An Isaac Newton Institute Workshop

Noncommutative Geometry and Physics: Fundamental Structure of Space and Time

A noncommutative closed Friedman world model

Authors: Michael HELLER (Vatican Observatory), Leszek PYSIAK (Department of Mathematics and Information Science, Warsaw University of Technology), Wieslaw SASIN (Department of Mathematics and Information Science, Warsaw University of Thechnology)

Abstract

In J. Math. Phys. 46, 2005, 122501, we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra A defined on a groupoid having the framer bundle over space-time as its base space. The generalized Einstein equation is defined in terms of the algebra A and its derivations; matter sources are assumed to vanish. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, additional components of the generalized Einstein equation turn out to be identical with the components of the energy-momentum tensor for the usual Friedman model and the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space-time geometry. These results will be briefly discussed.