Causal future ($J^{+}(S)$)

The causal future of $S$ consists of points reached from $S$ by future-directed causal curves, including points of $S$ itself. In a globally hyperbolic spacetime, the causal future of a compact set is closed.

The future horismos is the part of the causal future outside the chronological future. For a compact set in a globally hyperbolic spacetime, it equals the boundary of the chronological future. Its normal null geodesic generators cannot have passed a conjugate point to a spacelike surface when the initial set is that surface.