Momentum/Complexity Duality and the Black Hole Interior

TL;DR

This paper establishes a duality linking the growth rate of quantum operator complexity with a radial momentum component inside black holes, valid at late times and across different entropy regimes.

Contribution

It introduces a Momentum/Complexity duality connecting operator complexity growth to bulk momentum, extending the understanding of black hole interior dynamics.

Findings

Duality holds at late times after scrambling.

Linear complexity growth correlates with frozen interior momentum.

The formula applies broadly beyond the specific model used.

Abstract

We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and a radial component of bulk momentum for a test system falling into a black hole. In systems of finite entropy, our map remains valid for arbitrarily late times after scrambling. The asymptotic regime of linear complexity growth is associated to a frozen momentum in the interior of the black hole, measured with respect to a time foliation by extremal codimension-one surfaces which saturate without reaching the singularity. The detailed analysis in this paper uses the Volume-Complexity (VC) prescription and an infalling system consisting of a thin shell of dust, but the final PC duality formula should have a much wider degree of generality.