Generalized Fermat's principle and action for light rays in a curved spacetime

TL;DR

This paper formulates a generalized Fermat's principle for light in curved spacetime using optimal control theory, deriving an effective Hamiltonian that reproduces null geodesic equations.

Contribution

It introduces a novel Hamiltonian formulation for light rays in curved spacetime based on Pontryagin's minimum principle.

Findings

Derived an effective Hamiltonian for null geodesics.

Demonstrated the Hamiltonian reproduces null geodesic equations.

Discussed alternative forms of the light ray action.

Abstract

We start with formulation of the generalized Fermat's principle for light propagation in a curved spacetime. We apply Pontryagin's minimum principle of the optimal control theory and obtain an effective Hamiltonian for null geodesics in a curved spacetime. We explicitly demonstrate that dynamical equations for this Hamiltonian correctly reproduce null geodesic equations. Other forms of the action for light rays in a curved spacetime are also discussed.