Proposition: Failure of global hyperbolicity

The regular product (eq:host) has a noncompact causal diamond and is not globally hyperbolic.

Proof. Fix a regular spatial point of radius $r_{\ast }>0$, and two events there separated by more than $2r_{\ast }/c$. At an intermediate time, points on the same radial ray with radii decreasing to zero lie in the causal diamond between the two events. They have no subsequence converging in the regular product.