AdS$_5$-Schwarzschild deformed black branes and hydrodynamic transport coefficients

Deformed AdS$_5$-Schwarzschild black branes are here derived, employing the membrane paradigm and the ADM procedure. AdS/CFT near-horizon methods are then implemented to compute the shear viscosity-to-entropy ratio of the deformed AdS$_5$-Schwarzschild black branes. It provides constraints for the deformed black brane free parameter, generating new black brane solutions.

parameters into these new solutions.
Previously, we have explored the technique employed here to derive a family of solutions that consists of a deformation in the AdS 4 -Reissner-Nordström background, and its potential applications to AdS/CMT [17]. By embedding the brane into a higher dimensional bulk, we were able to mimic the Hamiltonian and momentum constrains from the ADM formalism for static configurations of the metric field [18,19]. These equations turn out to be a weaker condition on the metric functions, allowing for a family of deformations of solutions from classical GR. In the present work we apply a similar procedure to the AdS 5 -Schwarzschild black brane [20,21].
The paper is organized as follows: in Sect. II the relevant results of linear response theory and fluid dynamics are briefly presented within the hydrodynamics formalism, followed by a presentation of the AdS/CFT duality and its methods in Sect. II B. Sect. III is then devoted to derive the AdS 5 -Schwarzschild deformed gravitational background and to discuss the calculation of the η/s ratio, whose explicit computation is carried out for the family AdS 5 -Schwarzschild deformed black branes in Sect. IV. The saturation of η/s therefore is shown to constrain the free parameter AdS 5 -Schwarzschild deformed black brane, driving the family of deformed branes to two unique solutions: the standard AdS 5 -Schwarzschild black brane and a new black brane solution. The concluding remarks are then presented.

II. HYDRODYNAMICS AND LINEAR RESPONSE THEORY
The so called hydrodynamic limit is characterized by the long-wavelength, low-energy regime [22], and is often applicable to describe conserved quantities. As an effective description of field theory, hydrodynamics naturally does not contain the details of a microscopic theory. These are encoded into the transport coefficients, among which the shear viscosity, η, plays a prominent role.
The macroscopic variables encoded in the energymomentum stress tensor, T µν , along with its conservation law, ∂ µ T µν = 0, describe a simple fluid. In general, one introduces a constitutive equation by determining the form of T µν in a derivative expansion, given in terms of the normalized fluid velocity field u µ (x ν ), its pressure field p(x µ ) and its rest-frame energy density ρ(x µ ).
To first order in the derivative expansion, the stress tensor is expressed as [3,22] where τ µν , the term which is first-order in derivatives, carries dissipative effects. In the local rest frame, its spatial components are The constitutive equation for a viscous fluid, as defined above, yields both the continuity and Navier-Stokes equations, thus closing the system of equations of motion. The shear and bulk viscosities, respectively η and ζ, were introduced to account for dissipative effects.
For a theory described by an action functional S, the coupling of an operator O to an external source ϕ (0) reads [23] One is often interested in determining the response in O, which is given by where O(t, x) S denotes the ensemble average (onepoint function) of the operator O in the presence of ϕ (0) . The study of such response up to first order in ϕ (0) is known as linear response theory. In momentum space the one-point function (4) reads [23] δ where is the retarded Green's function associated to O in Fourier space [24]. As seen in Eq. (5), the one-point function, δ O , is reduced to the determination of G O,O R (ω, q). The Kubo formula relates the retarded Green's function to a transport coefficient. Computing the shear viscosity, η, the transport coefficient associated to the viscous fluid that is dual to the deformed AdS 5 -Schwarzschild black brane will restrict the possible values of the deformed black brane free parameter.
A. The shear viscosity and its Kubo formula By coupling a fictitious gravitational field to the stress tensor of the fluid one is able to derive the shear viscosity in agreement with the idea from GR that fluctuations in the stress tensor induce spacetime fluctuations and vice-versa [2]. This approach, although seemingly a pure analogy, has a natural interpretation in the context of AdS/CFT, as we will further detail in Sec. II B.
The response of T µν under gravitational fluctuation is determined by the introduction of an off-diagonal perturbation term, h µν . In the {t, x, y, z} coordinate system g (0) µν is: The perturbation is off-diagonal so that its response induced in T µν accounts for the shear effects encoded in η. The perturbation (6) requires the extension of the constitutive equation (1) to curved spacetime, where the dissipative term reads [2,3] beging ∇ µ the covariant derivative with respect to g (0) µν . In order to write the constitutive equation in a covariant way, the projection tensor Π µν = g µν(0) + u µ u ν is introduced.
Taking the appropriate covariant derivatives of the velocity, and assuming that u i = u i (t), leads to the response in Fourier space [23] δ τ xy (ω, q = 0) = iωηh (0) xy .
The general result from linear response theory expressed in Eq. (5) is, in this case, Comparing this with the result in Eq. (8), and solving for η, yields the Kubo formula It is given in the ω → 0 limit, since η does not depend on ω or q. Now, with Eq. (10), it is clear that η is fully determined once the retarded Green's function G xy,xy R is found. Computation of the retarded Green's function is straightforwardly achieved in this context applying the AdS/CFT duality.

B. AdS/CFT and the GKP-Witten relation
The AdS/CFT duality [4], in simple terms, relates a gravity theory defined on an asymptotically AdS spacetime, in D dimensions, to gauge theory in D − 1 spacetime dimensions. A further development of such duality is the so called GKPW relation [25,26], which relates the partition functions of both gravity and gauge theories. Explicitly, the GKPW formula reads where ϕ is a field in the gravitational bulk theory;S is the on-shell action; · denotes the ensemble average; and ϕ (0) = ϕ| u=0 , in coordinates such that the AdS boundary is at u = 0, which is where the gauge theory is realized.
The gauge theory is said to live on the boundary of the bulk. In fact, one obtains the on-shell action by evaluating the integral for a field ϕ which is the solution of the equations of motion subject to conditions imposed at the AdS boundary, ϕ| u=0 = ϕ (0) . In this case,S reduces to a surface term on the AdS boundary, which, for D = 5, reduces the 5D action (left-hand side of Eq. (11)) to a 4D quantity (right-hand side of Eq. (11), identified with the partition function of the boundary gauge theory, when an external source ϕ (0) is added). In fact, from the 4D gauge theoretical point of view, ϕ (0) is an external source, whereas from the 5D gravitational point of view, ϕ is seen as a field propagating across the bulk. Therefore, in the context of AdS/CFT, one can say that a bulk field behaves as an external source of an operator in the boundary theory.
In this framework, the GKPW relation yields the following expression for the one-point function, related to the response of a system when an external source is added [26,27], The one-point function in the absence of the external source is obtained by simply evaluating the expression above for ϕ (0) = 0, One considers the bulk theory to be GR in 5D with negative cosmological constant, Λ 5 . Therefore the action is simply the Einstein-Hilbert one, added to matter fields where S mat is specified by the boundary theory of interest. Regarding massless scalar field yields The solution to the 5D bulk action will be asymptotically AdS spacetime. A particular case of interest is the AdS 5 -Schwarzschild spacetime, where f (u) = 1 − u 4 , with u = r 0 /r defining the radial coordinated such that u = 1 locates the horizon, whereas u = 0 is the spacetime boundary. Also, one notices that the AdS radius is set to unity, L = 1. For u → 0, Eq. (16) reads Since the Einstein-Hilbert term in Eq. (14) clearly does not depend on the scalar field, the one-point function, Eq. (12), depends only on the matter contribution when it comes to computing the on-shell action,S. Then, in what follows one effectively considers S = S mat . Assuming that the scalar field is static and homogeneous along the spatial directions of the boundary, i.e. ϕ = ϕ(u), and considering the asymptotic behaviour of the metric, Eq. (17), the action for the massless scalar field at the boundary becomes assuming that the scalar field vanishes at the horizon. Notice that the second term is just the EOM for the scalar field, whose asymptotic solution reads The second term in Eq. (18) vanishes, and the on-shell action reduces to the surface term on the AdS boundary (u = 0). Finally, substituting the asymptotic form of the scalar field, Eq. (19), one obtains the on-shell action, The one-point function, Eq. (12), reads Notice that the absence of the external source implies that Eq. (13) vanishes, so that Relating this result to Eq. (5) one determines the retarded Green's function, where the Green's function does depend neither on ω nor on q, since ϕ (1) also does not. As it was mentioned in the beginning of this subsection, η is related to a gravitational perturbation. In fact, all the results obtained for the scalar field here apply in this case because, formally, the gravitational perturbation reduces to the EOM governing a massless scalar field [28]. Thus, one can use the results obtained in this section to compute δ τ xy from Eq. (22).

III. THE ADS5-SCHWARZSCHILD DEFORMED BLACK BRANE
The general solution to 5D vacuum Einstein gravity with a negative cosmological constant depends on the horizon metric H ij and an integration constant, k. Provided that the constraint R ij = 3kH ij holds, the solution for k = 0, leading to a planar horizon i.e. H ij = δ ij , is the AdS 5 -Schwarzschild black brane [29] ds 2 = −r 2 f (r)dt 2 + 1 r 2 f (r) dr 2 +r 2 dx 2 +dy 2 +dz 2 , (24) with f (r) = 1 − r 4 0 r 4 . The black brane entropy density is defined from the area law as s = a/(4G 5 ), where the horizon area density, a = A/V , is written in terms of the horizon area, Therefore, a = A/V can be evaluated directly by substituting Eq. (24) into (25), and the entropy density of the AdS 5 -Schwarzschild black brane reads The theory dual to AdS 5 -Schwarzschild black brane is a conformal fluid [30]. Hence its stress-energy tensor is traceless, fixing the bulk viscosity [1, 3], ζ = 0, leaving the shear viscosity η as the only non-trivial transport coefficient. To evaluate it, one first considers a gravitational perturbation and then compute the response to the stress-energy tensor, by solving the perturbation equation within the hydrodynamic limit [23,28].
We will fully present the arguments and a similar calculation in the next section, when considering the deformed AdS 5 -Schwarzschild black brane as the gravitational background. The calculation will not be exactly the same, but analogous. The saturation of the η/s ratio in the AdS 5 -Schwarzschild black brane gravitational background reads [31] One does not need discuss specific bulk features, as the existence of solutions to the higher-dimensional Einstein's equations describing gravity is undertaken by the Campbell-Magaard embedding theorems [32]. Therefore, considering a brane with finite tension embedded in an AdS bulk, the Gauss-Codazzi equations yield the electric part of the Weyl tensor.
In an AdS bulk with cosmological constant Λ, a solution must satisfy the effective Einstein's equations Projecting Eq. (28) onto the brane, which is timelike and has codimension 1, in Gaussian coordinates (x µ , z), where z = 0 corresponds to the brane itself, one obtains constraints for Λ denoting the brane cosmological constant. Eqs. (29) mimics constraints in the ADM procedure [33]. The Hamiltonian constraint is equivalent to R µν = E µν .
One supposes a general metric, performing the coordinate change u = r 0 /r and sets the AdS radius to unity, where r 0 is the horizon radius. By demanding that the ADM constraint leads to the AdS 5 -Schwarzschild metric when β → 1, the Hamiltonian constraint reads, where the function f (r, r 0 , β) is given by Eq. (A1) in the Appendix A. The constraint (31) is satisfied by The constant β parameter is referred to as a deformation parameter. In the next section we will investigate how the shear-viscosity-to-entropy density ratio can drive specific values for β.

IV. η/s FOR THE ADS5-SCHWARZSCHILD DEFORMED BLACK BRANE
We now consider the AdS 5 -Schwarzschild deformed black brane (30,32,33), and derive the η/s ratio in this gravitational background. An important remark is the use of some results which are only valid as long as the gravitational action takes the form of the Einstein-Hilbert action. In fact, the AdS 5 -Schwarzschild deformed metric arises as a deformation of the AdS 5 -Schwarzschild one [11]. Hence the same action-dependent results may be applied to the AdS 5 -Schwarzschild deformed black brane.
The metric determinant, g, is such that where, from now on, N and A refer respectively to N (u) and A(u), unless otherwise specified.
Let one considers a bulk perturbation h xy , so that: where ds 2 AdS5−SD denotes the AdS 5 -Schwarzschild deformed black brane metric, Eq. (30). Now, one considers Eq. (8), for h (0) xy being the perturbation added to the boundary theory, which is asymptotically related to h xy , the bulk perturbation, by according to Eq. (19). Notice that one can directly use the results for a massless scalar field, as g xx h xy obeys the EOM for a massless scalar field [23,28]. Besides, the AdS 5 -Schwarzschild deformed black brane has the same asymptotic behavior of the AdS 5 -Schwarzschild black brane (namely, Eq. (17)). In the context of what was discussed in Sec. II B, one can identify g xx h xy as the bulk field, ϕ, which plays the role of an external source of a boundary operator, in this case τ xy . Therefore, one can directly obtain the response δ τ xy , from Eq. (22), where it is now convenient to reintroduce the 1/16πG 5 factor. Comparing Eqs. (8) and (36) yields Since both metrics, the AdS 5 -Schwarzschild and its deformation, are obtained from the Einstein-Hilbert action, the entropy density is the same c.f. Eq. (26). Plugging this result into Eq. (37) yields Now, h (1) xy is found by solving the EOM for the perturbation g xx h xy ≡ ϕ, which is that of a massless scalar field [23,28] Considering a stationary perturbation of the form ϕ = φ(u)e −iωt , the perturbation equation reduces to a second-order ODE for φ(u), To derive the solution of Eq. (40), two boundary conditions are imposed: the incoming wave boundary condition in the near-horizon region, corresponding to u → 1, and a Dirichlet boundary condition at the AdS boundary, To incorporate the near-horizon incoming wave boundary condition, Eq. (40) is first solved in the limit u → 1. After a straightforward computation one finds the following solution As discussed in Ref. [24], this solution has a natural interpretation using tortoise coordinates, which allows one to identify this solution to a plane wave. The positive exponent represents the wave outgoing from the horizon, whereas the negative one describes the wave incoming to the horizon, which, according to the near-horizon boundary condition, allows us to fix, in the u → 1 regime, Now Eq. (40) will be solved for all u ∈ [0, 1], with a power series in ω, up to O(ω), as we are interested in a solution in the hydrodynamic limit, ω → 0: Therefore, in the hydrodynamic limit, the second term in Eq. (40), which is proportional to ω 2 , is not considered. By direct integration of the equation the solution reads for C i and K i the integration constants and i = 0, 1. Thus, according to Eq. (43), we have In the u → 0 and u → 1 limits, in order to impose the boundary conditions, one can expand the integral in (45) around these extremal values. It yields, up to leading order in the respective expansions, The boundary condition at the AdS boundary then fixes the first pair of integration constants, as implying that (C 0 + ωC 1 ) = φ (0) . Near the horizon, u → 1, one has Expanding the near-horizon solution, Eq. (42), up to O(ω), yields It is straightforward to see that Eq. (48) fixes the proportionality according to Comparing Eqs. (48) and (50) allows us to fix the second pair of integration constants: Then the full solution reads Accordingly, the full time-dependent perturbation is asymptotically given by: (54) Comparing Eq. (54) to Eq. (35), and identifying yields Thus, substituting the above result into Eq. (38) implies that For computations in gravitational backgrounds which are solutions of the EOM from the Einstein-Hilbert action with Λ < 0, the ratio bound η/s ≥ 1/4π holds [31]. Both the AdS 5 -Schwarzschild and the AdS 5 -Schwarzschild deformed black branes satisfy this bound. In this case, the deformation parameter can attain the values The saturation η/s = 1/4π, corresponding to N c → ∞, then implies β = 1. This result has been expected, as this case recovers the AdS 5 -Schwarzschild black brane (16). On the other hand, the saturation of the shear viscosityto-entropy density ratio also yields β = 3/4, generating a new black brane solution. In this case, the AdS 5 -Schwarzschild deformed black brane is given by Eq. (30), with This new solution can be an interesting result worthy further investigation, mainly in the AdS/QCD correspondence.

V. CONCLUDING REMARKS AND PERSPECTIVES
The ADM procedure was used to derive a family of AdS 5 -Schwarzschild deformed gravitational backgrounds, involving a free parameter, β, in the black brane metric (30,32,33). Computing the η/s ratio for this family provided two possible values to β. The first one, β = 1, was physically expected, corresponding to the AdS 5 -Schwarzschild black brane. The another one, β = 3/4, generates a new AdS 5 -Schwarzschild-like deformed black brane (30,59,60). Besides the importance of the result itself, in particular for the membrane paradigm of AdS/CFT, it has a good potential for relevant applications, mainly in AdS/QCD.
As large-N c gauge theories considered by AdS/CFT are good approximations to QCD, one could expect that the result of Eq. (27) may be applied to the quark-gluon plasma (QGP), which is a natural phenomenon in QCD, when at high enough temperature the quarks and gluons are deconfined from protons and neutrons to form the QGP [34]. In fact, experiments in the Relativistic Heavy Ion Collider (RHIC) have shown that the QGP behaves like a viscous fluid with very small viscosity, which implies that the QGP is strongly-coupled, thus discarding the possibility of using perturbative QCD to the study of the plasma. Therefore, the new AdS 5 -Schwarzschild deformed black brane (30,59,60) can be widely used to probe additional properties in the AdS/QCD approach. As in the holographic soft-wall AdS/QCD the AdS 5 -Schwarzschild black brane provides a reasonable description of mesons at finite temperature [20], we can test if using the AdS 5 -Schwarzschild deformed black brane derives a more reliable meson mass spectra for the mesonic states and their resonances, better matching experimental results. Besides, the new AdS 5 -Schwarzschild deformed black brane can be also explored in the context of the Hawking-Page transition and information entropy [35,36].