Gravity Models

Abstract

We shall describe three possible approaches to the construction of gravity models in noncommutative geometry which, while agreeing for the canonical triple associated with an ordinary manifold (and reproducing the usual Einstein theory), seem to give different answers for more general examples. As a general remark, we should like to mention that a noncommutative recipe to construct gravity theories (at least the usual Einstein one) has to include the metric as a dynamical variable which is not a priori given. In particular, one should not start with the Hilbert space $\mathcal{H}$ = L ² (M, S) of spinor fields whose scalar product uses a metric on M which, therefore, would play the role of a background metric. The beautiful result by Connes [35] which we recall in the following Section goes exactly in the direction of deriving all geometry a posteriori.