On Fermat's principle for causal curves in time oriented Finsler spacetimes

TL;DR

This paper extends Fermat's principle to causal curves in time-oriented Finsler spacetimes, deriving second variations, analyzing critical points, and establishing a Morse index theorem in this geometric context.

Contribution

It introduces a Fermat's principle for causal curves in Finsler spacetimes, computes second variations, and proves a Morse index theorem, advancing geometric analysis in Finsler spacetime theory.

Findings

Fermat's principle holds for causal curves in Finsler spacetimes.

Second variation of the time arrival functional is explicitly calculated.

A Morse index theorem is established for Finsler spacetime geodesics.

Abstract

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.