Distortions of images of Schwarzschild lensing

Abstract: We model the supermassive dark object $M87^*$ as a Schwarzschild lens and study the variations in tangential, radial, and total (the product of tangential and radial) magnifications of images (primary, secondary, and relativistic) against the changes in angular source position and the ratio of lens-source to the observer-source distance. Further, we study the behavior of partial derivatives (with respect to the angular source position) of total magnifications of images against the angular source position. Finally, we model supermassive dark objects at centers of 40 galaxies as Schwarzschild lenses and study the variations in tangential, radial, and total magnifications of images against the change in the ratio of mass of the lens to its distance. These studies yield many nonintuitive results which are likely to be significant for next generation Event Horizon Telescope observations. We {\em hypothesize} that there exists a distortion parameter such that their signed sum of all images of singular gravitational lensing of a source identically vanishes. We test this with images of Schwarzschild lensing in weak and strong gravitational fields and find that this esthetically appealing hypothesis succeeds with flying colors.