Globally hyperbolic flat space–times

Abstract

We consider (flat) Cauchy-complete GH space–times, i.e., globally hyperbolic flat Lorentzian manifolds admitting some Cauchy hypersurface on which the ambient Lorentzian metric restricts as a complete Riemannian metric. We define a family of such space–times—model space–times—including four subfamilies: translation space–times, Misner space–times, unipotent space–times, and Cauchy-hyperbolic space–times (the last family—undoubtful the most interesting one—is a generalization of standard space–times defined by G. Mess). We prove that, up to finite coverings and (twisted) products by Euclidean linear spaces, any Cauchy-complete GH space–time can be isometrically embedded in a model space–time, or in a twisted product of a Cauchy-hyperbolic space–time by flat Euclidean torus. We obtain as a corollary the classification of maximal GH space–times admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model space–time.