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

TL;DR

This paper proves the exact equivalence of two holographic methods for computing butterfly velocity in higher-derivative gravity theories, confirming their consistency and extending understanding of chaos in holography.

Contribution

It establishes a precise match between shockwave and entanglement wedge approaches for butterfly velocity in higher-derivative gravity, generalizing to all such theories.

Findings

The two holographic butterfly velocities match exactly in higher-derivative theories.

Evidence suggests the equivalence holds universally across all gravitational theories.

New results on shockwave spacetimes are derived.

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 out-of-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.