 Solution

For radial null geodesics in a de Sitter static patch, the conserved time quantity and the stated normalization at the origin give $\alpha \dot{t}=1$. The null condition then givesThe emitted outgoing ray selects the positive branch, soThe ingoing branch would have $\dot{r}=-c$ instead. Set the affine parameter to zero at emission. Then $r=c\lambda$, and integrating $dt/dr=1/[c(1-r^2/r_S^2)]$ givesThe coordinate time diverges as $r\to r_S$, although the outgoing affine parameter reaches $r_S/c$ there. This is a coordinate-horizon effect: a local stationary observer inside the patch measures radial distance $dr/\sqrt{\alpha }$ and proper time $\sqrt{\alpha }dt$, whose ratio along the ray is still $c$.