Analytical solution to geodesic equation in non-commutative gauge theory of gravity

In this paper, we employ the non-commutative (NC) gauge theory of gravity to construct the deformed metric $\hat{g}_{\mu\nu}(r,\Theta)$ corresponding to the Schwarzschild black hole. This construction is performed using the Seiberg–Witten map and the Moyal–Weyl star product, which systematically incorporate the effects of non-commutativity through the parameter $\Theta$. The resulting deformed metric provides a geometric framework to study the geodesic of a massive test particle in the presence of NC gravitational corrections. We begin by deriving the correction to the effective potential of the massive test particle, considering terms up to the second order in the NC parameter $\Theta$. Subsequently, we obtain the geodesic equation for the massive test particle with second order in NC corrections. An analytical solution to this modified geodesic equation is presented using an approximation method suitable for weak-field and slow-motion limits. In addition, we plot several representative orbital trajectories of the massive test particle around the NC-deformed Schwarzschild black hole and analyze their physical behavior. All our results consistently reduce to the well-known commutative case in the limit $\Theta \rightarrow 0$, thereby preserving the coherence and consistency of the theory. Furthermore, we demonstrate that NC effects manifest only at the perihelion of the test particle's orbital motion.