A variational principle for time of arrival of null geodesics

TL;DR

This paper develops a variational principle to determine which null geodesics from an extended source in a gravitational field arrive first at an observer, extending Fermat's principle to curved spacetime.

Contribution

It introduces a novel variational approach to identify earliest arriving light rays in curved spacetime, considering extended sources and gravitational effects.

Findings

Normal rays from the source arrive first at the observer.

Two proofs confirm the variational principle's validity in gravitational fields.

The approach extends classical optics principles to general relativity.

Abstract

Normally the issue or question of the time of arrival of light rays at an observer coming from a given source is associated with Fermat's Principle of Least Time which yields paths of extremal time. We here investigate a related but different problem. We consider an observer receiving light from an extended source that has propagated in an arbitrary gravitational field. It is assumed from the start that the propagation is along null geodesics. Each point of the extended source is sending out a light-cones worth of null rays and the question arises which null rays from the source arrive first at the observer. Stated in an a different fashion, a pulse of light comes from the source with a wave-front as the leading edge, which rays are associated with that leading edge. In vacuum flat-space we have, from Huygen's principle, that the rays normal to the source constitute the leading edge and hence arrive first at an observer. We here investigate this issue in the presence of a gravitational field. Though it is not obvious, since the rays bend and are focused by the gravitational field and could even cross, in fact it is the normal rays that arrive earliest. We give two proofs both involving the extemization of the time of arrival, one based on an idea of Schrodinger for the derivation of gravitational frequency shifts and the other based on V.I. Arnold's theory of generating families.