# Light propagation in a plasma on Kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow

## TL;DR

This paper analyzes how light propagates in a plasma around a Kerr black hole, deriving conditions for separability of the Hamilton-Jacobi equation, determining photon regions, and calculating the black hole's shadow analytically.

## Contribution

It establishes the necessary and sufficient conditions for plasma density to allow Hamilton-Jacobi equation separation in Kerr spacetime and derives an analytical formula for the black hole shadow boundary.

## Key findings

- Separable plasma density conditions are identified.
- Photon regions with spherical light rays are characterized.
- An analytical formula for the black hole shadow boundary is derived.

## Abstract

We consider light propagation in a non-magnetized pressureless plasma around a Kerr black hole. We find the necessary and sufficient condition the plasma electron density has to satisfy to guarantee that the Hamilton-Jacobi equation for the light rays is separable, i.e., that a generalized Carter constant exists. For all cases where this condition is satisfied we determine the photon region, i.e., the region in the spacetime where spherical light rays exist. A spherical light ray is a light ray that stays on a sphere $r = \mathrm{constant}$ (in Boyer-Lindquist coordinates). Based on these results, we calculate the shadow of a Kerr black hole under the influence of a plasma that satisfies the separability condition. More precisely, we derive an analytical formula for the boundary curve of the shadow on the sky of an observer that is located anywhere in the domain of outer communication. Several examples are worked out.