Lemma: Geometric interpretation

Formula (eq:Gcoordinates) describes all regular null geodesic segments. The additional off-diagonal limits in ${\mathcal{G}}^{+}$ occur at $|{\theta }_k|=\pi$.

Proof. A geodesic away from the apex develops into a straight line in the transverse plane, together with an affine longitudinal coordinate. A straight transverse segment avoiding the origin has a continuous angular change strictly between $-\pi$ and $\pi$. Its length is $R_k$ for the corresponding lift. The null condition is $|T|=R_k$. Differentiating its squared separation gives the parallel endpoint covectors in (eq:Gcoordinates).

A limit with angular change $\pi$ or $-\pi$ passes through the transverse origin. Conversely, those segments are limits of straight segments avoiding the origin. The local finiteness of the allowable lifts excludes other off-diagonal limits.