Chaotic fast scrambling at black holes

Fast scramblers process information in characteristic times scaling logarithmically with the entropy, a behavior which has been conjectured for black hole horizons. In this paper we use the AdS/CFT context 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 behaves like 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.

Figure 1

Diagram showing that the Schwarzschild time between $P$ and $P^{\prime }$ equals the reflection time of a photon from the stretched horizon, i.e. the piecewise-null trajectory $P{Q}^{\prime }\cup Q^{\prime }{P}^{\prime }$, where $Q$ and $Q^{\prime }$ lie, respectively, at the inner and outer edges of the stretched horizon, i.e. on the hypersurfaces $X^{+}{X}^{-}=\pm {\ell }_{\mathrm{P}}^2$.

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 (16). Free paths between successive collisions are circular arcs, with maximal radius $\Delta y_{\max }\sim \beta$, 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.