Geometry and General Relativity in the Groupoid Model with a Finite Structure Group

Abstract

In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a transformation groupoid $\Gamma$ determined by the action of the Lorentz group on the frame bundle $(E, \pi_M, M)$ over space-time $M$. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group $G$ and the frame bundle is trivial $E=M\times G$. The model is fully computable. We define the Einstein-Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space-time (giving the "general relativistic limit"), the extra terms that appear due to our generalization can be interpreted as "matter terms", as in Kaluza-Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their "Friedmann-like" solutions for the special case when $G =\mathbb{Z}_2$. One of them gives the flat Minkowski space-time (which, however, is not static), another, a hyperbolic, linearly expanding universe.