Necessary Conditions for the Existence of the Carter Constant in a General Axially Symmetric Stationary Spacetime in Plasma and its Applications

Abstract

When dealing with light rays propagating around compact relativistic objects (e.g. a black hole), usually a vacuum background is assumed. However, in many astrophysical applications this assumption is not sufficient.

More precise picture is obtained when a medium (plasma) surrounding the black hole is considered. The problem is typically solved in terms of the separation of the Hamilton-Jacobi equation.

To be able to find a solution of such system, a Carter constant has to exist. We derive general conditions for both axially symmetric stationary metric and plasma density function which are required in order to define the Carter constant.

Furthermore, we gain general formulas for photon region and black hole shadow for light rays around an axially symmetric stationary object in plasma. Obtained general expressions are then applied to find the boundary of photon region and shadow for particular examples, such as Kerr metric or Teo metric.