The theory of transparent gravitational lenses
Abstract
The theory of transparent gravitational lenses is presented in this paper. All calculations are done within the framework of the linearized Einstein gravitation theory, the geometrical optics approximation, and the small-angle scattering approximation. We present a general expression for the bending angle of a light ray that goes around or through a bounded mass. This expression is used to calculate the locations, intensifications, distortions, orientations, and velocities of image seen around or through such masses. The above formulae are evaluated for deflectors with spheroidal volume mass densities and deflectors with elliptical surface mass densities. A simple volume mass density proportional to 1/a is used to illustrate the multiple imaging capabilities of a spheroidal deflector.