Black hole scrambling from hydrodynamics

TL;DR

This paper links black hole scrambling in holographic theories to hydrodynamic sound modes, revealing that the same imaginary frequency and momentum characterize both phenomena and connect to the butterfly effect.

Contribution

It establishes a direct relation between shock wave computations of scrambling and hydrodynamic sound modes in holography, including the role of imaginary frequency and momentum.

Findings

Scrambling rate relates to hydrodynamic sound modes.

Imaginary frequency and momentum characterize both scrambling and sound dispersion.

Holographic theories exhibit properties akin to classical dilute gases.

Abstract

We argue that the gravitational shock wave computation used to extract the scrambling rate in strongly coupled quantum theories with a holographic dual is directly related to probing the system's hydrodynamic sound modes. The information recovered from the shock wave can be reconstructed in terms of purely diffusion-like, linearized gravitational waves at the horizon of a single-sided black hole with specific regularity-enforced imaginary values of frequency and momentum. In two-derivative bulk theories, this horizon "diffusion" can be related to late-time momentum diffusion via a simple relation, which ceases to hold in higher-derivative theories. We then show that the same values of imaginary frequency and momentum follow from a dispersion relation of a hydrodynamic sound mode. The frequency, momentum and group velocity give the holographic Lyapunov exponent and the butterfly velocity. Moreover, at this special point along the sound dispersion relation curve, the residue of the retarded longitudinal stress-energy tensor two-point function vanishes. This establishes a direct link between a hydrodynamic sound mode at an analytically continued, imaginary momentum and the holographic butterfly effect. Furthermore, our results imply that infinitely strongly coupled, large-$N_c$ holographic theories exhibit properties similar to classical dilute gasses; there, late-time equilibration and early-time scrambling are also controlled by the same dynamics.