Null directions and horismos

The selected conformal class determines null directions. For $g^{\prime }={\Omega }^{2}g$, the connections satisfy

If $k$ is tangent to a $g$-null geodesic, the final term vanishes and the difference is tangent to the curve. Reparametrization therefore preserves the unparametrized null geodesic.

To identify an abstract $J\setminus I$ with geometric horismos, one must additionally identify $J$ with Lorentzian causal reachability; a realization of $I$ alone does not verify an arbitrarily supplied $J$. A null geodesic also need not remain an achronal boundary generator throughout its entire extension. No global identification of all null geodesics with all horismotic pairs is asserted.