding membrane argument [14], we define a cutoff dependent tensor response function
(27)Gxy(uc,k)=---gguu∂uhxy2κ2hxy(uc,k).We define the shear viscosity as(28)η(uc,k):=Gxy(uc,k)iω.At zero momentum limit, the flow equation is given by(29)∂ucη(uc,ω)=iω(2κ2η2(uc,ω)-gguu--ggtt2κ2).The shear viscosity is requested to be(30)η(r0)=r032κ2L3(1+Q2r02),
because of the horizon regularity. Because the entropy density was given in (14), we can easily check the shear viscosity to entropy density ratio
(31)ηs=14π.This result agrees with [35–37] and obeys the KSS bound [38].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 [11], 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 [11]
(32)∂uS=0,
where S=sV3 and V3=e3A is the u-dependent volume. We can derive u-dependent entropy density from the thermodynamical relation (15). Multiplying (15) with the volume V3 and considering the derivative of the total entropy along the radial direction, we obtain
(33)∂uS=Vp16πGr0TH(h′′+(4A′-B′+2u)h′)eA-Bu2-VpTHTμνζμζν.The right hand side of the above equation is exactly a component of Einstein equation. HereTμν is the bulk matter stress tensor and ζμ is any null vector tangent to the cutoff, Tμνζμζν≥0 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
(34)s(uc)=uc34G5(Q2+r02)(Q2uc2+r02).We will follow [11] and assume that η/s=1/4π (obeys the KSS bound [38]) 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 [16]). Under such an assumption, we find the shear viscosity
(35)η(uc)=uc32κ2(Q2+r02)(Q2uc2+r02).


## 2.2.2. Shear Modes:htx(t,z,u), hzx(t,z,u), and Ax(t,z,u)

The linear perturbative Einstein equation can be read off from the(t,x), (u,x), and (x,z) components, respectively,
(36)0=htx′′+(2u+4A′-B′)htx′-r02u4he2B-2A(ωkhzx+k2htx)+ϕ′e4α-2AAx′,(37)0=khhzx′+ωhtx′+ϕ′e4α-2AωAx,(38)0=hzx′′+(2u+4A′-B′+h′h)hzx′+r02u4h2e2B-2A(ω2hzx+ωkhtx),
where the prime denotes the derivative with respect to u. 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 x-component gives
(39)0=Ax′′+(2u+2A′-B′+4α′+h′h)Ax′+r02u4h2e2B-2A(ω2-k2h)Ax+ϕ′hhtx′.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 fieldhtx and hzx as follows
(40)jμ=-1geff2(u)-gfuμ,fμν=∂μhνx-∂νhμx.We also introduce the effective couplingq2(u) and geff2(u)(41)G=-ggeff2(u),1q2(u)=e4α2κ2,1geff2(u)=gxx2κ2.The vector part off-shell action inAdS5 charged dilatonic black hole can be written as follows
(42)S=∫d5x-g(-14q2(u)FμνFμν-14geff2(u)fμνfμν+1q2(u)htxAx′Fut).Note thatFμν is the strength of the Maxwell fields Aμ and should not be confused with the effective strength of the shear modes of gravity fμν.One the other hand, we must introduce a new current related toAx(43)jλ=-1q2(u)-gFutAx.Then we can define the shifted current(44)j~t≡jt+jλ.In terms of the defined “current” and “strength,” the equations of motion (36)-(38) can be recast as
(45)∂tj~t+∂zjz=0,(46)∂uj~t+Ggttgzz∂zfzt=0,∂ujz-Ggttgzz∂tfzt=0.The Bianchi identity holds as(47)-guugzzG∂tjz+guugttG∂zjt+∂ufzt=0.For the same reason, we can definejx and j~x as
(48)jx≡-1q2(u)-gFux,j~x≡jx+jρ,
where
(49)jρ=1q2(u)-gFuthtx.The equation of motion forAx can be written as
(50)-∂uj~x+1q2(u)-ggxx(gtt∂tFtx+gzz∂zFzx)=0.In the zero momentum limit, the equation of motion forAx decouples from htx(51)∂u(1q2(u)-gguugxx∂uAx)-1q4(u)-ggeff2(u)guugtt(ϕ′)2Ax-1q2(u)-ggxxgttω2Ax=0.The relevant on-shell action forAx at boundary u=uc can be written as
(52)Son-shell=∫u=ucj~xAx.


## 2.2.3. DC Electric Conductivity

By defining(53)σx=jxEx=jxFxt=jxiωAx,
we have
(54)∂uσx=∂ujxFxt-jx∂u(Fxt)Fxt2,σx2=(jx)2(Fxt)2.So the flow equation (51) for electric conductivity can be rewritten as
(55)∂uσxiω-q2(u)σx2-gguugxx+geff2(u)-gguugttϕ′2ω2q4(u)-1q2(u)-ggxxgtt=0.The regularity condition at the horizon requires(56)σx(u=1)=1q2(u)-gguugttgxx|u=1=r02κ2L(1+Q2r02).On the other hand, the DC conductivity can be evaluated by using the Kubo formula(57)σDC=-lim⁡ω→0Im⁡Gx,xR(ω,k=0)ω.The retarded Green’s function is given by [35, 37]
(58)Gx,xR(ω,k=0)=-i∫dtdxeiωtθ(t)〈[Jx(x),Jx(0)]〉,
where Jμ is the conformal field theory (CFT) current dual to the bulk gauge field Aμ. It is convenient to define the radial momentum as [16]
(59)Jkx=1κ2K(u)Ax′(u,k).The equation of motion (51) for Ax can be rewritten as [43]
(60)∂uJkx=1κ2L(u)Ax(u,k),
where
(61)K(u)=--gq2(u)gxxguu,L(u)=ω2-gq2(u)gxxgtt-geff2-ggttguuϕ′2q4(u).From (56), we know the regularity at the horizon corresponds to
(62)Jkx(1)=-iωK(u)guu-gttAx(1).The DC conductivity can be calculated by using the following relation [43]:
(63)σDC=-K(u)guu-gtt|u=1Ax(1)Ax(1)Ax(uc)Ax(uc),
where Ax(u) is the solution of (51). We can solve Ax(u) by imposing boundary condition at u=0 and set ω to zero, which leads to
(64)Ax(u)=Ax(0)(r02+Q2)2(r02+Q2uc2).Inserting (64) into (63), finally we can obtain the DC conductivity at the cutoff surface (see Figure 1)
(65)σDC(uc)=r02κ2L(1+Q2r02)-1(1+Q2uc2r02)2.Figure 1
The DC conductivity as a function of the radial coordinateu with different charges, where we set L=r0=1=κ2=1. The dashed line corresponds to Q=0.At the horizonuc=1, the above formula becomes
(66)σ(u=1)=r02κ2L(1+Q2r02),
which agrees with (56). In the boundary uc→0, the DC conductivity is reduced to
(67)σ(u=0)=r02κ2L(1+Q2r02)-1.Without the chemical potential, (65) reduces to the formula for DC conductivity given in [14].We can see from Figure1 that the DC conductivity on the boundary depends on the charge Q. The slope of lines is proportional to the charge Q.As a side note, we will check whether the transport coefficients calculated satisfies the Einstein relationσDC/χ|ω=0=Dc. By using the relation which the value of susceptibility is defined by (21), we find the expression for σDC/χ can be written as
(68)σDC(0)χ=r032(Q2+r02)(3Q2+r02).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 potentialμ=0 (i.e., Q=0), the above equation becomes σDC/χ|ω=0=Dc with Dc=1/2πT.As a byproduct, we calculate the thermal conductivity by using the relation(69)κT=(ϵ+PTHρ)2THσDC(0)=πr04κ2L3Q2(1+Q2r02).The ratioκTμ2/ηTH can be computed as
(70)κTμ2ηTH=4π2.


## 2.2.4. Diffusion Coefficient

Now we are going to calculate the diffusion coefficient. Let us evaluate the “conductivity” introduced by the metric perturbationhzx at any momentum. The conductivity is defined as
(71)σh:=jzfzt.In the absence of the momentum, the decoupled flow equation forσh yields
(72)∂ucσh-iω+σh2guugzzG+Ggttgzz=0.Again, the regularity condition at the event horizon gives(73)σh(1)=r032κ2L3(1+Q2r02).Actually, the value of this conductivity is equivalent to shear viscosity. Noting that from (71) and (73), we can recast the the conductivity at the horizon as
(74)jz(1)=σh(1)fzt(1).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 onuc, charge, and dilaton field. The equation of motion of σh is coupled with other modes when momentum kz is nonzero. To proceed, we take the scaling limit for temporal and spatial derivatives as
(75)∂t~ϵ2,∂z~ϵ,fzt~ϵ3(fzt(0)+ϵfzt(1)+⋯).The in-falling boundary condition at the horizon impliesjz is linearly related to fzt. So, we have
(76)jz~ϵ3(jz(0)+ϵjz(1)+⋯).Through the charge conservation equation (45), we have
(77)j~t~ϵ2(j~t(0)+ϵj~t(1)+⋯),jt~ϵ2(jt(0)+ϵjt(1)+⋯).To zeroth order, (46) can be reduced to
(78)∂uj~t(0)=0,∂ujz=0.The first equation of (78) figures that
(79)j~t(0)=jt(0)-1q2-gguugtt∂uAtAx(0)=C0,
where C0 is a constant. The equation (45) implies
(80)jz(0)=j~t(0)ωk.The zeroth order of the Bianchi identity becomes(81)∂ufzt(0)+guugttG∂zjt(0)=0.Integrating the above equation fromu to the horizon u=1, we obtain
(82)fzt(0)(u)=fzt(0)(u)|u=1+2κ2∫u1du∂zjt(0)guugtt-ggxx.For the gauge perturbationAx(0), the Maxwell equation in the scaling limit becomes
(83)∂u(1q2(u)-gguugxx∂uAx(0))=1q2(u)-gguugtt∂uAthtx′(0).Following [11], imposing the boundary condition Ax(0)(uc)=0, and solving equation (79) and (83), we obtain
(84)Ax(0)(u)=Qr0(2Q2uc2+r02/u2+r02uc2)42(Q2+r02)2(u2uc2-1),
and substitute it into (79). Consequently, we have
(85)fzt(u)=f(0)(1)+ik(1-uc2)(2Q2uc2+(1+uc2)r02)r0L24(Q2uc2+r02)(Q2+r02).Following the sliding membrane paradigm [14], we defined the momentum conductivity by the current and electric fields as
(86)σh(uc)=jz(0)(uc)fzt(0)(uc)=ωkj~t(0)(uc)fzt(0)(uc).By using (85) and the boundary condition given in (74), we find the expression of the conductivity σh satisfying
(87)1σh(uc)=1σh(1)-k2iωD(uc)σh(1),
where the diffusion constant is given by
(88)D(uc)=r0L2(2Q2uc2+(1+uc2)r02)4(Q2uc2+r02)(Q2+r02)(1-uc2).Note that the dimensionalD(uc) 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 frequencyωc and the proper momentum kc on the hypersurface u=uc(89)ωc≡ω-gtt,kc≡kgii,
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 u=uc which is Tc(uc)=T/-gtt.We define a dimensionless diffusion constantD-(uc) which is coordinate-invariant as
(90)D-(uc)=D(uc)Tcgzz-gtt.Thus, in terms of the normalized momentum and the diffusion constant, the conductivity can be expressed as(91)σh(1)σh(uc)=1-kc2iωcD-(uc)Tc,
where the dimensionless diffusion coefficient is given by (see Figure 2)
(92)D-(uc)=14π(Q2uc2+r02)(Q2+r02).Figure 2
The dimensionless diffusion coefficient as a function of radial coordinateu with L=r0=1=κ2=1.Figure2 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 1/4π when Q=0. On the boundary, different charge gives different diffusion constants while at the horizon u=1, all these lines end at the same value D-=1/4π. Note that the slope of these lines is also proportional to the charge Q.


## 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(93)σDC(uc)=Ω(1)Ϝ2(uc),
one can easily obtain the DC conductivity. For d+2-dimensional charged black holes, the dimensionless diffusion coefficient can be written in a simple form
(94)D-(uc)=14πΞ(1)Ϝ(uc),
where
(95)Ξ(1)=-gxx3gtt′-g|u=1,Ω(1)=(gxx(5d-2)/4gtt′q(u)-g)2|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.On an arbitrary cutoff surfaceu=uc outside the horizon, there is a thermodynamic description of the fluid dual to the background configuration (5). The Brown-York tensor is defined as [44]
(96)Tij=18πG5(Θγij-Θij-Cγij),
with the extrinsic curvature
(97)Θij=-12(∇inj+∇jni),i,j=t,xi.The induced metricγμν can be expressed by bulk metric gμν and the normal vector in u direction, so
(98)γμν=gμν-nμnν,
where nu=u2h/r0eB is a unit normal vector and the trace of extrinsic curvature is given by Θ≡γijΘij. One may note that the indices μ,ν of the bulk and the indices i,j live on the cutoff surface. By calculating with the metric and induced metric, we obtain
(99)Θtt=-hu2r0eB(h′2h+A′)γtt,Θxx=-hu2A′r0eBγxx,Θ=-hu2r0eB(h′2h+4A′).The stress tensor at an arbitrary cutoff surface can be written as(100)Tijdxidxj=18πG5[hhu2e2Ar0eB3A′dt2-hu2e2Ar0eB(3A′+h′2h)dxidxi-Cdsc2hhu2e2Ar0eB].On the other hand, the stress-energy tensor of a relativistic fluid in equilibrium is(101)Tij=(ε+P)uiuj+Pγij,
where ε denotes the energy density, P is the pressure, and ui={e-A/h,0,0,0} is the normalized fluid four-velocity. For simplicity, we can redefine the energy density and pressure
(102)P⟶P-C8πG5,ϵ⟶ε+C8πG5,
which leaves the combination ε+P invariant. In this sense, we can omit the constant C in the Brown-York tensor (Note that C is the counterterm of the action and essential for the regularity of Tij as uc→0). Comparing (100) with (101), we find
(103)ε+P=116πG5nu(γtt′γtt-γxx′γxx)=-uc2h′2κ2hr0eB|u=uc.Notice that the entropy density is given by(104)sc=AH4G5V3=2πr0mκ2L2e3A|u=uc.The local Hawking temperature is given by(105)Tc=TH-gtt=r0πL2heA|u=uc,TH=14π-gtt′-gttguu|u=1.We can express thermodynamic variables in terms of metric components(106)scTcε+P=-4r03muc2L4h′e4A-B|u=uc.By detailed calculation, we can obtain the value(107)scTcε+P=Q2uc2+r02Q2+r02.Apparently, one can verify the following identity(108)scTcε+P=Ξ(1)Ϝ(uc),
by introducing the quantities
(109)Ξ(1)=-gxx3gtt′-g|u=1,Ω(1)=(gxx(5d-2)/4gtt′q(u)-g)2|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.Finally, we verify that(110)σDC(uc)=Ω(1)Ϝ2(uc).In concrete, it can be written as(111)σDC(uc)=r02κ2L(1+Q2r02)-1(1+Q2uc2r02)2.The above result agrees with (65).Moreover, we can verify the diffusion constant by using(112)D-(uc)=ηε+PTc=14πscTcε+P,
where η(uc) 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
(113)D-(uc)=Ξ(1)4πϜ(uc).For our case, the diffusion coefficient is given by(114)D-(uc)=14π(Q2uc2+r02)(Q2+r02).This is consistent with (92). As uc→1, the diffusion constant becomes D-(uc)=1/4π.


## 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 [49]
(115)L-g=R+2L2V-L28H4/3F2-13H-2gμν∂μH∂νH,
where the potential for the scalar field H and V is
(116)V=2H2/3+4H-1/3.The single-charged metric and gauge fields are given by(117)ds52=(πT0L)2uH1/3(-fHdt2+dx2+dy2+dz2)+L24fu2H1/3du2,Aμ=πT02κ1(1+κ1)uH(dt)μ,
where
(118)H=1+κ1u,f=(1-u)(1+(1+κ1)u),T=1+κ1/21+κ1T0,T0=r0πL2,
with κ1 denoting the R-charge while T0 denotes the Hawking temperature of neutral black hole. The energy density, pressure, and entropy density are given by
(119)ϵ=3π2N2T048(1+κ1),P=ϵ3,s=π2N2T0321+κ1.The charge density and conjugated chemical potential are(120)ρ=πN2T0382κ1+κ1,μ=πT02κ1+κ1.The Newton constant is given byG5=πL3/N2. The system of the gauged supergravity equations of motion for the fields gμν, Aμ, and H reads
(121)□H=H-1gμν∂μH∂νH+L24H7/3F2-3L2H2∂V∂H,∂μ(-gH4/3Fμν)=0,Rμν=L24H4/3FμγFνγ+13H-2∂μH∂νH-gμν[23L2V+L224H4/3F2].The vector type perturbation takes the form(122)htx=gxx(u)htx(u)e-iwt+iKz,hzx=gxx(u)hzx(u)e-iwt+iKz,ax=μ2Ax(u)e-iwt+iKz,
where ω=w/2πT0 and q=K/2πT0 are dimensionless frequency and momentum. We are interested in gravitational fluctuations of the shear type, where the only nonzero components of hμν are hta, hza, a=x,y. One can show that fluctuations of all other fields except Aa(r,t,z), a=x,y, can be consistently set to zero. Introduce the new variables
(123)htx≡Hta=gxxhta,hzx≡Hza=gxxhza,Ax=2Aaμ.The linearized equations derived from (121) can be written as [45](124a)htx′+qfωHhzx′+κ1u2HAx=0,(124b)htx′′+uH′-HuHhtx′-ωqfuhzx-q2fuhtx+κ1u2HAx′=0,(124c)hzx′′+uf′-fufhzx′+ω2Hf2uhzx+qωHf2uhtx=0,(124d)(HfAx′+2(1+κ1)htx)′-q2HuAx+ω2H2fuAx=0.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
(125)htx′=-uκ12HAx.Substituting (125) into (124d), we obtain
(126)Ax′′+(f′f+H′H)Ax′+ω2Huf2Ax-uκ1(1+κ1)fH2Ax=0.Now we introducejμ and fμν to denote the current and strength for the vector modes
(127)jμ=-1geff2(u)-gfuμ,fμν=∂μhνx-∂νhμx,G=-ggeff2(u),1q2(u)=L2H4/34κ2,1geff2(u)=(πT0L)2H1/32κ2u.By further define(128)jλ=-1q2(u)-g4κ1u3L4H5/3Ax.j~t=jt+jλ,
we can recast the equations of motion as
(129)∂tj~t+∂zjz=0,∂uj~t+Ggttgzz∂zfzt=0,∂ujz-Ggttgzz∂tfzt=0.The Bianchi identity holds as(130)-guugzzG∂tjz+guugttG∂zjt+∂ufzt=0.Similarly we can definejx and j~x as
(131)j~x=jx+jρ,jx=-1q2(u)-gFux,
where
(132)jρ=1q2(u)-g8u3(1+κ1)(πT0)2L4H5/3htx′.The equation of motion forAx can be written as
(133)-∂uj~x+-gq2(u)gxx(gtt∂tFtx-gzz∂zFzx)=0.One can see thatAx decouples from htx in the q→0 limit
(134)∂u(1q2(u)-gguugxx∂uAx)--gq2(u)8u4κ1(1+κ1)(πT0)2L4H8/3Ax--gq2(u)gxxgttω2Ax=0.


### 3.1. DC Conductivity

By defining(135)σx=jxiωAx,
we rearrange (134) as the flow equation for electric conductivity
(136)∂uσxiω-q2(u)σx2-gguugxx+-gq2(u)8u4κ1(1+κ1)ω2(πT0)2L4H8/3-1q2(u)-ggxxgtt=0.We can immediately write down the regularity condition at the horizon(137)σx(u=1)=1q2(u)-gguugttgxx|u=1=r016πG5L(1+κ1)3/2.It is convenient to define the radial momentum as(138)Jkx=1κ2K(u)Ax′(u,k).The equation of motion forAx then takes the form [16]
(139)∂uJkx=1κ2L(u)Ax(u,k),
where
(140)K(u)=--gq2(u)gxxguu,L(u)=ω2-gq2(u)gxxgtt--gq2(u)8u4κ1(1+κ1)(πT0)2L4H8/3.The regularity at the horizonu=1 corresponds to
(141)Jkx(1)=-iωK(u)guu-gttAx(1).According to the following relation [16]:
(142)σDC=-K(u)guu-gtt|u=1Ax(1)Ax(1)Ax(uc)Ax(uc),
where Ax(u) is the solution of (124d) at zero momentum. We can solve Ax(u) by imposing some regularity condition at the horizon and setting ω to zero, which leads to
(143)Ax(u)=Ax(0)2+ucκ12(1+ucκ1).Finally we can obtain the DC conductivity at the cutoff surface (see Figure3)
(144)σDC(uc)=r016πG5L(1+κ1)3/2(2+κ12+ucκ1)2(1+ucκ11+κ1)2.Figure 3
The DC charge conductivity as a function of the radial coordinateu. The dashed line represents the zero charge case.At the horizonuc=1, the above equation becomes
(145)σDC=116πG5r0L(1+κ1)3/2,
which is consistent with (137). As one goes to the boundary uc→0, the DC conductivity is reduced to
(146)σDC=116πG5r0L(1+κ1)-1/2(1+κ12)2,
in good agreement with [45].Figure3 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(147)∂μ(-g∂μhxy)=0.The flow equation for shear viscosity is the same as in the previous section(148)∂ucη(uc,ω)=iω(2κ2η2(uc,ω)-gguu--ggtt2κ2).On the horizon, the regularity gives(149)η(r0)=r032κ2L3(1+κ1)1/2.The entropy density is(150)s=2πr03κ2L3(1+κ1)1/2.So the shear viscosity to entropy density ratioη/s=1/4π. Following [11], we assume that this ratio does not change with cutoff surface. Under this assumption, we obtain the shear viscosity of any cutoff surface as
(151)ηc(uc)=uc3/22κ2(1+κ1)1/2(1+ucκ1)1/2.On any cutoff surface, the entropy density can be expressed as(152)sc(uc)=uc3/24G5(1+κ1)1/2(1+ucκ1)1/2.


### 3.3. Diffusion Coefficient

In order to obtain the diffusion coefficient, we need to evaluate the “conductivity” introduced by the metric perturbationhzx at any momentum. The conductivity in this case can be defined as
(153)σh:=jzfzt.In the zero momentum limit, the decoupled flow equation forσh is given by
(154)∂ucσh-iω+σh2guugzzG+Ggttgzz=0.The regularity condition at the event horizon gives(155)σh(1)=r032κ2L3(1+κ1)1/2.It is worth noting that (153) and (155) indicate that the conductivity at the horizon obeys
(156)jz(1)=σh(1)fzt(1).We take the scaling limit for temporal and spatial derivatives as(157)∂t~ϵ2,∂z~ϵ,fzt~ϵ3(fzt(0)+ϵfzt(1)+⋯).With regard to the lowest order, we have∂uj~t(0)=0, ∂ujz=0. The equation of j~t(0) indicates that
(158)j~t(0)=jt(0)-1q2(u)-g4κ1u3L4H5/3Ax=C0.Note thatC0 is a constant. For the gauge perturbation Ax(0), the Maxwell equation becomes
(159)∂u(1q2(u)-gguugxx∂uAx(0))=1q2(u)-g8u3(1+κ1)(πT0)2L4H5/3htx′(0).The zeroth order Bianchi identity yields(160)fzt(0)(u)=fzt(0)(u)|u=1+2κ2∫u1du∂zjt(0)guugtt-ggxx.After imposing the boundary conditionAx(0)(uc)=0, we solve (159) and obtain the solution for A(0)x(161)Ax(0)(u)=(1-uuc)[(2+κ1)2(4uc-κ12uc2u)(2+uκ1)2(2+ucκ1)23u+8(1+κ1)uc-(2+κ1)2(4uc-κ12uc2u)(2+uκ1)2(2+ucκ1)2].Then substituting it into (163), we have
(162)fzt(u)=f(0)(1)+ikL2(2+κ1)(1+uκ1)(1+u+uκ1)2r0(1+κ1)3/2(2+uκ1)2(1-uc).Following the sliding membrane paradigm [13], we define the momentum conductivity by the current and electric fields as
(163)σh(uc)≔jz(0)(uc)fzt(0)(uc)=ωkj~t(0)(uc)fzt(0)(uc).By further using the boundary condition given in (156), we find the expression for the conductivity σh from (162)
(164)1σh(uc)=1σh(1)-k2iωD(uc)σh(1),
where the dimensional diffusion is
(165)D(uc)=L2(2+κ1)(1+uκ1)(1+u+uκ1)2r0(1+κ1)3/2(2+uκ1)2(1-uc).The dimensionless diffusion constantD-(uc) can be defined as
(166)D-(uc)=D(uc)Tcgzz-gtt,
where Tc=T/-gtt. Thus the conductivity can be written in terms of the normalized momentum and the diffusion constant as follows:
(167)σh(1)σh(uc)=1-kc2iωcD-(uc)Tc,
where the dimensionless diffusion coefficient is given by (see Figure 4)
(168)D-(uc)=(2+κ1)(2+ucκ1)(1+ucκ1)4π(1+κ1).Figure 4
The dimensionless diffusion coefficient of single charge runs on the cutoff surface with different charges. The dashed line corresponds to the chargeless case.Whenuc→0, the diffusion coefficient reduces to
(169)D-(0)=2+κ18π(1+κ1),
which exactly agrees with [45]. As uc→1, the diffusion constant becomes
(170)D-(1)=14π.From Figure4 we can see that while the diffusion coefficientd is different at the boundry, they all approach to the same value at u=1. Also different charges can affect the diffusion coefficient seriously. When the charge becomes larger, the ratio of diffusion coefficient to the cutoff radius u becomes bigger.


### 3.4. Transport Coefficients from Brown-York Stress Tensor

In the following, we present a consistent check by using the Brown-York tensor. The Brown-York tensor is defined as(171)Tij=18πG5(Θγij-Θij-Cγij).The induced metricγμν can be expressed by bulk metric gμν and the normal vector in u direction. The norm vector is nu=2fu/LH1/6 and the trace of extrinsic curvature is given by Θ≡γijΘij. By calculating with the metric and induced metric, we obtain
(172)Θtt=fuLH1/6(2H′3H+1u-f′f)γtt,Θxx=fuLH1/6(1u-H′3H)γxx,Θ=fuLH1/6(4u-f′f-H′3H).
Thus, we obtain
(173)Tijdx→idx→j=18πG5(ff(πT0L)2LH5/6(2H′3H+1u-f′f)dt2-f(πT0L)2LH-1/6(3u-f′f)dx→adx→a).
Comparing the above stress tensor with the energy-momentum tensor of the fluid
(174)Tij=(ε+P)uiuj+Pγij,
with ui={fπT0L/uH1/3,0,0,0} the normalized fluid four-velocity, we obtain
(175)ε+P=116πGnu(γtt′γtt-γxx′γxx)=fucκ2LH1/6(H′H-f′f)|u=uc.Notice the entropy density(176)sc=2πuc3/2κ2H1/2(1+κ1)1/2|u=uc.On the cutoff surface, the local Hawking temperature(177)Tc=ucH1/32πLfκ1-2(1+κ1)1/2|u=uc.So we can obtain the result according to the previous section (108)
(178)scTcε+P=Ξ(1)Ϝ(uc)=2+κ12+ucκ1(1+ucκ1)(1+κ1),
where
(179)Ξ(1)=-gxx3gtt′-g|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.
In the end, we can find that the formulation of DC conductivity can be obtained from the combination of (137) and (178)
(180)σ(uc)=1q2(u)-gguugttgxx|u=1(scTcε+p)2.
It identifies with (93)
(181)σDC(uc)=Ω(1)Ϝ2(uc).The diffusion coefficient is given by(182)D-(uc)=ηε+PTc=14πΞ(1)Ϝ(uc),
where η(uc) is given in (151). According to (175), (176), and (177) and the relation η/s=1/4π, we can obtain
(183)D-(uc)=14π2+κ12+ucκ1(1+ucκ1)(1+κ1).
This is consistent with the result (168).


## 3.1. DC Conductivity

By defining(135)σx=jxiωAx,
we rearrange (134) as the flow equation for electric conductivity
(136)∂uσxiω-q2(u)σx2-gguugxx+-gq2(u)8u4κ1(1+κ1)ω2(πT0)2L4H8/3-1q2(u)-ggxxgtt=0.We can immediately write down the regularity condition at the horizon(137)σx(u=1)=1q2(u)-gguugttgxx|u=1=r016πG5L(1+κ1)3/2.It is convenient to define the radial momentum as(138)Jkx=1κ2K(u)Ax′(u,k).The equation of motion forAx then takes the form [16]
(139)∂uJkx=1κ2L(u)Ax(u,k),
where
(140)K(u)=--gq2(u)gxxguu,L(u)=ω2-gq2(u)gxxgtt--gq2(u)8u4κ1(1+κ1)(πT0)2L4H8/3.The regularity at the horizonu=1 corresponds to
(141)Jkx(1)=-iωK(u)guu-gttAx(1).According to the following relation [16]:
(142)σDC=-K(u)guu-gtt|u=1Ax(1)Ax(1)Ax(uc)Ax(uc),
where Ax(u) is the solution of (124d) at zero momentum. We can solve Ax(u) by imposing some regularity condition at the horizon and setting ω to zero, which leads to
(143)Ax(u)=Ax(0)2+ucκ12(1+ucκ1).Finally we can obtain the DC conductivity at the cutoff surface (see Figure3)
(144)σDC(uc)=r016πG5L(1+κ1)3/2(2+κ12+ucκ1)2(1+ucκ11+κ1)2.Figure 3
The DC charge conductivity as a function of the radial coordinateu. The dashed line represents the zero charge case.At the horizonuc=1, the above equation becomes
(145)σDC=116πG5r0L(1+κ1)3/2,
which is consistent with (137). As one goes to the boundary uc→0, the DC conductivity is reduced to
(146)σDC=116πG5r0L(1+κ1)-1/2(1+κ12)2,
in good agreement with [45].Figure3 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(147)∂μ(-g∂μhxy)=0.The flow equation for shear viscosity is the same as in the previous section(148)∂ucη(uc,ω)=iω(2κ2η2(uc,ω)-gguu--ggtt2κ2).On the horizon, the regularity gives(149)η(r0)=r032κ2L3(1+κ1)1/2.The entropy density is(150)s=2πr03κ2L3(1+κ1)1/2.So the shear viscosity to entropy density ratioη/s=1/4π. Following [11], we assume that this ratio does not change with cutoff surface. Under this assumption, we obtain the shear viscosity of any cutoff surface as
(151)ηc(uc)=uc3/22κ2(1+κ1)1/2(1+ucκ1)1/2.On any cutoff surface, the entropy density can be expressed as(152)sc(uc)=uc3/24G5(1+κ1)1/2(1+ucκ1)1/2.


## 3.3. Diffusion Coefficient

In order to obtain the diffusion coefficient, we need to evaluate the “conductivity” introduced by the metric perturbationhzx at any momentum. The conductivity in this case can be defined as
(153)σh:=jzfzt.In the zero momentum limit, the decoupled flow equation forσh is given by
(154)∂ucσh-iω+σh2guugzzG+Ggttgzz=0.The regularity condition at the event horizon gives(155)σh(1)=r032κ2L3(1+κ1)1/2.It is worth noting that (153) and (155) indicate that the conductivity at the horizon obeys
(156)jz(1)=σh(1)fzt(1).We take the scaling limit for temporal and spatial derivatives as(157)∂t~ϵ2,∂z~ϵ,fzt~ϵ3(fzt(0)+ϵfzt(1)+⋯).With regard to the lowest order, we have∂uj~t(0)=0, ∂ujz=0. The equation of j~t(0) indicates that
(158)j~t(0)=jt(0)-1q2(u)-g4κ1u3L4H5/3Ax=C0.Note thatC0 is a constant. For the gauge perturbation Ax(0), the Maxwell equation becomes
(159)∂u(1q2(u)-gguugxx∂uAx(0))=1q2(u)-g8u3(1+κ1)(πT0)2L4H5/3htx′(0).The zeroth order Bianchi identity yields(160)fzt(0)(u)=fzt(0)(u)|u=1+2κ2∫u1du∂zjt(0)guugtt-ggxx.After imposing the boundary conditionAx(0)(uc)=0, we solve (159) and obtain the solution for A(0)x(161)Ax(0)(u)=(1-uuc)[(2+κ1)2(4uc-κ12uc2u)(2+uκ1)2(2+ucκ1)23u+8(1+κ1)uc-(2+κ1)2(4uc-κ12uc2u)(2+uκ1)2(2+ucκ1)2].Then substituting it into (163), we have
(162)fzt(u)=f(0)(1)+ikL2(2+κ1)(1+uκ1)(1+u+uκ1)2r0(1+κ1)3/2(2+uκ1)2(1-uc).Following the sliding membrane paradigm [13], we define the momentum conductivity by the current and electric fields as
(163)σh(uc)≔jz(0)(uc)fzt(0)(uc)=ωkj~t(0)(uc)fzt(0)(uc).By further using the boundary condition given in (156), we find the expression for the conductivity σh from (162)
(164)1σh(uc)=1σh(1)-k2iωD(uc)σh(1),
where the dimensional diffusion is
(165)D(uc)=L2(2+κ1)(1+uκ1)(1+u+uκ1)2r0(1+κ1)3/2(2+uκ1)2(1-uc).The dimensionless diffusion constantD-(uc) can be defined as
(166)D-(uc)=D(uc)Tcgzz-gtt,
where Tc=T/-gtt. Thus the conductivity can be written in terms of the normalized momentum and the diffusion constant as follows:
(167)σh(1)σh(uc)=1-kc2iωcD-(uc)Tc,
where the dimensionless diffusion coefficient is given by (see Figure 4)
(168)D-(uc)=(2+κ1)(2+ucκ1)(1+ucκ1)4π(1+κ1).Figure 4
The dimensionless diffusion coefficient of single charge runs on the cutoff surface with different charges. The dashed line corresponds to the chargeless case.Whenuc→0, the diffusion coefficient reduces to
(169)D-(0)=2+κ18π(1+κ1),
which exactly agrees with [45]. As uc→1, the diffusion constant becomes
(170)D-(1)=14π.From Figure4 we can see that while the diffusion coefficientd is different at the boundry, they all approach to the same value at u=1. Also different charges can affect the diffusion coefficient seriously. When the charge becomes larger, the ratio of diffusion coefficient to the cutoff radius u becomes bigger.


## 3.4. Transport Coefficients from Brown-York Stress Tensor

In the following, we present a consistent check by using the Brown-York tensor. The Brown-York tensor is defined as(171)Tij=18πG5(Θγij-Θij-Cγij).The induced metricγμν can be expressed by bulk metric gμν and the normal vector in u direction. The norm vector is nu=2fu/LH1/6 and the trace of extrinsic curvature is given by Θ≡γijΘij. By calculating with the metric and induced metric, we obtain
(172)Θtt=fuLH1/6(2H′3H+1u-f′f)γtt,Θxx=fuLH1/6(1u-H′3H)γxx,Θ=fuLH1/6(4u-f′f-H′3H).
Thus, we obtain
(173)Tijdx→idx→j=18πG5(ff(πT0L)2LH5/6(2H′3H+1u-f′f)dt2-f(πT0L)2LH-1/6(3u-f′f)dx→adx→a).
Comparing the above stress tensor with the energy-momentum tensor of the fluid
(174)Tij=(ε+P)uiuj+Pγij,
with ui={fπT0L/uH1/3,0,0,0} the normalized fluid four-velocity, we obtain
(175)ε+P=116πGnu(γtt′γtt-γxx′γxx)=fucκ2LH1/6(H′H-f′f)|u=uc.Notice the entropy density(176)sc=2πuc3/2κ2H1/2(1+κ1)1/2|u=uc.On the cutoff surface, the local Hawking temperature(177)Tc=ucH1/32πLfκ1-2(1+κ1)1/2|u=uc.So we can obtain the result according to the previous section (108)
(178)scTcε+P=Ξ(1)Ϝ(uc)=2+κ12+ucκ1(1+ucκ1)(1+κ1),
where
(179)Ξ(1)=-gxx3gtt′-g|u=1,Ϝ(uc)=-gguu(ln⁡-gttgxx)′|u=uc.
In the end, we can find that the formulation of DC conductivity can be obtained from the combination of (137) and (178)
(180)σ(uc)=1q2(u)-gguugttgxx|u=1(scTcε+p)2.
It identifies with (93)
(181)σDC(uc)=Ω(1)Ϝ2(uc).The diffusion coefficient is given by(182)D-(uc)=ηε+PTc=14πΞ(1)Ϝ(uc),
where η(uc) is given in (151). According to (175), (176), and (177) and the relation η/s=1/4π, we can obtain
(183)D-(uc)=14π2+κ12+ucκ1(1+ucκ1)(1+κ1).
This is consistent with the result (168).


## 4. Conclusions

In summary, we verified the conjectured ansatz for DC conductivity and diffusion coefficient by exploring the RG flows of charged dilatonic and single-charged STU black holes. The DC conductivity as well as the diffusion coefficient shows its nontrivial flow from the IR horizon to the UV boundary. The results indicate that black hole thermodynamics evaluated on arbitraryu can provide solutions to the perturbed equation of motion for the gauge fields. We also proposed a unified version of retarded Green functions for shear modes in terms of black hole thermodynamic variables. A quick check shows that the formulae also work for RN-AdS black holes. The black hole thermodynamics is truly useful to describe the strongly coupled quark gluon plasma and more [35, 50].It would be interesting to extend our work to the sound modes of charged black holes and conductivity of anisotropic and inhomogeneous holographic background, because the sound modes of hydrodynamics and conductivity of holographic lattice are very complicated and difficult to solve [51, 52]. It would also be interesting to discuss the relation between the holographic RG flow and the instability problem as noted in [30, 34].



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