Black hole scrambling from hydrodynamics

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 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 approximate group velocity give the holographic Lyapunov exponent and the butterfly velocity. This establishes a direct link between a hydrodynamic instability of a driven sound mode and the holographic butterfly effect. Thus, infinitely strongly coupled, large-$N$ holographic theories exhibit properties similar to classical dilute gasses in which late-time equilibration and early-time scrambling are controlled by the same dynamics.