Light deflection in axially symmetric stationary spacetimes filled with a moving medium

The deflection of light rays near gravitating objects can be influenced not only by gravity itself but also by the surrounding medium. Analytical studies of such effects are possible within the geometrical optics approximation, where the medium introduces additional light bending due to refraction. These studies typically assume a cold nonmagnetized plasma, for which light propagation is independent of the medium's velocity. In this paper, we extend the analysis to the general case of dispersive refractive media in motion and study its influence on light deflection. We consider an axially symmetric stationary spacetime filled with a moving medium, motivated by the interplay between rotational effects originating from the spacetime and those induced by the medium's motion. We begin by analyzing light deflection in the equatorial plane of a rotating object in the presence of a radially moving and rotating medium. Assuming a specific form of the refractive index enables a fully analytic treatment. In the particular cases of either pure radial or pure rotational motion, we obtain explicit expressions for the deflection angle. Next, we analyze the case of a slowly moving medium and identify two particularly interesting results. First, we show that to the first order in the medium's velocity, the radial motion does not affect the light deflection. Second, assuming slow rotation of the gravitating object, we demonstrate that the black hole rotation and the medium motion can produce equivalent observational signatures. We find the quantitative condition under which these effects compensate each other. This relation becomes particularly clear for a Kerr black hole, discussed as an example.