Chaotic Fast Scrambling At Black Holes

TL;DR

The paper addresses how black holes scramble information at the fastest possible rate without violating causality. Using AdS/CFT and an optical metric with hyperbolic spatial sections, it recasts near-horizon dynamics as chaotic ballistic motion, deriving the causality-bound time tau_* ≈ β log(S_cell). It then argues that scrambling within a single thermal cell occurs in O(tau_*), and that across many cells the full scrambling time scales as tau_s ~ tau_* (n_cell)^{2/d} = β log(N_eff) (S/N_eff)^{2/d}, effectively linking fast scrambling to classical chaos. The results suggest that chaos in the optical geometry is a key ingredient to reconcile fast horizon scrambling with large-N locality, while also outlining limitations and avenues for future work in non-conformal or de Sitter contexts.

Abstract

Fast scramblers process information in characteristic times scaling logarithmically with the entropy, a behavior which has been conjectured for black hole horizons. In this note we use the AdS/CFT fold to argue that causality bounds on information flow only depend on the properties of a single thermal cell, and admit a geometrical interpretation in terms of the optical depth, i.e. the thickness of the Rindler region in the so-called optical metric. The spatial sections of the optical metric are well approximated by constant-curvature hyperboloids. We use this fact to propose an effective kinetic model of scrambling which can be assimilated to a compact hyperbolic billiard, furnishing a classic example of hard chaos. It is suggested that classical chaos at large N is a crucial ingredient in reconciling the notion of fast scrambling with the required saturation of causality.

Figures (2)

• Figure 1: Diagram showing that the Schwarzschild time between $P$ and $P'$ equals the reflection time of a photon from the stretched horizon, i.e. the piecewise-null trajectory $P\,Q' \cup Q' \,P'$, where $Q$ and $Q'$ lie respectively at the inner and outer edges of the stretched horizon, i.e. on the hypersurfaces $X^+ X^- = \pm \ell_{\rm P}^2$.
• Figure 2: Near-horizon random walk of a localized probe by scattering at the stretched horizon, pictured in the Poincaré coordinates of the spatial optical metric (\ref{['hyps']}). Free paths between successive collisions are circular arcs, with maximal radius $\Delta y_{\rm max} \sim \beta$, since longer glides are reflected back by the asymptotic AdS potential well which starts at the edge of the Rindler region, represented in this picture by a dashed line.