Abstract
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.
Highlights
Quantum chaos is naturally characterized by the commutator 1⁄2Wðt; xÞ; Vð0Þ which measures the dependence of a later operator Wðt; xÞ on an earlier perturbation Vð0Þ
In the holographic approach the butterfly velocity is identified by the velocity of the shock wave which describes how the perturbation spreads in space [8,10,11]
In Appendix B, we present some details in deriving our formula of the butterfly velocity in Einstein-GaussBonnet gravity with arbitrary matter fields
Summary
Quantum chaos is naturally characterized by the commutator 1⁄2Wðt; xÞ; Vð0Þ which measures the dependence of a later operator Wðt; xÞ on an earlier perturbation Vð0Þ. In the second part of the paper, we consider the butterfly velocity for the GaussBonnet gravity with arbitrary matter fields. Velocity in the higher-derivative gravity, including the Gauss-Bonnet term without matter fields, has been discussed in [8] and [18]. The Gauss-Bonnet gravity is the simplest correction of the Einstein theory, without introducing derivatives higher than the second, appearing in the field equation. We add the matter fields and apply our general formula to evaluate the butterfly velocity in several interesting holographic systems. We calculate the butterfly velocities in the Einstein-Gauss-Bonnet-Maxwell theory with spherical or hyperbolic black holes. In Appendix B, we present some details in deriving our formula of the butterfly velocity in Einstein-GaussBonnet gravity with arbitrary matter fields
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