The Possibility of Relativistic Finslerian Geometry

V. I. Noskov

Contemporary Mathematics. Fundamental Directions, Geometry, Tome 22 (2007), pp. 73-99

Résumé

Foundations of Finslerian geometry are investigated that are of interest for solving the problem of geometrization of classical electrodynamics in metric four-dimensionality. It is shown that parametrization of the interval — the basic aspect of geometry — is carried out non-relativistically. Relativistic way of parametrization is suggested, and the corresponding variant of the geometry is constructed. The equation for geodesic of this variant of geometry, aside from the Riemannian, has a generalized Lorentz term, connection contains an additional Lorentz tensorial summand, the first schouten is different from zero. Some physical consequences of the new geometry are considered: 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 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.