The possibility of relativistic Finslerian geometry. (English. Russian original) Zbl 1166.53306

Noskov, V. I.

J. Math. Sci., New York 153, No. 6, 799-827 (2008); translation from Sovrem. Mat., Fundam. Napravl. 22, 73-99 (2007).

The paper under review deals with foundations of Finslerian geometry that, in the author’s opinion, are of interest for solving the problem of geometrization of classical electrodynamics in metric four-dimensionality. A relativistic way of parametrization of the interval is suggested, and the corresponding variant of the geometry is constructed. It is shown that the equation for the geodesic of this variant of geometry has a generalized Lorentz term and the connection contains an additional Lorentz tensorial summand. Certain physical consequences of the new geometry are considered. Firstly, the non-measurability of the generalized electromagnetic potential in the classical case and its measurability on quantum scales (the Aharonov-Bohm effect) is discussed. Then 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 also shown that the experimental results could uniquely determine the choice between the standard Riemannian and relativistic Finslerian models of space-time.

Reviewer: Adrian Sandovici (Piatra Neamt)

MSC:

53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)
53B50	Applications of local differential geometry to the sciences

Keywords:

Finslerian parametrization; homogeneous function; Cartan tensor; the geodesic equation; parallel translation; connection; curvature

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