  In most analytical studies of light ray propagation in curved spacetimes
around a gravitating object surrounded by a medium, it is assumed that the
medium is a cold nonmagnetized plasma. The distinctive feature of this
environment is that the equations of motion of the rays are independent of the
plasma velocity, which, however, is not the case in other media. In this paper,
we consider the deflection of light rays propagating near a spherically
symmetric gravitating object in a moving dispersive medium given by a general
refractive index. The deflection is studied when the motion of the medium is
defined either as a radially falling onto a gravitating object (e.g., black
hole), or rotating in the equatorial plane. For both cases the deflection
angles are obtained. These examples demonstrate that fully analytic expressions
can be obtained if the Hamiltonian for the rays takes a rather general form as
a polynomial in a given momentum component. The general expressions are further
applied to three specific choices of refractive indices and these cases are
compared. Furthermore, the light rays propagating around a gravitating object
surrounded by a generally moving medium are further studied as a small
perturbation of the cold plasma model. The deflection angle formula is hence
expressed as a sum of zeroth and first order components, where the zeroth order
term corresponds to the known cold plasma case and the first order correction
can be interpreted as caused by small difference in the refractive index
compared to the cold plasma. The results presented in this paper allow to
describe the effects caused by the motion of a medium and thus go beyond cold
nonmagnetized plasma model.
