Global properties of gravitational lens maps in a Lorentzian manifold setting

TL;DR

This paper uses differential topology to analyze the global properties of gravitational lens maps in Lorentzian manifolds, providing insights into image counting and illustrating with various spacetime models.

Contribution

It introduces a topological framework for understanding gravitational lens maps in general-relativistic spacetimes, extending previous quasi-Newtonian approaches.

Findings

Mapping degree helps determine the number of images.

Global properties vary with spacetime geometry.

Illustrations include static strings and cosmological models.

Abstract

In a general-relativistic spacetime (Lorentzian manifold), gravitational lensing can be characterized by a lens map, in analogy to the lens map of the quasi-Newtonian approximation formalism. The lens map is defined on the celestial sphere of the observer (or on part of it) and it takes values in a two-dimensional manifold representing a two-parameter family of worldlines. In this article we use methods from differential topology to characterize global properties of the lens map. Among other things, we use the mapping degree (also known as Brouwer degree) of the lens map as a tool for characterizing the number of images in gravitational lensing situations. Finally, we illustrate the general results with gravitational lensing (a) by a static string, (b) by a spherically symmetric body, (c) in asymptotically simple and empty spacetimes, and (d) in weakly perturbed Robertson-Walker spacetimes.