Black hole scrambling from hydrodynamics

TL;DR

The results imply that infinitely strongly coupled, large-N_{c} holographic theories exhibit properties similar to classical dilute gases; there, late-time equilibration and early-time scrambling are also controlled by the same dynamics.

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.

Figures (1)

• Figure 1: Dispersion relations of the hydrodynamic sound modes, plotted for imaginary dimensionless $\mathfrak{w} \equiv \omega / 2 \pi T$ and $\mathfrak{q} \equiv k / 2\pi T$. The blue lines depict the third-order hydrodynamic result Grozdanov:2015kqa and the red crosses the numerically computed $\mathfrak{w}^*_\pm (\mathfrak{q})$. Dashed lines indicate the values of $\omega = i \lambda_L$ and $k = i \mu$. The dotted line is the linear dispersion relation $\mathfrak{w} = v_B \mathfrak{q}$. The inlay depicts a zoomed-in plot around $k = i \mu$.