Abstract
Along with the construction of non-Lorentz-invariant effective field theories, recent studies which are based on geometric models of Finsler space-time become more and more popular. In this respect, the Finslerian approach to the problem of Lorentz symmetry violation is characterized by the fact that the violation of Lorentz symmetry is not accompanied by a violation of relativistic symmetry. That means, in particular, that preservation of relativistic symmetry can be considered as a rigorous criterion of the viability for any non-Lorentz-invariant effective field theory. Although this paper has a review character, it contains (with few exceptions) only those results on Finsler extensions of relativity theory, that were obtained by the authors.
Highlights
Nowadays, the program of geometrization and algebraization of the fundamental laws of nature which was formulated at the early stage of GR development is still not fulfilled
The Finslerian approach to the problem of Lorentz symmetry violation is characterized by the fact that the violation of Lorentz symmetry is not accompanied by a violation of relativistic symmetry
Despite the abstract character of the manifolds studied in modern physics and mathematics and of a lot of additional structures which geometrically describe the laws of nature, some of these structures still remain rather conservative
Summary
The program of geometrization and algebraization of the fundamental laws of nature which was formulated at the early stage of GR development is still not fulfilled. As for the possibility of spontaneous emergence of the complete local anisotropy of space-time with the Abelian homogeneous group of local relativistic symmetry and the corresponding generalized Finslerian Berwald-Moor metric, the answer to this question will depend on the threeparticle correlation function, whose measurement is already planned by the CMS collaboration. Much more meaningful from the physical point of view is the special relativistic theory of the locally anisotropic space-time [5,6,7], based on Finsler metric (1), which describes a family of flat relativistically invariant spaces of events with partially broken 3D isotropy, and with broken Lorentz symmetry. The corresponding symmetry is more frequently called DISIMb(2) symmetry (where b is the new designation of the parameter r), and the theory itself [5,6,7] is more frequently called General Very Special Relativity (GVSR)
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