Generalized products and Lorentzian length spaces

We construct a Lorentzian length space with an orthogonal splitting on a product IxX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\times X$$\end{document} of an interval and a metric space and use this framework to consider the relationship between metric and causal geometry, as well as synthetic time-like Ricci curvature bounds. The generalized Lorentzian product carries a natural Lorentzian length structure but can fail the push-up condition in general. We recover the push-up property under a log-Lipschitz condition on the time variable and establish sufficient conditions for global hyperbolicity. Moreover, we formulate time-like Ricci curvature bounds without push-up and regularity assumptions and obtain a partial rigidity of the splitting under a strong energy condition.