modified geometrical optics in a curved spacetime

In this article, the propagation of high-frequency monochromatic beam of circularly polarized electromagnetic waves in a curved spacetime has been studied. at first, the standard geometrical optics is investigated; that is, we consider the waves with infinitely high frequency. in this step, it is found that the trajectories of light are null geodesics. secondly, the geometrical optics is modified. for this end, by considering the polarization of light, helicity-dependent correction on the geometrical optics is included. as a result, we realize that the modified wave vector is null. furthermore, the trajectories of light are null non-geodesic paths.