The notions of distance and velocity modulus in the linear Finsler spaces 2005jaz  Garas'ko G. I., Pavlov D. G.
The formulas for the 3dimensional distance and the velocity modulus in the
4dimensional linear space with the BerwaldMoor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary polylinear Finsler space in which the timelike component could be chosen. The constructed here modulus of the 3dimensional velocity in the space with the BerwaldMoor metrics coincides with the corresponding expression in the Galilean space at small (nonrelativistic) 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 3dimensional 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.
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