Comment on "The relativistic Doppler effect: when a zero-frequency ..."

TL;DR

This paper critiques a previous work on the relativistic Doppler effect, identifying and correcting errors related to the equations used, thereby clarifying the correct interpretation of frequency shifts for approaching sources.

Contribution

It provides a correction to prior faulty conclusions by addressing errors in the equations used in the original analysis of the relativistic Doppler effect.

Findings

Corrects the erroneous equations in the previous paper

Clarifies the conditions for zero-frequency and red shifts in relativistic Doppler effect

Ensures accurate interpretation of frequency shifts for approaching sources

Abstract

In the paper "The relativistic Doppler effect: when a zero-frequency shift or a red shift exists for sources approaching the observer, Ann. Phys. (Berlin) 523, No. 3, 239-246 (2011), DOI 10.1002/andp.201000099 by C. Wang the use of an erroneous equation ended up at a number of faulty conclusions which are corrected in the present Comment.

Equations

$$k^{\mu }=[\frac{\gamma {\omega }^{\prime }}{c}(1+\beta \cos {\varphi }^{\prime }),\frac{\gamma {\omega }^{\prime }}{c}(\cos {\varphi }^{\prime }+\beta ),\frac{{\omega }^{\prime }}{c}\sin {\varphi }^{\prime },0].$$

$$\omega =\gamma {\omega }^{\prime }(1+\beta \cos {\varphi }^{\prime })$$

$$\cos \varphi =\frac{\cos {\varphi }^{\prime }+\beta }{1+\beta \cos {\varphi }^{\prime }}$$

$$\sin \varphi =\frac{\sin {\varphi }^{\prime }}{\gamma (1+\beta \cos {\varphi }^{\prime })}.$$

$$\Delta ={\varphi }^{\prime }-\varphi$$

$${\omega }^{\prime }=\gamma \omega (1-\beta \cos \varphi )$$

$$\cos {\varphi }^{\prime }=\frac{\cos \varphi -\beta }{1-\beta \cos \varphi },$$

$$\cos {\varphi }^{\prime }=\frac{\beta -\cos \varphi }{1-\beta \cos \varphi }.$$

$$\cos \varphi =\frac{-\beta -\cos {\varphi }^{\prime }}{1+\beta \cos {\varphi }^{\prime }}$$

Full text

Keywords: Doppler effect, special relativity theory, zero-frequency shift, aberration of light.

In this Comment are corrected some results that are obtained in wang . Instead of using the phase invariance and the time dilation in the derivation of the expressions for the Doppler shift and the aberration as in wang we use the Lorentz transformations (LT) of the four-dimensional (4D) wave vector. Otherwise, all notations are kept the same as in wang .

Let us assume that the spectrograph is at rest in the laboratory inertial frame of reference (IFR) $S$. The light source is at rest in the IFR $S^{\prime}$ which is moving with velocity ${\mathbf{v}}$ relative to $S$ along the common $x,x^{\prime}$–axis. The components of the wave 4-vector in $S$ are $k^{\mu}=(\omega/c)(1,\cos\phi,\sin\phi,0)$ and in $S^{\prime}$ they are $k^{\prime\mu}=(\omega^{\prime}/c)(1,\cos\phi^{\prime},\sin\phi^{\prime},0)$ for which it holds $k^{\mu}k_{\mu}=k^{\prime\mu}k^{\prime}_{\mu}=0$. Here, $\phi$ ($\phi^{\prime}$) is the angle between the wave direction of propagation and $x$– ($x^{\prime}$–)axis, i.e. relative to ${\mathbf{v}}$. The components of $k^{\mu}$ can be obtained by the LT of the components $k^{\prime\mu}$ which yields

[TABLE]

Hence, the Doppler shift is given as

[TABLE]

whereas the equations that describe the change in the direction of wave propagation are

[TABLE]

and

[TABLE]

In fact, if the observation of the unshifted line (i.e. of the frequency $\omega^{\prime}=\omega_{0}$ from the atom at rest) is performed at an observation angle $\phi^{\prime}$ in $S^{\prime}$, the rest frame of the emitter, then the same light wave (from the same but now moving atom) will have the shifted frequency $\omega$ and will be seen at an observation angle $\phi$ (generally different from $\phi^{\prime}$) in $S$, the rest frame of the spectrometer. In astronomy the angular shift

[TABLE]

is dubbed aberration.

The inverted relations between IFRs $S$ and $S^{\prime}$ are obtained by mere interchange of $\phi$ and $\phi^{\prime}$ and $\beta$ by $-\beta$. Thus, the inverted Doppler effect and cosine, Eqs. (2) and (3), read

[TABLE]

and

[TABLE]

respectively. We emphasize that Eqs. (6) and (7) have been derived by Einstein in his fundamental work on special relativity theory (SRT) ein905 and may also be found in many textbooks on SRT like e.g. pauli .

In contrast to the above Eq. (7) the cosine Eq. (6) in wang reads

[TABLE]

This erroneous equation gives unphysical results that are presented in Fig. 3 of Ref. wang and repeated by the full-line curves in Fiq. 1 here. These curves pretend to describe the dependence of $\phi$, $\phi^{\prime}$, and $\phi+\phi^{\prime}$ as a function of the ratio $\omega^{\prime}/\omega$ for $\gamma=2$. Let us firstly examine the case of the light wave which is head-on approaching the observer, i.e. the case of $\phi=0$ in IFR $S$. Eq. (8) suggests an obviously unphysical behavior of such a wave because in the emitter IFR $S^{\prime}$ it would have to recede with $\phi^{\prime}=\pi$. Equally unphysical is the case of a light wave which is receding from the observer at $\phi=\pi$ because it would have to move towards the observer in IFR $S^{\prime}$ with $\phi^{\prime}=0$. Further consequence of the erroneous Eq. (8) is that the Doppler zero-frequency shift (zfs), i.e. $\omega^{\prime}=\omega$ occurs when the two position angles $\phi$ and $\phi^{\prime}$ are equal or $\phi_{\rm zfs}=\phi_{\rm zfs}^{\prime}$. The author states …there is no frequency shift in such a case, although the light aberration must exist ($\phi+\phi^{\prime}\neq\pi$ for $\phi^{\prime}=\phi$) … wang . It is unclear from where the assertion that the light aberration vanishes for $\phi^{\prime}+\phi=\pi$ does come? In fact, it will be shown below (see Fig. 2) that for $\phi^{\prime}+\phi=\pi$ the aberration $\Delta$ is at maximum. Also, he claims the zero shift taking place at $\phi=\phi_{\rm zfs}$, where the aberration reaches a maximum wang which is an entirely contradictory statement because for $\phi=\phi^{\prime}$ the aberration, Eq. (5), vanishes. For $\phi$ and $\phi^{\prime}$ of Eq. (8) the resulting $\Delta$ is shown by the dashed curve. Because $\Delta$ changes the sign for $\omega^{\prime}>\omega$, in order to fit into the figure frame, in Fig. 1 is displayed the absolute value $|\Delta|$.

According to the principle of relativity the physical reality should not depend on the concrete IFR and coordinate basis used in describing it. The most simple way to verify the correctness of an expression is to interchange the IFRs, i.e. $S$ and $S^{\prime}$. In that case Eq. (8) becomes

[TABLE]

and its predictions for $\phi$, $\phi^{\prime}$, and $\phi+\phi^{\prime}$ are shown by the dash-dotted line curves in Fig. 1. All three considered physical quantities $\phi$, $\phi^{\prime}$, and $\phi+\phi^{\prime}$ display an entirely different feature: these $\phi$ and $\phi^{\prime}$ are mirror symmetric about $\pi/2$ relative to their previous graphs while their sum is symmetric about the angle $\pi$. One again has $\phi_{\rm zfs}=\phi_{\rm zfs}^{\prime}$ but its value is $\pi$ minus the previous one that was obtained from Eq. (8). Although the aberration curve seems to be unchanged that is due to its absolute value. Namely, with Eq. (9) $\Delta>0$ for $\omega^{\prime}>\omega$ and $\Delta<0$ for $\omega^{\prime}<\omega$.

Figure 2 displays the correct observables $\phi$, $\phi^{\prime}$, $\Delta$, and $\phi+\phi^{\prime}$ obtained by using Eqs. (3), (7), and (5), respectively. Inverting the role of the IFRs $S$ and $S^{\prime}$ gives the identical results as it should be (the full, Eq. (3), and dash-dotted curves, Eq. (7), are laying over each other). Because $\phi$ and $\phi^{\prime}$ are monotonically increasing functions of $\omega^{\prime}/\omega$ such is also $\phi+\phi^{\prime}$. The aberration $\Delta$ is indeed maximal at $\omega^{\prime}=\omega$ where $\phi+\phi^{\prime}=\pi$ and $\phi_{\rm zfs}^{\prime}=\pi-\phi_{\rm zfs}$.

Another way to verify the correctness of Eqs. (1) to (4) is to use a geometric approach to SRT from ive02l . As seen from Sec. 7.2 in ive02l an abstract coordinate-free wave vector $k^{\rm a}$ is represented in $S$ ($S^{\prime}$) by the coordinate-based geometric quantity (CBGQ) $k^{\mu}e_{\mu}$ ($k^{\prime\mu}e^{\prime}_{\mu}$) comprising both the components $k^{\mu}$ and the 4D basis vectors $e_{\mu}$. Any CBGQ is an invariant 4D quantity under the LT since the components transform by the LT and the basis vectors by the inverse LT leaving the whole CBGQ unchanged; it is the same physical quantity for relatively moving inertial observers. It can be easily seen that with (1) it holds that $k^{\mu}e_{\mu}$=$k^{\prime\mu}e^{\prime}_{\mu}$, which proves the validity of (1), i.e. of Eqs. (2), (3) and (4) and at the same time it disproves Eq. (8).

The author is indebted to Dr. T. Ivezić for initiating this study, his constant encouragement, and the critical reading of the manuscript. This work has been supported in part by Croatian Science Foundation under the Project No. 7194 and in part by the Scientific center of excellence for advance materials and sensors.