The possibility of relativistic Finslerian geometry

Abstract

Foundations of Finslerian geometry that are of interest for solving the problem of geometrization of classical electrodynamics in metric four-dimensionality are investigated. It is shown that parametrization of the interval—the basic aspect of geometry—is carried out non-relativistically. A relativistic way of parametrization is suggested, and the corresponding variant of the geometry is constructed. The equation for the geodesic of this variant of geometry, aside from the Riemannian, has a generalized Lorentz term, the connection contains an additional Lorentz tensorial summand, and the first schouten is different from zero. Some physical consequences of the new geometry are considered: the non-measurability of the generalized electromagnetic potential in the classical case and its measurability on quantum scales (the Aharonov-Bohm effect); it is shown that in the quantum limit the hypothesis of discreteness of space-time is plausible. The linear effect with respect to the field of the “redshift” is also considered and contemporary experimental possibilities of its registration are estimated; it is shown that the experimental results could uniquely determine the choice between the standard Riemannian and relativistic Finslerian models of space-time.