Proposition: Failure of global hyperbolicity

The regular product (eq:host) is not globally hyperbolic.

Proof. Fix a regular spatial point of radius $r_{\ast }>0$. Take two events there separated in $T$ by $4r_{\ast }/c_{\mathrm{light}}$. At their middle time, points on the same radial ray with radii tending to zero lie in the causal diamond between the two events. They have no convergent subsequence in the regular product. The diamond is not compact.