×

Gravity as a Finslerian metric phenomenon. (English) Zbl 1243.83047

Summary: 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. Gravitation 25, No. 11, 1139–1158 (1993; Zbl 0790.53023)].

MSC:

83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories

Citations:

Zbl 0790.53023
PDFBibTeX XMLCite
Full Text: DOI

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) · Zbl 0637.53081
[2] Asanov, G.S.: Finsler Geometry, Relativity and Gauge Theories, Fundamental Theories of Physics. Reidel, Dordrecht (1985) · Zbl 0576.53001
[3] Asanov, G.S.: Finslerian metric and tetrads in static spherically-symmetric case of gravitational field. Rep. Math. Phys. 39(1), 69–75 (1997) · Zbl 0884.53051 · doi:10.1016/S0034-4877(97)81471-1
[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) · doi:10.1023/A:1020908309910
[6] Baki, T.: A possible electromagnetic singularity in Finslerian space-times. Afr. J. Sci. Technol. 7(2), 87–94 (2006)
[7] Barthel, W.: Nichtlineare zusammenhange und deren holonomiegruppen. J. Reine Angew. Math. 212, 120–149 (1963) · Zbl 0115.39704
[8] Beil, R.G.: Finsler geometry and relativistic field theory. Found. Phys. 33(7), 1107–1127 (2003) · doi:10.1023/A:1025689902340
[9] Budden, T.: A star in the Minkowskian sky: anisotropic special relativity. Stud. Hist. Phil. Mod. Phys. 28(3), 325–361 (1997) · Zbl 1222.83021 · doi:10.1016/S1355-2198(97)80894-5
[10] Cartan, E.: Les espaces de Finsler, Actualités Scientifiques et Industrielles, vol. 79. Hermann, Paris (1934) · Zbl 0008.41805
[11] Clarke, C.J.S.: The Analysis of Space-Time Singularities. Cambridge University Press, Cambridge (1993) · Zbl 0835.53093
[12] Dazord, P.: Tores finslériens sans points conjugués. Bull. Soc. Math. France 99, 171–192 (1971); erratum, ibid., 397 · Zbl 0215.23504
[13] Dazord, P.: Sur la formule de Gauss-Bonnet en géométrie finslérienne. C.R. Acad. Sci. Paris, Sér. A-B 270, A1241–A1243 (1970) · Zbl 0192.59005
[14] Dazord, P.: Variétés finslériennes en forme de sphères. C.R. Acad. Sci. Paris, Sér. A-B 267, A353–A355 (1968) · Zbl 0159.51501
[15] Dazord, P.: Variétés finslériennes de dimension paire {\(\delta\)}-pincées. C.R. Acad. Sci. Paris, Sér. A-B 266, A496–A498 (1968) · Zbl 0153.23402
[16] Dazord, P.: Variétś finslériennes à géodésiques fermées. C.R. Acad. Sci. Paris, Sér. A-B 266, A348–A350 (1968) · Zbl 0157.52304
[17] Dazord, P.: Connexion de direction symétrique associée à un ”spray” généralisé. C.R. Acad. Sci. Paris, Sér. A-B 263, A576–A578 (1966) · Zbl 0172.23403
[18] Dazord, P.: Sur une généralisation de la notion de ”spray”. C.R. Acad. Sci. Paris Sér. A-B 263, A543–A546 (1966) · Zbl 0146.44001
[19] Dazord, P.: Tenseur de structure d’une G-structure dérivée. C.R. Acad. Sci. Paris 258, 2730–2733 (1964) · Zbl 0122.16907
[20] Dragomir, S.: p-Distributions on differentiable manifolds. Analele Ştiinţ. Univ. Al.I. Cuza, Iaşi XXVIII, 55–58 (1982)
[21] Eddington, A.S.: A comparison of Whitehead’s and Einstein’s formulae. Nature 113, 192 (1924) · JFM 50.0623.02 · doi:10.1038/113192a0
[22] Einstein, A.: The foundation of the general theory of relativity. Ann. Phys. 49, 769–822 (1916) · JFM 46.1293.01 · doi:10.1002/andp.19163540702
[23] Finsler, P.: Über Kurven und Flächen in Allgemeinen Räumen. Birkhäuser, Basel (1951). Reprint of the 1918 dissertation (with a bibliography by H. Schubert) · Zbl 0044.37003
[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) · Zbl 0328.53015
[25] Grifone, J.: Transformations infinitésimales conformes d’une variété finslérienne,. C.R. Acad. Sci. Paris, Sér. A-B 280, A519–A522 (1975) · Zbl 0311.53070
[26] Grifone, J.: Sur les transformations infinitésimales conformes d’une variété finslérienne. C.R. Acad. Sci. Paris, Sér. A-B, 280, A583–A585 (1975) · Zbl 0311.53071
[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) · Zbl 0218.53042
[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) · Zbl 0644.53070
[30] Kilmister, D.A., Stephenson, G.: An axiomatic criticism of unified field theories, I-II. Nuovo Cimento Suppl. 11, 91–105, 118–140 (1954) · Zbl 0056.21610 · doi:10.1007/BF02780905
[31] Lackey, B.: On the Gauss-Bonnet formula in Riemann-Finsler geometry. Bull. Lond. Math. Soc. 34, 329–340 (2002) · Zbl 1039.53083 · doi:10.1112/S002460930100892X
[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) · JFM 68.0443.02
[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) · Zbl 0060.38106
[36] Lichnerowicz, A.: Quelques théorèmes de géométrie différentielle globale. Comment. Math. Helv. 22, 271–301 (1949) · Zbl 0039.17501 · doi:10.1007/BF02568060
[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) · Zbl 0594.53001
[39] Mignemi, S.: Doubly special relativity and Finsler geometry. Phys. Rev. D 76, 047702 (2007)
[40] Rund, H.: The Differential Geometry of Finsler Spaces. Springer, Berlin (1959) · Zbl 0087.36604
[41] Rutz, S.F.: A Finsler generalization of Einstein’s vacuum field equations. Gen. Relativ. Gravit. 25(11), 1139–1158 (1993) · Zbl 0790.53023 · doi:10.1007/BF00763757
[42] Rutz, S.F.: Theorems of Birkhoff type in Finsler spaces. Preprint CBPF-NF-014/98, Centro Brasileiro de Pesquisas Fisicas (1998) · Zbl 0997.83053
[43] Rutz, S.F., McCarthy, P.J.: A Finsler perturabtion of the Poincaré metric. Gen. Relativ. Gravit. 2(25), 179–187 (1992)
[44] Skakala, J., Visser, M.: Bi-metric pseudo-Finslerin spacetimes. J. Geom. Phys. 61, 1396–1400 (2011) · Zbl 1218.53024 · doi:10.1016/j.geomphys.2011.03.003
[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) · Zbl 0256.05105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.