Geodesic equation in $\kappa$-Minkowski spacetime

In this paper, we derive corrections to the geodesic equation due to the $\kappa$-deformation of curved spacetime, up to the first order in the deformation parameter $a$. This is done by generalizing the method from our previous paper [Phys. Rev. D 84, 085020 (2011)] to include curvature effects. We show that the effect of $\kappa$-noncommutativity can be interpreted as an extra drag that acts on the particle while moving in this $\kappa$-deformed curved spacetime. We have derived the Newtonian limit of the geodesic equation and using this, we discuss possible bounds on the deformation parameter. We also derive the generalized uncertainty relations valid in the nonrelativistic limit of the $\kappa$-spacetime.