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Gravity as a Finslerian Metric Phenomenon

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Abstract

We give a description of the effect of the gravitational field by using the geodesic equation of motion with respect to a first order Finslerian approximation of the Minkowski metric. This motivates linking the physical force of gravity to the non flat nature of space in the Finslerian setting and leads to an anisotropic version of the red shift formula. We solve the linearized Finslerian field equations proposed by S.F. Rutz (Gen. Relativ. Gravit. 25(11):1139–1158, 1993).

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References

  1. Amici, O., Casciaro, B., Dragomir, S.: On the cohomology of Finsler manifolds. Colloq. Math. Soc. J. Bolyai 46, 57–82 (1984). (Topics in Differential Geometry, Debrecen, Hungary, 1984)

    MathSciNet  Google Scholar 

  2. Asanov, G.S.: Finsler Geometry, Relativity and Gauge Theories, Fundamental Theories of Physics. Reidel, Dordrecht (1985)

    Book  MATH  Google Scholar 

  3. Asanov, G.S.: Finslerian metric and tetrads in static spherically-symmetric case of gravitational field. Rep. Math. Phys. 39(1), 69–75 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Asanov, G.S.: Finslerian anisotropic relativistic metric function obtainable under breakdown of rotational symmetry. arXiv:gr-qc/0204070v1 (2002)

  5. Asanov, G.S.: Finslerian extension of Lorentz transformations and first-order censorship theorem. Found. Phys. Lett. 15(2), 199–207 (2002)

    Article  MathSciNet  Google Scholar 

  6. Baki, T.: A possible electromagnetic singularity in Finslerian space-times. Afr. J. Sci. Technol. 7(2), 87–94 (2006)

    Google Scholar 

  7. Barthel, W.: Nichtlineare zusammenhange und deren holonomiegruppen. J. Reine Angew. Math. 212, 120–149 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beil, R.G.: Finsler geometry and relativistic field theory. Found. Phys. 33(7), 1107–1127 (2003)

    Article  MathSciNet  Google Scholar 

  9. Budden, T.: A star in the Minkowskian sky: anisotropic special relativity. Stud. Hist. Phil. Mod. Phys. 28(3), 325–361 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cartan, E.: Les espaces de Finsler, Actualités Scientifiques et Industrielles, vol. 79. Hermann, Paris (1934)

    MATH  Google Scholar 

  11. Clarke, C.J.S.: The Analysis of Space-Time Singularities. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  12. Dazord, P.: Tores finslériens sans points conjugués. Bull. Soc. Math. France 99, 171–192 (1971); erratum, ibid., 397

    MathSciNet  MATH  Google Scholar 

  13. Dazord, P.: Sur la formule de Gauss-Bonnet en géométrie finslérienne. C.R. Acad. Sci. Paris, Sér. A-B 270, A1241–A1243 (1970)

    MathSciNet  Google Scholar 

  14. Dazord, P.: Variétés finslériennes en forme de sphères. C.R. Acad. Sci. Paris, Sér. A-B 267, A353–A355 (1968)

    MathSciNet  Google Scholar 

  15. Dazord, P.: Variétés finslériennes de dimension paire δ-pincées. C.R. Acad. Sci. Paris, Sér. A-B 266, A496–A498 (1968)

    MathSciNet  Google Scholar 

  16. Dazord, P.: Variétś finslériennes à géodésiques fermées. C.R. Acad. Sci. Paris, Sér. A-B 266, A348–A350 (1968)

    MathSciNet  Google Scholar 

  17. Dazord, P.: Connexion de direction symétrique associée à un “spray” généralisé. C.R. Acad. Sci. Paris, Sér. A-B 263, A576–A578 (1966)

    MathSciNet  Google Scholar 

  18. Dazord, P.: Sur une généralisation de la notion de “spray”. C.R. Acad. Sci. Paris Sér. A-B 263, A543–A546 (1966)

    MathSciNet  Google Scholar 

  19. Dazord, P.: Tenseur de structure d’une G-structure dérivée. C.R. Acad. Sci. Paris 258, 2730–2733 (1964)

    MathSciNet  MATH  Google Scholar 

  20. Dragomir, S.: p-Distributions on differentiable manifolds. Analele Ştiinţ. Univ. Al.I. Cuza, Iaşi XXVIII, 55–58 (1982)

    Google Scholar 

  21. Eddington, A.S.: A comparison of Whitehead’s and Einstein’s formulae. Nature 113, 192 (1924)

    Article  ADS  Google Scholar 

  22. Einstein, A.: The foundation of the general theory of relativity. Ann. Phys. 49, 769–822 (1916)

    Article  MATH  Google Scholar 

  23. Finsler, P.: Über Kurven und Flächen in Allgemeinen Räumen. Birkhäuser, Basel (1951). Reprint of the 1918 dissertation (with a bibliography by H. Schubert)

    MATH  Google Scholar 

  24. Grifone, J.: Sur les connexions induite et intrinsèque d’une sous-variété d’une variété finslérienne. C.R. Acad. Sci. Paris, Sér. A-B 282(11), A599–A602 (1976)

    MathSciNet  Google Scholar 

  25. Grifone, J.: Transformations infinitésimales conformes d’une variété finslérienne,. C.R. Acad. Sci. Paris, Sér. A-B 280, A519–A522 (1975)

    MathSciNet  Google Scholar 

  26. Grifone, J.: Sur les transformations infinitésimales conformes d’une variété finslérienne. C.R. Acad. Sci. Paris, Sér. A-B, 280, A583–A585 (1975)

    MathSciNet  Google Scholar 

  27. Grifone, J.: Sur les connexions d’une variété finslérienne et d’un système mécanique. C.R. Acad. Sci. Paris, Sér. A-B, 272, A1510–A1513 (1971)

    MathSciNet  Google Scholar 

  28. Hassan, B.T.M.: The theory of geodesics in Finsler spaces. Thesis, Southampton (1967)

  29. Ikeda, S., Dragomir, S.: On the field equations in the theory of the gravitational fields in Finsler spaces. Tensor (N.S.) 44, 157–163 (1987)

    MathSciNet  MATH  Google Scholar 

  30. Kilmister, D.A., Stephenson, G.: An axiomatic criticism of unified field theories, I-II. Nuovo Cimento Suppl. 11, 91–105, 118–140 (1954)

    Article  MathSciNet  Google Scholar 

  31. Lackey, B.: On the Gauss-Bonnet formula in Riemann-Finsler geometry. Bull. Lond. Math. Soc. 34, 329–340 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, X., Chang, Z.: Toward a gravitation theory in Berwald-Finsler space. arXiv:0711.1934v1 [gr-qc]

  33. Li, X., Chang, Z.: Modified Newton’s gravity in Finsler space as a possible alternative to dark matter hypothesis. arXiv:0806.2184v2 [gr-qc] (2008)

  34. Lichnerowicz, A.: Sur une généralisation des espaces de Finsler. C.R. Acad. Sci. Paris 214, 599–601 (1942)

    MathSciNet  Google Scholar 

  35. Lichnerowicz, A.: Sur une extension de la formule d’Allendoerfer-Weil à certaines variétés finslériennes. C.R. Acad. Sci. Paris 223, 12–14 (1946)

    MathSciNet  MATH  Google Scholar 

  36. Lichnerowicz, A.: Quelques théorèmes de géométrie différentielle globale. Comment. Math. Helv. 22, 271–301 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, J.-M.: On local structure of gravity-free space and time. arXiv:physics/9901001 (1999)

  38. Matsumoto, M.: Foundations of Finsler Geometry and Special Finsler Spaces. Kasheisa Press, Kyoto (1982)

    Google Scholar 

  39. Mignemi, S.: Doubly special relativity and Finsler geometry. Phys. Rev. D 76, 047702 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  40. Rund, H.: The Differential Geometry of Finsler Spaces. Springer, Berlin (1959)

    MATH  Google Scholar 

  41. Rutz, S.F.: A Finsler generalization of Einstein’s vacuum field equations. Gen. Relativ. Gravit. 25(11), 1139–1158 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Rutz, S.F.: Theorems of Birkhoff type in Finsler spaces. Preprint CBPF-NF-014/98, Centro Brasileiro de Pesquisas Fisicas (1998)

  43. Rutz, S.F., McCarthy, P.J.: A Finsler perturabtion of the Poincaré metric. Gen. Relativ. Gravit. 2(25), 179–187 (1992)

    MathSciNet  Google Scholar 

  44. Skakala, J., Visser, M.: Bi-metric pseudo-Finslerin spacetimes. J. Geom. Phys. 61, 1396–1400 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Voicu, N.: New considerations on Einstein equations in anisotropic spaces. arXiv:0911.5034v1 [gr-qc] (2009)

  46. Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972)

    Google Scholar 

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Acknowledgements

The authors are grateful to the referees for their valuable comments.

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  1. Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, Campus Macchia Romana, Via dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Elisabetta Barletta & Sorin Dragomir

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  1. Elisabetta Barletta
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  2. Sorin Dragomir
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Correspondence to Sorin Dragomir.

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Barletta, E., Dragomir, S. Gravity as a Finslerian Metric Phenomenon. Found Phys 42, 436–453 (2012). https://doi.org/10.1007/s10701-011-9614-8

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