On Butterfly effect in Higher Derivative Gravities

TL;DR

This work analyzes chaos propagation in gravity theories with higher-derivative terms, specifically actions containing $R^2$ and $R^{\mu\nu}R_{\mu\nu}$, and in the 3D Topologically Massive Gravity (TMG). Using shock-wave techniques near AdS black hole horizons, the authors derive a fourth-order equation that factorizes into two second-order equations, yielding two butterfly velocities $v_B^{(1)}$ and $v_B^{(2)}$ associated with two boundary spin-2 operators; in TMG these reduce to $v_B^{(1)}=1$ and $v_B^{(2)}=1/(\mu\ell)$, coinciding at the critical point $\mu\ell=1$. A dual CFT perspective via four-point functions and conformal block decomposition confirms that the velocities are determined by operator dimensions $\Delta$ and spins $s$ through $v_B(\Delta,s)=\frac{s-1}{\Delta-1}$, with two contributing operators when bulk gravitons are present. The paper also discusses Gauss-Bonnet-like cases where only a single velocity persists, highlighting the role of boundary gravitons, and suggests broader implications for holographic chaos in higher-derivative theories.

Abstract

We study butterfly effect in $D$-dimensional gravitational theories containing terms quadratic in Ricci scalar and Ricci tensor. One observes that due to higher order derivatives in the corresponding equations of motion there are two butterfly velocities. The velocities are determined by the dimension of operators whose sources are provided by the metric. The three dimensional TMG model is also studied where we get two butterfly velocities at generic point of the moduli space of parameters. At critical point two velocities coincide.