The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Inspired by the proof for smooth spacetimes (Beem et al. in Global differential geometry and global analysis 1984, Springer, pp. 1–13, 1985), we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces as proven in Burago et al. (A course in metric geometry, vol 33, American Mathematical Society, Providence, 2001) and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible by a result in Grant et al. (Ann Glob Anal Geom 55(1):133–147, 2019), which globalises the local result.