A Gravitational Lens need not produce an Odd Number of Images

TL;DR

This paper discusses the mathematical properties of gravitational lensing, challenging the common assumption that such lenses produce an odd number of images, and suggests that this may not always be the case.

Contribution

It introduces a new perspective on gravitational lensing by analyzing the mapping of null geodesics and questions the general applicability of the odd image theorem.

Findings

The odd image theorem may not hold universally in gravitational lensing.

The condition for an odd number of images is likely too restrictive in general.

Calculating specific mappings can provide examples or counterexamples to the theorem.

Abstract

Given any space-time $M$ without singularities and any event $O$, there is a natural continuous mapping $f$ of a two dimensional sphere into any space-like slice $T$ not containing $O$. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere $S^2$ and the map $f$ sends a point in $S^2$ to the point in $T$ which is the intersection of the corresponding geodesic with $T$. To require that $f$, which maps a two dimensional space into a three dimensional space, satisfy the condition that any point in the image of $f$ has an odd number of preimages, is to place a very strong condition on $f$. This is exactly what happens in any case where the odd image theorem holds for a transparent gravitational lens. It is argued here that this condition on $f$ is probably too restrictive to occur in general; and if it appears to hold in a specific example, then some $f$ should be calculated either analytically or numerically to provide either an illustrative example or counterexample.