Gravitational lensing by Elliptical Galaxies, and the Schwarz Function

TL;DR

This paper explores gravitational lensing effects of elliptical galaxies with specific mass distributions, revealing limits on the number of images and the shapes of Einstein rings using advanced mathematical techniques.

Contribution

It introduces a novel application of quadrature domain theory and the Schwarz function to analyze lensing phenomena in elliptical galaxies, establishing new bounds and geometric constraints.

Findings

Maximum of 5 images for certain mass distributions

Einstein rings are either circles or ellipses

Mathematical techniques provide new insights into lensing geometry

Abstract

We discuss gravitational lensing by elliptical galaxies with some particular mass distributions. Using simple techniques from the theory of quadrature domains and the Schwarz function (cf. \cite{Sh}) we show that when the mass density is constant on confocal ellipses, the total number of lensed images of a point source cannot exceed 5 (4 bright images and 1 dim image). Also, using the Dive--Nikliborc converse of the celebrated Newton's theorem concerning the potentials of ellipsoids, we show that ``Einstein rings'' must always be either circles (in the absence of a tidal shear), or ellipses.