Abstract
A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.
Similar content being viewed by others
References
Shirokov, M. F., and Fisher, I. Z. (1962).Astron. Zh. 39, 899 (in Russian).
Ellis, G. F. R. (1984). InGeneral Relativity and Gravitation, Bertotti, B., de Felici, F., and Pascolini, A., eds. (Reidel, Dordrecht).
Carfora, M., and Marzuoli, A. (1984).Phys. Rev. Lett. 53, 2445.
de Groot, S. R., and Suttorp, L. G. (1972).Foundations of Electrodynamics (North-Holland, Amsterdam).
Mitskievich, N. V. (1969).Proc. P. Lumumba Peoples' Friendship Univ. 44, 137 (in Russian).
Szekeres, P. (1971).Ann. Phys. (N.Y.) 64, 599.
Nelson, A. H. (1972).Mon. Not. R. Astr. Soc. 158, 159.
Noonan, T. W. (1984).Gen. Rel. Grav. 16, 1103.
Noonan, T. W. (1985).Gen. Rel. Grav. 17, 535.
Isaacson, R. A. (1968).Phys. Rev. 166, 1272.
Rosen, G. (1980).Nuovo Cimento 57B, 125.
Marochnik, L. S., Pelikhov, N. V., and Vereshkov, G. M. (1975).Astrophys. Space Sci. 34, 249.
Marochnik, L. S. (1980).Astron. Zh. 57, 903 (in Russian).
Bailey, I., and Israel, W. (1975).Commun. Math. Phys. 42, 65.
Arifov, L. Ya., and Zalaletdinov, R. M. (1989). “Averaging out tensor fields on Riemannian manifolds according to Lorentz. I. Averaging scheme and algebra of the averages.” Preprint, Institute of Nuclear Physics, Uzbek Academy of Sciences, No. R-12-450 (Tashkent),1 (in Russian).
Arifov, L. Ya., Shein, A. V., and Zalaletdinov, R. M. (1990). “Averaging out tensor fields on Riemannian manifolds according to Lorentz. II. Commutation formulae for the averaging and derivation (absolute and covariant).” Preprint, Institute of Nuclear Physics, Uzbek Academy of Sciences, No. R-12-480 (Tashkent),1 (in Russian).
Arifov, L. Ya., Shein, A. V., and Zalaletdinov, R. M. (1990). “Averaging out tensor fields on Riemannian manifolds according to Lorentz. III. Analysis of the averages.” Preprint, Institute of Nuclear Physics, Uzbek Academy of Sciences, No. R-12-499 (Tashkent),1 (in Russian).
Synge, J. L. (1960).Relativity: The General Theory (North-Holland, Amsterdam).
Arifov, L. Ya., and Zalaletdinov, R. M. (1988). (1988). InAbstracts of the V Marcel Grossman Meeting, Perth, D. Blair, M. J. Buckingham, eds. (University of Western Australia Press, Perth).
Arifov, L. Ya., and Zalaletdinov, R. M. (1989). “An averaging of a pseudo-Riemannian space-time according to Lorentz.” Preprint, Institute of Nuclear Physics, Uzbek Academy of Sciences, No. R-12-451 (Tashkent),1 (in Russian).
Christensen, S. M. (1976).Phys. Rev. D 14, 2450.
Schouten, J. A., and Struik, D. J. (1935).Einfūhrung in die neueren Methoden der Differentialgeometrie (Noordhoff, Gröningen-Batavia), vol. I.
Kramer, D., Stephani, H., MacCallum, M. A. H., and Herlt, E. (1980).Exact Solutions of Einstein's Field Equations (VEB Deutscher Verlag der Wissenschaften, Berlin / Cambridge University Press, Cambridge).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zalaletdinov, R.M. Averaging out the Einstein equations. Gen Relat Gravit 24, 1015–1031 (1992). https://doi.org/10.1007/BF00756944
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00756944