Geometry and general relativity in the groupoid model with a finite structure group

In a series of papers (M. Heller et al. J. Math. Phys. 38, 5840 (1997). doi:10.1063/1.532186; M. Heller and W. Sasin. Int. J. Theor. Phys. 38, 1619 (1999). doi:10.1023/A:1026617913754; M. Heller et al. Int. J. Theor. Phys. 44, 619 (2005). doi:10.1007/s10773-005- 3992-7) 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, ^AΓ, defined on a transformation groupoid Γ determined by the action of the Lorentz group on the frame bundle (E, π_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 x 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 'Friedman-like' solutions for the special case when G = ^Z2. One of them gives the flat Minkowski space-time (which, however, is not static), another, a hyperbolic, linearly expanding universe. (author)