 Static-emitter redshift in a de Sitter patch

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.