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Geometrical Models of the Locally Anisotropic Space-Time ()

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“Transilvania” University of Bra?ov, Bra?ov, Romania.

Institute of Hypercomplex Systems in Geometry and Physics, Fryazino, Russia.

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia.

State University of Civil Aviation, St. Petersburg, Russia.

University Politehnica of Bucharest, Bucharest, Romania.

Institute of Hypercomplex Systems in Geometry and Physics, Fryazino, Russia.

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia.

State University of Civil Aviation, St. Petersburg, Russia.

University Politehnica of Bucharest, Bucharest, Romania.

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.

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Cite this paper

Balan, V. , Bogoslovsky, G. , Kokarev, S. , Pavlov, D. , Siparov, S. and Voicu, N. (2012) Geometrical Models of the Locally Anisotropic Space-Time.

*Journal of Modern Physics*,**3**, 1314-1335. doi: 10.4236/jmp.2012.329170.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | H. Rund, “Differential Geometry of Finsler Spaces,” Springer, Berlin, 1959. |

[2] | D. Bao, S. S. Chern and Z. Shen, “An Introduction to Finsler Geometry,” Springer-Verlag, New York, 2000. HUdoi:10.1007/978-1-4612-1268-3U |

[3] | G. I. Garas’ko, “Elements of Finsler Geometry for Physicists,” Tetru, Moscow, 2009. |

[4] | G. Yu. Bogoslovsky, “On a Special Relativistic Theory of Anisotropic Space-Time,” Doklady Akademii Nauk SSSR, Vol. 213, No. 5, 1973, pp. 1055-1058. |

[5] | G. Yu. Bogoslovsky, “A Special Relativistic Theory of the Locally Anisotropic Space-Time, Part I. The Metric and Group of Motions of the Anisotropic Space of Events,” Nuovo Cimento B, Vol. 40, No. 1, 1977, pp. 99-115. HUdoi:10.1007/BF02739183UH |

[6] | G. Yu. Bogoslovsky, “A Special Relativistic Theory of the Locally Anisotropic Space-Time, Part II. Mechanics and Electrodynamics in the Anisotropic Space,” Nuovo Cimento B, Vol. 40, No. 1, 1977, pp. 116-134. HUdoi:10.1007/BF02739184U |

[7] | G. Yu. Bogoslovsky, “Theory of Locally Anisotropic Space-Time,” Moscow State University Press, Moscow, 1992. |

[8] | D. A. Kirzhnits and V. A.Chechin, “Ultra-High Energy Cosmic Rays and Possible Generalization of the Relativistic Theory,” Yadernaya Fizika, Vol. 15, No. 5, 1972, pp. 1051-1059. |

[9] | T. G. Pavlopoulos, “Breakdown of Lorentz Invariance,” Physical Review, Vol. 159, No. 5, 1967, pp. 1106-1110. HUdoi:10.1103/PhysRev.159.1106U |

[10] | S. V. Siparov, “Theory of Zero Order Effect that Can Be Used to Investigate the Space-Time Geometrical Properties,” Hypercomplex Numbers in Geometry and Physics, Vol. 3, No. 2, 2006, pp. 155-173. |

[11] | S. V. Siparov, “Theory of Zero Order Effect Suitable to Investigate the Space-Time Geometrical Properties,” Acta Mathematica APN, Vol. 24, No. 1, 2008, pp. 135-144. |

[12] | S. V. Siparov, “On the Problem of Anisotropy in Geometrodynamics,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 64-74. |

[13] | S. V. Siparov, “Gravitation Law and Source Model in the Anisotropic Geometrodynamics,” Hypercomplex Numbers in Geometry and Physics, Vol. 6, No. 2, 2009, pp. 140-161. |

[14] | S. V. Siparov, “Anisotropic Metric for the Gravitation Theory: New Ways to Interpret the Classical GRT Tests.” In: S. V. Siparov, Ed., BSG Proceedings, Geometry Balkan Press, Bucharest, 2010, pp. 205-218. |

[15] | S. V. Siparov, “Anisotropic Geometrodynamics in Cosmological Problems,” In: S. V. Siparov, Ed., AIP Conference Proceedings, Melville, New York, 2010, pp. 222-231. |

[16] | B. S. DeWitt, “Relativity, Groups and Topology,” Gordon and Breach, New York, 1964. |

[17] | C. Brans and R. H. Dicke, “Mach’s Principle and a Relativistic Theory of Gravitation,” Physical Review, Vol. 124, No. 3, 1961, pp. 925-935.HUdoi:10.1103/PhysRev.124.925U |

[18] | P. D. Mannheim and D. Kazanas, “Newtonian Limit of Conformal Gravity and the Lack of Necessity of the Second Order Poisson Equation,” General Relativity and Gravitation, Vol. 26, No. 4, 1994, pp. 337-345. HUdoi:10.1007/BF02105226U |

[19] | J. W. Moffat, “Nonsymmetric Gravitational Theory,” Physics Review Letters B, Vol. 355, No. 3-4, 1995, pp. 447-452. HUdoi:10.1016/0370-2693(95)00670-GU |

[20] | M. Milgrom, “A Modification of the Newtonian Dynam- ics as a Possible Alternative to the Hidden Mass Hypothesis,” Astrophysical Journal, Vol. 270, 1983, pp. 365- 370. HUdoi:10.1086/161130U |

[21] | J. D. Bekenstein, “Relativistic Gravitation Theory for the Modified Newtonian Dynamics Paradigm,” Physics Review D, Vol. 70, No. 8, 2004, pp. 1-28. |

[22] | J. Lense and H. Thirring, “On the Influence of the Proper Rotation of Central Bodies on the Motions of Planets and Moons According to Einstein’s Theory of Gravitation,” Physikalische Zeitschrift/Physical Journal, Vol. 19, 1918, pp. 156- 163. |

[23] | M. L. Ruggiero and A. Tartaglia, “Gravitomagnetic Effects,” Preprint, 2002. arXiv:gr-qc/0207065v2 |

[24] | H. Muller, S. W. Chiow, S. Herrmann, S. Chu and K. Y. Chung, “Atom Interferometry Tests of the Isotropy of Post-Newtonian Gravity,” Physics Review Letters, Vol. 100, No. 3, 2008, pp. 1-4. |

[25] | The CMS Collaboration, “Observation of Long-Range Near-Side Angular Correlations in Proton-Proton Collisions at the LHC,” JHEP, Vol. 9, 2010, pp. 1-37;. |

[26] | D. Colladay and V. A. Kostelecky, “Lorentz-Violating Extension of the Standard Model,” Physics Review D, Vol. 58, No. 11, 1998, pp. 1-23. |

[27] | Q. G. Bailey and A. Kostelecky, “Signals for Lorentz Violation in Post-Newtonian Gravity,” Physics Review D, Vol. 74, No. 4, 2006, pp. 1-46. |

[28] | V. A. Kostelecky, “CPT and Lorentz Symmetry,” World Scientific, Singapore, 1999. |

[29] | V. A. Kostelecky, “CPT and Lorentz Symmetry II,” World Scientific, Singapore, 2002. |

[30] | V. A. Kostelecky, “CPT and Lorentz Symmetry III,” World Scientific, Singapore, 2005. |

[31] | V. A. Kostelecky, “CPT and Lorentz Symmetry IV,” World Scientific, Singapore, 2008. |

[32] | V. A. Kostelecky, “CPT and Lorentz Symmetry V,” World Scientific, Singapore, 2011. |

[33] | D. Blas and S. Sibiryakov, “Technically Natural Dark Energy from Lorentz Breaking,” Preprint, 2011. arXiv:1104.3579v1 [hep-th] |

[34] | G. Yu. Bogoslovsky and H. F. Goenner, “Finslerian Spaces Possessing Local Relativistic Symmetry,” General Rela- tivity and Gravitation, Vol. 31, No. 10, 1999, pp. 1565-1603. HUdoi:10.1023/A:1026786505326U |

[35] | J. Patera, P. Winternitz and H. Zassenhaus, “Continuous Subgroups of the Fundamental Groups of Physics. II. The Similitude Group,” Journal of Mathematical Physics, Vol. 16, No. 8, 1975, pp. 1615-1624. HUdoi:10.1063/1.522730U |

[36] | P. Winternitz and I. Fris, “Invariant Expansions of Relativistic Amplitudes and Subgroups of the Proper Lorentz Group,” Yadernaya Fizika, Vol. 1, No. 5, 1965, pp. 889- 901. |

[37] | G. Yu. Bogoslovsky, “Lorentz Symmetry Violation without Violation of Relativistic Symmetry,” Physics Letters A, Vol. 350, No. 1-2, 2006, pp. 5-10. HUdoi:10.1016/j.physleta.2005.11.007U |

[38] | A. G. Cohen and S. L. Glashow, “Very Special Relativity,” Physics Review Letters, Vol. 97, No. 2, 2006, pp. 1- 3. |

[39] | A. G. Cohen and S. L. Glashow, “A Lorentz-Violating Origin of Neutrino Mass?” Preprint, 2006. arXiv:hep-ph/0605036v1. |

[40] | G. W. Gibbons, J. Gomis and C. N. Pope, “General Very Special Relativity is Finsler Geometry,” Physics Review D, Vol. 76, No. 8, 2007, pp. 1-5. |

[41] | G. W. Gibbons, J. Gomis and C. N. Pope, “Deforming the Maxwell-Sim Algebra,” Physics Review D, Vol. 82, No. 6, 2010, pp. 1-15. |

[42] | G. Yu. Bogoslovsky and H. F. Goenner, “Concerning the Generalized Lorentz Symmetry and the Generalization of the Dirac Equation,” Physics Letters A, Vol. 323, No. 1-2, 2004, pp. 40-47. HUdoi:10.1016/j.physleta.2004.01.040U |

[43] | G. Yu. Bogoslovsky, “Some Physical Displays of the Space Anisotropy Relevant to the Feasibility of Its Being Detected at a Laboratory,” Preprint, 2007. arXiv:0706.2621v1 [gr-qc] |

[44] | G. Yu. Bogoslovsky, “Finsler Model of Space-Time,” Physics of Particles and Nuclei, Vol. 24, No. 3, 1993, pp. 354-379. |

[45] | G. Yu. Bogoslovsky, “A Viable Model of Locally Anisotropic Space-Time and the Finslerian Generalization of the Relativity Theory,” Fortschritte der Physik/Progress of Physics, Vol. 42, No. 2, 1994, pp. 143-193. HUdoi:10.1002/prop.2190420203U |

[46] | G. Yu. Bogoslovsky and H. F. Goenner, “On the Possibility of Phase Transitions in the Geometric Structure of Space-Time,” Physics Letters A, Vol. 244, 1998, pp. 222-228. |

[47] | G. Yu. Bogoslovsky, “Dynamic Rearrangement of Vacuum and the Phase Transitions in the Geometric Structure of Space-Time,” International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 1, 2012, pp. 1-20. doi:10.1142/S0219887812500077 |

[48] | S. V. Siparov and N. Brinzei, “Space-Time Anisotropy: Theoretical Issues and the Possibility of an Observational Test,” Preprint, 2008. arXiv:0806.3066 [gr-qc] |

[49] | N. Brinzei and S. V. Siparov, “On the Possibility of the OMPR Effect in Spaces with Finsler Geometry, Part I,” Hypercomplex Numbers in Geometry and Physics, Vol. 4, No. 2, 2007, pp. 41-53. |

[50] | S. V. Siparov and N. Brinzei, “On the Possibility of the OMPR Effect in Spaces with Finsler Geometry, Part II,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 56-64. |

[51] | N. Voicu, “New Considerations on Einstein Equations in Anisotropic Spaces,” In: N. Voicu, Ed., AIP Conference Proceedings, Melville, New York, 2010, pp. 249-258. |

[52] | A. Aguirre, C. P. Burgess, A. Friedland and D. Nolte, “Astrophysical Constraints on Modifying Gravity at Large Distances,” Classical and Quantum Gravity, Vol. 18, No. 23, 2001, pp. R223-R232. |

[53] | D. G. Pavlov and S. S. Kokarev, “Conformal Gauges of the Berwald-Moor Geometry and Nonlinear Symmetries Induced by Them,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 3-14. |

[54] | S. S. Kokarev, “Is It Really That Different Geometries Are Different?” In: M. C. Duffy, V. O. Gladyshev, A. N. Morozov and P. Rowlands, Eds., Proceedings of XV International Meeting PIRT-2009, BMSTU Press, Moscow, 2009, pp. 116-123. |

[55] | D. G. Pavlov and S. S. Kokarev, “Riemannian Metrics Osculating to the 3-Dimentional Berwald-Moor Finsler metric,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 15-24. |

[56] | D. G. Pavlov and S. S. Kokarev, “Additive Angles in H3,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 25-43. |

[57] | D. G. Pavlov and S. S. Kokarev, “Polyangles and Their Symmetries in H3,” Hypercomplex Numbers in Geometry and Physics, Vol. 6, No. 1, 2009, pp. 42-67. |

[58] | N. P. Sokolov, "Spatial Matrices and Their Applications, GIFML, Moscow, 1960 (in Russian). |

[59] | S. S. Kokarev, “Isometry Classification of Cubic Homogeneous 3-Dimensional Forms,” Symmetry: Culture and Science, Symmetry Festival, Vol. 20, No. 1-4, 2009, pp. 371-391. |

[60] | D. G. Pavlov and S. S. Kokarev, “h-Holomorphic Functions of Double Variable and Their Applications,” Hyper-complex Numbers in Geometry and Physics, Vol. 7, No. 1, 2010, pp. 44-77. |

[61] | D. G. Pavlov and S. S. Kokarev, “Hyperbolic Theory of Field on the Plane of Double Variable,” Hypercomplex Numbers in Geometry and Physics, Vol. 7, No. 1, 2010, pp. 78-126. |

[62] | D. G. Pavlov and S. S. Kokarev, “Algebraic Unified Theory of Space-Time and Matter on the Plane of Double Variable,” Hypercomplex Numbers in Geometry and Physics, Vol. 7, No. 2, 2010, pp. 11-37. |

[63] | Gh. Atanasiu and M. Neagu, “On Cartan Spaces with the m-th Root Metric ,” Hypercomplex Numbers in Geometry and Physics, Vol. 6, No. 2, 2009, pp. 67-73. |

[64] | V. Balan, “Spectral Properties and Applications of Numerical Multilinear Algebra of m-Root Structures,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 101-108. |

[65] | V. Balan, “Notable Submanifolds in Berwald-Moor Spaces,” BSG Proceedings of 17th (ISI Proceedings), Geometry Balkan Press, Bucharest, 2010, pp. 21-30. |

[66] | V. Balan, “Numerical Multilinear Algebra of Symmetric m-Root Structures. Spectral Properties and Applications,” Symmetry: Culture and Science, Symmetry Festival 2009, Symmetry in the History of Science, Art and Technology. Part 2. Geometric Approaches to Symmetry, Vol. 21, No. 1-3, 2010, pp. 119-131. |

[67] | V. Balan and S. Lebedev, “On the Legendre Transform and Hamiltonian Formalism in Berwald-Moor Geometry,” Differential Geometry-Dynamical Systems, Vol. 12, No. 1, 2010, pp. 4-11. |

[68] | V. Balan and A. Manea, “Leafwise 2-Jet Cohomology on Foliated Finsler Manifolds,” BSG Proceedings 16 (ISI Proceedings), Geometry Balkan Press, Bucharest, 2009, pp. 28-41. |

[69] | V. Balan and M. Neagu, “Jet Geometrical Extension of the KCC-Invariants,” Balkan Journal of Geometry and Its Applications, Vol. 15, No. 1, 2009, pp. 8-16. |

[70] | V. Balan and I-R. Nicola, “Berwald-Moor Metrics and Structural Stability of Conformally-Deformed Geodesic SODE,” Differential Geometry-Dynamical Systems, Vol. 11, 2009, pp. 19-34. |

[71] | V. Balan and N. Perminov, “Applications of Resultants in the Spectral m-Root Framework,” Applied Science, Vol. 12, 2010, pp. 20-29. |

[72] | V. Balan and A. Pitea, “Symbolic Software for Y-Energy Extremal Finsler Submanifolds,” Applied Science, Vol. 1, 2009, pp. 41-53. |

[73] | N. Brinzei (Voicu) and S. V. Siparov, “Equations of Electromagnetism in Some Special Anisotropic Spaces,” Hypercomplex Numbers in Geometry and Physics, Vol. 5, No. 2, 2008, pp. 45-55. |

[74] | N. Brinzei (Voicu), “On Cubic Berwald Spaces,” Bulletin of the Calcutta Mathematical Society, Vol. 17, No. 1-2, 2009, pp. 75-84. |

[75] | N. Brinzei (Voicu), “Projective Relations for m-th Root Metric Spaces,” Journal of the Calcutta Mathematical Society, Vol. 5, No. 1-2, 2009, pp. 21-35. |

[76] | A. Manea, “The Vaisman Connection of the Vertical Bundle of a Finsler Manifold,” Bulletin of the Transilvania University of Brasov, Series III, Vol. 2, No. 51, 2009, pp. 199-206. |

[77] | A. Manea, “Some New Types of Vertical 2-Jets on the Tangent Bundle of a Finsler Manifold,” Scientific Bulletin—University Politehnica of Bucharest, Series A, Vol. 172, No. 1, 2010, pp. 177-194. |

[78] | O. Pasarescu, “Curves on Some Irrational Scrolls,” Romanian Academy of Sciences, GAR 4/2009. |

[79] | O. Pasarescu, “Curves on Rational Surfaces with Hyperelliptic Hyperplane Sections,” in press. arXiv:1101.0577v1 [math.AG]. |

[80] | N. Voicu (Brinzei), “Anisotropy and Analogies between Gravity and Electromagnetism,” In: M. C. Duffy, V.O. Gladyshev, A. N. Morozov and P. Rowlands, Eds., Proceedings of XV International Meeting PIRT-2009, BMSTU Press, Moscow, 2009, pp. 124-132. |

[81] | N. Voicu-Brinzei and S. Siparov, “A New Approach to Electromagnetism in Anisotropic Spaces,” BSG Proceedings 17th International Conference on Differential Geometry and Dynamical Systems, Bucharest, 2009 pp. 235-245. |

[82] | N. Voicu-Brinzei and S. Siparov, “Space-Time Anisot- ropy—Mathematical Formalism and the Possibility of an Experimental Test,” AIP Proceedings of 4th Gamow International Conference on Astrophysics and Cosmology after Gamow: Recent Progress and New Horizons, Odessa, 2009, pp. 152-162. |

[83] | S. Siparov, “Introduction to the Anisotropic Geometrodynamics,” World Scientific, Singapore, 2011. |

[84] | S. Molinari, et al., “A 100-Parsec Elliptical and Twisted Ring of Cold and Dense Molecular Clouds Revealed by HERSCHEL around the Galactic Center,” Preprint, 2011. arXiv:1105.5486v1, [astro-ph.GA] |

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