Relativistic Doppler effect

Classically, the Doppler effect is where the frequency of sound or light changes depending on the relative velocity of the source and the observer. If the source is moving toward you, the wave appears higher frequency, and vice versa.

The derivation is simple. Consider a source at rest and a receiver moving away at speed $u$ . The source emits a light beam with frequency $\nu$ in the direction of the receiver.

When the first crest reaches the receiver, the second crest will be a distance $1/\nu=\lambda$ behind. But in the time it crosses this distance, the observer will have already moved.

Let $t_o$ be the period of light crests arriving at the observer as measured in the source frame. We can say

\begin{align*} \lambda + u t_o &= c t_o \\ \lambda &= ct_o (1-v/c) \\ t_o &= \frac{1}{\nu (1-v/c)}. \end{align*}

We use time dilation to find the period $t'$ and frequency $\nu'$ in the receiver’s frame.

\begin{align*} t' &= \frac{t_o}{\gamma} \\ \nu' &= \frac{1}{t'} = \frac{\nu (1-\beta)}{\sqrt{1-\beta^2}} = \nu \sqrt{\frac{1-\beta}{1+\beta}} \\ \text{where } \beta &= \frac vc. \end{align*}

A similar effect can be observed for a source and receiver moving perpendicular to each other.