Christian Doppler in 1942 predicted the Doppler effect for all kinds of waves, not only sound waves. The Doppler effect for light was demonstrated in 1948 by Fizeau. In conjunction with the article on the classical Doppler effect, as it occurs e.g. for sound waves, we will here examine the relativistic Doppler effect which is responsible for frequency shifts in electromagnetic waves such as light.

Einstein’s special relativity theory postulates that, as long as we are dealing with uniform motion, there is no frame of reference preferable over another. Also, all the laws of physics (and thus electromagnetism) must be valid irrespective of the chosen reference frame. One immediate conclusion then must be that we need not differentiate between a moving source (with stationary receiver) and a moving receiver (with stationary source) as in the classical case with a medium.

Again, we start with the plane wave, represented by its phasor $A(t)$, as in the classical case.

$$A(\mathbf{r},t)=A_{0}\exp\bigl(i\omega t-i\mathbf{k}\cdot\mathbf{r}\bigr) \label{plane-wave}$$

where $A_0$ describes the (constant) amplitude and phase of the emitted signal, $\omega$ is its frequency at the source and $\mathbf{k}$ is the wave vector, describing the spatial frequency of the wave and its direction.

In the reference frame of the source, a moving receiver can be described similarly to the classical case, with the wave field at the location of the receiver moving with velocity $\mathbf{v}$ (in the source frame of reference) given by

$$A(\mathbf{r},t)=A_{0}\exp\Bigl[i\omega t-i\mathbf{k}\cdot\bigl(\mathbf{r}_{0}+\mathbf{v} t\bigr)\Bigr] \label{field-R}$$

In the classical case, the received frequency was determined by differentiation of the phase of the field with respect to time,

$$\begin{aligned}

\omega^\prime_{R} &= \partial_t \arg\bigl[A(\mathbf{r},t)\bigr] \vphantom{\biggl(\biggr)}\\

& =\omega-\mathbf{k}\cdot\mathbf{v}\\

& =\omega\biggl(1-\frac{\mathbf{\hat{k}}\cdot\mathbf{v}}{c}\biggr)

\end{aligned}\label{phase-differentiation}$$

In relativistic physics, however, in order to obtain the frequency observed by the receiver we have to do the phase differentiation in the reference frame of the receiver. Einstein postulated that we need to apply the Lorentz transformation to convert the measurements of time and space in a reference frame which is defined to be at rest (source) into the measurements of the same quantities that would be obtained in another reference frame (receiver) which is moving with velocity $\mathbf{v}$ relative to the frame at rest. The Lorentz transforms for time and space are

$$\begin{gathered}

t^\prime = \gamma \biggl(t - \frac{\mathbf{v}\cdot \mathbf{r}}{c^2} \biggr)\\

\mathbf{r}^\prime = \mathbf{r} + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}}{\mathbf{v} \cdot \mathbf{v}} - \gamma \, \mathbf{v} \, t

\end{gathered}\label{Lorentz-transformation}$$

where the primed variables denote the quantities in the moving reference frame and

$$\gamma = \biggl(1 - \frac{\mathbf{v}\cdot\mathbf{v}}{c^2}\biggr)^{-\frac{1}{2}}$$

is the Lorentz factor. The inverse transforms are

$$\begin{gathered}

t = \gamma \biggl(t^\prime + \frac{\mathbf{v}\cdot \mathbf{r}^\prime}{c^2} \biggr)\\

\mathbf{r} = \mathbf{r}^\prime + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}^\prime}{\mathbf{v} \cdot \mathbf{v}} + \gamma \, \mathbf{v} \, t^\prime

\end{gathered}\label{inverse-Lorentz-transformation}$$

The difference in the forward and inverse transforms is merely in the sign of $\mathbf{v}$ in agreement with the postulated equivalence of all reference frames. It is equally valid to define the receiver reference frame to be at rest with the source frame moving in the opposite direction $-\mathbf{v}$ (assuming the coordinate systems in both cases to have the same orientation). Hence the inverse transform can only differ in the sign of the velocity vector.

We can write $\eqref{plane-wave}$ in terms of the receiver coordinates $t^\prime$ and $\mathbf{r}^\prime$ by inserting $\eqref{inverse-Lorentz-transformation}$ and regrouping the arguments

$$\begin{align}

A(\mathbf{r},t)&=A_{0}\exp\biggl(i \omega \gamma \biggl[t^\prime + \frac{\mathbf{v}\cdot \mathbf{r}^\prime}{c^2} \biggr] - i\mathbf{k}\cdot \Bigl[ \mathbf{r}^\prime + \mathbf{v} \bigl(\gamma - 1 \bigr) \frac{\mathbf{\hat{v}} \cdot \mathbf{r}^\prime}{\mathbf{v} \cdot \mathbf{v}} + \gamma \, \mathbf{v} \, t^\prime \Bigr] \biggr)\notag\\

&= A_{0}\exp\bigl( i \omega^\prime t^\prime - i \mathbf{k}^\prime \cdot \mathbf{r}^\prime \bigr)\vphantom{\biggl(\biggr)}

\end{align}$$

with

$$\omega^\prime = \gamma \bigl(\omega - \mathbf{k}\cdot\mathbf{v}\bigr) \quad \text{and} \quad \mathbf{k}^\prime = \mathbf{k} - \frac{\omega \gamma}{c^2} \mathbf{v} + \frac{\mathbf{k} \cdot \mathbf{v}}{\mathbf{v}\cdot\mathbf{v}} \bigl(\gamma - 1\bigr) \mathbf{\hat{v}}\label{frequency-R}$$

We see that the frequency detected by a receiver at a fixed location $\mathbf{r}^\prime$ within its own reference frame, $\omega^\prime$, is just the frequency $\omega^\prime_R$ from $\eqref{phase-differentiation}$, describing the receiver motion through the wave field in the source reference frame, multiplied by the Lorentz factor $\gamma$. Intuitively, this is sensible since the Lorentz factor also describes the relation between the rates of passage of time in both reference frames, cf. $\eqref{Lorentz-transformation}$. Unlike the classical Doppler effect, time dilation causes a “true” frequency shift in the receiver reference frame since the very rate of passage of time is now different.

To determine which of the two is the dominating effect at some velocity, we plot the relative frequency shift due to time dilation, which is given by $\gamma - 1$, and the relative detected frequency shift due to $\eqref{phase-differentiation}$, which is given by $\mathbf{\hat{k}}\cdot\mathbf{v}/c$, versus $v$ and $\mathbf{\hat{k}}\cdot\mathbf{v}$, respectively. For the case that $\mathbf{k}$ and $\mathbf{v}$ are (anti-) parallel, $\bigl|\mathbf{\hat{k}}\cdot\mathbf{v}\bigr| = v$.

For $\mathbf{\hat{k}}\cdot\mathbf{v} = \pm 1$, the effect of time dilation is less than the geometric phase $\eqref{phase-differentiation}$ for all velocities not very close to light speed. However, for some applications like GPS the relativistic correction significantly increases accuracy. For applications like vibration measurement via the Doppler effect for light the relativistic contribution can usually be neglected.

The geometric frequency shift in $\eqref{phase-differentiation}$ depends on the angle between wave and receiver motion while the time dilation does not. When the receiver moves nearly orthogonally to the wave, Doppler shifte due to time dilation can dominate at comparatively low speeds, as is shown in Figure 2.

Depending on the angle between the wave and the receiver motion, both effects can offset each other. In particular, for any $v$ so that $0 \lt v \lt c$ there exists a direction with $\mathbf{\hat{k}} \cdot \mathbf{v} \lt 0$ so that $\omega^\prime = \omega$. This is shown exemplary for $v = 10^6$m/s – and thus an angle of $\cos^{-1} \bigl(\mathbf{\hat{k}} \cdot \mathbf{\hat{v}}\bigr) = 89.90^\circ$ – in Figure 3 which plots the compound Doppler shift due to both effects versus velocity.

We also see in $\eqref{frequency-R}$ that the direction of the wave in the receiver reference frame, described by $\mathbf{k}^\prime$, changes when $\mathbf{k}$ and $\mathbf{v}$ are not (anti-) parallel. This effect is referred to as aberration and is significant e.g. in astronomy when determining the location of stars.

Equation $\eqref{frequency-R}$ is written in very general terms and valid for arbitrary directions of $\mathbf{k}$ and $\mathbf{v}$. In the “standard” case of both being (anti-) parallel, it simplifies to

$$\omega^\prime = \omega\biggl(1-\frac{v_{R}}{c}\biggr) \bigg/ \sqrt{1 - \frac{v_R^2}{c^2}} = \omega \sqrt{\frac{c-v_R \vphantom{v_R^2}}{c+v_R}}$$

which may look more familiar from physics books.

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