Generalized derivations and general relativity

Abstract

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra ${\cal A}$. Such a derivation, introduced by Bresar in 1991, is given by a linear mapping $u: {\cal A} \rightarrow {\cal A}$ such that there exists a usual derivation $d$ of ${\cal A}$ satisfying the generalized Leibniz rule $u(a b) = u(a) b + a \, d(b)$ for all $a,b \in \cal A$. The generalized geometry "is tested" in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study 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 nonzero term, the action reduces to the O'Hanlon action that is the 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 noncompact extra dimension. This opens a possibility to consider Kaluza-Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space-time dimension but appears because of the generalization of the derivation concept.