On the formulations of the Fermat principle in general relativity and beyond

TL;DR

This paper surveys the Fermat principle in general relativity, discussing its mathematical formulation, challenges, and extensions to various spacetime settings and geometries.

Contribution

It provides a comprehensive overview of the Fermat principle's evolution, addresses variational difficulties, and explores extensions beyond classical formulations.

Findings

The space of lightlike curves lacks a smooth manifold structure due to the null cone.

The quadratic arrival time functional helps establish multiplicity results for light rays.

Extensions include applications to extended sources, arbitrary arrival curves, and Finsler spacetimes.

Abstract

This paper presents a survey of the Fermat principle within the framework of general relativity, tracing its evolution from classical optics to its modern variational formulation in Lorentzian geometry. In particular, we provide its proof in the framework of smooth lightlike curves. We also analyze the mathematical difficulties inherent in the relativistic setting, specifically demonstrating that the space of lightlike curves in the Sobolev topology does not admit a smooth manifold structure due to the cone nature of the null condition. To address these variational obstacles, we discuss alternative frameworks highlighting the role of the quadratic arrival time functional in establishing multiplicity results for light rays. Furthermore, we explore significant extensions of the principle, such as its application to extended sources and receivers, arbitrary arrival curves, timelike geodesics with prescribed proper time, Finsler spacetimes, or settings with a non-continuous interface giving rise to a Snell law.