M. Heller, T. Miller, L. Pysiak, W. Sasin
We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra A. Such a derivation, introduced by Bresar in 1991, is given by a linear mapping u: A -> A such that there exists a usual derivation d of A satisfying the generalized Leibniz rule u(a b) = u(a) b + a d(b) for all a,b in A. The generalized geometry "is tested" in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study the generalized general relativity. We define the Einstein-Hilbert action and deduce from it Einstein's field equations. We show that for a special class of metrics containing, besides the usual metric components, only one non-zero term, the action reduces to O'Hanlon action that is a Brans-Dicke action with potential and with the parameter \omega equal to zero. We also show that the generalized Einstein equations (with zero energy-stress tensor) are equivalent to those of the Kaluza-Klein theory satisfying a "modified cylinder condition" and having a non-compact extra dimension. This opens a possibility to consider Kaluza-Klein models with a non-compact extra dimension that remains invisible for a macroscopic observer. In our approach this extra dimension is not an additional physical space-time dimension but it appears due to the generalization of the derivation concept.
View original:
http://arxiv.org/abs/1301.0910
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