A Tale of Two Butterflies: An Exact Equivalence in Higher-Derivative Gravity

Abstract

We prove the equivalence of two holographic computations of the butterfly velocity in higher-derivative theories with Lagrangian built from arbitrary contractions of curvature tensors. The butterfly velocity characterizes the speed at which local perturbations grow in chaotic many-body systems and can be extracted from the outof-time-order correlator. This leads to a holographic computation in which the butterfly velocity is determined from a localized shockwave on the horizon of a dual black hole. A second holographic computation uses entanglement wedge reconstruction to define a notion of operator size and determines the butterfly velocity from certain extremal surfaces. By direct computation, we show that these two butterfly velocities match precisely in the aforementioned class of gravitational theories. We also present evidence showing that this equivalence holds in all gravitational theories. Along the way, we prove a number of general results on shockwave spacetimes.

Pole skipping in holographic theories with bosonic fields

“…Furthermore, there are now three ways of computing the butterfly velocity: (i) using entanglement wedge, (ii) using shockwave and (iii) using pole-skipping. Evidence for the equivalence between the first two was presented in [7,11], and here we proved the equivalence between the last two.…”

“…where the second line follows from the vanishing of most Christoffel components on the horizon. Therefore, (11) shows the minimum number of such factors it has to contain for the weight to match. We now discuss the general conditions for poleskipping.…”

“…In general higher derivative gravity and for a shockwave along V = 0, the only non-trivial component of δE µ ν perturbed by a local source is δE U V [11]. For a general perturbation, δg vv , translating to ingoing Eddington-Finkelstein coordinates, this component is given by…”

“…For classical bulk gravitational theories, λ L saturates the chaos bound λ L ≤ 2πT [6], so they are said to be maximally chaotic. The butterfly velocity, however, depends on the theory [2,[7][8][9][10][11].…”