Finsler spaces with mutually opposite preferred directions and their groups of isometries

Abstract: It is shown that in Minkowski space there exist transformations of the coordinates of events alternative to the 3-parameter Lorentz boosts. However, in contrast to the boosts, they constitute a 3-parameter noncompact group which, in turn, is a subgroup of the homogeneous 6-parameter Lorentz group. Moreover, in the same space, there exists another 3-parameter noncompact group isomorphic to above-mentioned one. As we shall see, these two 3-parameter noncompact groups are rudiments of the 3-parameter groups of relativistic symmetry of the axially symmetric Finslerian spaces with the preferred directions $\nu$ and $-\nu$, respectively. Finally, it will be also demonstrated that inversion of the preferred direction $\nu$ in the axially symmetric Finslerian space-time does not change the Lobachevski geometry of 3-velocity space. However, this leads to an inversion of the corresponding family of horospheres of the space.

Classification (MSC2000): 53C60; 53C80, 83A05, 81T13, 81R40

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