The Relativistic Doppler Effect

The sound of a train's horn shifts in frequency as the train passes by due to the relative motion of the train and the one who hears it. Similarly, the frequency of light shifts due to relative motion of the source and observer, even without relativity. Relativity modifies this Doppler Effect due to time dilation.

Consider a source at rest at the origin with an observer moving in the $x$ direction. We will consider the possibility that the observer is at some distance in $y$. The beginning of one wavelength is at $t_1=0$ and $x_1=y_1=0$. The end of the wave is emitted at $t_2=\tau$ and still at $x_2=y_2=0$. This transforms to the observers frame to be at

$$ct'_1=\gamma(ct_1-\beta x)=0$$

$$x'_1=\gamma(x-\beta ct)=0$$

$$y'_1=y_1=0$$

$$ct'_2=\gamma(c\tau-\beta x)=\gamma c \tau$$

$$x'_2=\gamma(x-\beta c\tau)=-\beta\gamma c\tau$$

$$y'_2=y_2=0$$

$$\tau'=\gamma\tau$$

The time to emit the wave in the observers frame is dilated which decreases the frequency. If the wave travels to the observer in the $y$ direction, the travel time is essentially the same for the beginning and the end of the wave so the frequency is not affected. That is the transverse Doppler effect gives a red-shift

$$\nu'_\perp= {\nu\over\gamma}$$

which is entirely a relativistic effect.

If the observer is moving directly away from the source we have the additional effect of the distance to the observer increasing with time which gives rise to the parallel Doppler effect . The time at which the beginning and end of the wave arrive at the observer is

$$t'_{1o}=t'_1-{x'_1\over c}=0$$

$$t'_{2o}=t'_2-{x'_2\over c}=\gamma\tau+\beta\gamma\tau=\gamma(1+\beta)\tau$$

$$\tau'_o=\gamma(1+\beta)\tau={(1+\beta)\over\sqrt{(1+\beta)(1-\beta)}}\tau=\sqrt{1+\beta\over 1-\beta}\tau$$

$$\nu'_\parallel=\sqrt{1-\beta\over 1+\beta}\nu$$

$\beta$ is positive for the observer moving away from the source and negative if the observer is moving toward the source.