Quantum space–time and classical gravity

A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space–time is considered here. It is found that there is a natural differential calculus over this version of quantized space–time using which the only possible torsion-free, metric-compatible, linear connection has zero curvature. It is then the noncummutative version of Minkowski space–time. Perturbations of this calculus are shown to give rise to nontrivial gravitational fields.