Lemma: Causality of the metric cone product

For $p=(T,x)$, $q=(T^{\prime },y)$,

This relation is reflexive and transitive, including at the axis.

Proof. Integration of the speed inequality proves necessity. The cone is a proper length space: bounded closed sets are compact. A minimizing sequence of spatial paths can be parametrized with uniformly bounded speed in a common compact ball; uniform convergence gives a length-minimizing path. Parametrize that path at speed at most one in the available time, inserting a wait if necessary. This proves sufficiency. The triangle inequality gives transitivity.