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ErratumFree Access

Erratum: QED-based derivation of the general forms of the relativistic Doppler effect and of the relativistic aberration for uniformly rotating frames and for uniformly accelerated frames

https://doi.org/10.1142/S0217732323920013Cited by:0 (Source: Crossref)
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Abstract

The current paper corrects an error that occurred in a previously published paper. The error starts from Sec. 3, Eq. (3.2) whereby the scalar value of the wave-vector needs to be corrected. The net result is that the final formulas for the relativistic Doppler effect and for aberration get simplified. The source of the error is an extra multiplication factor by “c”. We need to replace the incorrect k=C′⋅k′c+a44k′ in [A. Sfarti, MPLA 37, 2250238 (2022)] with the correct expression k=C′⋅k′+a44k′.

[Modern Physics Letters A, Vol. 37, Nos. 37 & 38, 2250238 (2022), https://doi.org/10.1142/S0217732322502388]

PACS: 03.30.+p, 52.20.Dq, 52.70.Nc

3. Application: The General Formulas for the Relativistic Doppler Effect and for Relativistic Aberration

We will derive the most general formula of the relativistic Doppler effect for the case of source and receiver moving with arbitrary velocities vs,vr with respect to an inertial frame F. The non-inertial frame F′(τ) is moving with velocity u with respect to the inertial frame F.

Let fr be the frequency of the electromagnetic wave detected by the receiver R. Let Er=hfr be the total energy of the electromagnetic wave measured by the receiver R. In the non-inertial frame F′(τ) the frequency detected by the receiver is f′r.

Let fs be the frequency of the electromagnetic wave emitted by the source S. Let Es=hfs be the total energy of the electromagnetic wave emitted by the source S. In the non-inertial frame F′(τ) the frequency emitted by the source is f′s. We are going to establish the relationship between f′s and f′r, that is, we will establish the expression of the relativistic Doppler effect in the non-inertial frame. Likewise, we will establish the expression of the relativistic aberration in the non-inertial frame F′(τ).

From QED we know that

(p,E∕c)=h(k,f).(3.1)
Substituting (3.1) into (2.10) we obtain
k=B′k′+D′k′,k=C′⋅k′+a44k′.(3.2)
The aberration formula is obtained from (3.2)
k=B′k′+D′k′.(3.3)
The alternative equation, as measured from the rotating or accelerating frame, is
B′k′−D′(C′.k′)1a44=k−D′ka44.(3.4)
We know from16 that in the inertial frame
frfs=1−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(3.5)
Substituting (3.5) into (3.2) we obtain the following Doppler relationship in the rotating frame :
frfs=krks=C′⋅k′rc+a44f′rC′⋅k′sc+a44f′s=1−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(3.6)
The above can be re-written as
C⋅ˆkr+α44C⋅ˆks+α44f′rf′s=1−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(3.7)
Since it is obvious that ˆkr=ˆks=ˆk it follows that
f′rf′s=1−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(3.8)

4. Alternative Formulation

Let

A−1=[α11α12α13α14α21α22α23α24α31α32α33α34α41α42α43α44],B=[α11α12α13α21α22α23α31α32α33],C=(α41α42α43),D=[α14α24α34].(4.1)
Then, we can replace (3.2) with
k′=Bk+Dk,k′=C⋅k+α44k.(4.2)
From (4.2) we obtain immediately the unit vector
ˆk′=Bk+DkC⋅k+α44k=Bˆk+DC⋅ˆk+α44.(4.3)
Then, the aberration formula in the non-inertial frame F′(τ) is
cosθ′=ˆk′.ββ=(Bˆk+D).ˆβC⋅ˆk+α44,β=uc.(4.4)
The second formula of (3.2) gets replaced with
f′=C⋅kc+α44f.(4.5)
The Doppler formula in the non-inertial frame is
f′rf′s=C⋅krc+α44frC⋅ksc+α44fs=C⋅ˆkr+α44C⋅ˆks+α441−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(4.6)
Since it is obvious that ˆkr=ˆks=ˆk it follows that
f′rf′s=1−βr.ˆk1−βs.ˆkγ(vr)γ(vs).(4.7)
One can appreciate that the second approach produces more elegant formulas, all at the expense of inverting matrix A in order to obtain the terms necessary for the computation of aberration.

ORCID

A. Sfarti    https://orcid.org/0000-0002-2973-7140.