Within the synthetic‐geometric framework of Lorentzian (pre‐)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54 (2018), no. 3, 399–447) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts such as timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non‐branching result for spaces with timelike curvature bounded below.

We introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for CAT(k) spaces, which roughly states that gluing is compatible with upper curvature bounds. Due to the absence of a notion of spacelike distance in Lorentzian pre-length spaces we can only formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces.

We prove a synthetic Bonnet-Myers rigidity theorem for globally hyperbolic Lorentzian length spaces with global curvature bounded below by K < 0 and an open distance realizer of length L = π √ |K| . In the course of the proof, we show that the space necessarily is a warped product with warping function cos : (− π 2 , π 2 ) → R + .