5.7 Doppler Effect for Light

As discussed in the chapter on sound, if a source of sound and a listener are moving farther apart, the listener encounters fewer cycles of a wave in each second, and therefore lower frequency, than if their separation remains constant. For the same reason, the listener detects a higher frequency if the source and listener are getting closer. The resulting Doppler shift in detected frequency occurs for any form of wave. For sound waves, however, the equations for the Doppler shift differ markedly depending on whether it is the source, the observer, or the air, which is moving. Light requires no medium, and the Doppler shift for light traveling in vacuum depends only on the relative speed of the observer and source.

The Relativistic Doppler Effect

Suppose an observer in S sees light from a source in $S\prime$ moving away at velocity v (Figure 5.22). The wavelength of the light could be measured within $S\prime$—for example, by using a mirror to set up standing waves and measuring the distance between nodes. These distances are proper lengths with $S\prime$ as their rest frame, and change by a factor $\sqrt{1-v^2\text{/}{c}^2}$ when measured in the observer’s frame S, where the ruler measuring the wavelength in $S\prime$ is seen as moving.

Figure 5.22 (a) When a light wave is emitted by a source fixed in the moving inertial frame $S\prime ,$ the observer in S sees the wavelength measured in $S\prime .$ to be shorter by a factor $\sqrt{1-v^2\text{/}{c}^2}.$ (b) Because the observer sees the source moving away within S, the wave pattern reaching the observer in S is also stretched by the factor$\left(c\text{Δ}t+v\text{Δ}t\right)\text{/}\left(c\text{Δ}t\right)=1+v\text{/}c.$

If the source were stationary in S, the observer would see a length $c\text{Δ}t$ of the wave pattern in time $\text{Δ}t.$ But because of the motion of $S\prime$ relative to S, considered solely within S, the observer sees the wave pattern, and therefore the wavelength, stretched out by a factor of

$$\frac{c\text{Δ}{t}_{\text{period}}+v\text{Δ}{t}_{\text{period}}}{c\text{Δ}{t}_{\text{period}}}=1+\frac{v}{c}$$

as illustrated in (b) of Figure 5.22. The overall increase from both effects gives

$${\lambda }_{\text{obs}}={\lambda }_{\text{src}}\left(1+\frac{v}{c}\right)\sqrt{\frac{1}{1-\frac{v^2}{c^2}}}={\lambda }_{\text{src}}\left(1+\frac{v}{c}\right)\sqrt{\frac{1}{\left(1+\frac{v}{c}\right)\left(1-\frac{v}{c}\right)}}={\lambda }_{\text{src}}\sqrt{\frac{\left(1+\frac{v}{c}\right)}{\left(1-\frac{v}{c}\right)}}$$

where ${\lambda }_{\text{src}}$ is the wavelength of the light seen by the source in $S\prime$ and ${\lambda }_{\text{obs}}$ is the wavelength that the observer detects within S.