The optical-mechanical analogy for stationary metrics in general relativity

Abstract

The optical-mechanical analogy allows for a common description of the motion of particles in mechanics and for light in geometrical optics. In a recent series of articles in this journal, it has been shown that the optical-mechanical analogy can be extended to general relativity for the case of static metrics expressible in isotropic coordinates. In this paper, we extend the optical-mechanical analogy in general relativity to the case of stationary metrics. A variational principle for the trajectories of both photons and particles is derived which takes the form of Fermat’s principle or the principle of Maupertuis. When the (suitably defined) spatial portion of the metric is written as or restricted to an isotropic form, exact equations of motion for both massless and massive particles are obtained in the form of Newtonian mechanics, describing objects moving in a medium with a spatially varying index of refraction. Such restrictions of the metric commonly occur, for example, when orbital motion is considered in a plane perpendicular to the axis of rotation of a rotating black hole or spacetime. The Newtonian form of the equations of geodesic motion are illustrated by applications to a uniformly rotating reference system and a rotating black hole (the Kerr metric).