Horizon frequency comparison (${\omega }_2/{\omega }_1=e^{-\kappa (v_2-v_1)}$)

Let $K={\partial }_v$ generate a Killing horizon with constant surface gravity $\kappa$, so ${\nabla }_{K}K=\kappa K$. In geometric optics, an affinely transported light wavevector on the same generator is $k=q(v)K$. The geodesic equation gives $q^{\prime }+\kappa q=0$. 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.