GRAVITY IN NONCOMMUTATIVE GEOMETRY

We study general relativity in the framework of non-commutative differ ential geometry. As a prerequisite we develop the basic notions of non -commutative Riemannian geometry, including analogues of Riemannian me tric, curvature and scalar curvature. This enables us to introduce a g eneralized Einstein-Hilbert action for non-commutative Riemannian spac es. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-com mutative Riemannian geometry, and derive its generalized Einstein-Hilb ert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geo metrically interpreted as describing the distance between the two poin ts in the internal space.