# Fuzzy Classical Dynamics as a Paradigm for Emerging Lorentz Geometries

## TL;DR

This paper reveals that classical particle dynamics on fuzzy spaces can be understood through Lorentz geometries, linking noncommutative effects to standard space-time structures like Minkowski space and magnetic monopole backgrounds.

## Contribution

It establishes a geometric framework connecting fuzzy space dynamics with Lorentzian geometries and identifies the underlying space-time structures emerging from noncommutative effects.

## Key findings

- Classical equations of motion on fuzzy spaces relate to Lorentz geometries.
- Noncommutative dynamics induce standard space-time metrics with corrections.
- Equivalence between fuzzy sphere dynamics and charged particle in magnetic monopole field.

## Abstract

We show that the classical equations of motion for a particle on three dimensional fuzzy space and on the fuzzy sphere are underpinned by a natural Lorentz geometry. From this geometric perspective, the equations of motion generally correspond to forced geodesic motion, but for an appropriate choice of noncommutative dynamics, the force is purely noncommutative in origin and the underpinning Lorentz geometry some standard space-time with, in general, non-commutatuve corrections to the metric. For these choices of the noncommutative dynamics the commutative limit therefore corresponds to geodesic motion on this standard space-time. We identify these Lorentz geometries to be a Minkowski metric on $\mathbb{R}^4$ and $\mathbb{R} \times S ^2$ in the cases of a free particle on three dimensional fuzzy space ($\mathbb{R}^3_\star$) and the fuzzy sphere ($S^2_\star$), respectively. We also demonstrate the equivalence of the on-shell dynamics of $S^2_\star$ and a relativistic charged particle on the commutative sphere coupled to the background magnetic field of a Dirac monopole.