Doppler Effect for Light — Question 4

Question 4

Transverse Doppler shifts depend on which frame defines transverse

A source moves at $v=0.60c$. Compare photons emitted at laboratory angle $\theta=90^\circ$ with photons emitted at source-frame angle $\theta\prime=90^\circ$. For a source of rest frequency $f_0$ moving at +v, use $f=f_0/[\gamma(1-\beta\cos\theta)]$, where $\beta=v/c$ and $\theta$ is the photon propagation angle from +x in the laboratory.

1. Find the frequency ratio for laboratory-transverse emission.

2. Using $f=\gamma f_0(1+\beta\cos\theta\prime)$, find the ratio for source-transverse emission and explain the difference.

Question 4 - Solution

Transverse Doppler shifts depend on which frame defines transverse

(a) Method and calculation

$$\theta=90^\circ:\quad\boxed{f/f_0=1/\gamma=0.800}.$$$$\cos\theta\prime=\frac{\cos\theta-\beta}{1-\beta\cos\theta}=-0.600.$$

(b) Interpretation and further calculation

$$\theta\prime=90^\circ:\quad\boxed{f/f_0=\gamma=1.250}.$$$$\cos\theta=\frac{\cos\theta\prime+\beta}{1+\beta\cos\theta\prime}=0.600.$$

Physical check

The two cases refer to different photons and different angles. The red and blue factors are therefore consistent.