Revisit the relationship between spread complexity rate and radial momentum

TL;DR

This work investigates the proposed link between the boundary spread complexity rate $dC_K/dt$ and bulk radial momentum within AdS/CFT. By revisiting massive and massless probe scenarios and reconciling two independent approaches, the authors show that the proper radial momentum observed by a stationary bulk observer coincides with the spread complexity rate in the boundary CFT for particles of arbitrary mass. A general argument, plus a concrete 3D AdS example, demonstrates that $dC_K/dt$ is equivalent to the bulk radial momentum, and an optical-geometry method is proposed to compute $C_K$ from $P_r$ without explicit geodesic calculations. The results deepen the holographic understanding of complexity and offer a practical route to relate boundary Krylov-like complexity to bulk dynamical quantities, while raising questions about a formal proof and broader holographic applicability.

Abstract

This article discusses the relationship between the boundary spread complexity rate and the radial momentum in the bulk within the framework of AdS/CFT. We demonstrate that the radial momentum of a freely falling particle, as measured by a stationary observer in the bulk, is equal to the spread complexity rate of the boundary conformal field theory. For a massive particle (no matter what the specific mass is), the particle is located at the asymptotic boundary with zero velocity at $t=0$. Additionally, we provide a simple method for obtaining spread complexity from radial momentum using optical geometry.

Figures (1)

• Figure 1: $C_K(t)$ for different values of $\frac{m}{2z_0}$.