The general relativity dynamics in the Eisenhart geometry

It has been proposed a long time ago by Wheeler and deWitt to look at the evolution of three-metrics as a geodesic flow on the superspace. Since then a lot of attention has been paid towards better understanding the geometric structure of the superspace. In particular it has been appreciated that the minisuperspace can in a natural way be equipped with the Jacobi metric. However the Jacobi metric is degenerated on certain codimension one hypersurfaces (boundary sets) leading to severe difficulties. In this contribution we propose to use the so-called Eisenhartʹs principle as an alternative geometrical construction on minisuperspace. Then the dynamics of general relativity, represented by a constrain Hamiltonian system, is mapped onto a geodesic flow on a smooth manifold without boundary. Hence Eisenhartʹs proposal seems to be the right way to desingularization of motion in Jacobi metric (e.g. the dynamics of homogeneous cosmological models near the initial singularity) which is nontractable in the Jacobi picture. Different methods of desingularizing of the Jacobi metric through the isometric embedding into a flat space with the Lorentzian signature will also be presented. The extension of the Fermat principle for the case of timelike trajectories is also given.