The space generated by metric and torsion tensors, derivation of Einstein-Hilbert equation
DOI:
https://doi.org/10.15673/2072-9812.2/2014.29622Kulcsszavak:
Metric tensor, torsion tensor, curvature tensor, Ricci - Jacobi identity, geodesic equation, tangent bundle, covariant derivative, tensor densities, affine transformation, principal homogeneous spaceAbsztrakt
This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space are generated jointly and agreed by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor obtained analog Ricci - Jacobi identity; was evaluated gap that occurs at the transition from the original to the image and vice versa, in the case of an infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor π_αβ which is similar to the second fundamental tensor of hypersurfaces Y^n-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself in the small. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).
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