Globally Hyperbolic Flat Spacetimes

Abstract

We consider (flat) Cauchy-complete GH spacetimes, 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 spacetimes - model spacetimes - including four subfamilies: translation spacetimes, Misner spacetimes, unipotent spacetimes, and Cauchy-hyperbolic spacetimes (the last family - undoubtfull the most interesting one - is a generalization of standart spacetimes defined by G. Mess). We prove that, up to finite coverings and (twisted) products by euclidean linear spaces, any Cauchy-complete GH spacetime can be isometrically embedded in a model spacetime, or in a twisted product of a Cauchy-hyperbolic spacetime by flat euclidean torus. We obtain as a corollary the classification of maximal GH spacetimes admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model spacetime.