A synthetic Lorentzian Cartan-Hadamard theorem
Darius Erös, Sebastian Gieger
TL;DR
This work develops a synthetic Lorentzian Cartan–Hadamard theorem by leveraging local concavity of the time-separation, global hyperbolicity, and future one-connectedness to obtain existence, uniqueness, and continuous dependence of timelike geodesics between timelike related points. It establishes local existence/uniqueness within comparison neighborhoods and then globalizes these results to produce a unique, continuously varying geodesic map for globally hyperbolic, locally concave Lorentzian pre-length spaces, with applications to zero upper timelike curvature bounds. The results generalize Beem–Ehrlich’s smooth Lorentzian Cartan–Hadamard theorem to the synthetic Lorentzian setting and extend Gromov-type local-to-global principles to non-smooth spacetimes, providing foundational tools for non-smooth Lorentzian geometry and potential implications for general relativity beyond smooth manifolds.
Abstract
We formulate and prove a synthetic Lorentzian Cartan-Hadamard theorem. This result both transfers the corresponding statement for locally convex metric spaces established by S. Alexander and R. Bishop to the Lorentzian setting, and simultaneously generalizes the smooth Lorentzian theorem discussed by J. Beem and P. Ehrlich to the recently established framework of synthetic Lorentzian geometry. Our approach is based on an appropriate notion of local concavity for Lorentzian (pre-)length spaces, which allows us to establish existence and uniqueness of timelike geodesics between any pair of timelike related points under the additional assumptions of global hyperbolicity and future one-connectedness. We also provide a globalization result for our notion of concavity in the setting of Lorentzian length spaces, and apply our results to obtain a globalization statement for nonnegative upper timelike curvature bounds.