Butterfly velocity in quadratic gravity

We present a systematic procedure of finding the shock wave equation in anisotropic spacetime of quadratic gravity with Lagrangian Λ = R + λ + αR_{μ vδ p} R^{μ vδ p}+ βR_{μ v}γR_{μ v} + ΛR² + λ_matter. The general formula of the butterfly velocity is derived. We show that the shock wave equation in the planar, spherical or hyperbolic black hole spacetime of EinsteinGaussBonnet gravity is the same as that in Einstein gravity if the space is isotropic. We consider the modified AdS spacetime deformed by the leading correction of the quadratic curvatures and find that the fourth order derivative shock wave equation leads to two butterfly velocities if 4α+β < 0. We also show that the butterfly velocity in the D = 4 planar black hole is not corrected by the quadratic gravity if 4α+β = 0, which includes the R2 gravity. In general, the correction of butterfly velocity by the quadratic gravity may be positive or negative, depending on the values of α, β, γand temperature. We also investigate the butterfly velocity in the Gauss-Bonnet massive gravity.