Comment on "The relativistic Doppler effect: when a zero-frequency ..."
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Comment on ”The relativistic Doppler effect: when a zero-frequency. . . ”
Z. Basrak ∗ Ruder Boˇskovi´c Institute, Zagreb, Croatia (Dated: April 23, 2019)In the paper The relativistic Doppler effect: when a zero-frequency shift or a red shift existsfor sources approaching the observer, Ann. Phys. (Berlin) 523, No. 3, 239-246 (2011) / DOI10.1002/andp.201000099 by C. Wang the use of an erroneous equation ended up at a number offaulty conclusions which are corrected in the present Comment.
Keywords:
Doppler effect, special relativity theory, zero-frequency shift, aberration of light.
In this Comment are corrected some results that are ob-tained in [1]. Instead of using the phase invariance andthe time dilation in the derivation of the expressions forthe Doppler shift and the aberration as in [1] we usethe Lorentz transformations (LT) of the four-dimensional(4D) wave vector. Otherwise, all notations are kept thesame as in [1].Let us assume that the spectrograph is at rest in thelaboratory inertial frame of reference (IFR) S . Thelight source is at rest in the IFR S ′ which is mov-ing with velocity v relative to S along the common x, x ′ –axis. The components of the wave 4-vector in S are k µ = ( ω/c )(1 , cos φ, sin φ,
0) and in S ′ they are k ′ µ = ( ω ′ /c )(1 , cos φ ′ , sin φ ′ ,
0) for which it holds k µ k µ = k ′ µ k ′ µ = 0. Here, φ ( φ ′ ) is the angle between the wavedirection of propagation and x – ( x ′ –)axis, i.e. relative to v . The components of k µ can be obtained by the LT ofthe components k ′ µ which yields k µ = [ γω ′ c (1+ β cos φ ′ ) , γω ′ c (cos φ ′ + β ) , ω ′ c sin φ ′ , . (1)Hence, the Doppler shift is given as ω = γω ′ (1 + β cos φ ′ ) (2)whereas the equations that describe the change in thedirection of wave propagation arecos φ = cos φ ′ + β β cos φ ′ (3)and sin φ = sin φ ′ γ (1 + β cos φ ′ ) . (4)In fact, if the observation of the unshifted line (i.e. ofthe frequency ω ′ = ω from the atom at rest) is performedat an observation angle φ ′ in S ′ , the rest frame of theemitter, then the same light wave (from the same butnow moving atom) will have the shifted frequency ω andwill be seen at an observation angle φ (generally different ∗ [email protected] F ' F ' FF '- F -( F '- F )-( F '- F ) F '- FF + F ' F + F ' w '/ w F , F ' , F + F ' , | F ' - F | ( i n p r a d ) FIG. 1. (Color online.) Angles φ (blue), φ ′ (red), and φ + φ ′ (black) as well as the absolute value of the aberration | φ ′ − φ | as a function of ω ′ /ω . Results due to Eq. (8) are shown by thefull-line curves while those of Eq. (9) by the dash-dotted-linecurves. The aberration, Eq. (5) is shown by the dashed-linecurves, in magenta for IFR S and in cyan for IFR S ′ . from φ ′ ) in S , the rest frame of the spectrometer. Inastronomy the angular shift∆ = φ ′ − φ (5)is dubbed aberration.The inverted relations between IFRs S and S ′ are ob-tained by mere interchange of φ and φ ′ and β by − β .Thus, the inverted Doppler effect and cosine, Eqs. (2)and (3), read ω ′ = γω (1 − β cos φ ) (6)and cos φ ′ = cos φ − β − β cos φ , (7)respectively. We emphasize that Eqs. (6) and (7) havebeen derived by Einstein in his fundamental work on spe-cial relativity theory (SRT) [2] and may also be found inmany textbooks on SRT like e.g. [3]. F ' FF '- F F '- FF + F ' w '/ w F , F ' , F + F ' , F ' - F ( i n p r a d ) FIG. 2. (Color online.) Same as Fig. 1 but by using Eqs. (3)and (7) for φ and φ ′ as well as (5) for ∆. In contrast to the above Eq. (7) the cosine Eq. (6) in[1] reads cos φ ′ = β − cos φ − β cos φ . (8)This erroneous equation gives unphysical results that arepresented in Fig. 3 of Ref. [1] and repeated by the full-linecurves in Fiq. 1 here. These curves pretend to describethe dependence of φ , φ ′ , and φ + φ ′ as a function of theratio ω ′ /ω for γ = 2. Let us firstly examine the case ofthe light wave which is head-on approaching the observer,i.e. the case of φ = 0 in IFR S . Eq. (8) suggests anobviously unphysical behavior of such a wave because inthe emitter IFR S ′ it would have to recede with φ ′ = π .Equally unphysical is the case of a light wave which isreceding from the observer at φ = π because it wouldhave to move towards the observer in IFR S ′ with φ ′ =0. Further consequence of the erroneous Eq. (8) is thatthe Doppler zero-frequency shift (zfs), i.e. ω ′ = ω occurswhen the two position angles φ and φ ′ are equal or φ zfs = φ ′ zfs . The author states . . . there is no frequency shiftin such a case, although the light aberration must exist( φ + φ ′ = π for φ ′ = φ ) . . . [1]. It is unclear fromwhere the assertion that the light aberration vanishes for φ ′ + φ = π does come? In fact, it will be shown below(see Fig. 2) that for φ ′ + φ = π the aberration ∆ is atmaximum. Also, he claims the zero shift taking place at φ = φ zfs , where the aberration reaches a maximum [1]which is an entirely contradictory statement because for φ = φ ′ the aberration, Eq. (5), vanishes. For φ and φ ′ of Eq. (8) the resulting ∆ is shown by the dashed curve.Because ∆ changes the sign for ω ′ > ω , in order to fitinto the figure frame, in Fig. 1 is displayed the absolutevalue | ∆ | .According to the principle of relativity the physical re-ality should not depend on the concrete IFR and coordi-nate basis used in describing it. The most simple way toverify the correctness of an expression is to interchangethe IFRs, i.e. S and S ′ . In that case Eq. (8) becomescos φ = − β − cos φ ′ β cos φ ′ (9)and its predictions for φ , φ ′ , and φ + φ ′ are shown bythe dash-dotted line curves in Fig. 1. All three consid-ered physical quantities φ , φ ′ , and φ + φ ′ display an en-tirely different feature: these φ and φ ′ are mirror sym-metric about π/ π . One againhas φ zfs = φ ′ zfs but its value is π minus the previous onethat was obtained from Eq. (8). Although the aberrationcurve seems to be unchanged that is due to its absolutevalue. Namely, with Eq. (9) ∆ > ω ′ > ω and ∆ < ω ′ < ω .Figure 2 displays the correct observables φ , φ ′ , ∆, and φ + φ ′ obtained by using Eqs. (3), (7), and (5), respec-tively. Inverting the role of the IFRs S and S ′ gives theidentical results as it should be (the full, Eq. (3), anddash-dotted curves, Eq. (7), are laying over each other).Because φ and φ ′ are monotonically increasing functionsof ω ′ /ω such is also φ + φ ′ . The aberration ∆ is indeedmaximal at ω ′ = ω where φ + φ ′ = π and φ ′ zfs = π − φ zfs .Another way to verify the correctness of Eqs. (1) to (4)is to use a geometric approach to SRT from [4]. As seenfrom Sec. 7.2 in [4] an abstract coordinate-free wave vec-tor k a is represented in S ( S ′ ) by the coordinate-based ge-ometric quantity (CBGQ) k µ e µ ( k ′ µ e ′ µ ) comprising boththe components k µ and the 4D basis vectors e µ . AnyCBGQ is an invariant 4D quantity under the LT since thecomponents transform by the LT and the basis vectorsby the inverse LT leaving the whole CBGQ unchanged; itis the same physical quantity for relatively moving iner-tial observers. It can be easily seen that with (1) it holdsthat k µ e µ = k ′ µ e ′ µ , which proves the validity of (1), i.e. ofEqs. (2), (3) and (4) and at the same time it disprovesEq. (8).The author is indebted to Dr. T. Ivezi´c for initiatingthis study, his constant encouragement, and the criticalreading of the manuscript. This work has been supportedin part by Croatian Science Foundation under the ProjectNo. 7194 and in part by the Scientific center of excellencefor advance materials and sensors. [1] C. Wang , Ann. Phys. (Berlin) , 239 (2011).[2]
A. Einstein , Ann. Phys. (Leipzig) , 891 (1905); in ThePrinciple of Relativity (Methuen and Co., London, 1923),p. 56. [3] W. Pauli , Theory of Relativity (Pergamon Press, Londonand New York, 1958), p. 19.[4]
T. Ivezi´c , Foundations Phys. Lett.75