Relativistic Doppler Effect

Consider the two inertial reference frames, $S$ and $S'$, discussed in Section 3.3.6. Suppose that we place a radiation source at the origin of reference frame $S$. Let the source emit plane waves of angular frequency $\omega$ that travel in the positive $x$-direction. It follows that the wavevector of the radiation in $S$ is ${\bf k} = (k_x$, 0, 0), where

$$\omega=k_x\,c.$$ (3.205)

[See Equation (3.199).]

Consider an observer located at the origin of frame $S'$. To this observer, the radiation source appears to move at the speed ${\bf v}= - v\,{\bf e}_x$. Let ${\bf k}' = (k_x'$, 0, 0) and $\omega'$ be the wavevector and angular frequency of the radiation measured by our observer. It follows from Equations (3.201), (3.204), and (3.205) that

$$k_x' = \gamma\left(k_x-\frac{v\,\omega}{c^2}\right) = \gamma\left(1-\frac{v}{c}\right)k_x,$$ (3.206)

$$\omega' = \gamma\,(\omega-v\,k_x) = \gamma\left(1-\frac{v}{c}\right)\omega.$$ (3.207)

Let $f=\omega/(2\pi)$ and $\lambda=2\pi/k_x$ be the frequency (in hertz) and wavelength, respectively, of the radiation emitted by the source in its rest frame, and let $f'=\omega'/(2\pi)$ and $\lambda'=2\pi/k_x'$ be the frequency (in hertz) and wavelength, respectively, of the radiation measured by an observer in the frame $S'$, in which the source recedes directly away from the observer at the speed $v$. Given that $\gamma=(1-v^2/c^2)^{-1/2}$, we deduce that

$$\lambda' = \left(\frac{1+v/c}{1-v/c}\right)^{1/2}\lambda,$$ (3.208)

$$f' = \left(\frac{1-v/c}{1+v/c}\right)^{1/2}f.$$ (3.209)

It follows that if a radiation source recedes from an observer (or vice versa, because, in the absence of a medium through which electromagnetic waves propagate, all motion of sources and observers is relative) then the wavelength and frequency of the radiation measured by the observer will be larger and smaller, respectively, than the corresponding values measured in the rest frame of the source. This shift in the wavelength and frequency of electromagnetic radiation due to the relative motion of the observer and source is known as the relativistic Doppler effect. [Note that the non-relativistic Doppler effect for sound waves takes a different form to Equations (3.208) and (3.209) because motion of the source and the observer can be distinguished from one another in the presence of a medium though which the waves travel.]

By analogy with the previous two formulae, if the source moves directly toward the observer with the speed $v$ then

$$\lambda' = \left(\frac{1-v/c}{1+v/c}\right)^{1/2}\lambda,$$ (3.210)

$$f' = \left(\frac{1+v/c}{1-v/c}\right)^{1/2}f.$$ (3.211)

In this case, the wavelength and frequency of the radiation measured by the observer are smaller and larger, respectively, than the corresponding values measured in the rest frame of the source. Thus, we can write the composite formulae

$$\lambda' = \left(\frac{1\pm v/c}{1\mp v/c}\right)^{1/2}\lambda,$$ (3.212)

$$f' = \left(\frac{1\mp v/c}{1\pm v/c}\right)^{1/2}f,$$ (3.213)

where the upper/lower signs correspond to the source moving directly away from/toward the observer (or vice versa).