Generalized products and Lorentzian length spaces

Abstract We construct a Lorentzian length space with an orthogonal splitting on a product $$I\times X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> 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.