Causal diamond

Definition

For events $p,q$ in a time-oriented Lorentzian manifold, the causal diamond from $p$ to $q$ is

where $J^\pm$ are the causal future and past. It consists of the events that can be reached causally from $p$ and from which $q$ can be reached causally. The diamond is empty when $q\notin J^+(p)$.

Compactness of all causal diamonds, together with strong causality, defines a globally hyperbolic spacetime in the convention used here.