= Horizon frequency comparison
{title2=$\omega_2/\omega_1=e^{-\kappa(v_2-v_1)}$}
Let $K=\partial_v$ generate a <Killing horizon> with constant <surface gravity> $\kappa$, so $\nabla_KK=\kappa K$. In <geometric optics>, an affinely transported <light> <wavevector> on the same generator is $k=q(v)K$. The <geodesic equation> gives $q'+\kappa q=0$. Observers related by the <isometry> generated by $K$ have the same $-K\cdot W$ at their crossings, where $W$ is their <four-velocity>, so their measured <angular frequencies> $\omega=-k\cdot W$ obey the displayed relation. A <degenerate Killing horizon> gives ratio one for equally prepared observers; changing their crossing velocities can produce a separate <Doppler effect>.
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