# Complexity and Newton's Laws

## TL;DR

This paper explores how gravitational attraction can be understood as a consequence of quantum operator growth and complexity increase, providing a new perspective on Newton's laws via holography and the SYK model.

## Contribution

It introduces an improved size-momentum correspondence and demonstrates that Newton's laws emerge from quantum complexity dynamics in holographic models.

## Key findings

- Operator size relates to complexity and gravity.
- Newton's laws follow from quantum operator growth.
- The improved correspondence is justified by symmetry arguments.

## Abstract

In a recent note I argued that the holographic origin of gravitational attraction is the quantum mechanical tendency for operators to grow under time evolution. In a followup the claim was tested in the context of the SYK theory and its bulk dual---the theory of near-extremal black holes. In this paper I give an improved version of the size-momentum correspondence and show that Newton's laws of motion are a consequence. Operator size is closely related to complexity. Therefore one may say that gravitational attraction is a manifestation of the tendency for complexity to increase. The improved version of the size-momentum correspondence can be justified by the arguments of Lin, Maldacena, and Zhao constructing symmetry generators for the approximate symmetries of the SYK model.