ON FERMAT PRINCIPLE IN GENERAL-RELATIVITY .2. THE CONFORMALLY STATIONARY CASE

On a conformally stationary spacetime, Fermat's principle, and hence the equations of motion of light rays, can be formulated in three-dimensional (purely spatial) terms. This reduction from four-dimensional spacetime to three-dimensional space is very similar to the well known reduction formalism of Kaluza-Klein theory. The resulting equation of motion of light rays in 3-space is formally identical with the Lorentz force equation for charged particles of unit specific charge and fixed energy in a magnetostatic field on a Riemannian 3-manifold. This analogy, in which the magnetostatic field corresponds to the rotation of the timelike conformal Killing vector field, has the following consequence. To every magnetostatic electron lens there can be constructed mathematically an analogous gravitational lens.