A synthetic Lorentzian Cartan-Hadamard theorem

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 [Enseign. Math. (2) 36 (1990), pp. 309–320] to the Lorentzian setting, and simultaneously generalizes the smooth Lorentzian theorem discussed by J. Beem, P. Ehrlich, and K. Easley [ Global Lorentzian geometry , Marcel Dekker, Inc., New York, 1996] 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 non-negative upper timelike curvature bounds.