Butterfly Velocity in Quadratic Gravity

Abstract

We present a systematic procedure of finding the shock wave equation in anisotropic spacetime of quadratic gravity with Lagrangian ${\cal L}=R+ \Lambda+\alpha R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}+\beta R_{\mu\nu}R^{\mu\nu}+\gamma R^2+{\cal L}_{\rm matter}$. 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 Einstein-Gauss-Bonnet gravity is the same as that in Einstein gravity if 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 $4\alpha+\beta<0$. We also show that the butterfly velocity in a D=4 planar black hole is not corrected by the quadratic gravity if $ 4\alpha+\beta=0$, which includes the $ R^2$ gravity. In general, the correction of butterfly velocity by the quadratic gravity may be positive or negative, depends on the values of $\alpha$, $\beta$, $\gamma$ and temperature. We also investigate the butterfly velocity in the Gauss-Bonnet massive gravity.