  In arXiv:1105.5629, equations of the modified geometrical optics for
circularly polarized photon trajectories in a stationary spacetime are derived
by using a (1+3)-decomposed form of Maxwell's equations. We derive the same
results by using a four-dimensional covariant description. In our procedure,
the null nature of the modified photon trajectory naturally appears and the
energy flux is apparently null. We find that, in contrast to the standard
geometrical optics, the inner product of the stationary Killing vector and the
tangent null vector to the modified photon trajectory is no longer a conserved
quantity along light paths. This quantity is furthermore different for left and
right handed photon. A similar analysis is performed for gravitational waves
and an additional factor of 2 appears in the modification due to the spin-2
nature of gravitational waves.
