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Supplement 2. Substantiation of gross mistakes in the relativistic derivation of expression for the transverse Doppler effect, which we revealed in the paper and in the first supplement to it

As we can see, if the researcher was undemanding to the basic logic of mathematical formalism, if he had no wish to refer his mathematical operations with physical meaning on which the modelling equations were grounded, this can allow Born to reduce his clearly artificial equations (12) to Einstein's formulas for the transverse Doppler effect, but on the other hand this demonstrated the stipulations, under which such solutions can be yielded. And not in vain we analysed namely Born's derivation. Other authors did not dare to substantiate this derivation even in such form and often confine themselves simply to a record of final solution [3] either to the longitudinal Doppler effect [4]. For example, Landau, when considering the speed transform in the item 5, chapter 1 of his "Field theory" [6, p. 26], gave as an application only the derivation of relativistic formula for the light aberration, but fully omitted Doppler effect, giving neither derivation nor final expression.

But if speaking of the authors who tried to add some interrelation to the relativistic formula of Doppler shift, we will for certain consider the way, how Levich [7, p. 262- 265] solved it.

"Introducing formally the 4-vector   _a

which is called the 4D wave vector, we can write down the phase of wave as a scalar product of two 4-vectors  _a  and  _a ,

As the phase is invariant, the last formula shows that formally determined value _a  really is a 4-vector.

The law of transformation of components of 4-vectors allows to find the law for the frequency transformation in the theory of relativity. Namely, it follows from the definition of 4D wave vector that its fourth component is transformed as

Expressing  k    through the frequency , we find

Writing down k' as

where  cos '  is the directing cosine of wave vector, we find

[7, p. 262- 263].

We would like to mark the elegance of Levich's derivation. There are absent in it those immediately seen unobvious re-determinations which we saw in Born's derivation. At the same time, noting the said above, we see that in this derivation the radius-vector  _a  also does not involve the light beam propagation from the source to receiver but describes the receiver's location in the considered reference frames. Consequently, there arises a simple question: which angles are meant under the symbols   and ' ? And another question: what concern the relativistic rotation in the plane (x, ) has to Doppler effect? In accordance with Einstein's method, Doppler effect has to be calculated from the phase equality in the stationary and moving reference frames, not from the condition of relativistic transformation of the wave vector. Symbols used by Levich in Einsteinian formalism will take the form

And it is easy to make sure in Pauli derivation that just from (29) we immediately yield the relativistic solutions for Doppler effect. Outwardly (29) does not correspond to the condition of relativistic transformation of vectors which Levich used and which is [7, p. 215- 218]

In fact, there is no discrepancy between (29) and (30). The matter is, so-called rotation in the plane (x, ) is nothing else than Lorentz transformation for coordinates (not rotation by some mythic imaginary angle ), which we can interpret as the projections of radius-vector. In the main derivation, in transition from the Einstein conditions to the solution, Lorentz transformations are applied namely to the dotted part of equality. After this, the left-part and right-part terms are grouped and the selected parts equalised. We have three such parts, two of them - say, time and y-component - factually determine the solution of problem. The time component in the formal record corresponds to (28).

Thus, in essence, the method by Levich is not opposed to Einstein's stipulations but is aimed to avoid their problems and to come to the same result. Namely for this sake in his statement of problem, Levich retains the radius-vector as a warranty that his derivation will come to the necessary result. The retaining of this common mistake that predestines the needed result makes Levich's derivation the same wrong as the main derivation.

At the same time, to these discrepancies in Levich's approach, there is added the no less problematic issue, how much legal is to limit the phenomenology of Doppler effect to the space-time transformation of wave vector. It is easy to show this limitation inadmissible.

Actually, Doppler effect is known to be experimentally observed not only in light beams propagation but in hydro- and aerodynamics. Neither for gas-like nor for liquid media the 4D interval conserves. In these media, just as in solid bodies, the phase velocity of wave process is strongly constant relatively the medium and is determined by its parameters. If we now proceed from the fact that Doppler effect is caused exceptionally by the transformation of space-time metric, in absence of such transformations the effect has to absent. The practice shows, this is not so. Hence, Doppler effect basically cannot be determined exceptionally by the transformation of metric. But Levich yielded his result without any additional premises either account of some additional factors. This evidences only that his elegant derivation is nothing else than a successful fitting the formulas for the regularity yielded in frames of classical formalism. Just so other authors who represented their derivations of relativistic formula for Doppler effect adhered, nonetheless, Einstein's conditions, although Einstein seemingly yielded his formula just in the way which presented Levich. Now we see the resulting discrepancy. On one hand, the formula which relativists need is yielded in a very simple way, but with a full neglect of physics. On the other hand, at least partially noting the physics of processes, it is impossible to come through the space-time transformation to the expression comparable with that classical, - and this makes necessary to introduce unobvious re-determinations either to omit the derivation.

As a result of this little study we see, the derivation of relativistic formula for Doppler effect is grounded on the gross violations in the statement of problem and in choice of the methods of its solving. Namely so in the derivations they often are compelled to dare to clear juggling in order to corroborate the result published by Einstein. And the fact that in the course of solution relativists pursuit not to provide the scrupulous study of model but to bring the analysis to the solutions given beforehand by the coryphaei of Relativity, only reveals the dogmatic inclinations of adherents of space-time metric transformation. And this is the objective estimation of relativistic derivation.

June 27, 2005

References:

1. Einstein, A. On the electrodynamics of moving bodies. Collection of scientific works, vol. 1, p. 7. Nauka, Moscow, 1965 (Russian)

2. Born, M. Einstein's Theory of relativity. Dover Publications, New York, 1962 (cited after Russian edition)

3. Pauli, W. Theory of relativity. OGIZ- Gostechizdat, Moscow- Leningrad, 1942 (Russian)

4. Zisman, G.A. and Todes, O.M. The course of general physics, vol. 3. Nauka, Moscow, 1970 (Russian)

5. Einstein, A. On the relativity principle and its corollaries. Collection of scientific works, vol. 1, p. 65. Nauka, Moscow, 1965 (Russian)

6. Landau, L.D. and Lifshiz, E.M. Field theory. Nauka, Moscow, 1965 (Russian)

7. Levich, V.G. Theoretical physics, vol. 1. Physmathgiz, Moscow, 1962 (Russian)