Revisit the relationship between spread complexity rate and radial momentum

TL;DR

This paper explores the connection between boundary spread complexity rate and bulk radial momentum in AdS/CFT, showing their equivalence and providing a method to compute complexity from radial momentum via optical geometry.

Contribution

It establishes a direct relationship between spread complexity rate and radial momentum, including a simple method to derive complexity from radial momentum in AdS/CFT.

Findings

Radial momentum equals spread complexity rate in AdS/CFT.

Method to compute spread complexity from radial momentum using optical geometry.

Massive particles at the boundary start with zero velocity at t=0.

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.