Time Delay and Lensing Maps

Abstract

We shall develop two useful perspectives that lie at the heart of the mathematical theory of gravitational lensing. One viewpoint uses light ray arrival times, that is, time delay functions, and the other employs ray tracing, which involves lensing maps. The goal of this chapter is to precisely define these mappings (including subsidiary concepts like light rays, lensed images, magnification, etc.) for single and multiplane lensing, relate them mathematically, and discuss their relative advantages. In the process, we shall also extend part of earlier work by A.P. [Pet91, Pet96b], where first steps were taken towards expressing the core concepts of weak-field gravitational lensing in precise mathematical form.

Philosophy is written in that great book which ever lies before our gaze — I mean the universe — but we cannot understand if we do not first learn the language and grasp the symbols in which it is written. The book is written in the mathematical language, and the symbols are triangles circles and other geometrical figures, without the help of which it is impossible to conceive a single word of it, and without which one wanders in vain through a dark labyrinth. Galileo Galilei