Can A Gravitational Quadruple Lens Produce 17 images?

Abstract

Gravitational lensing can be by a faint star, a trillion stars of a galaxy, or a cluster of galaxies, and this poses a familiar struggle between particle method and mean field method. In a bottom-up approach, a puzzle has been laid on whether a quadruple lens can produce 17 images. The number of images is governed by the gravitational lens equation, and the equation for $n$-tuple lenses suggests that the maximum number of images of a point source potentially increases as $n^2+1$. Indeed, the classes of $n=1, 2, 3$ lenses produce up to $n^2+1 = 2, 5, 10$ images. We discuss the $n$-point lens system as a two-dimensional harmonic flow of an inviscid fluid, count the caustics topologically, recognize the significance of the limit points and discuss the notion of image domains. We conjecture that the total number of positive images is bounded by the number of finite limit points $2(n-1): n>1$ (1 limit point at $\infty$ if $n=1$). A corollary is that the total number of images of a point source produced by an $n$-tuple lens can not exceed $5(n-1):n>1$. We construct quadruple lenses with distinct finite limit points that can produce up to 15 images and argue why there can not be more than 15 images. We show that the maximum number of images is bounded from below by $3(n+1): n \ge 3$. We also comment on "thick Einstein rings" that can have one or more holes.