Transverse Doppler Effect

Figure 3.13: Relativistic Doppler effect.

Consider the situation illustrated in Figure 3.13. A radiation source that is located at the origin of reference frame $S$ emits electromagnetic radiation of angular frequency $\omega$, whose direction of propagation lies in the $x$-$y$ plane and subtends an angle $\theta$ with the $x$-axis. It follows that the wavevector of the radiation has the non-zero components

$$k_x = \frac{\omega}{c}\,\cos\theta,$$ (3.214)

$$k_y = \frac{\omega}{c}\,\sin\theta.$$ (3.215)

Suppose that the radiation is observed in a frame $S'$ that moves with velocity ${\bf v} = v\,{\bf e}_x$, and is in a standard configuration, with respect to $S$. In $S'$, let $\omega'$ be the angular frequency of the radiation, and let $\theta'$ be the angle subtended by its direction of propagation and the $x'$-axis. It follows that, in $S'$, the wavevector of the radiation has the non-zero components

$$k_x' = \frac{\omega'}{c}\,\cos\theta',$$ (3.216)

$$k_y' = \frac{\omega'}{c}\,\sin\theta'.$$ (3.217)

The previous four equations can be combined with Equations (3.201)–(3.204) to give

$$\omega'\,\cos\theta' = \gamma\left(\cos\theta-\frac{v}{c}\right)\omega,$$ (3.218)

$$\omega'\,\sin\theta' = \omega\,\sin\theta,$$ (3.219)

$$\omega' = \gamma\left(1-\frac{v}{c}\,\cos\theta\right)\omega.$$ (3.220)

Given that $\omega\propto f$, we deduce that

$$f'= \gamma\left(1-\frac{v}{c}\,\cos\theta\right)f,$$ (3.221)

which is the generalization of formula (3.209) to the case where the radiation is not propagating parallel to the relative velocity of the source and the observer. In particular, if $\theta=\pi/2$, so that the radiation is propagating in a direction that is perpendicular to the relative velocity of the source and the observer, then

$$f' = \gamma\,f.$$ (3.222)

In other words, in this case, the observer always measures an increased frequency of the radiation, relative to the frequency measured in the rest frame of the source. This effect is known as the transverse Doppler effect, and is a purely relativistic effect (i.e., it has no concomitant in Newtonian dynamics). The transverse Doppler effect was experimentally verified by Ives and Stilwell in 1938.

Finally, Equations (3.218) and (3.219) can be combined to give

$$\tan\theta' = \frac{\sin\theta}{\gamma\,(\cos\theta+v/c)}.$$ (3.223)

However, this formula is identical to the relativistic aberration formulae, (3.131), that we derived previously.