Holographic Butterfly Velocities in Brane Geometry and Einstein-Gauss-Bonnet Gravity with Matters

In the first part of the paper we generalize the butterfly velocity formula to anisotropic spacetime. We apply the formula to evaluate the butterfly velocities in M-branes, D-branes and strings backgrounds. We show that the butterfly velocities in M2-branes, M5-branes and the intersection M2$\bot$M5 equal to those in fundamental strings, D4-branes and the intersection F1$\bot$D4 backgrounds, respectively. These observations lead us to conjecture that the butterfly velocity is generally invariant under a double-dimensional reduction. In the second part of the paper, we study the butterfly velocity for Einstein-Gauss-Bonnet gravity with arbitrary matter fields. A general formula is obtained. We use this formula to compute the butterfly velocities in different backgrounds and discuss the associated properties.


Introduction
Quantum chaos is naturally characterized by the commutator [W (t, x), V (0)] which measures the dependence of a later operator W (t, x) on an earlier perturbation V (0). The strength of the associated butterfly effect can be described by [1,2,3] [W (t, x), V (0)] 2 T ∼ e where t * is a time scale called the scrambling time at which the commutator grows to be O (1). The buttery velocity v B characterizes speed at which the perturbation grows. The Lyapunov exponent λ measures the rate of growth of chaos, and it is bounded by temperature T: λ ≤ 2πβ where β = 1 T . The inequality saturates for the thermal systems that have a dual holographic black hole described by Einstein gravity [4].
Butterfly effects in quantum chaos have been extensively studied recently in the holographic theories . In the holographic approach the butterfly velocity is identified by the velocity of shock wave which describes how the perturbation spreads in space [8,10,11]. The method of finding the shock wave velocity for the general spacetime with matters had been described in many years ago [27], and was used to obtain the butterfly velocity. In this paper we consider futher generalizations. In the first part of this paper we extend the known butterfly velocity formula to the anisotropic spacetime. We will apply our formula to several brane systems. Our computations lead us to conjecture that the butterfly velocity is a quantity that is invariant under a double-dimensional reduction. In the second part of the paper, we consider the butterfly velocity for the Gauss-Bonnet gravity with arbitrary matter fields. Butterfly velocity in the higher-derivative gravity including Gauss-Bonnet term without matter fields had been discussed in [8] and [18]. The Gauss-Bonnet gravity is a simplest correction of the Einstein theory without introducing derivatives higher than second appearing in the field equation. In the AdS/CFT correspondence, the Gauss-Bonnet term in the bulk corresponds to next-to-leading order corrections in the 1/N expansion of the dual CFT [28]. Here we add the matter fields and apply our general formula to evaluate the butterfly velocity in several interesting holographic systems.
After reviewing the previous method, we extend the butterfly velocity formula [8,10,11] to the anisotropic spacetime in section 2. We present a rather simple way, i.e., without too many complicated tensor calculations, to derive a general formula given in (2.30). In section 3 we apply this formula to calculate the butterfly velocities in M2-branes, M5 branes, Dp-branes and also string backgrounds. We find that the butterfly velocity in M2 equals to that in string background. Also, the velocity in M5 equals to that in D4. Since the spacetime of string and D4 are those through double dimensional reduction from M2 and M5 backgrounds respectively, we conjecture that the butterfly velocity is invariant under a double-dimensional reduction. One more example involving the intersections F1⊥D4 and M2⊥M5, which are related also through a double-dimensional reduction, also supports our claim. In section 4, we develop a butterfly velocity formula of Gauss-Bonnet gravity with arbitrary matter fields. We present the detailed tensor analysis and obtain a compact and general formula given in (4.18). For the special cases of planar, spherical or hyperbolic black holes, the formula becomes a simpler form described in (4.22). In section 5 we first check that our formula reproduces the previous result in the literature without matter field [8,18]. We next apply our formula to the Einstein-Gauss-Bonnet-Maxwell theory and Einstein-Gauss-Bonnet-scalar theory. Furthermore, we calculate the butterfly velocities in the Einstein-Gauss-Bonnet-Maxwell theory with spherical or hyperbolic black hole. We discuss future works in the last section. In appendix A, we briefly describe the Kruskal coordinate in general geometry and derive some relations which are useful in computing butterfly velocity. In appendix B, we present some details in deriving our formula of the butterfly velocity in Einstain-Gauss-Bonnet gravity with arbitrary matter fields.

Shock Wave Equation and Butterfly Velocity in
Anisotropic Spacetime

Shock Wave Geometry and Shock Wave Equation
We will derive the formula of butterfly velocity in the following anisotropic background: where horizon locates at r = r H . Note f (r H ) = 0 while a(r H ) = 0, b(r H ) = 0. In the original derivation [27], Sfetsos considered an isotropic background representing the special case S=1. The authors in [17,[22][23][24][25][26] considered the anisotropic case in flat space where g ij = δ ij . In our study, the relevant line element (for example g (2) ij (x)dx i (2) dx j (2) ) will be able to describe curved surface where g (2) ij = δ ij . Note the associated temperature of above black hole or black brane is given by Since the holographic geometry of the chaos covers two sides, we consider the line element expressed in the Kruskal coordinate : 1 in which r * is the tortoise coordinate defined by dr * = .
To proceed we follow [5,10,11,27] to add a small null perturbation of asymptotic energy E. At later time the perturbation will follow null trajectories very close to the (past) horizon, where the trajectories become exponentially blue-shifted and the perturbation grow exponentially large, in contrast to what was expected in an earlier study [29]. After solving the associated Einstein's equation one obtain shock wave geometry shown in figure  1. We first describe the computation scheme of obtaining the shock wave equation in spacetime (2.1). This will enable us to find butterfly velocities in M-branes, D-branes and also string backgrounds. The scheme was clearly described in [27], and it had been applied to study the holographyic butterfly velocity with matter field in [10] recently.
The metric (2.1) describes the solution of Einstein equation with general form of stress 1 In appendix A we describe the more properties of Kruskal coordinate. We also present several useful relations.
tensor. We can express it as where G is the Einstein tensor. Along the arguments of Dray and G. t'Hooft [29], for U < 0 the spacetime is described by (2.5). After adding a small null perturbation, for U > 0 the spacetime is still described by (2.5) but V is shifted by as shown in figure 1 . The function α(x) will be determined by a shock wave equation which is to be determined. The resulting metric and energy momentum tensor are where Θ = Θ(U ) is a step function and As described by Sfetsos [27], in terms of the new coordinateŝ the metric and stress tensor can be expressed by the following simple forms: T matter = 2 TÛV − TVVαδ dÛ dV (2.14) Following [8,27], we add an extra stress tensor to produce the shock wave geometry. Now we have to solve the Einstein equation (we will drop the hat notation in what follows.) The key point is that using un-perturbed Einstein equation V V = 0 in the model spacetime. Therefore, T matter remains only one term that is linear in perturbation: −2TÛVαδ dÛ 2 . The shock wave equation can be written as U U is the first-order correction of the Einstein tensor from the metric (2.13).

Butterfly Velocity Formula in Anisotropic Spacetime
According to the scheme described above, the main problem is to calculate the tensors G (1) For the case of S=1, the above two tensors were presented in [27]. Here we are interested in a more general case.
We first need the following tensor properties in un-perturbed spacetime (2.5): They be found through straightforward algebra calculations. Note dim(S) = g (S)ij g (S) ij arises from the tensor contraction from different species. We have considered values on the horizon, U = 0. In contrast to performing straightforward but tedious algebra calculations, we will calculate G (1) U U starting from the following basic relation: Now from δg ab = −2A α(x)δ(U )δ aU δ bU and g (0) Note that in the above relations the covariant derivative acts on un-perturbed background. It is interesting that the term 1 2 KR (0) is canceled by 1 2 A(0) R (0) in (2.20), according to the shock wave equation (2.18). To proceed we find where g ab is the metric given in (2.5). The Laplacian is To obtain above result we have used δ (U ) = −δ(U )/U . Collecting all the above results, we finally obtain Note R (0) parts are canceled out. The shift function α(t, x) is determined by the perturbation function a(x) through the shock wave equation To obtain a simple formula of butterfly velocity we can consider the case in which the local source is a( where we have used which are derived in appendix A. Note that our formula (2.30) reduces to [17,[22][23][24][25][26] and [27] in the flat anisotropic space and curved isotropic space, respectively. In the next section we will use our new formulae to evaluate the butterfly velocities in the brane geometry.
3 Butterfly Velocity in Brane Geometry

M5 and D4 Backgrounds
In this section we use our general butterfly velocity formula to calculate the butterfly velocities in M-branes, D-branes and string backgrounds. The N 5 black M5-branes solution is given by [30,31] where H is the harmonic function defined by The function f (r) specified by the horizon at r H is We can approximate H ∼ N 5 r 3 in the "near-horizon" limit. On the other hand, the spacetime of a stack of N 4 black D4-branes (in the Einstein frame) is given by In the "near-horizon" limit, H ∼ N 4 r 3 . Using these metrices, we calculate the black-brane temperature, T (r H ), the butterfly velocities v B along (x 1 · · · x 5 ) and along Ω 4 in M5. We can also compute the velocities along (x 1 · · · x 4 ) and along Ω 4 in D4. We collect the relevant functions in the table 1. It seems striking that although the parameter functions for M5 and D4 are quite different, the resulting butterfly velocities and temperature, given below, are the same. Let us make following comments : 1. For temperature to be the same, one needs the same number of branes for M5 and D4. The butterfly velocities are the same independent of N.
2. While the butterfly velocity v B (M ) satisfies upper bound v B (Ω) violates it. This violation does not contract to [4] since that paper assumes space isotropy [15,17].
3. Notice that through a double-dimensional reduction M5 becomes D4. 4. For completeness we present in below the butterfly velocity in Dp brans background  We again observe that although the parameter functions for M2 and F1 are different, the temperature and butterfly velocities, shown in below, have the same behaviour. Since that through a double-dimensional reduction M2 (M5) becomes F1 (D4) one tempts to conjecture that the butterfly velocity is generally invariant under a double dimensional reduction. Let us consider one more example in the next section.

Intersections M2⊥M5 and F1⊥D4 Backgrounds
We consider the brane intersection M2⊥M5 system where M2-branes lie in (w, x) and M5-branes locate at (w, y 1 ...y 4 ) i.e., Upon a double-dimensional reduction, the space w is wrapped, and the geometry becomes the intersection F1⊥D4, i.e., t x y 1 y 2 y 3 y 4 z 1 z 2 z 3 z 4 The metric of black intersection M2⊥M5 is [30,31] where In the Einstein frame, the metric of the intersection F1⊥D4 is where In the "near horizon" limit, the relevant functions are collected in the table 3.
again have same behaviors. Thus, the butterfly velocity is again invariant under a doubledimensional reduction.

Formula of Butterfly velocity in Einstein-Gauss-
Bonnet Gravity with Matter Fields

Formula in Arbitrary Spacetime
The Lagrangian we consider will contain a curvature scalar, a cosmological constant, a Gauss-Bonnet term and matter fields: where AdS is the AdS radius. The associated gravity equation is given by where we denote The Gauss-Bonnet term is a special combination of curvatures in which its field question contains only second-order derivatives. All higher-derivatives terms are cancelled out. In four-dimensions, the Gauss-Bonnet term is a topological invariant so it does not enter dynamics. We will consider the theory in d+2 dimensions 2 . We assume that the solution expressed in Kruskal coordinates could be written as Above metric lead to and condition T U U conponent, which is linear in α as described in (2.13). In Gauss-Bonnet gravity, according to the scheme described in section 2, the shockwave equation becomes Using the definitions g U V R (0) are canceled out in section 2. The shock wave equation for Gauss-Bonnet gravity now has a very simple form: Now we only have to calculate H U V in (2.13) for the Einstein gravity theory. To proceed, we find the following general relations: where, up to a factor δ(U ) The covariant derivative ∇ (d) i and Laplacian ∆ (d) are defined in the d-dimensional metric, ds 2 = g ij dx i dx j . R (0) is the 2+d dimensional Ricci scalar calculated by un-perturbative metric (4.4). Note that all values are evaluated at the horizon. It will require further analysis to derive the relation (R U cU d R cd ) (1) , which we present them in the appendix B. It is interesting to see that in the Einstein gravity the term containing derivatives of α(x) are formed as the Laplacian, ∆ (d) α(x), while in the Gauss-Bonnet gravity it appears a new form Collect above calculations, the shock wave equation of Einstein-Gauss-Bonnet gravity from (4.14) reduces to the following formula: the butterfly velocity formula has a simple form a(x) (4.18) in which R (d)ij and R (d) are defined in the d-dimensional metric ds 2 = g ij dx i dx j ,

Formula in Planar, Spherical and Hyperbolic Black Holes
To further simplify (4.18), let us consider 2+d dimensional planar, spherical or hyperbolic black holes. The general metric is Substituting these relations into (4.18) and with a help of appendix A we obtain an amazingly simple expression for the shock-wave equation of Einstein-Gauss-Bonnet gravity with arbitrary matters: Let us summarize and make some remarks. 1. Shock wave equation (4.18) can be directly applied to Einstein-Gauss-Bonnet theory with arbitrary matters once the black hole metric being expressed as (4.4). For planar, spherical or hyperbolic black holes we have a simple formula (4.22).
2. The case of k = 0 was studied in [8] In the next section we consider some examples to illustrate our formulae.

Planar Black Hole
In this subsection we consider the case in which the butterfly is propagating in the planar black hole background

Einstein-Gauss-Bonnet Gravity
Consider first the simplest theory without any matter. The planar black hole solution is [32][33][34] We let the AdS radius AdS = 1. The horizon locates on r H . As discussed in [8], the presence of the parameter N in the metric rescales time so the temperature is β = 4πf (1)/N . This rescaling implies that the butterfly velocities in Einstein gravity and Einstein-Gauss-Bonnet gravity without matters are respectively. These results were first found in [8].

Einstein-Gauss-Bonnet-Maxwell Gravity
For the Einstein-Gauss-Bonnet Gravity theory with Maxwell field, one adds a matter Lagrangian The associated planar black hole solution then is [32][33][34] The parameters q and µ denote the strength of Maxwell fields. The values of N and λ are defined before.
We find the butterfly velocities in the Einstein-Maxwell Gravity theory and Einstein-Gauss-Bonnet-Maxwell Gravity theory are given by v (q) respectively. When q = 0 above results reduce to previous case. Comparing (5.6) and (5.14) we observe that the ratio of butterfly velocities between that with and without Gauss-Bonnet term is in fact "universal", in the sense that it does not depend on the strength of Maxwell field.

Einstein-Gauss-Bonnet-Scalar Gravity
Consider the Einstein-Gauss-Bonnet Gravity theory with scalar fields. The matter field Lagrangian is Choosing the particular values of parameters , where Λ is the cosmological constant, the exact black hole solution can be described by [35,36] We find the butterfly velocity is Note that in this model the coupling of Gauss-Bonnet term γ GB is fixed in (5.16).

Spherical and Hyperbolic Black Holes
Next we consider non-planar black hole backgrounds, including spherical and hyperbolic geometry.

Einstein Gravity
Consider first the simplest case. The metric of a black hole in Einstein Gravity theory is where k = 0, 1, −1 describe flat, spherical and hyperbolic black hole, respectively. The relation between horizon radius and the black hole temperature is more complicated and is given by The butterfly velocity becomes where the first relation was found earlier in [8].

Einstein-Gauss-Bonnet-Maxwell Gravity
Finally, we consider 2+d dimensional planar, spherical or hyperbolic black hole solutions in Einstein-Gauss-Bonnet-Maxwell gravity. After taking a proper limit, i.e. k = 0 and/or q = 0, the result obtained in what follows reproduces all previous results, as they must be. The metric is To find the butterfly velocity we solve (5.29) to express the horizon radius r H in terms of the black hole temperature T , and then substitute it into (4.24). Since the exact expression is complicated we consider the high-temperature expansion. We find Let us make following comments : 1. From above results we can see how λ and k affect the butterfly velocity. 2. At high temperature the butterfly velocities in both sphere and hyperbolic black holes reduce to that in the planar black hole.
3. The ratio of the butterfly velocities between that with and without Gauss-Bonnet term is also universal likes as planar case mentioned before.

Conclusion
In the first part of this paper we have investigated the butterfly velocity in the anisotropic spacetime. We have derived a general formula given in (2.30). We used this formula to study the butterfly velocities in brane systems. We have conjectured that the butterfly The metric can be written in Kruskal coordinate as follows. and B Derivation of (4.11) We first rewrite (R U cU d R cd ) (1) as U cU d R (0)cd (B.1) The first term can be easily calculated using where Laplacian ∆ (d) α(x) is defined in the d-dimensional metric ds 2 = g ij dx i dx j . We next rewrite the second term as U V U V = 0. To calculate the remaining term, R U iU j , we write it as and g (1) The final step is to analyze (R V iU j ) (1) . We find Note that, in the second line, the first three terms are zero, while last two terms are non-zero but they are canceled to each other and therefore only the forth term left. Using δg ab = −2A(U, V )α(x)δ(U )δ aU δ bU we obtain a simple relation is defined in a d-dimensional metric, ds 2 = g ij dx i dx j . Collecting the above results we find the relation of (R U cU d R cd ) (1) given in (4.11).