Gravity From Noncommutative Geometry

We introduce the linear connection in the noncommutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, de ne the metric connection and calculate the curvature. We de ne also the Ricci tensor and the scalar curvature. We nd that the latter di ers from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant and apart from that we nd no other dynamical elds in the model. Finally we discuss an example solution of at linear connection, with the nontrivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as con rmed by the Standard Model, with the scaling factor corresponding to the Weinberg angle. TPJU 1/94 hep-th/9401145 January 1994 Partially supported by KBN grant 2 P302 168 4 E-mail: [email protected]