Timelike conjugate points in Lorentzian length spaces

Abstract

We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Er\"{o}s--Gieger. In the appendix, we give a detailed treatment of the Fr\'{e}chet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.