GRAVITY FROM NONCOMMUTATIVE GEOMETRY

We introduce the linear connection in the non-commutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant, and apart from that we find no other dynamic fields in the model. Finally we discuss an example solution of flat linear connection, with the non-trivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the standard model, with the scaling factor corresponding to the Weinberg angle.