Chaos in the black hole S-matrix

TL;DR

Polchinski extends the link between black hole chaos and horizon dynamics from eternal cavities to black holes that form and evaporate, and derives a precise S-matrix identity that connects amplitudes with and without an added infalling particle via near-horizon shock interactions. The result translates horizon-induced chaotic sensitivity into a calculable modification of the S-matrix, valid up to the scrambling time, and ties the exponential redshift to Lyapunov growth in the S-matrix. This work situates horizon chaos within a unitary scattering framework and engages with 't Hooft's ideas about horizon shifts and information transfer, while also confronting firewall-related puzzles inherent to combining S-matrix assumptions with effective field theory outside the horizon. Overall, it highlights the need for a UV-complete understanding of black hole microphysics to fully reconcile chaos, information, and horizon dynamics in holographic contexts.

Abstract

Recent work by Shenker, Stanford, and Kitaev has related the black hole horizon geometry to chaotic behavior. We extend this from eternal black holes to black holes that form and then evaporate. This leads to an identity for the change in the black hole S-matrix (over times shorter than the scrambling time) due an addition infalling particle, elaborating an idea of 't Hooft.

Figures (2)

• Figure 1: Phase space of a thermalizing system. a) Initial classical phase space region. b) A few thermalization times later, the region is well-mixed into the full phase space (subject to conservation laws). The initial marked points are now well separated. c) At small $\hbar$, the initial Wigner distributions still have little overlap.
• Figure 2: Commutator-squared versus time. The exponential Lyapunov growth and Ruelle decay are noted.