Geodesic Equation in Noncommutative Space: A Field Theory Perspective

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 FNCμ=−12gμν∂νM2(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−θ) twist applied to spherically symmetric backgrounds, we obtain explicit expressions demonstrating that the leading correction to geodesic motion appears at order Θ2 and is proportional to the probe particle’s mass, while massless particles remain unaffected.