= Radial null geodesics in a de Sitter static patch
{title2=$r=r_S\tanh(ct/r_S)$}
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 $t=(r_S/c)\operatorname{artanh}(r/r_S)$ 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 $c$.
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