Deflection of light rays by spherically symmetric black hole in dispersive medium

TL;DR

This paper generalizes the gravitational deflection of light to arbitrary dispersive media around spherically symmetric black holes, deriving formulas for deflection angles, circular orbits, and black hole shadows, extending previous plasma-focused studies.

Contribution

It introduces a generalized framework for light deflection in arbitrary dispersive media around black holes, beyond the cold plasma case, including new formulas and analysis.

Findings

Refractive index scaling does not affect deflection or orbit radius.

Dispersion alters light trajectories even in homogeneous media.

Calculated black hole shadow modifications due to dispersive media.

Abstract

The influence of the medium on the gravitational deflection of light rays is widely discussed in literature for the simplest non-trivial case: cold non-magnetized plasma. In this article, we generalize these studies to the case of an arbitrary transparent dispersive medium with a given refractive index. We calculate the deflection angle of light ray moving in a general spherically symmetric metric in the presence of medium with the spherically symmetric refractive index. The equation for the radius of circular light orbits is also derived. We discuss in detail the properties of these results and various special cases. In particular, we show that multiplying the refractive index by a constant does not affect the deflection angle and radius of circular orbits. At the same time, the presence of dispersion makes the trajectories different from the case of vacuum even in spatially homogeneous medium. As one of the applications of our results, we calculate the correction to the angle of vacuum gravitational deflection for the case when a massive object is surrounded by homogeneous but dispersive medium. As another application, we present the calculation of the shadow of a black hole surrounded by medium with arbitrary refractive index. Our results can serve as a basis for studies of various plasma models beyond the cold plasma case.