On the universality of thermodynamics and $\eta/s$ ratio for the charged Lovelock black branes

We investigate general features of charged Lovelock black branes by giving a detailed description of geometrical, thermodynamic and holographic properties of charged Gauss-Bonnet (GB) black branes in five dimensions. We show that when expressed in terms of effective physical parameters, the thermodynamic behaviour of charged GB black branes is completely indistinguishable from that of charged Einstein black branes. Moreover, the extremal, near-horizon limit of the two classes of branes is exactly the same as they allow for the same AdS$_2\times R_3$, near-horizon, exact solution. This implies that, although in the UV the associated dual QFTs are different, they flow in the IR to the same fixed point. The calculation of the shear viscosity to entropy ratio $\eta/s$ confirms these results. Despite the GB dual plasma has in general a non-universal temperature-dependent $\eta/s$, it flows monotonically to the universal value $1/4\pi$ in the IR. For negative (positive) GB coupling constant, $\eta/s$ is an increasing (decreasing) function of the temperature and the flow respects (violates) the KSS bound.


Introduction
Since its first formulation, Lovelock gravity [1] has been a fruitful and widely explored subject [2][3][4][5]. The peculiarity of the theory is to be a higher curvature gravity theory with second-order field equations for the metric. This nice feature not only allows to avoid some of the shortcomings of generic higher-derivative theories (such as ghosts in the linearized excitation spectrum and ill-posed Cauchy problem) but also enables us to derive exact black hole (and black brane) solutions of the theory. As a consequence, the thermodynamics of bound is violated if translation invariance is broken [30][31][32][33]. If translation symmetry is preserved in the IR, η/s tends to a constant as T → 0, whereas it scales as a positive power of T when translation symmetry is broken. Although the breaking of translation symmetry prevents a purely hydrodynamic interpretation of η, this result strongly indicates that bounds on η/s are completely determined by IR physics and insensitive to the UV regime of the theory.
A promising way to tackling this kind of problems is to consider gravitational backgrounds in which η/s flows as a function of the temperature and for which an IR fixed point exists at T = 0. Following this indication, in this paper, we will focus on the charged 5d GB black brane solutions (BB) for which it is known that the ratio η/s flows as a function of the temperature [34].
We will start by investigating the general Lovelock BB solution as a thermodynamic system. We will show that, when expressed in terms of effective physical parameters, the thermodynamic behavior of charged Lovelock BB is completely indistinguishable from that of charged Einstein BB. We then proceed by focusing on the 5d GB case and investigating in detail the geometrical properties of the charged GB black brane. We show that the theory allows for two branches of solutions continuously connected trough a branch-point singularity. Holographically they represent flows between two different CFTs through a singularity. Moreover, we show that at extremality, in the near horizon regime, the charged GB black brane has exactly the same AdS 2 × R 3 geometry of the Einstein charged black brane. In fact, in the near horizon regime the contributions of the higher-curvature terms to the field equations vanish and the AdS 2 × R 3 solution of Einstein-Maxwell gravity in 5d is also the exact solution of GB gravity in 5d.
In terms of the dual QFT description this means that, although in the UV the associated dual QFTs for Einstein and GB gravity are different, in the IR they flow to the same fixed point. We then calculate the shear viscosity to entropy ratio η/s for the extremal and nonextremal case, using the simple method recently proposed in Refs. [30,31,35]. Whereas in the non-extremal case we find a non-universal, monotonically increasing (for negative GB coupling parameter) or decreasing (for positive GB coupling parameter) temperaturedependent expression for η/s, in the extremal case we find the universal value 1/4π. Thus, charged Gauss-Bonnet gives an example of a higher curvature gravity theory in which the IR behaviour of the dual theory respects the universal bound for η/s and is completely independent from the UV regime.
The structure of the paper is as follows. In Sect. 2 we briefly review some relevant features of black brane solutions of Lovelock-Maxwell gravity and show the universality of their thermodynamic behaviour. In Sect. 3 we review the Reissner-Nordström (RN) BB solutions of 5d Einstein-Maxwell, including its extremal limit and its AdS 2 × R 3 , extremal, near horizon geometry. In Sect. 4 we discuss the charged black brane solution of GB gravity, paying particular attention to the geometry of the solution and the extremal, near horizon regime. In Sect. 5 we discuss the charged GB black brane thermodynamics, and we consider in detail the thermodynamic behaviour at small and large temperature. In Sect. 6 we discuss the shear viscosity to entropy ratio for the GB plasma and compute the value both for T = 0 and T = 0. We also present a discussion about the large T and small T behaviour. Finally, in Sect. 7 we draw our conclusions. In the Appendix A we briefly discuss the black hole solutions of the GB theory, i.e., the solution with spherical horizons.

Black brane solutions of Lovelock gravity
Let us consider black branes that are solutions of Lovelock higher curvature gravity in ddimensional spacetime. To describe the static, electrically charged, radially symmetric AdS Lovelock BB, we use the following line element and electromagnetic (EM) field where dΣ 2 d−2 denotes the (d − 2)-dimensional space with zero curvature and planar topology, whereas L is related to the cosmological constantα (0) by Notice that the metric in Eq.(2.1) differs from that in the usual Schwarzschild gauge by a (constant) rescaling t → N t of the time coordinate t. As we will see later in this paper this rescaling is necessary in order to have a unit speed of light in the dual CFT. Using the rescaled Lovelock coupling constants where G N is the d-dimensional Newton's constant and each of the Einstein-like tensors G independently satisfies a conservation law ∇ a G (k) a b = 0. The higher-curvature terms contribute to the equations of motion only for d > 2k. For d = 2k the higher-curvature corrections are topological, and they vanish identically in lower dimensions. Settingα (k) = 0 for k ≥ 2, one can recover the standard form of general relativity. In the notation (2.2), the field equations (2.3) reduce to the requirement that f (r) solves the following polynomial equation of degree k max = d−1 2 (see e.g., [2][3][4][5][36][37][38]) Here M ADM is the ADM mass of the black brane and ω d is 2.1 Universality of black brane thermodynamics in Lovelock gravity Interestingly, even without knowing f = f (r) in Eq.(2.5) explicitly, it is possible to find the thermodynamic quantities characterizing the Lovelock black brane solution [2,39,40]. Let r + denotes the radius of the event horizon, determined as the largest root of f (r) = 0.
Introducing the effective mass M and temperature T related to the usual ADM mass M ADM and Hawking temperature T H by the relations the black brane mass M , the temperature T , the entropy S, and the gauge potential Φ are given by [2,41] (2.11) The rescaling of the physical parameters (2.8) of the Lovelock BB having the dimensions of energy is essentially due to the presence of the constant N 2 in the metric. The two time coordinates t and N t correspond to using two different units to measure the energy. However, when we deal with Einstein-Hilbert branes the rescaling of the time coordinate is not necessary and we will simply set M = M ADM and T = T H . Notice that the area-law for the entropy S always hold for the generic Lovelock black brane. A striking feature of these thermodynamic expressions is that they do not depend on the Lovelock coupling constants α k for k ≥ 2 but only on α 0 and α 1 , i.e., they depend only on the cosmological constant and on Newton constant. This means that the thermodynamic behaviour of the BB in Lovelock theory is universal, in the sense that it does not depend on the higher order curvature terms but only on the Einstein-Hilbert term, the cosmological constant and the matter fields content (in our case the EM field). This implies, in turn, that as thermodynamic system the charged BBs of Lovelock gravity are indistinguishable from the Reissner-Nordström BBs of Einstein-Hilbert gravity. Notice that this feature is not shared by the black hole solutions of the theory, i.e., solutions with spherical or hyperbolic horizons. In fact, in the Lovelock thermodynamic expressions (see Refs. [2,41]) the dependence on the Lovelock coupling constants α k≥2 is introduced by the dependence on the curvature κ of the (d − 2)-dimensional spatial sections. This dependence drops out when κ = 0.
We remark, however, that the universal thermodynamic behaviour of charged Lovelock black branes is strictly true only when we choose N = 1 in the metric (2.1). As we will see later in this paper, the parameter N has to be fixed in terms of the Lovelock coupling constants α k≥2 . Hence, the ADM mass and the Hawking temperature of the Lovelock BB will depend on α k≥2 . The universality of the Lovelock BB thermodynamics is recovered simply by rescaling the units we use to measure the energy, i.e., by using in Eqs. (2.9) and (2.10) the effective parameters M and T instead of M ADM and T H .
In the following, we provide a detailed calculation for the case k max = 2, i.e., GB gravity in five spacetime dimensions, which is the most interesting case from the AdS/CFT point of view. However, we expect that most of our considerations can be easily generalized to every charged BB solution of Lovelock gravity in generic dimensions.
where ω 5 is given by Eq. (2.6) and G N is the five dimensional Newton's constant. Performing the asymptotic limit r → ∞, the function (3.1) reduces to f = r 2 /L 2 , i.e., AdS 5 with AdS length L 2 = α −1 0 . The ratio L 3 /G N is proportional to the central charge c of the dual CFT. The central charge c can be defined as the coefficient of the large temperature expansion of the free energy (see Sect. 5.1). The condition for the validity of classical AdS gravity in the bulk is c >> 1. In most of the established examples of the AdS/CFT correspondence c ∝ N , where the limit c >> 1 is referred to as the large N limit.
Setting r 2 = Y in Eq. (3.1), the RN BB horizons are determined by the cubic equation This equation has two positive roots for which gives the extremal (BPS [42,43]) bound for the RN black brane in 5d. In general, we will have an inner and outer horizon, when the bound is saturated the two horizons merge at r 0 and the RN BB becomes extremal. In the extremal case, Eq. (3.2) has a double root at Y 0 = ω 5 M L 2 /3 and f (r) can be factorized in the following way The extremal near-horizon geometry can be determined expanding the metric near r 0 and keeping only the leading term in the metric a simple translation of the radial coordinate r → r + r 0 gives the AdS 2 × R 3 extremal near-horizon geometry with AdS 2 lenght l The extremal solution given in Eq. (3.4) is a soliton interpolating between the asymptotic AdS 5 geometry in the UV and the AdS 2 × R 3 geometry (3.6) in the IR.
(2.5) reduces to a quadratic equation from which one obtains two possible solutions, f ± . In the following, we will refer to the solution f − as the 'Einstein branch' because it approaches the Einstein case when the Gauss-Bonnet coupling α 2 goes to zero and to f + as the 'Gauss-Bonnet branch' [10]. The quadratic Eq. (4.1) gives the following necessary condition requirement for the existence of f ± for large r: When this inequality is violated, the space becomes compact because of the strong nonlinear curvature [10]. Therefore, there is no asymptotic 'AdS region' and consequently no proper black hole with standard asymptotics.

5d GB black brane
In this subsection, we discuss the special case of 5d GB BB (κ = 0). Moreover, from now on we set α 1 = 1 in order to recover the usual Newtonian limit. It is easy to check that that for d = 5 and κ = 0, then Eq. (4.1) reduces to the following equation and the two branches are respectively In case of positive GB coupling α 2 > 0 that satisfy the condition (4.2), the two branches describe two asymptotically AdS 5 spacetimes, however, from Eq. (4.4) one can see that f + has no zeroes, hence the f + -branch does not describe a BB but a solution with no event horizon. Thus, only the f − -branch describes a BB solution.
Let us now study the asymptotic geometry of the GB BB. At leading order for r → ∞ the metric coefficient In order to have the boundary of the asymptotic AdS d conformal to (d − 1)-Minkowski space with speed of light equal to 1, ds 2 ≈ α 0 r 2 (−dt 2 + dΣ 2 3 ), the constant N 2 has to be chosen as where we have the + sign for the f − branch, the BB solution, while the − sign has to be used when we consider the f + branch.
In the AdS/CFT correspondence, the central charge c of the dual CFT is determined by the AdS length. Thus, the CFTs dual to GB gravity in both branches have central charge different from the RN case. Only in the α 2 → 0 limit the central charge of the f − -branch coincides with that of the CFT dual to the RN theory. However, naive computation of the central charge in terms of the AdS length does not work in this case because of the rescaling of the time coordinate. We will compute c in Sect. 5 using the scaling law of the mass and entropy as a function of the temperature.
For α 2 < 0, only the f − branch is asymptotically AdS. Conversely, the f + branch describes a spacetime which is asymptotically de Sitter (dS) and can be therefore relevant as a cosmological solution.

Singularities
To determine the position of the singularities of the spacetime we calculate the scalar curvature for both the f ± branches: where the ± sign refers respectively to the f ± branches. To simplify expressions we used (here and after) the following notation (4.8) There are curvature singularities at r = 0 and at the zeroes of the argument of the square root in Eq. (4.7) (branch-point singularities). The position of the physical singularities of the spacetime is therefore determined by the pattern of zeroes of the function g(Y ), with (4.9) The singularity will be located at the biggest positive zero Y 1 of g(Y ) or at r = 0 when g(Y ) has no zeroes for positive Y . The singularity at Y = Y 1 is a branch point singularity.
The pattern of zeroes of g(Y ) is determined by the signs of the coefficients ρ, σ and the discriminant ∆ = ρ 2 2 + σ 3 3 .
• For σ > 0, the function g(Y ) is a monotonic increasing function of Y with a single zero which, depending on the sign of ρ, will be positive Y = Y 1 (ρ > 0) or negative (ρ < 0). The physical spacetime singularity will be therefore located at r = √ Y 1 for ρ, σ > 0 and at r = 0 for ρ < 0, σ > 0.
• For σ < 0, the function g(Y ) is an oscillating function with a maximum at negative Y and a minimum at positive Y , it may therefore have one, two or three zeros. For σ < 0, ρ > 0, g(Y ) has at least a positive zero. For σ < 0, ρ < 0 we have a positive zero for ∆ ≤ 0 and no positive zeros for ∆ > 0. For ∆ = 0 we have a double zero of g(Y ) so that Y 1 is not anymore a branch point singularity. In this latter case the singularity is at r = 0.

f − -Branch
In this subsection, we study in detail the horizons of the f − -branch, solution of Eq. (4.4), describing the GB black brane. In general the BB will have an inner (r = r − ) and outer (r = r + ) event horizon. The BB becomes extremal when r + = r − . Using the notation (4.8), (4.9), one finds that the necessary condition for the existence of the BB is the positivity of the argument in the square root in Eq. (4.4), i.e., g(Y ) ≥ 0. The position of the event horizon(s) is determined by the positive roots of the cubic equation (4.10) We will first consider the case α 2 > 0, which corresponds to σ, ρ, e > 0 (since also α 0 > 0). The condition for the existence of real roots of the function h(Y ) can be easily found: The function h(Y ) has a maximum (minimum) for, respectively also, h(Y = 0) = ρ > 0, hence the cubic equation (4.10) always has a negative root. The existence of other roots is determined by the sign of h(Y m ). We will have two (one) positive roots hence a BB with two (one) event horizons for h(Y m ) ≤ 0, i.e., for Using Eq. (4.8), the previous inequality can be written in terms of the charge Q and the effective mass M and gives the same Bogomol'nyi-Prasad-Sommerfield (BPS) bound (3.3) found in the RN case. However, the BPS bound is modified when we instead express it in terms of the ADM mass: (4.13) When the bound is saturated, the inner and outer horizon merge at r − = r + , the BB becomes extremal, and the solution describes a soliton. The striking feature of the BPS bound (4.12) is that the BPS bound of 5d Gauss-Bonnet BB does not depend on the Lovelock coupling constant, and it is exactly the same one gets for GR (α 2 = 0), i.e., for the 5d Reissner-Nordström BB. When M does not satisfy the inequality (4.12), the spacetime describes a naked singularity. For α 2 > 0, the condition M > 0 implies σ, ρ > 0 and the function g(Y ) is a monotonic increasing function which cuts the Y -axis at the point Y 1 , and, in view of the previous discussion, it also gives the position of the singularity. Since, the position of the event horizon Y h is determined by the equation , this checks that in the region where the bound (4.12) holds the condition g(Y ) > 0 is always satisfied and that the physical singularity is always shielded by two (one in the extremal case) event horizons. The behaviour of the metric function f − for α 2 , M > 0 and selected values of the other parameters is shown in Fig. 1. The solid red, green and brown lines describe respectively a naked singularity, extremal and two-horizon BB geometry. The solid blue line represents a zero-charge, BB solution with single horizon. The case α 2 < 0, M > 0 gives exactly the same BPS bound. Now, we have σ, ρ, e < 0. The function h(Y ) in Eq. (4.10) always has a negative root and a minimum (maximum) for The conditions for the existence of two positive roots become |ρ| ≤ 2 3 |σ| σ 3e leading to the same BPS bound (4.12). However, there is a crucial difference from the α 2 > 0 case. When α 2 < 0, the condition M > 0 implies σ, ρ < 0. Taking into account that |e| < 1 owing to (4.2), we see that the condition ∆ < 0 implies the BPS bound (4.12). This means that the two horizons are separated by a region in which the solution does not exist. The spacetime breaks into two disconnected parts. The physical part, having an asymptotic AdS region, describes a BB with singularity shielded by a single event horizon. The behaviour of the metric function f − for α 2 < 0 and selected values of the other parameters is shown in Fig.  2. The solid red, green and brown lines describe respectively a naked singularity, extremal and single-horizon BB geometry. The solid blue line represents a zero-charge, BB solution with horizon.

Near horizon extremal solution
When the bound (4.12) is saturated, the BB becomes extremal and the metric function (4.4) has a double zero at thus, the solution f − can be factorized as This solution represents the extremal GB soliton.
Let us now consider the near-horizon geometry. In this regime, the solution (4.17) can be expanded around r = r 0 = σ 3e 1/4 . At the leading order the Einstein branch reads (4.18) Translating the radial coordinate r → r + r 0 and rescaling the time coordinate as t → t/N we get the extremal, near-horizon geometry: i.e., AdS 2 ×R 3 with the AdS 2 length l being determined uniquely by α 0 . Thus, the extremal near-horizon geometry does not depend on α 2 and fully coincides with the extremal nearhorizon geometry (3.6) one gets in the RN case.

Near horizon metric as exact Solution of equations of motion
In this section, we will show that the near-horizon solution given in Eq. (4.20) We note that, since the Eq. (4.19) describes a spacetime with AdS 2 × R 3 geometry, the contribution to the curvature tensors coming from the planar geometry R 3 vanishes. Thus, the EOM includes only the contribution of the AdS 2 part of the metric which is a two dimensional maximally symmetric space. For a generic n-dimensional maximally symmetric space with R = Λ the two terms in Eqs. (4.20), that are quadratic in the curvature tensors, are given respectively by These relations are consequence of the fact that the GB term in the action is topological for d = 4 and identically vanishes for d = 2 and d = 3. The previous equations imply that in the case of the AdS 2 × R 3 geometry, the contributions given by the GB terms to the EOM vanish; therefore, the near horizon metric (4.19) is an exact solution of both Einstein and GB EOM. In particular, the latter reduces to the usual Einstein-Maxwell equations in 5d.
Summarising, we have seen that the AdS 2 × R 3 geometry is not only a near horizon approximation but it is an exact solution of the field equations of GB-Maxwell gravity. The presence of two exact extremal solutions (the extremal soliton interpolating through a throat region the AdS 2 ×R 3 geometry with the asymptotic AdS geometry and the AdS 2 ×R 3 geometry itself) is a typical feature of extreme black branes describing BPS states (see e.g. Refs. [44,45]). The two solutions correspond to two different extremal limits. As we will see in Sect. 5, the presence of two different extremal, exact, solutions give rise to a non-trivial extremal thermodynamic behaviour.

f + Branch
This branch does not describe a BB but a spacetime with a singularity for every value of the parameters Q = 0, M = 0. Depending on the value of the parameter α 2 we have either a spacetime with a naked singularity (for α 2 > 0) or a cosmological, asymptotically de Sitter (dS) solution with a singularity (for α 2 < 0.) This follows from the above discussion of the singularities of the scalar curvature (4.7). In the f + branch the spacetime always has a singularity, which can be located at r = 0 or r = √ Y 1 depending on the values of the parameters. This is consistent with the results of Ref. [36], according to which the f + branch is unstable and contains ghosts 1 .
For M, α 2 > 0, the metric functions for the f + branch are the dashed lines shown in Figs. 1. An interesting, peculiar feature is that in this case, all the solutions of the f − branch are continuously connected with the solution of the f + -branch passing trough the singularity. This feature has a simple analytic explanation. In the cases under consideration the singularities are the zeros of the function g(Y ) and when g(Y ) = 0 then f + = f − . This fact can have interesting holographic implications: we have two CFTs with different central charges connected through the same singularity.
For M > 0 and α 2 < 0, the f + branch describes a cosmological solution with a singularity. The corresponding metric functions are shown (dashed lines) in Fig. 2. Also in this case the solutions of the f − branch with an horizon are continuously connected with the solution of the f + -branch passing trough the singularity. We have now an asymptotically AdS solution continuously connected through a cosmological singularity to a late de Sitter geometry. On the other hand, the solutions of the f − branch describing a naked singularity are disconnected from the cosmological solutions. For α 2 , M < 0, the f + branch describes a cosmological solution with a singularity with late de Sitter behaviour, whereas the f − branch describes an asymptotically AdS spacetime with a naked singularity. However, here the two branches are disconnected. The metric functions for this case are shown in Fig. 3.
It should be stressed that in the Q = 0 case, the f + branch has ghosts in the spectrum [36]. We naturally expect this to extend to the charged case and is consistent with the intrinsic instability of these branch of solutions connected with the presence of naked singularities.

Charged GB black brane thermodynamics
In this section, we will study the thermodynamics of the GB BB solutions, i.e., solutions in the f − branch and make a comparison with the Reissner-Nordström black branes. The effective thermodynamic potentials M = M ADM /N, S, Φ and the temperature T = T H /N can be written as functions of the horizon radius r + and the charge Q by specializing Eqs. (2.9),(2.10),(2.11) to d = 5. We obtain the following equations Let us now consider separately the two limits of interest: T → ∞ and T → 0.

Large temperature
The limit T → ∞ corresponds to large radius BB, i.e., r + → ∞. In this regime, the temperature scales linearly with r + T r + πL 2 (5.3) and, at the leading order, we get for M and S This is exactly the scaling behaviour one expects for a UV fixed point described by a CFT 4 . Because of the universality of the thermodynamic behaviour, the relations (5.4) hold for both the RN and the GB BB. In the former case, Eqs.

Small temperature
The T → 0 thermodynamic behaviour corresponds to extremal BBs in which the BPS bound (4.13) is saturated. This is achieved at non vanishing, constant value of the BB radius that corresponds, as expected for BPS states, to the extremal brane T = 0 state with non vanishing mass and entropy given by We can now expand in Taylor series the temperature near r 0 to obtain 8) and the behaviour of M and S near T = 0 is of the form Again, universality of the thermodynamic behaviour imply that the relations in Eq. (5.9) hold both for the RN and for the GB BB. For the RN case, the relations take the same form with M = M ADM and T = T H . For the GB case, when we express the relations (5.9) in terms of ADM mass and Hawking temperature we get (5.10)

Excitations near extremality and near-horizon limit
An important feature of the RN BB, which in view of the previous results extends to the charged GB BB, is that the semiclassical analysis of its thermodynamic behavior breaks down near extremality [44,45]. In fact, the energy of an Hawking radiation mode is of order T H and the semiclassical description breaks down when this energy is comparable with the energy above extremality M − M ext given by Eq. (5.9). This results in an energy gap for excitations above extremality [44], which in the case under consideration is E gap ∼ (N ω 5 )/L 2 r 2 0 . The fact that the extremal limit is singular, can be also understood in geometrical terms. It has been observed that at extremality the geometry splits into two spacetimes: an extremal black hole and a disconnected AdS space [46].
The presence of this energy gap has important consequences for what concerns the spectrum of BB excitations near extremality. In particular, whereas in the extremal case the near-horizon geometry is given, as shown in Sect. 4.3.1, by AdS 2 × R 3 , finite energy excitations of AdS 2 × R 3 are suppressed. Analogously to the RN case in 4d [44], one can consider near-horizon limits not restricted to zero temperature and excitation energy. These limits are obtained by letting the 5d Planck length L P go to zero, holding fixed some of the other physical parameters of the BB (energy, charge and temperature).

Shear viscosity to entropy ratio
The universality of the shear viscosity η to entropy density s ratio for Einstein-Hilbert gravity represents a very important result of the gauge/gravity correspondence. First found for the hydrodynamic regime of the QFT dual to black branes and black holes of the Einstein-Hilbert theory [13,21,22], the KSS bound η/s ≥ 1/4π has been extended to a variety of cases. These include Einstein-Hilbert gravity with all possible matter terms in the action, hence, among others the QFT dual to Reissner-Nordström 5d gravity [21,22] and the important case of the quark-gluon plasma (see e.g. [47]). It has been also conjectured that the KSS bound holds for any fluid in nature. For a detailed discussion on the shear viscosity to entropy ratio see Refs. [13,21,22,24,34,41,[47][48][49][50].
The KSS bound seems to lose its universality when one introduces, in the Einstein-Hilbert action, higher powers of the curvature tensors. This is, for instance, the case of Lovelock (and Gauss-Bonnet) gravity we are discussing in this paper. In particular, the KSS bound depends on the coupling constant for the higher curvature terms [24].
Following the notation of [24], we rewrite the GB BB solution (4.4) as follows where α 0 α 2 = α 2 /L 2 = λ. In 5d Gauss-Bonnet gravity, the shear viscosity to entropy ratio is [24] The KSS bound still holds if λ ≤ 0 but is violated for 0 < λ ≤ 1/4 (the upper bound follows from Eq. (4.2)). The dependence of the bound from the coupling constant λ makes the bound not anymore universal as in the Einstein-Hilbert theory. In terms of the dual gauge theory, the curvature corrections to the Einstein-Hilbert action correspond to finite N and λ tH effects. It has been argued that the universality of the KSS bound strictly holds in the limit N → ∞ whereas, in general, finite N effects will give lower bounds for η/s [17]. A crucial issue is that the relation (6.2) seems to allow for arbitrary violations of the KSS bound. However, consistency of the QFT dual to bulk GB gravity as a relativistic field theory constrains the allowed values of λ. For instance, in [27][28][29] it was found that causality and positivity of the energy for the dual QFT describing the Gauss-Bonnet plasma require −7/36 < λ < 9/100 implying 4πη/s > 16/25 , a bound lower then the KSS bound. On the other hand, the hydrodynamic description of the dual GB plasma is valid in the IR regime, i.e., for ω, k << T , whereas causality is determined by the propagation of modes in the ω, k > T , UV regime. Thus, the existence of lower bounds for η/s implies a higher nontrivial relationship between the transport properties in the IR and causality requirements in the UV regime of the QFT dual to GB gravity.
Recent investigations have shown that if translation symmetry is broken in the IR then one may have strong violation of the KSS bound even in the context of Einstein gravity, in the form of η/s ∼ T 2ν , ν ≤ 1 [30,51,52]. Although, for these backgrounds, the breaking of translational invariance prevents an hydrodynamical interpretation of the viscosity, this behaviour of η/s is clearly related to the emergence of extremely interesting physics in the far IR.
A way to shed light on these questions is to investigate the behaviour of η/s in the case of a gravitational bulk background for which there is a temperature flow of η/s. The charged GB BB represents a nice example of this behaviour, particularly in view of the universality of the IR AdS 2 × R 3 fixed point. This will be the subject of the next three subsections.

η/s for the charged GB black brane
A standard way to calculate the shear viscosity for a QFT is by using the Kubo formula for the transverse momentum conductivity where G R xy is the retarded Green function for the T xy component of the stress-energy tensor. The application of the usual AdS/CFT procedure for the computation of correlators gives for the U (1)-charged Gauss-Bonnet black brane in five dimensions [34,48] , and s is the entropy density S/V following from (5.1).
A drawback of the usual computation of the shear viscosity is that it does not work in the extremal T = 0 case because the metric function has a double zero at the horizon. For this reason, η in the case of extremal BB cannot be simply computed by taking the T H = 0 limit in Eq. (6.4). Building on [53], a method of dealing with this problem has been developed in [14]. Recently, a very simple and elegant formula for computing correlators of the form (6.3) in QFTs dual to a gravitational bulk theory has been proposed in [35] (see also [30,31]). This method also works in the extremal case; thus, in the following, we will use it to compute η for the charged GB BB.
Considering perturbations g ab = g (0) ab + h ab of the background (6.1), at the linear level the field equations (4.20) give for the h y x (t, r) = φ(r)e −iωt component of the perturbation where γ(r) = (r/L) 3 is the determinant of the spatial metric, f − (r) is given by Eq. (6.1) and F = N 1 − λL 2 r ∂ r f − (r) . Notice that in the background (6.1), the component h y x decouples from the other perturbation modes. Let us first consider the non extremal black brane. The extremal case will be discussed in Sect. 6.3. Following Ref. [35] we now denote with φ 0 (r) the time independent solution of (6.5) which is regular on the horizon r = r + and such that φ 0 → 1 as r → ∞. The other linearly independent solution φ 1 (r) of Eq. (6.5) behaves as 1/r 4 for r = ∞ and can be computed using the Wronskian method, Expanding near the horizon r = r + we get at leading order , where T H is the Hawking temperature of the BB and a is defined as in Eq. (6.4). Solving now Eq. (6.5) near the horizon with infalling boundary conditions and for small ω, one gets at leading order in ω Comparing Eq. (6.7) with Eq. (6.8) and expanding near the r → ∞ boundary of AdS, one gets The usual AdS/CFT rules for computing boundary correlators tell us that the retarded Green function is 1/(16πG N ) the ratio between normalizable and non-normalizable modes, so that we have (6.10) Because φ 0 (r) goes to 1 as r → ∞ and must be regular on the horizon, we have φ 0 (r + ) = 1 and Eq. (6.10) reproduces correctly the previous result (6.4). Now, the second Eq. (5.1) can be used to define, implicitly, the horizon radius as a function of the BB Hawking temperature and the electric charge, thus allowing us to write also the shear viscosity (6.4) as a function of T H and Q . (6.11) In the same way, the entropy density in Eq. (5.1) can be written as a mere function of T H and Q, so that we can write the shear viscosity to entropy ratio in the form It is also of interest to write explicitly the dependence of η/s from the normalization constant N : When the electric charge is set to zero, the ratio T H /r + in Eq. (6.12) is N/(πL 2 ) and η/s reaches the value in Eq. (6.2), as one expects. On the other hand, the dependence of η/s on T H and N in the generic case is rather puzzling.
In view of the universality of the thermodynamic behaviour of GB BB described in the previous sections one would naively expect also the shear viscosity to entropy ratio to be universal, i.e., that Eq. (6.13) becomes the same as in the RN case just by using the effective temperature T = T H /N instead of T H . This is not the case. Only for N = 1, which corresponds to α 2 = 0, i.e., exactly the RN case, η/s assumes the universal value 1/4π, while for N generic we have a quite complicated dependence on N and T H . This indicates strongly that the transport features of the dual QFT in the hydrodynamic regime contain more information about the underlying microscopic theory than that contained in the thermodynamic description. An investigation on the behaviour of η/s at large and small T H can shed light on this issue. In fact, as we have seen in the previous sections, in these limits the BB allows for a simple thermodynamic description. We, therefore, expect this to be true also for the shear viscosity to entropy ratio. This will be the subject of the next sections.

η/s in the large and small T H regime
The behavior of the shear viscosity (6.11) for large and small temperatures can be investigated in a way similar to that used for the BB thermodynamics.

Large T H
For large T H , the Hawking temperature is given by Eq. (5.3), thus leading to the following expression for the shear viscosity in Eq. (6.11), 14) The shear viscosity at large T H scales as T 3 H . In this limit, the entropy density also depends on the temperature as T 3 H (see Eq. (5.4)), the shear viscosity to entropy density ratio approaches Eq. (6.2) and reduces to the universal value 1/4π when λ → 0. This is rather expected, because at large T H the contribution of the electric charge can be neglected.

Small T H
To investigate the small T H behaviour we invert Eq. (5.8) and we write the horizon radius as where r 0 is defined by Eq. (5.6). At small temperature the subleading term in the shear viscosity scales linearly in T H The behavior of the entropy density in the small T H regime is given by the second equation in (5.9). Hence, in this limit, also the subleading term of the shear viscosity to entropy density ratio scales linearly η s The result η/s = 1/4π for T H = 0 has been already found and discussed in the literature in the case of the RN solution [14,53]. It has been argued that at small temperatures, the dual QFT behaves as a "strange RN metal". The optical conductivity exhibits the generic perfect-metal behaviour, but although we have a non-vanishing ground-state entropy, for the strange metal hydrodynamics continues to apply and energy and momentum can diffuse. In the limit T H = 0, the ratio becomes η/s = 1/4π attaining the universal value one expects from the KSS bound. This result is what one naturally expects in view of the fact that at T H = 0 the near-horizon solution of the GB brane gives exactly the same AdS 2 × R 3 geometry of the RN solution. However, extra care is needed when one takes the T H → 0 limit in Eq. (6.12). Taking T H → 0 directly in Eq. (6.12) is not save for several reasons. First, as discussed in Sect. (5.3) the semiclassical description for the BB breaks down at small temperature when the energy gap above extremality prevents excitations with finite energy. Second, as noted by Cai [34], although the T H → 0 limit is well defined, the usual computation of the shear viscosity to entropy ratio fails in the extremal case because the metric function as a double zero at the horizon. Third, also the computations of Sect. 6.1 do not hold for T H = 0 because the expressions (6.7) and (6.8) are ill defined for T H = 0. However, the general method based on [35] and used in Sect. 6.1 for calculating η, works also for extremal BB.

η/s in the extremal case
Let us now extend the calculations of η described in Sect. 6.1 to the case of the extremal brane. In the extremal case the function f − given by Eq. (6.1) and its first derivative vanish when evaluated on the horizon. We have therefore at leading order near the horizon where k is some non zero constant. Using the previous expression in (6.6) one gets .
On the other hand the near-horizon, small ω expansion gives now . (6.20) Comparing Eqs. (6.19) and (6.20), near the r → ∞ boundary of AdS 5 we find the expansion from which follows the shear viscosity Using the same argument used in Sect. (6.1) to infer that φ 0 (r + ) = 1, we get for the shear viscosity to entropy ratio of the extremal GB black brane the universal value It is interesting to notice that the universality of η/s for the extremal GB BB is a direct consequence of the universality of the AdS 2 × R 3 , extremal, near-horizon geometry. In fact the extremal, near-horizon metric background (4.19) does not depend on λ. The other source for a λ-or Q-dependence of η is the function F in Eq. (6.5). However, this contribution, hence the dependence of η from λ and Q, is removed by the condition f (r + ) = 0, which implies that near the horizon the two-dimensional sections of the metric behave as AdS 2 .
To conclude, let us now discuss the global behaviour of η/s as a function of the temperature in order to gain some insight about the η/s bounds. Taking into account that r + (T H ) is a monotonically increasing function, one easily finds that also the function P (T H ) = πL 2 T H /(N r + ) = 1 − 2πG N Q 2 L 2 /(3r 6 + ) in Eq. (6.12) is a monotonically increasing function of T H , with P (0) = 0 and P (∞) = 1. The global behaviour of η/s in Eq. (6.12) therefore is ruled by the sign of λ. For λ < 0, η/s is a monotonically increasing function of T H , which raises from its minimum value 1/4π at T H = 0 to its maximum value (1 + 4|λ|)/4π for T H = ∞, in full agreement with the KSS bound. On the other hand, for 0 < λ < 1/4, η/s is a monotonically decreasing function of T H , which drops from its maximum value 1/4π at T H = 0 to its minimum value (1 − 4λ)/4π for T H = ∞, violating the KSS bound.

Summary and outlook
In this paper, we have discussed in detail geometrical, thermodynamic and holographic properties of charged 5d GB black branes. Although our discussion has been mainly confined to the GB case, we expect that most of our results can be generalised to Lovelock gravity theories in any spacetime dimensions.
We have shown that the particular combination of GB higher curvature terms added to the Einstein gravity action have three main effects: (1) They introduce a new branch of brane solutions, which are however not black branes but describe naked singularities. The global structure of the RN geometry of Einstein gravity is preserved only for α 2 > 0. For α 2 < 0 the spacetime splits into two disconnected regions (an inner and outer region), with the external region having a single event horizon also in the non-extremal case. An interesting feature is that the solutions of the two branches may be, in some cases, continuously connected one with the other through the singularity. When this is the case, they describe transitions of the kind: AdS 5 → singularity → AdS 5 , AdS 5 -black brane → singularity → AdS 5 or AdS 5 -black brane → singularity → dS 5 . Although, it is known that one of the two branches of the solution (f + ) is unstable [36] one expects that the first two of these transitions have a holographic interpretation as the flow between two CFTs of different central charge through a singularity.
(2) The thermodynamic behaviour of charged GB black brane is universal, i.e., when expressed in terms of effective mass and temperature is indistinguishable from that of the RN black brane.
(3) Higher curvature terms modify the asymptotics (the AdS length) of the 5d AdS-RN gravity leaving unchanged the AdS 2 × R 3 , extremal near-horizon geometry of the RN black brane. At thermodynamic level, when expressed in terms of M ADM and T H a dependence on the normalization factor N of the metric is introduced but not for the extremal, nearhorizon geometry AdS 2 × R 3 . In terms of the dual CFTs, this property can be described as a deformation of the CFT which changes the UV behaviour but leaves unchanged the IR. This behaviour is very similar to the attractor mechanism found in supergravity theories [54][55][56][57], where the AdS 2 × R n (or AdS 2 × S n ) geometry is always the same irrespectively from the asymptotic values of the scalar fields.
We have also computed the shear viscosity to entropy density ratio for the GB charged black brane both for the non-extremal and the extremal case. We have found that consistently with the geometrical and thermodynamic picture, universality of η/s is lost in the UV but is restored in the IR. The ratio η/s has a non-universal temperature-dependent behaviour for non-extremal black branes but attains the universal 1/4π value at extremality. This result implies that η/s is completely determined by the IR behaviour and is completely insensitive to the UV regime of the dual QFT. On the one hand, this is largely expected because transport features in the hydrodynamic regime should be determined by IR physics. On the other hand, it is not entirely clear if this result has a general meaning or it is a just a consequence of the peculiarities of the charged GB black brane (higher curvature corrections vanish on the AdS 2 × R 3 background).
Although the lesson to be drawn from our results is that probably it is not wise to look at the UV physics to infer about bounds on η/s, the question about the possible existence of bounds on η/s lower than the KSS one remains still open. We have found that η/s is a smooth monotonic function of the temperature. Going to small temperatures, it always flows to the universal value 1/4π but this value is a minimum for λ < 0 and maximum for λ > 0. Thus, the QFT dual to GB-Maxwell gravity with λ < 0 gives a nice example of temperature-flow of η/s always bounded from below by 1/4π. On the other hand, the KSS-bound-violating flow we obtain in the theory for 0 < λ < 1/4 remains open to further investigations. Due to the presence of the curvature (κ = 1), now the mass depends explicitly both on the AdS radius, L 2 = α −1 0 and on the GB coupling constant, α 2 . The other important aspect which makes black holes different from black branes is that also temperature and entropy depend explicitly from α 2 through the coupling with the curvature since all the higher curvature corrections (like the Gauss Bonnet one) enter in the expression for the temperature trough a coupling with κ. As found by Cai [2], for a charged 5d GB black hole one gets We see that since M, T, S depend explicitly on the GB coupling constant α 2 , differently from the black branes case, it is not anymore true that the thermodynamic behaviour of the Reissner-Nordström and Gauss-Bonnet black hole is the same. From the previous equation one can also realize that for the entropy, the area law no longer holds and that it receives a correction from α 2 .
Let us now consider the extremal of the GB black hole. In the BB case we have found the remarkable property that the extremal, near-horizon solution of the charged GB black brane is exactly the same as the RN black brane. One can easily show that this is not the case for the extreme, near-horizon GB black hole. In the RN case the extremal, nearhorizon, solution, which actually is an exact solution of the field equation is the AdS 2 × S 3 geometry (S 3 is the three sphere), i.e the direct product of two maximally symmetric spaces, respectively with negative curvature R (2) = −2/l 2 and positive curvature R (3) = Λ, where l and Λ can be written in terms of the 5d cosmological constant and the U (1) charge Q.
Using Eqs. (4.21) one can show that the individual contributions of the AdS 2 and S 3 spaces, to the two terms in Eq. (4.20) that are quadratic in the curvature tensors vanish. Nevertheless there are still some cross-product contributions arising from the mixing of AdS 2 and S 3 terms. Splitting the 5d indices (a, b) into µ, ν = 0, 1 (running on AdS 2 ) and i, j = 1, 2, 3 (running on S 3 ) we find a contribution to the µ, ν components of the field equations given by 2α 2 Λ/l 2 g µν and a contribution 4α 2 Λ/3l 2 g ij to the ij components of the field equations.
We see that the AdS 2 × S 3 solution of the RN field equations cannot be also solution of the GB field equations. Obviously, this not prevents the existence of a different AdS 2 × S 3 solution, i.e a solution with different curvatures for AdS 2 and S 3 . However, from the structure of the field equations and from Eqs. (4.21) one can infer that these solutions, if existing, imply a dependence of l and/or Λ not only from the 5d cosmological contant and from the black hole charge Q but also from the GB coupling constant α 2 .