  We study general relativity in the framework of non-commutative differential
geometry. In particular, we introduce a gravity action for a space-time which
is the product of a four dimensional manifold by a two-point space. 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 geometrically interpreted as describing the
distance between the two points in the internal space.
