Abstract
This paper gives a generalisation of Einstein's vacuum field equations for Finsler metrics. The given generalised field equation reproduces the Einstein equations for Riemannian metrics, and also admits non-Riemannian solutions. This is shown in detail by deriving a first order Finsler perturbation, solving the new field equation, of the Schwarzschild metric. This perturbation turns out to be time independent. The effects of the perturbation on the three Classical Tests of General Relativity are derived, and used to give limits on the size of the perturbation parameter involved.
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References
Riemann, G. F. B. (1854). “Über die Hypothesen welche der Geometrie zu Grunde liegen,” Habilitation thesis, University of Göttingen.
Finsler, P. (1918). “Über Kurven und Flächen in Allgemeinen Räumen,” Dissertation, University of Göttingen.
Cartan, É. (1934).Les Espaces de Finsler (Actualités 79, Paris).
Berwald, L. (1947).Ann. Math. (2)48, 755.
Rund, H. (1959).The Differential Geometry of Finsler Spaces (Springer-Verlag, Berlin).
Matsumoto, M. (1986).Foundations of Finsler Geometry and Special Finsler Spaces (Kaiseisha Press, Shigaken).
Kilmister, D. A., and Stephenson, G. (1954).Nuovo Cimento 11, Suppl. 91, 118.
Asanov, G. S. (1979).Nuovo Cimento 49, 221; Bogoslovsky, G. Yu. (1977).Nuovo Cimento 40B, 99,116;id. (1992).Class. Quant. Grav. 9, 569; Ishikawa, H. (1981).J. Math. Phys. 22, 995; Matsumoto, M. (1975).Rep. Math. Phys. 8 103; Takano, Y. (1974).Lett. Nuovo Cimento 10, 747.
Roxburgh, I. W. (1992).Gen. Rel. Grav. 24, 419; Roxburgh, I. W., and Tavakol, R. K. (1979).Gen. Ret. Grav. 10, 307.
Tavakol, R. K., and Van der Bergh, N. (1986).Gen. Rel. Grav. 18, 849.
Pirani, F. A. E. (1965). In Lectures on General Relativity (Brandeis Summer Inst. in Theoretical Physics 1964) S. Deser and K. W. Ford, eds. (Prentice-Hall, Englewood Cliffs, N.J.), vol. 1, lecture 2.
Carathéodory, C. (1935).Variationsrechnung und partielle Differentialgleichungen erster Ordnung (Teubner Press, Leipzig).
Rutz, S. F., McCarthy, P. J. (1993).Gen. Rel. Grav. 25, 179.
Hearn, A. (1991). REDUCEUser's Manual, version 3.4 (Rand Corporation, Santa Monica, California).
Rutz, S. F. (1993). “Symmetry and Gravity in Finsler Spaces”, PhD thesis, Queen Mary & Westfield College, University of London.
McCarthy, P. J., Rutz, S. F. (1993).Gen. Rel. Grav. 25, 589.
Landau, L. D., and Lifshitz, E. M. (1962).The Classical Theory of Fields (2nd. revised ed., Pergamon, Oxford).
Weinberg, S. (1972).Gravitation and Cosmology (Wiley, New York).
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Rutz, S.F. A Finsler generalisation of Einstein's vacuum field equations. Gen Relat Gravit 25, 1139–1158 (1993). https://doi.org/10.1007/BF00763757
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DOI: https://doi.org/10.1007/BF00763757