Gravity/Fluid Correspondence and Its Application on Bulk Gravity with $U(1)$ Gauge Field

As the long wavelength limit of the AdS/CFT correspondence, the gravity/fluid correspondence has been shown to be a useful tool for extracting properties of the fluid on the boundary dual to the gravity in the bulk. In this paper, after briefly reviewing the algorithm of gravity/fluid correspondence, we discuss the results of its application on bulk gravity with a $U(1)$ gauge field. In the presence of a $U(1)$ gauge field, the dual fluid possesses more interesting properties such as its charge current. Furthermore, an external field $A_\mu^{ext}$ could affect the charge current, and the $U(1)$ Chern-Simons term also reinduces extra structures to the dual current giving anomalous transport coefficients.


I. INTRODUCTION
The AdS/CFT correspondence [1][2][3][4] provides a remarkable connection between a gravitational theory and a quantum field theory. According to this correspondence, the gravitational theory in an asymptotically AdS spacetime can be reformulated in terms of a quantum field theory on its boundary. In particular, the dynamics of a classical gravitational theory in the bulk can be mapped into a strongly coupled quantum field theory on the boundary. Therefore, the AdS/CFT correspondence provides a useful tool of investigating the strongly coupled field theory from the dual classical gravitational theory [5,6].
Since the discovery of the AdS/CFT correspondence, there has been much work studying the hydrodynamical behavior of the dual quantum field theory using this correspondence [7][8][9][10]. A particular example is the computation of shear viscosity . The reason that the correspondence applies here is because hydrodynamics is an effective description of interacting quantum field theory in the long wavelength limit, i.e. when the length scales under consideration are much larger than the correlation length of the quantum field theory. Later, the long wavelength limit of the AdS/CFT correspondence has been developed as the gravity/fluid correspondence [44,45], which could provide a systematic mapping of bulk gravity and boundary fluid. Through the gravity/fluid correspondence, one can construct the stress-energy tensor of the fluid order by order from the derivative expansion of the gravity solution, and then extract certain transport coefficients of the fluid. For example, the shear viscosity η, entropy density s, and thus their ratio η/s can be calculated from the first-order stress-energy tensor [46][47][48][49][50][51][52][53][54]60]. In addition, in the presence of extra gauge fields in the bulk, the correspondence also allows us to construct the conserved charge current in the dual fluid, and thereby allows us to extract the thermal and electrical conductivities of the dual fluid [46][47][48]. It should be pointed out that the presence of an external field A ext µ is needed if one wants to obtain the electrical conductivity [46,49,53,54]. Furthermore, it has also been shown that introducing the topological Chern-Simons term in the bulk gravity brings interesting features such as the chiral magnetic effect (CME) and the chiral vortical effect (CVE) into the dual fluid [49,50,[53][54][55][56]. The Chern-Simons term was first discussed in Maxwell theory in three dimensions where it renders the gauge theory massive [57]. Such terms also affect the phase transition of holographic superconductors in four dimensions [58] and stability of the Reissner-Nordstrom (RN) black holes in AdS space in five dimensions [59]. In this paper, we will discuss the results for transport coefficients of dual boundary fluid in the presence of a U(1) gauge field in the bulk.
The rest of this paper is organized as follows. We start Sec.II by briefly reviewing the algorithm of gravity/fluid correspondence. In Sec. III, we discuss the application of gravity/fluid correspondence to studying fluid dynamics dual to bulk gravity in the presence of a U(1) gauge field. One can see that both the stress tensor and the conserved charge current of the dual fluid can be extracted. Moreover, adding the external field A ext µ and Chern-Simons term in the bulk also introduces extra structures to the charge current of dual fluid. The conclusion and discussion are given in Sec. IV.

II. ALGORITHM OF GRAVITY/FLUID CORRESPONDENCE
After several years' development, there exists now a well-formulated algorithm of how to extract information on the boundary fluid from the dual gravity using the gravity/fluid correspondence. In this section, we just give a brief introduction, and the more details could be seen in [44,45]. For illustrative purposes, let us take the 5-dimensional Einstein gravity in the bulk as an example. Its action can be written as where R is the Ricci scalar, and Λ is a negative cosmological constant Λ = −6/ℓ 2 . For later convenience, we set ℓ = 1 and 16πG = 1. The equation of motion then reads The starting point is the black brane solution written in the Eddington-Finkelstein coordinate where we have assumed β i (0) = 0, we find the first-order source terms by inserting the metric (2.4) with (2.7) into W IJ . By choosing an appropriate gauge like the background field gauge in [44] (G represents the metric) and taking into account the spatial SO(3) symmetry preserved in the background metric (2.3), the first-order correction terms around x µ = 0 can be written as The first order perturbative solution can then be obtained by requiring a cancelation of the source terms and the correction terms. Note that, after obtaining the solution around x µ = 0, one could convert it into a covariant form so that it applies to other spacetime points [44].
Given the first order perturbative solution for the bulk gravity, we are able to extract information of the dual fluid using the gravity/fluid correspondence. According to the correspondence, the stress tensor of dual fluid τ µν can be obtained from the following relation [61] where h µν is the background metric upon which the dual field theory resides, γ µν is the boundary metric obtained from the well-known ADM decomposition and T µν is the boundary stress tensor defined as where is the Gibbons-Hawking surface term, K is the trace of the extrinsic curvature K µν of the boundary, which is given by K µν = − 1 2 (∇ µ n ν + ∇ ν n µ ) with n µ the normal vector of the constant hypersurface r = r c pointing toward increasing r direction. In addition, is the boundary counterterm, and R is the curvature scalar associated with the induced metric on the boundary γ ab [62][63][64][65].
From Eq. (2.12), the boundary stress tensor is which can be used to compute the boundary stress tensor of the first order perturbative black brane solution.
The background metric upon which the dual field theory resides usually is chosen as [61], and is the Minkowski metric. From Eq. (2.10) and the result for the boundary stress tensor T µν , the first order stress tensor of the dual fluid τ µν can be found to be from which one immediately reads off the pressure and viscosity of the dual fluid.

III. APPLICATIONS ON THE BULK GRAVITY WITH U (1) GAUGE FIELD
In the previous section, we illustrate the algorithm of the gravity/fluid correspondence with a simple pure gravity model, which yields a viscosity for the dual fluid on the boundary.
In general, the gravity/fluid correspondence could be applied to studying fluid dynamics dual to various bulk gravity configurations with matter fields in the bulk gravity, and it has been found that after adding the matter fields to the bulk gravity, more interesting properties of the dual fluid could be extracted. We discuss here the impact on the dual boundary fluid of adding a U(1) gauge field to the bulk.
A. The simplest case The action with a U(1) gauge field can be written as [47,48] which gives the following equations of motion Following the general algorithm described in Sec. II, we start with the five-dimensional charged RN-AdS black brane solution [66][67][68] where The outer horizon of the black brane is located at r = r + , where r + is the largest root of f (r) = 0, and its Hawking temperature is Writting the above black brane solution in the Eddington-Finkelstin coordinate system, one has ds 2 = −r 2 f (r)dv 2 + 2dvdr + r 2 (dx 2 + dy 2 + dz 2 ) , (3.6) where v = t + r * , and r * is the tortoise coordinate satisfying dr * = dr/(r 2 f ). The boosted solution can then be written as where velocities β i , M, Q are constants.
The procedure of solving for the first-order perturbative solution is essentially the same as in the previous section. Now the tensors inducing the source terms become Letting the parameters be x µ -dependent and expanding them around x µ = 0 to the firstorder, one has and the corresponding required first order correction terms around x µ = 0 are then given by ds (1) 2 = k(r) r 2 dv 2 + 2h(r)dvdr + 2 Note that the gauge a r (r) = 0 has been chosen. Therefore, the first-order perturbative gravitational and Maxwell solution could be achieved by requiring a cancelation of the source terms and the correction terms.
The solutions then allow one to compute the boundary stress tensor. According to the dictionary of fluid-gravity correspondence, the first-order stress tensor of the dual fluid τ µν then reads with the pressure and viscosity given by where the coupling 16πG and ℓ are recovered for comparison purposes.
Following the AdS/CFT correspondence, the dual operator ofÃ µ (hereÃ µ is the boundary value of A µ projected on the boundary) is the charge current of the dual fluid. One can thus extract the charge current of dual fluid as where the factor r 4 comes from the conformal transformation. From this equation and the result for the first-order perturbative solution, one obtains the charge current with the zeroth-order particle number current 18) and the first-order charge current is the chemical potential, and κ = π 2 T 3 r 7 + 4g 2 M 2 is the thermal conductivity. From this simplest case, it is obvious that after adding the U(1) gauge field in the bulk, the charge current of dual fluid could also be extracted.
B. In the presence of an external field A ext µ It has been found that in the presence of an external field A ext µ , the charge current of dual fluid has extra structures, whereas this external field does not change the stress tensor of dual fluid [46,49,53,54]. The key point is that a constant external field A ext µ could be added into (3.8) like [46]  After requiring the external field A ext µ to be a function of x µ , we can expand it around x µ = 0 to the first order as in (3.12), The corresponding correction terms around x µ = 0 could be same as (3.13). One finds that the external field A ext µ changes the coefficients a i (r), j i (r) in Eq. (3.13), and hence change the charge current of dual fluid [46,53,54]. Finally, the result of the first-order charge current now becomes where κ = π 2 T 3 r 7 are the thermal and electrical conductivity, respectively. Obviously, a new structure related to the electrical conductivity appears in the first order charge current.  24) and the equations of motion become Although the Maxwell equation is different from Eq. (3.2), it turns out that the fivedimensional charged RN-AdS black brane solution (3.7)(3.8) still solves the above equations (3.25). In the following we also insert an external field A ext µ into the bulk solution. Now the tensors inducing the source terms become One can solve the first-order perturbative gravitational and Maxwell solution as before. Note that the Chern-Simons term appears in the second term of W A in (3.27), while it does not change W IJ in (3.26). As a consequence, h(r), k(r), α ij are the same as in the case without the Chern-Simons term, while a i (r) and j i (r) are different [53,54]. Therefore, the Chern-Simons term will not change the first-order stress tensor of the dual fluid τ µν . However, in the presence of Chern-Simons term, the charge current of dual fluid can be computed as Although the zeroth-order particle number current is the same as Eq. (3.18), the first-order charge current becomes [47-49, 53, 54] where κ = π 2 T 3 r 7 Obviously, the Chern-Simons term changes the charge current of dual fluid, as can be seen from the last three terms in Eq. (3.29), whose coefficients are proportional to κ cs . In addition, it should be emphasized that the external field A ext µ and its boundary value are also important, as there are three terms related to the external field A ext µ , and if one chooses A ext µ (0) = 0 in Eq. (3.21), the first-order charge current becomes which is the well-known result related to triangle anomalies [49]. Moreover, if one chooses A ext µ (0) = (C, 0, 0, 0), although the first order current Eq. (3.29) becomes the same as Eq. (3.31), the chiral magnetic conductivity is given as the second term is the extra contribution from the last term in Eq. (3.29) [54,56].

IV. CONCLUSION AND DISCUSSION
In this paper, we illustrated the algorithm of gravity/fluid correspondence to extracting transport coefficients of dual fluid from the bulk gravity with a simple pure gravity configuration, and discussed its application to various bulk gravity configurations in the presence of a U(1) gauge field. On the fluid side, the transport coefficients can be obtained by perturbing the system away from equilibrium. On the gravity side, this can be achieved by promoting the parameters in the boosted black brane solution to functions of boundary coordinates.
Extra correction terms are then needed to render the boundary-coordinate-dependent metric a solution of Einstein equations. This determines the extra correction terms and thus the perturbative solution of the bulk gravity, from which the transport coefficients of dual fluid can be determined using the dictionary of gravity/fluid correspondence.
As expected, adding a U(1) gauge field to the bulk gravity induces charge current for the dual fluid. One also finds that the external field A ext µ and U(1) Chern-Simons term in the bulk could affect the charge current. Some recent work investigated the dual fluid on a finite cutoff surface and showed that the electric and magnetic conductivity could still show up even without the external field A ext µ . A simple interpretation is that the projection of A µ onto the cutoff surface naturally introduces its boundary valueÃ µ as the external field A ext µ [60]. In addition, it should be emphasized that the introduction of Chern-Simons term in the bulk usually can lead to extra structures which generate anomalous transport coefficients [69]. For example, after adding the Chern-Simon term of the Maxwell field in the bulk, the conserved current J µ contains additional terms related to the anomalous magnetic and vortical effects [47-49, 53, 54]. Moreover, it has been shown that if the gravitational Chern-Simons term is added in the bulk, a new term related to the Hall viscosity will appear in the stress tensor of the dual fluid [70][71][72]. Therefore, it will be interesting to further investigate the effects of Chern-Simon terms in other modified gravity configurations.