Results on Lorentzian metric spaces
E. Minguzzi
TL;DR
The paper develops a purely metric framework for Lorentzian geometry by defining Lorentzian metric spaces (LMS) from the Lorentzian distance $d$, under minimal axioms that recover global hyperbolicity in a synthetic setting. It constructs the canonical topology, causal relation $J$, and notions of isocausal curves and (pre)length spaces, and establishes stability under Gromov-Hausdorff convergence alongside a canonical quasi-uniform structure, including a quasi-metric for sequenced LMS. A central contribution is a constructive proof that every countably generated LMS admits a Cauchy time function, avoiding volume-based methods and using only the properties of $d$ and compactness of diamonds. The results unify discrete and continuous Lorentzian geometry, connect LMS to causets and optimal transport, and provide tools applicable to low-regularity or quantum-gravity-inspired spacetime models. Overall, the work offers a robust, intrinsic, and scalable framework for synthetic Lorentzian geometry with concrete limit and time-function results.
Abstract
We provide a short introduction to ``Lorentzian metric spaces" i.e., spacetimes defined solely in terms of the two-point Lorentzian distance. As noted in previous work, this structure is essentially unique if minimal conditions are imposed, such as the continuity of the Lorentzian distance and the relative compactness of chronological diamonds. The latter condition is natural for interpreting these spaces as low-regularity versions of globally hyperbolic spacetimes. Confirming this interpretation, we prove that every Lorentzian metric space admits a Cauchy time function. The proof is constructive for this general setting and it provides a novel argument that is interesting already for smooth spacetimes.