Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime

I characterize the Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime (i.e., the Lorentzian submanifolds of Minkowski spacetime that are also closed subsets). Moreover, I prove that the Lorentzian manifolds that can be properly conformally embedded in Minkowski spacetime coincide with the globally hyperbolic spacetimes. Finally, by taking advantage of the embedding, I obtain an infinitesimal version of the distance formula.