The general relativity dynamics in the Eisenhart geometry
The Eisenhart's principle is used as an alternative geometrical construction on minisuperspace. It is shown that this principle maps the trajectories of Hamiltonian systems into geodesic on certain extended configuration space. The Eisenhart geometry appears as the right way to desingularization of motion in the Jacobi metric. The author presents different methods of desingularization of the Jacobi metric through the isometric embedding into a flat space with the Lorentzian signature. The extension of the Fermat principle to the case of timelike trajectories is also studied. A theorem, which establishes one-to-one correspondence between trajectories of simple definite systems and trajectories of fictitious particles in spacetime with Eisenhart geometry, is proven. It is concluded that the geometrization according to Eisenhart's principle enables to build geometrical models of dynamics in different spaces: Euclidean, Riemannian, Weyl affine, Einstein-Cartan with torsion and Finsler spaces.
scientific article; zbMATH DE number 1486538
scientific article