The Notions of Distance and Velocity Modulus in the Some Linear Finsler Spaces

TL;DR

This paper derives formulas for distance and velocity modulus in 4D linear Finsler spaces with Berwald-Moor metrics, extending concepts from special relativity and providing new transformation relations in such geometries.

Contribution

It introduces new formulas for distance and velocity in Berwald-Moor Finsler spaces and derives velocity transformation laws analogous to Lorentz transformations, extending relativistic concepts.

Findings

Velocity modulus matches Galilean space at low speeds

Velocity modulus equals unity at maximal velocities

Derived velocity transformation laws generalize Lorentz transformations

Abstract

The formulas for the 3-dimensional distance and the velocity modulus in the 4-dimensional linear space with the Berwald-Moor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary poly-linear Finsler space in which the time-like component could be chosen. The constructed here modulus of the 3-dimensional velocity in the space with the Berwald-Moor metrics coincides with the corresponding expression in the Galilean space at small (non-relativistic) velocities, while at maximal velocities, i.e. for the world lines lying on the surface of the cone of future, this modulus is equal to unity. To construct the 3-dimensional distance, the notion of the surface of the relative simultaneity is used which is analogous to the corresponding speculations in special relativity. The formulas for the velocity transformation when one pass from one inertial frame to another are obtained. In case when both velocities are directed along one of the three selected straight lines, the obtained relations coincide with the analogous relations of special relativity, but they differ in other cases. Besides, the expressions for the transformations that play the same role as Lorentz transformations in the Minkowski space are obtained. It was found that if the three space coordinate axis are straight lines along which the velocities are added as in special relativity, then taking the velocity of the new inertial frame collinear to the one of these coordinate axis, one can see that the transformation of this coordinate and time coordinate coincide with Lorentz transformations, while the transformations of the two transversal coordinates differ from the corresponding Lorentz transformations.