= Static-emitter redshift in a de Sitter patch
{title2=$1+z=\sqrt{\alpha_O/\alpha_E}$}
For stationary emitter and observer in the same static patch, repeated rays between fixed radii have the same coordinate travel delay. Since a stationary clock obeys $d\tau=\sqrt\alpha\,dt$, the received-to-emitted proper-period ratio is $\sqrt{\alpha_O/\alpha_E}$. At a central observer and emitter radius $r_E$, this is $(1-r_E^2/r_S^2)^{-1/2}$. Holding an emitter stationary requires unbounded proper acceleration as it approaches the horizon; the limiting redshift formula does not describe a timelike static worldline on the horizon itself.
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