Hausdorff-type metric geometry of the space of Cauchy hypersurfaces

Abstract

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric. For a timelike Cauchy complete, smooth Lorentzian manifold with a compact Cauchy hypersurface, we show that the resulting metric space is geodesic and proper. We further discuss extensions to more general synthetic Lorentzian settings. For this purpose, we generalize results on completeness properties of spacetimes due to Beem and Takahashi.

39 pages. New version includes a discussion on geodesics