On globally hyperbolic spacetimes
Abstract: Riemannian manifolds can be equipped with a natural metric space structure, thanks to which many results involving curvature bounds have been extended to metric (measure) spaces. A crucial ingredient in this process is the Hopf-Rinow Theorem which equates geodesic completeness with metric completeness. Lorentzian manifolds, in contrast, do not admit a canonical metric space structure and geodesic incompleteness is actually a desired feature. Still, the "best" Lorentzian manifolds mimic the good properties of complete Riemannian manifolds in many other ways. They are called globally hyperbolic spacetimes and are also of utmost importance in General Relativity (well-posedness of the initial value formulation of the Einstein equations, singularity theorems of Penrose and Hawking, splitting results etc.). In this talk, we present a surprising new characterization of global hyperbolicity. We show that globally hyperbolic spacetimes are precisely those Lorentzian manifolds for which the null distance is complete. This is joint work with Leonardo García-Heveling.