Black holes in 4D Einstein-Maxwell-Gauss-Bonnet gravity coupled with scalar fields

Einstein-Maxwell-Gauss-Bonnet-axion theory in 4-dimensional spacetime is studied in this paper with a"Kaluza-Klein-like"process. The dyonic black hole solution coupled with higher derivative terms is obtained. The behaviour of shear viscosity to entropy density ratio of uncharged black holes is found to be similar with that in 5-dimensional spacetime, violating the bound as well. In addition, the main features of this ratio remains almost unchanged in 4 dimensions, which is characterised by $(T/\Delta)^2$, at low temperature T with $\Delta$ proportional to the coefficient from scalar fields.


Introduction
Lovelock's theory suggests that Einstein gravity can be modified with higher derivative terms, with second order equations of motion [1,2]. One example of such a theory is the well-known Einstein-Gauss-Bonnet (EGB) gravity. Increasing interest has been put on this sort of gravity in 4-dimensional spacetime. Recent research [3] gives a method to realise it through rescaling Gauss-Bonnet coupling constantα → α/(D − 4), and taking D → 4 to obtain spherically symmetric 4D black hole solutions with non-vanishing Gauss-Bonnet term. The strategy is quite straightforward. Considering an action with the contribution from the Gauss-Bonnet term after the rescaling of Gauss-Bonnet constantα, one obtains The action then yields the equations of motion: where H µν is the Gauss-Bonnet tensor: As we know, H µν vanishes in 4 dimensions and the theory will reduce to Einstein's gravity. After being rescaled, however, the infinity caused by (D − 4) seems will leave us with a non-vanishing term. So it was suggested that taking D → 4 limit will give Gauss-Bonnet gravity in 4 dimensions: This work sheds a light on the investigation into higher derivative gravity in four-dimensional spacetime. However, arguments are also raised, claiming that the method performed [3] cannot actually provides Gauss-Bonnet gravity in four dimensions [4][5][6][7], i.e. it is not a "novel Einstein-Gauss-Bonnet gravity". Since the strategy proposed above is unable to give topologically non-trivial solutions, another method is supposed with "Kaluza-Klein-like" procedure [8,9], compactifying Ddimensional EGB gravity which is on a maximally symmetric (D − 4)-dimensional space.
With similar rescaling and D → 4 limit, one obtains a purely 4-dimensional EGB theory [8,10]. The resulting theory can also be viewed as a special Hondeski gravity or generalised Galileons [8,[11][12][13]. Meanwhile, as black holes have a good feature of thermodynamical properties [14], great attention has been paid in this area. During past decades, the AdS/CFT dictionary provides a powerful tool to investigate strongly coupled gauge theories, which are dual to black holes in AdS space. For such systems, there exists a universal bound, known as Kovtun-Starinets-Son (KSS) bound for shear viscosity to entropy density ratio [15][16][17][18][19]: KSS bound has been found to be violated when small corrections are added to Einstein's gravity. Coupled with Gauss-Bonnet term, a modified version of this bound reads [20] η s where α GB = −Λα/3, with Λ the cosmological constant. It has been shown that when considering non-vanishing electric charge, bound (1.6) will be violated for 4D Gauss-Bonnet gravity, and constraint on Gauss-Bonnet coupling could be obtained by analysing the causal structure in bulk [21]. It is of our research interest to check whether violation will happen and will there be a new constraint for the coupling constant if we include two scalar fields linear in all the spatial directions. In this paper, 4D black hole solution coupled with higher derivative terms with both electric and magnetic charges as well as axions will be given. The "Kaluza-Klein-like" procedure introduced in [8] will be used to find the solution. Then, shear viscosity to entropy density ratio for a neutral black hole with scalar fields is studied, which violates the KSS bound. This paper is organised as follows: In section 2, the Einstein-Maxwell-Gauss-Bonnet-axion gravity is found through "Kaluza-Klein-like" method, and a dyonic black hole solution coupled with scalar fields and higher derivative terms is obtained. The thermodynamics of this black hole will be briefly mentioned as well.
Removing electric and magnetic charges, in section 3 we will calculate the shear viscosity to entropy density ratio for the neutral black holes in 4D Gauss-Bonnet gravity with axions. It will be shown that the KSS bound will be violated because of the existence of scalar fields. Section 4 focuses on the difficulties one may meet when finding constraints on Gauss-Bonnet constant through causality analysis when black hole is uncharged. Finally, conclusion and outlook will be represented in section 5.
2 Dyonic black holes in four dimensions with Gauss-Bonnet coupling

Reduced action
Working in four dimensions, we first introduce two scalar fields. The general action in Ddimensional spacetime for Einstein-Maxwell-Gauss-Bonnet-axion (EMGBA) gravity reads where ϕ i are the scalar fields, and Gauss-Bonnet term takes the form Now we parameterise a D-dimensional metric: where φ is called "breathing scalar", depending only on external p-dimensional coordinates. Line elements dΣ 2 D−p,λ describs the internal maximally symmetric space, and λ gives the curvature of the internal spacetime. This is a diagonal reduction along Σ D−p,λ . When λ = 0, the "internal" space is flat. Here we only consider Abelian isometry group for the "internal" space, so the massive modes could be truncated. Now let us rescale Gauss-Bonnet coupling constant, and where = (D − p). The next step is to consider the cases where p < 5, and take the limit D → p. The resulting reduced p-dimensional action then reads [8,9] where G µν are Einstein tensors. This action is claimed to be the EMGBA theory in 4 dimensions, and it also works for D < 4. Constructed in a mathematically more rigorous way, (2.5) does not suffer from defects of the naïve D → 4 limit such as being ill-defined [4,5,10].

Dyonic Black hole solution
The next step is to find the solution in spherically symmetric spacetime. First of all, one could get equation of motion from the action (2.5), and in four dimensions (p=4) it can be written as where E µν is the equation of motion from metric variation, and E φ is that for "breathing scalar". In such a way, the dependence on φ can be removed. We assume that φ = φ(r), and apply planar symmetric ansatz Also, we choose the scalar fields to be linearly dependent on spatial coordinates, such that where β is a constant and x i = {x, y}. Electric and magnetic charges are added to the black hole in such a way: where q and h are electric and magnetic charges respectively, and Please find more details on equations of motion in the appendix. Now combining (2.6) together with (2.7), one obtains This equation is, however, not enough for us to find the solution for f (r). Substituting the metric (2.7) into eqn.(2.5), and removing total derivative terms, one finds the effective Lagrangian to be [8] where flat "internal" space is considered, which means λ = 0 and the theory is invariant under a constant shift of φ. Doing variation of (2.12) with respect to f (r), and making χ(r) to be zero, one finds and inserting (2.14) into δχ equation yields Take r H to be the black brane horizon, that is, f (r H ) = 0. One is now able to give the exact solution from (2.11) and (2.15):

Planar black brane in AdS space
Now rescale the parametres, where (2.20) The line element for planar black brane in AdS space can be written as with l the AdS radius, and where we use the fact that when D = 4, To find the value of constant N 2 , one needs to note that the geometry would reduce to the flat Minkowski metric conformally at the boundary. With r → ∞, one has As a result, This solution is a dyonic black hole in 4D EMGB gravity with linear axions. Through the "Kaluza-Klein-like" method, this sort of black hole now contains the contribution from higher derivative terms.

Thermodynamics
Since Gauss-Bonnet term only contains curvature terms, its black hole thermodynamics is the same as that of Schwarzchild-AdS. This property is preserved by our reduction frame. The Hawking temperature at the event horizon reads The black brane approaches extremal when T → 0, that is, The entropy density of the horizon is given by [24,25] where V 2 = dxdy represents the spatial 2-volume (or area), and the free energy F reads 3 Shear viscosity for neutral black holes

Weaker horizon formula
Before the shear viscosity is investigated, let us change the coordinates where u = r H /r, and (3.1) From now on, for simplicity, only black holes without electric or magnetic charge are considered, i.e. H = Q = 0, and The tensor type perturbation is considered where and Usually Kubo formula gives the shear viscosity in general cases: where T xy is the momentum-stress tensor. If translation invariance is not broken by any sources, then the momentum T ty is conserved, whose current corresponds to T xy . The KSS bound is true in momentum-conserving situation. It is weakened and becomes (1.6) when higher derivative terms are non-vanishing. Now with scalar fields, the momentum is no longer conserved, and the translation invariance is broken. As a consequence, the shear viscosity η does not have a hydrodynamic interpretation any more, but η/s is still closely associated with entropy production. In this situation, one cannot simply apply Kubo formula to compute shear viscosity η here, and it will not work if one tries to find it only via horizon data directly. Instead, a "weaker horizon formula" has been suggested to study shear viscosity to entropy density ratio η/s in such cases [22]. While the perturbation is massive, one has Substituting (3.4) into (2.6) and taking ω → 0, one obtains with boundary conditions We rewrite Hawking temperature (2.26) as As (3.7) is too complicated to be solved, neither analytically nor numerically, it needs to be solved at high and low temperatures separately.

High temperature expansion
At high temperature,β 2 → 0. Therefore, (3.7) could be perturbatively expanded around β 2 ∼ 0. In this way, h(u) will also be expanded as At 0th order, h 0 (u) turns out to be a constant function. With boundary condition (3.8), one has At second order, one finds At this order, one has The numerical solution for (3.14) is found and the ratio η/s as a function of ξ/T is illustrated, which is shown in Fig.(1) as a log-log plot. One finds that at high temperature, the bound (1.6) is violated when the temperature is getting lower.

Low temperature expansion
With (3.10), one hasβ 2 → 3 at low temperature. Similarly, As the equations of motion are rather complicated, and what is needed is only the value of h(1), we are not going to give the explicit form of it in this paper. However, one could find that at 0th order, and at the first order, According to formula (3.6), to first order, the shear viscosity to entropy density ratio at low temperatures reads Therefore, the problem reduces to finding out the value of h 0 (1). Though the equation at 0th is too cumbersome to be solved directly, one could follow the similar step performed in the previous work, solving h 0 (u) at u = 0 and u = 1 respectively, and matching the solutions to get h 0 (1) [27]. The strategy is 1) Solve h 0 (u) at u = 0 and u = 1, and these two solutions are labeled as h 00 (u) and h 01 (u) respectively. At this stage, the integral constant in h 01 (u) is still not fixed.
Following the steps, one finds that at u = 0, Similarly, at horizon where u = 1, Next thing to do is exactly the same as that in high-temperature case: write η/s as a function of ξ/T according to (3.10). Asβ 2 → 3, Therefore, at low temperature, (3.31) One finds from Fig.(2) that the KSS bound is violated as well.
In spite of the fact that "Kaluza-Klein-like" process gives us really different equation of motion from that in 5-dimensional spacetime, the behaviour η/s is quite similar with what has been found in five dimensions [26]. In our case, η/s ∼ (T /ξ) 2 when T /ξ → 0, which satisfies the conjecture [23] that η/s ∼ (T /∆) 2 as T /∆ → 0 with ∆ being some scale to be chosen. Here we take ξ to be ∆.

Discussion
It is obvious that at high temperature, the KSS bound in higher derivative gravity η/s ≥ (1− 4α GB ) is hardly violated. This is because high temperatures correspond to very smallβ 2 , where the contribution from axions is nearly neglectable. While low-temperature behaviour of the ratio violates the bound dramatically, since in this situation,β 2 is a large number comparing with that in high-temperature cases. Thus for fixed Gauss-Bonnet constant, the larger the mass of graviton, the bolder the violation would be, which evinces one's intuitive postulation.
One may find that (3.31) looks rather different with that in 5 dimensions [27], who contains confluent hypergeometric limit function related with Bessel functions. This results from the fact that our theory in four-dimensional spacetime is rather different from those in higher dimensions, since it also includes "breathing scalar" φ that brings a completely different Einstein equations. All calculation is based on these equations of motion, so it is natural that in this paper one would get a quite different form of the ratio. Nevertheless, the critical characteristics of η/s is almost the same in 4− and 5− dimensional spacetime. They both violates KSS bound markedly. Furthermore, in both cases, η/s ∼ (T /∆) 2 , and ∆ ∼ β, only differing by a coefficient that is influenced by the dimensionality. Although governed by different actions and therefore different equations, the main features characterising the shear viscosity to entropy density ratios in four and higher dimensions are actually very similar to each other.

Inability of "Kaluza-Klein-like" process in causality analysis
When introducing Gauss-Bonnet terms, one would find that the causality will be violated, and charge as well as scalar fields have effects on such violation [23][24][25][26][27]. Through analysis of causal structure, one is capable of finding restrictions on α GB . For example, in 5 dimensions, causality will be violated if α GB > 0.09 [23,24]. We would study the causal structure of the bulk, and here we continue the research on neutral black holes. According to the AdS/CFT correspondence, 4D AdS gravity we study here is dual to a 3D quantum field theory on the boundary. Usually, the procedure to study the causality in dimension D is: 1) Start with a D-dimensional metric Write the perturbation (which is the wave function of the transverse graviton) as 2) Then take large momentum limit, where k µ → ∞. The x m x n -component of equation of motion will reduce to k µ k ν g eff µν 0, where is the effective metric.
3) Find c g and the constraint by letting c 2 g − 1 ≤ 0.
It is important to mention that c g can be interpreted as the local speed of graviton on a constant r-hypersurface. Its dependence on dimensionality is given by [26] c 2 g (r) = Before doing so, let us come back to the "Kaluza-Klein-like" procedure [8,10] performed in this paper. It is expected that we could also make a similar perturbation like (4.2), and directly get the momentum term from (2.6), or just from E µν . Since we only have 4 dimensions, one should either take x i in (4.2) to be x or y. As a consequence, the momentum k will be in x (or y) direction. More precisely, for instance, one has h(x, u) = e −iωt+ikuu+ikx . However, if one tries to substitute (4.6) into (2.6), no momentum term could be found, neither in (2.6) nor in E µν . That is, it is impossible to analyse causal structure through "Kaluza-Klein-like" procedure as far as we are concerned currently.

Vanishing momentum terms in four dimensions
Turing back to (4.5), on the other hand, one is able to study the causality with this formula at the first sight. But one has to be careful when performing such a limitation. There is a rather simple way to express (3.3) more generally, which also tells why things get more complicated when we deal with 4 dimensions. It will be useful to rewrite the metric (4.1) as where z = 1 when D ≥ 5, and z = 0 for D ≤ 4. With (4.7), one obtains . (4.8) From (4.8), one finds that if one works with dimensions higher than 5, then z = 1, she or he will definitely recover (4.5). But if we take D = 4, we should also make z = 0 at the same time, where (4.5) works no more. What will be got is an infinite graviton velocity. In short, Moreover, the momentum term in D-dimensional equation of motion reads c 2 g,z k 2 /(N 2 f (r)). Obviously, z appears in denominator. When D is 4, k has to vanish in order that the momentum term would not diverge. One could turn back to the very beginning to see why this happens, or what makes 4D cases so special. The answer is quite simple: the momentum term containing k vanishes if x i = x m or x i = x n , in spite of how many dimensions one has. As a result, at least three spatial coordinates, i.e. five dimensions in total are required to construct the perturbation in (4.1) that leads to non-zero momentum. For example, in 5-dimensional spacetime, we often choose x m x n to be xy, and thus x i is z. However, this is just the cases for neutral black holes with axions. One may have another story when adding magnetic and electric charges back, where H = 0 and Q = 0. For charged black holes in 4D EGB gravity, there in a constraint such that α GB < 0 [21,28]. Causality is not only one way to get this result [21], and investigation into the completeness of the spacetime also yields a similar upshot [28]. Thus, it is reasonable to expect that based on "Kaluza-Klein-like" method, one could get a similar constraint on α GB .

Conclusion
In this paper, we obtained the dyonic black hole solution with linear axions in 4-dimensional higher derivative gravity through "Kaluza-Klein-like" process. Shear viscosity to entropy density ratio η/s is investigated after electric and magnetic charges are removed. It turns out that violation still happens when D = 4. The behaviour of η/s is rather similar with that in 5 dimensions, such that η/s ∼ (T /ξ) 2 when T /ξ is very small. One important outcome is that the main feature of the ratio is almost the same with what has been found in 5 dimensions [27]. The only difference comes from the different equations of motions brought by "breathing scalar", which is inevitable if one applies "Kaluza-Klein-like" process to get the four-dimensional theories. While the bulk causal structure of uncharged black holes is studied, it is found that the momentum term vanishes in equations of motion, while the velocity formula in D dimensions is only valid when D > 4. Therefore, neither "Kaluza-Klein-like" process nor naïve D → 4 limit can help in causality analysis, since the construction itself is only well-defined in dimensions no lower than 5. As is mentioned, it has been shown that non-vanishing electric or magnetic charge may lead to a different result. Since we only consider neutral black hole with tensor type perturbations in this paper, our next task may focus on charged black holes and different types of perturbations as well. The research work done previously implies a possibility that the momentum term will exist in these cases, which means that causality could be studied. We expect a constraint similar with α GB < 0 for charged black holes obtained from "Kaluza-Klein-like" procedure.
The dyonic black hole solution derived may be used as a tool to study the transport properties of the normal state of high-temperature superconductors. It is of our further interest as well to go further and explore more of its properties, such as transport behaviour like electric and thermal conductivity. Much interesting upshot is expected to appear for this is the first time for a 4d dyonic black hole to contain contributions from higher-derivative terms, which may bring an insight into the study on high-temperature superconductivity.

A Equations of motion from reduced action
Starting from (2.5), there are four sets of equations. The first is Klein-Gordon equation: These equations are naturally satisfied by choosing ϕ i = βx i . Then is the Maxwell equations which read implying A t = −q/r, and A y = hx. The variation with "breathing scalar" φ yields the equation [10] where G µν are Einstein tensors. Finally is the Einstein equation Combining the last two equations by g µν E µν + αE φ /2, one obtains an equation independent of φ.