Higher order corrections to deflection angle of massive particles and light rays in plasma media for stationary spacetimes using the Gauss-Bonnet theorem

The purpose of this article is twofold. First, we extend the results presented in [Gabriel Crisnejo and Emanuel Gallo, Phys.Rev.D 97, 124016 (2018)] to stationary spacetimes. Specifically, we show that the Gauss-Bonnet theorem can be applied to describe the deflection angle of light rays in plasma media in stationary spacetimes. Second, by using a correspondence between the motion of light rays in a cold non magnetized plasma and relativistic test massive particles we show that this technique is not only powerful to obtain the leading order behavior of the deflection angle of massive/massless particles in the weak field regime but also to obtain higher order corrections. We particularize it to a Kerr background where we compute the deflection angle for test massive particles and light rays propagating in a non homogeneous cold plasma by including third order corrections in the mass and spin parameters of the black hole.