## the Doppler effect (relativistic)

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, $\bigl|\mathbf{\hat{k}}\cdot\mathbf{v}\bigr| = v$.

Fig­ure 1: Con­tri­bu­tion to fre­quency shifts due to receiver motion in the source wave field and rel­a­tivis­tic time dila­tion.

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.

Fig­ure 2: Ratio of the con­tri­bu­tions due to time dila­tion and spa­tial phase accu­mu­la­tion for the receiver in motion. Col­ors denote dif­fer­ent angles between receiver motion and wave direc­tion.

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.

Fig­ure 3: Rel­a­tive fre­quency shift for a receiver mov­ing at var­i­ous angles to the plane wave emit­ted by the source.

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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