Moving-mirror Doppler shift (${\omega }_r={\omega }_i(1-\beta )/(1+\beta )$)

A mirror receding at velocity $u=\beta c$ from normally incident light produces two applications of the relativistic Doppler effect, one into the mirror rest frame and one back to the observer. In that rest frame, reflection preserves frequency. Consequently ${\omega }_r={\gamma }^2(1-\beta {)}^2{\omega }_i={\omega }_i(1-\beta )/(1+\beta )$. Equivalently, continuity of wave phase on the moving surface $z=ut$ gives ${\omega }_i-k_{i}u={\omega }_r+k_{r}u$, with $k_i={\omega }_i/c$ and $k_r={\omega }_r/c$. For $|u|\ll c$, the fractional frequency change is $-2u/c$ to first order.

For a photon emitted from the origin and arriving at receding mirror position $x(t)$, its emission time is $t_e=t-x(t)/c$. Thus $d{t}_e/dt=1-\beta$ and a steady source rate $R$ gives impact rate $R(1-\beta )$ in source-frame coordinate time. This is distinct from the emission rate and from a rate per mirror proper time.