[Jarvamundo (Alex)]

I fail to comprehend how the content does not already address your post.

I acknowledge the differences in gradients as you have described, but that is exactly what is derived here?

Researchers utilizing this tool seek an alternative simplification for the application of the cumbersome light bending rule of General Relative and may or may not be thoroughly familiar with the basic principle of light bending of General Relativity given by the equation

(1) http://www.extinctionshift.com/EquaGRtheta.gif

derived using General Relativity considerations! The geometry as well as the Physics of conservation of energy, the Physics of the minimum energy path, minimum time and the principle of reciprocity must all be considered for any correct interpretation of the astrophysical events.

From the above Figure, "Ray Geometry Technique for Gravitational Lensing", the angle

(2)

is derived. It is readily see that, as the source approaches the vicinity of the gravitational lens, where the adjustable parameter Dds is the distance between the source and the lens and Ds = 26,000 light years, the distance between the observer and the source, the lensing estimated by this equation (2), a modification of (1), approaches a very small value (approaches zero). This suggests incorrectly that the lensing effect described by (2) is minimized for sources in proximity to the gravitating mass, the lens.

It is important to note and recall that the principle of light bending of General Relativity given by Equation (1) and the Ray Geometry Technique for Gravitational Lensing given by Equation (2) are not the same equations, the main subject of the published paper.

The stellar sky presents vast opportunities to modern Astronomy and Astrophysics to allow for the detection of lensing events due to the large numbers of stellar objects that just happen to be positioned in a near perfect line-of-site to the earth based observers; again of course, assuming the validity of the light bending rule of General Relativity. With this in mind the entire celestial sky should be full of Einstein-rings.