Fermat Potentials for Non-Perturbative Gravitational Lensing

TL;DR

This paper introduces a novel variational principle based on null surface envelopes to derive gravitational lens equations in various spacetimes, providing a unified and geometric approach to lensing phenomena.

Contribution

It presents an alternative formulation of Fermat's principle using null surface envelopes, applicable to generic spacetimes and including moving lenses and cosmological models.

Findings

Derived lens equations for static asymptotically flat spacetimes.

Expressed bending angles for moving lenses in terms of static case.

Extended the approach to cosmological FRW spacetimes.

Abstract

The images of many distant galaxies are displaced, distorted and often multiplied by the presence of foreground massive galaxies near the line of sight; the foreground galaxies act as gravitational lenses. Commonly, the lens equation, which relates the placement and distortion of the images to the real source position in the thin-lens scenario, is obtained by extremizing the time of arrival among all the null paths from the source to the observer (Fermat's principle). We show that the construction of envelopes of certain families of null surfaces consitutes an alternative variational principle or version of Fermat's principle that leads naturally to a lens equation in a generic spacetime with any given metric. We illustrate the construction by deriving the lens equation for static asymptotically flat thin lens spacetimes. As an application of the approach, we find the bending angle for moving thin lenses in terms of the bending angle for the same deflector at rest. Finally we apply this construction to cosmological spacetimes (FRW) by using the fact they are all conformally related to Minkowski space.