Counting Lensed Images: Single-Plane Case

Abstract

A striking feature of gravitational lensing is the formation of multiple images of a background light source. Within our mathematical framework, lensed images of a light source at position y are solutions x of the lens equation (see (6.30) on page 200 and Section 3.2.3, page 77):

$$ y = x - {\alpha _1} \circ {\eta _1}\left( x \right) - \cdots - {\alpha _k} \circ {\eta _k}\left( x \right) $$

(11.1)

where α_i is the bending angle vector field on the ith lens plane and η_i the ith partial lensing map. For a gravitational lens consisting of a single point mass (e.g., star or black hole), there are exactly two lensed images produced of a light source not on the line of sight (Section 6.3.3, page 187). This result was known to Einstein as early as 1912 (before he completed the general theory of relativity); see page 6. For two point masses on the same lens plane, equation (11.1) can be solved directly (after a lot of tedious algebra) to show that there are either three or five lensed images for sources not on caustics (see Schneider and Weiss [Schn-W86]). If there are three or more point masses on one or several lens planes, it is practically not feasible to determine analytically the solutions of (11.1). In such cases, numerical methods are employed with simple gravitational lens potentials — a task that is also nontrivial and has the difficulty of knowing whether all lensed images have been found. We are then naturally led to the Lensed-Image Counting Problem: For a generic gravitational lens system, determine the number of lensed images of a light source not on a caustic.

“The light coming from a star A traverses the gravitational field of another star B,…. the observer will see A as two point-like light-sources, which are deviated from the true geometrical position of A …. there is no great chance of observing this phenomenon,….” Albert Einstein, 1936

“0957+561 A, B: twin quasistellar objects or gravitational lens?…. Their spectra leave little doubt that they are associated. Difficulties arise in describing them as two distinct objects and the possibility that they are two images of the same object formed by a gravitational lens is discussed.” Walsh, Carswell, and Weymann, 1979

“We believe that we have observed the gravitational lens that is responsible for producing the double quasar…. We conclude that the double quasar is almost certainly the multiple image of a single object produced by a gravitational lens.” Young, Gunn, Kristian, Oke, and Westphal, 1980