Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / B4

In the instantaneous mirror frame the incident frequency is $f_m=\gamma (1-\beta )f$. Ideal reflection reverses propagation without changing that frame's frequency. Transforming the backward photon to Earth gives another factor $\gamma (1-\beta )$:This uses a classical mirror with negligible recoil per individual photon.

A photon reaching spacecraft position $x(t)$ at Earth time $t$ was emitted at $t_e=t-x(t)/c$. Differentiating gives $d{t}_e/dt=1-\beta$. Since $Rd{t}_e$ photons arrive in $dt$,The emission rate and impact rate refer to different events, even though both are expressed in Earth coordinate time.

For proper-time acceleration by reflected laser light, the spacecraft receives momentum $(hf+h{f}_r)/c=2hf/[c(1+\beta )]$ per incident photon. Combining with the impact rate givesBecause $dt/d\tau =\gamma$ and $d(\gamma \beta )/d\beta ={\gamma }^3$,Integrating from rest gives $1/(1-\beta )-1=\tau /{\tau }_0$, henceThe mirror's rest mass is assumed constant and the laser illuminates it continuously; diffraction losses, incomplete interception and microscopic recoil are neglected.