An Interpretation for the Equivalence of Two Holographic Computations of the Butterfly Velocity with the Canonical Formalism of Gravity

TL;DR

The paper reframes two holographic computations of the butterfly velocity—the shockwave method and the entanglement wedge reconstruction method—within the canonical Hamiltonian formalism of gravity. By foliating spacetime with ADM Cauchy surfaces and enforcing the constraint equations, both computations are recast as evolutions of identical initial-data structures, with the shockwave profile $h(x)$ and the RT profile $\rho(x)$ satisfying the same underlying equation. Introducing a double shockwave ensures metric consistency and reveals the same constraint-derived dynamics for both methods, establishing their equivalence at a structural level. This work deepens the understanding of butterfly velocity in holography by highlighting how canonical gravity constraints encode the propagation of chaos, and it opens avenues for extensions to more general gravity theories and connections to modular Hamiltonians via JLMS.

Abstract

In this paper, we revisit the equivalence of two holographic computations of the butterfly velocity: the computation with the shock wave solution and the computation with the entanglement wedge reconstruction. We provide an interpretation for the equivalence of the two computations with the canonical formalism of gravity. Specifically, by taking use of the canonical formalism, we reformulate both computations into the ones with a similar form. Here, in both reformulated computations, the butterfly velocity is computed from applying a given set of initial data into the constraint equations. And the sets of initial data of both computations have a similar structure. We then interpret the equivalence of the two computations as from the similar form of the reformulated computations.

Figures (7)

• Figure 1: Interpretation of the butterfly cone. We denote the scrambling region by blue region, with the boundary $C_\beta\sim\mathcal{O}(1)$.
• Figure 2: An infalling particle with large $t_0$.
• Figure 3: An illustration of the growth of entanglement wedge including an infalling particle.
• Figure 4: An interpretation of double shockwave: two infalling particles into black hole.
• Figure 5: An illustration of an extremal surface and Cauchy surface.