Operator growth and black hole formation

TL;DR

The paper investigates microscopic criteria for black hole formation in AdS by recasting collisions as two competing chaotic perturbations on a boundary quantum circuit. It proposes a threshold based on the overlap of butterfly cones and introduces a diagnostic six-point correlator to detect the transition, with explicit calculations in $AdS_3$ and in a $2d$ CFT via an eikonal action and boundary reparametrizations. The key result is that black hole formation coincides with butterfly-cone overlap and reproduces the Gott condition in three bulk dimensions, linking operator growth and gravitational collapse. This work provides a calculable bridge between quantum chaos on the boundary and bulk BH dynamics, suggesting extensions to higher dimensions and dual CFT diagnostics, as well as numerical checks in SYK-like systems.

Abstract

When two particles collide in an asymptotically AdS spacetime with high enough energy and small enough impact parameter, they can form a black hole. Motivated by dual quantum circuit considerations, we propose a threshold condition for black hole formation. Intuitively the condition can be understood as the onset of overlap of the butterfly cones describing the ballistic spread of the effect of the perturbations on the boundary systems. We verify the correctness of the condition in three bulk dimensions. We describe a six-point correlation function that can diagnose this condition and compute it in two-dimensional CFTs using eikonal resummation.

Figures (12)

• Figure 1: Cartoon of the collision of two localized particles sent into an AdS-Rindler spacetime from the boundary. At high enough energies and small enough impact parameter, a black hole forms.
• Figure 2: Illustration of the causal cone (light red) and butterfly cone (dark red) describing the relativistic spread in space and the ballistic exponential growth of a small perturbation.
• Figure 3: Overlap of two causal cones spreading in opposite directions of the quantum circuit (i.e., one of them corresponds to a perturbation of the right system, and the other to a perturbation of the left system). Each constant $x$ slice can be thought of as a local quantum circuit describing the evolution of the thermofield double state shared between local matrix degrees of freedom in the two systems at location $x$.
• Figure 4: Collision setup in Rindler AdS and in the quantum circuit. We propose that a black hole forms precisely when the butterfly cones begin to overlap (right hand side).
• Figure 5: (a) Each transverse $x$-slice has a description in terms of a quantum circuit with all-to-all qubit interactions. (b) Exponential growth of the perturbation within this circuit can be approximated by a step function located at a time that is separated from the initial perturbation by $t_*(x)$. (c) We can further simplify notation and collapse the approximate diagram into a line and denote the intersection of the $x$-slice with the causal cone of the perturbation by a faint dot; we will use this notation in section \ref{['sec:diagnosis']}.