Butterfly velocity and bulk causal structure

Abstract

The butterfly velocity was recently proposed as a characteristic velocity of chaos propagation in a local system. Compared to the Lieb-Robinson velocity that bounds the propagation speed of all perturbations, the butterfly velocity, studied in thermal ensembles, is an "effective" Lieb-Robinson velocity for a subspace of the Hilbert space defined by the microcanonical ensemble. In this paper, we generalize the concept of butterfly velocity beyond the thermal case to a large class of other subspaces. Based on holographic duality, we consider the code subspace of low energy excitations on a classical background geometry. Using local reconstruction of bulk operators, we prove a general relation between the boundary butterfly velocities (of different operators) and the bulk causal structure. Our result has implications in both directions of the bulk-boundary correspondence. Starting from a boundary theory with a given Lieb-Robinson velocity, our result determines an upper bound of the bulk light cone starting from a given point. Starting from a bulk space-time geometry, the butterfly velocity can be explicitly calculated for all operators that are the local reconstructions of bulk local operators. If the bulk geometry satisfies Einstein equation and the null energy condition, for rotation symmetric geometries we prove that infrared operators always have a slower butterfly velocity that the ultraviolet one. For asymptotic AdS geometries, this also implies that the butterfly velocities of all operators are upper bounded by the speed of light. We further prove that the butterfly velocity is equal to the speed of light if the causal wedge of the boundary region coincides with its entanglement wedge. Finally, we discuss the implication of our result to geometries that are not asymptotically AdS, and in particular, obtain constraints that must be satisfied by a dual theory of flat space gravity.