Causality of black holes in 4-dimensional Einstein-Gauss-Bonnet-Maxwell theory

We study charged black hole solutions in 4-dimensional (4D) Einstein-Gauss-Bonnet-Maxwell theory to the linearized perturbation level. We first compute the shear viscosity to entropy density ratio. We then demonstrate how bulk causal structure analysis imposes a upper bound on the Gauss-Bonnet coupling constant in the AdS space. Causality constrains the value of Gauss-Bonnet coupling constant $\alpha_{GB}$ to be bounded by $\alpha_{GB}\leq 0$ as $D\rightarrow 4$.


Introduction
The AdS/CFT correspondence [1][2][3] provides a powerful tool for studying the physics of strongly coupled gauge theories and also can be used to examining alternative theories to the general relativity. Alternative theories to Einstein's General Relativity paradigm can be scrutinized by diverse approaches. The higher derivative gravity with α ′ corrections was also studied widely within the framework of the AdS/CFT correspondence (see [4][5][6][7][8][9][10][11][12] for an incomplete list). For Einstein-Gauss-Bonnet (EGB) theory, strong constraints can be imposed on the Gauss-Bonnet coupling constant from the analysis of the bulk causal structure. For 5-dimensional (5D) EGB theory, causality demands α GB ≤ 0.09 to avoid superluminal propagation of signals in the dual boundary field theory [5][6][7]12]. Recently, revived interests on EGB gravity in 4-dimensional spacetime have been first concerned in [13].
The 4-dimensional EGB gravity is realized by first rescaling the coupling constant α ′ → α ′ D−4 of the Gauss-Bonnet term and then take the limit D → 4 [13]. In this way, the Lovelock's theorem [14][15][16] can be bypassed and spherically symmetric 4D black hole solutions can be obtained in the presence of the Gauss-Bonnet term.
In this paper, we are going to investigate whether causality violation happens in the 4D EGB gravity and check the upper bound of the Gauss-Bonnet coupling constant. As shown in [5][6][7]9], higher derivative terms in the gravity action can result in superluminal propagation of gravitons outside the light cone of a given background geometry. The graviton cone in such case does not coincide with the standard null cone or light cone defined by the background metric. Utilizing the tool provided by AdS/CFT correspondence, firstly we will study the linearized perturbation of the black holes in 4D Einstein-Gauss-Bonnet-Maxwell (EGBM) gravity. Then we calculate the shear viscosity to entropy density ratio in this context. We then examine the causality constraint on the Gauss-Bonnet coupling constant. We are going to show that for 4D black holes in EGBM theory, if we take the limit D → 4 before the series expansion of the local speed of the transverse graviton on the boundary, the speed cannot reach the local speed of light (i.e. c = 1) on the boundary unless we choose α GB = 0. The local speed of graviton is smaller than the local speed of light for any positive value of α GB . Meanwhile, if the limit D → 4 is taken after the series expansion of the local speed of the transverse graviton near the boundary, no causality violation requires α GB ≤ 0.
We will show that the bulk graviton propagates faster than the local speed of light could leads to signals in the boundary theory propagate outside the light cone. According to the AdS/CFT correspondence, the boundary theory is non-gravitational. In a boosted frame, perturbations will propagate backward in time. Hence, these could lead to unambiguous signals of causality violation.
The structure of this paper is organized as follows. In section 2, we study the charged black hole solutions in 4D Einstein-Gauss-Bonnet-Maxwell theory. Then, in section 3, we compute the shear viscosity to entropy density ratio. In section 4, we discuss the bulk causal structure and its boundary consequences. The conclusion and discussions are provided in the last section. given by the action where α ′ is a (positive) Gauss-Bonnet coupling constant with dimension (length) 2 , the field The general D-dimensional static and maximally symmetric black hole can be described as and an electrostatic vector potential where κ = −1, 0, 1. Since all the functions are radially dependent only, by substituting (2.2) into the action, we obtain with the integral constant Q as the electric charge. The metric function f (r) can be obtained by defining a new variable ψ(r) In this form, the action reduces to 2 ) . Notice that ψ(r) satisfies the relation [17] ψ where M is the ADM mass. We then obtain the metric function and the scalar potential as follows (2.8) The sign + denotes perturbative branch in α ′ , while the − sign corresponds to the branch that the metric function f (r) goes to infinity as α ′ → 0. We choose the + sign hereafter.
For planar black branes in AdS space, the line elements can be written as (2.10) Note that α GB and α ′ are connected by a relation Taken the limit α ′ → 0, the solution recovers the metric of Reissner-Nordström-AdS black branes.
The constant N 2 in the metric (2.9) can be determined at the boundary whose geometry would reduce to the flat Minkowski metric conformaly, i.e. ds 2 ∝ −c 2 dt 2 + d x 2 . On the boundary with r → ∞, we have so that N 2 is fixed as Note that the boundary speed of light is specified to be unity c = 1.
The Hawking temperature at the event horizon is given by The black brane approaches extremal as a → D−1 D−3 (i.e. T → 0). The entropy density is given by [17] s = 1 4 In order to investigate the causality structure and the upper bound of the Gauss-Bonnet coupling constant in four-dimensional spacetime, we will take the D → 4 limit and analysis the shear viscosity to entropy density ratio first.

Shear viscosity
In this section, we are going to study the shear viscosity in the 4D Einstein-Maxwell Gauss-Bonnet gravity theory and examine the shear viscosity bound. Since we already took α ′ → α ′ D−4 in equation (2.1), we will compute the shear viscosity in general D dimensions and then take the limit D → 4 so as to circumvent the Lovelock theorem. It is convenient to introduce new coordinates in the following computation (3.14) We now study the tensor type perturbation h x i x j (t, x i , z) = φ(t, x i , z) with i = j on the black brane background of the form we can obtain the equation of motion for φ(z) from the Einstein-Gauss-Bonnet-Maxwell field equation where , , (3.16) and the prime denotes the derivative with respect to z. Note that the factors (D − 5) and (D − 6) in the expression of g 2 (z) comes from higher than 5-dimensional contribution of the Gauss-Bonnet theory.
The Green function related to the shear viscosity takes the form The shear viscosity can be defined as We can then recast equation (3.15) as a flow equation The shear viscosity can be computed by requiring horizon regularity The ratio of the shear viscosity to the entropy density for 4D charged black hole solutions in Gauss-Bonnet gravity is then (3.21) In the limit D → 4, we obtain We can see that for 4D Einstein-Gauss-Bonnet theory, the shear viscosity bound can still be violated. But as the black hole temperature approaches zero a → 3, one can recover the well-known result η/s ∼ 1/4π [35][36][37][38][39][40].

Bulk causal structure
According to the AdS/CFT correspondence, the physics in bulk 4D AdS gravity is dual to boundary 3D quantum field theory on its boundary. In this section, we study the bulk causal structure and show how a high-momentum metastable state in the bulk graviton wave equation that may have consequence for boundary causality.
Because of higher derivative terms in the gravity action, the equation (3.15) for the propagation of a transverse graviton differs from the standard Klein-Gordon equation of a minimally coupled massless scalar field propagating in the same background geometry.
Writing the wave function of the transverse graviton as and taking large momenta limit k µ → ∞, one can find that the equation of motion (3.15) reduces to k µ k ν g eff µν ≃ 0, (4.24) where the effective metric is Note that the function c g can be interpreted as the local speed of graviton on a constant r-hypersurface [7,41]: (4.26) The local speed of light defined by the background metric c 2 b = N 2 f (z) z 2 , which is 1 at the boundary z → ∞. In the bulk, the background local speed of light c b is smaller than 1 because of the redshift of the black hole geometry.
The causality problem arises because a graviton wave packet moving at speed c g in the bulk corresponds to perturbations of the stress tensor propagating with the same velocity in the boundary theory. Since the replacement α ′ → α ′ D−4 was done already, now we can expand the local speed of graviton c 2 g near the boundary z → ∞ in the limit D → 4, The first term in (4.27) is not equal to the speed of light. This is because higher dimension term with factors (D − 5) and (D − 6) also contribute to c 2 g at the leading order when expand near the boundary and modify the leading order term. The second term in the square brackets diverges as D → 4, which implies that the limiting procedure does not work well to the linearized perturbation level [18]. As the local speed of graviton should not be bigger than 1 (the local speed of light of the boundary CFT), the leading term in An alternative approach to the limit D → 4 is to expand c 2 g near the boundary z → ∞ for general D [7] The condition that the local speed of graviton should be smaller than 1 requires Note that this is significantly different from the result obtain in [7] α GB ≤ D 4 − 10D 3 + 41D 2 − 92D + 96 4(D 2 − 5D + 10) 2 , (4.31) which in the D → 4 limit leads to α GB ≤ 0. But as D = 5, one can recover the well-known result α GB ≤ 0.09 [6].
The bulk causal structure and its relation with the boundary theory can be discussed as follows. In the boundary theory, the local operators create bulk disturbances at infinity , (4.32) where ψ (z(r * )) and the potential is defined by Geodesics starting from the boundary can bounce back to the boundary. It has been proven that the quasiparticles can travel faster than the speed of light and violate causality [7].
From the geodesic equation of motion g eff µν dx µ ds dx ν ds = 0, (4.34) and the Bohr-Sommerfield quantization condition one can find that the group velocity of the test particle along the geodesic line is given by [6] v g = dω dk → c g . For 4D EGBM theory, the equation (4.27) shows that c g is slower than the speed of light for α GB ≥ 0, so do the group velocity of the test particle. However, equation (4.31) implies that causality impose the condition α GB ≤ 0 to avoid propagation of signals faster than the speed of light. A natural question is whether α GB can be negative from the holographic point of view. For black holes in 5D EGB gravity, α GB has a lower bound α GB ≥ −7/36 from the analysis of sound mode perturbations [12]. We leave the study on sound mode perturbations in 4D EGB gravity theory to a future work.

Conclusions and discussions
In summary, we studied the linearized metric perturbation of black holes in 4D Einstein-Gauss-Bonnet-Maxwell theory within the framework of the AdS/CFT correspondence. The charged black hole solutions were obtained for general D dimensions. We then study the shear viscosity to entropy density ratio by considering the planar black brane solution.
The ratio η ] turns out to be different from the Kovtun-Son-Starinets bound η/s = 1/4π if α GB is non-vanishing. We then investigated the bulk causal structure of the 4D charged black holes. In order to guarantee no violation of causality in the boundary field, the Gauss-Bonnet coupling α GB should be in the range α GB ≤ 0. Note that these results were obtained by following the procedure proposed in [13]: First define the Gauss-Bonnet coupling α ′ → α ′ D−4 and then take D → 4 limit. There are some subtleties in the bulk causal structure analysis. In (4.27), we expanded the local speed of graviton at boundary z → ∞ by setting D = 4. But the leading term does not match with the local speed of light and there is a divergent term in (4.27) at O(z −3 ) order. A way to bypass this situation is to expand c g near the boundary for general D dimensions as given in (4.29). In this case, causality requires α GB ≤ 0, which is a strong constraint on the reasonable value of α GB . It would be useful to consider a mathematically more rigorous definition for the D → 4 limit of EGB gravity at the linearized level.
If one adds a linear axion field into the action (2.1) and break the translational symmetry, then the bulk causal structure could be drastically changed. For example, in 5D EGB theory, causality violation still happens in the presence of the linear scalar field but with an effective mass of the graviton dependence [42][43][44]. If the effective mass of the graviton is large enough, then there will be no causality violation and hence no constraints for the Gauss-Bonnet coupling. For 4D EGB gravity with a linear axion field, one may expect the same result. We defer these discussions to a future study.