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 Incoming links: Proper-time acceleration by reflected laser light

Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / B4 / c / Solution 2026-10-20

For proper-time acceleration by reflected laser light, the spacecraft receives momentum

(h f + h f_{r}) / c = 2 h f / [c (1 + β)]

per incident photon. Combining with the impact rate gives

F = \frac{2 R h f}{c} \frac{1 - β}{1 + β}, \frac{d ( γ β )}{d t} = \frac{1}{τ _{0}} \frac{1 - β}{1 + β}, τ_{0} = \frac{M c ^{2}}{2 R h f} .

Because

d t / d τ = γ

and

d (γ β) / d β = γ^{3}

,

γ^{2} \frac{d β}{d τ} = \frac{1}{τ _{0}} \frac{1 - β}{1 + β} ⟹ \frac{d β}{d τ} = \frac{( 1 - β ) ^{2}}{τ _{0}} .

Integrating from rest gives

1/ (1 - β) - 1 = τ / τ_{0}

, hence

β (τ) = \frac{τ}{τ + τ _{0}} .

The mirror's rest mass is assumed constant and the laser illuminates it continuously; diffraction losses, incomplete interception and microscopic recoil are neglected.

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