  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.

High Energy Physics - Theory;Strongly Correlated Electrons;General Relativity and Quantum Cosmology;Quantum Physics