Radial null geodesics in a de Sitter static patch ($r=r_S\tanh (ct/r_S)$)

For lapse $\alpha =1-r^2/r_S^2$, conservation of static time momentum and an origin normalization give $\alpha \dot{t}=1$. The null geodesic condition gives $\dot{r}=\pm c$. The outgoing ray from the origin therefore obeys $t=(r_S/c)\mathrm{artanh}(r/r_S)$ and the title trajectory. Static-coordinate time diverges at the horizon, while the chosen affine parameter reaches it at $r_S/c$. Local stationary observers inside the patch still measure speed $c$.