Geodesic equation in noncommutative space: a field theory perspective

TL;DR

This paper derives a modified geodesic equation in noncommutative spacetime using a field-theoretic approach, revealing an effective mass function and additional force terms that depend on the quantum nature of spacetime.

Contribution

It introduces a new derivation of the noncommutative geodesic equation from a field theory perspective, incorporating an effective mass and external force due to noncommutativity.

Findings

Noncommutative effects are absorbed into a position-dependent mass function.

Leading corrections to geodesic motion appear at second order in noncommutativity parameter.

Massless particles are unaffected by noncommutative corrections.

Abstract

We derive the geodesic equation for point particles propagating in Moyal-type noncommutative spacetimes using a field-theoretic approach based on the quasi-classical limit of the noncommutative Klein-Gordon equation. Starting from a twisted-geometric construction of the covariant Laplace-Beltrami operator, we obtain the noncommutative Hamilton-Jacobi equation and show that all noncommutative effects are absorbed into an effective, position-dependent mass function $M(x)$ appearing in an otherwise standard relativistic dispersion relation. The corresponding particle dynamics then acquires an additional term in the geodesic equation that takes the form of a fixed external force $F_{\text{NC}}^\mu = -\frac{1}{2} g^{\mu\nu}\partial_\nu M^2(x)$, sourced entirely by the quantum nature of spacetime. We compute this effective mass perturbatively up to fourth order in the noncommutativity parameter for a general metric, proving that all odd-order corrections vanish identically. For the specific case of an $(r-\theta)$ twist applied to spherically symmetric backgrounds, we obtain explicit expressions demonstrating that the leading correction to geodesic motion appears at $\Theta^2$ order and is proportional to the probe particle's mass, while massless particles remain unaffected.