Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

In this paper, we return to the subject of Jacobi metrics for timelike and null geodesics in stationary spacetimes, correcting some previous misconceptions. We show that not only null geodesics but also timelike geodesics are governed by a Jacobi-Maupertuis type variational principle and a Randers-Finsler metric for which we give explicit formulas. The cases of the Taub-NUT and Kerr spacetimes are discussed in detail. Finally, we show how our Jacobi-Maupertuis Randers-Finsler metric may be expressed in terms of the effective medium describing the behavior of Maxwell's equations in the curved spacetime. In particular, we see in very concrete terms how the gravitational electric permittivity, magnetic permeability, and magnetoelectric susceptibility enter the Jacobi-Maupertuis Randers-Finsler function.