V.5 No 1

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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

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

In respond to Tim Shuba (see Supplement 1 ), above all other, we indicated a gross Einstein's mistake in the derivation of relativistic formula describing Doppler effect.

The experience of discussions with relativists that followed these our publications showed, they are unable to analyse independently the derivation by coryphaei of Relativity, we have to return to this issue and to consider it in a full sequence of mathematical operations and phenomenological substantiation.

It is known [1, paragraph 7, p. 35], [2, chapter 6, paragraph 10, p. 200], [3, chapter 2, paragraph 6, p. 36], [4, chapter 8, paragraph 28, p. 180] that in relativistic approach to the calculation of Doppler effect, the scientists use the principle formulated by Einstein - in particular, in his [5]. From his view, to determine the transformation of frequency received by a moving receiver from a stationary source (or the transformation of frequency received by a stationary receiver from a moving source), "it is enough to compare the vector proportional to

(where l, m, n are the directing cosines - Authors), i.e. the vector of plane light wave propagating in a void relatively the system S, with the vector proportional to

i.e., with the vector of the same wave relatively the system  S " [5, paragraph 7, p. 158].

Einstein gives such substantiation of this operation: "Consider an event, which is the following. At the given point of space and at the given instant of time, the components of field are zero. This fact does not depend on, in which reference frame the coordinates and time of event are presented. The event will remain invariable under Lorentz transformation. But zero field at the given values of coordinates and time is determined by the value of phase '. Hence, the phase of light wave remains invariable under Lorentz transformation" [4, p. 180- 181].

W. Pauli visualised this Einstein's stipulation as follows [3, p. 36]:

It is obvious from (3) that the phase equality of which Zisman said has to be satisfied at any instant of time and for any point of space through which the light beam passes. Let us draw our attention also that in the left and right parts of (3) we see the coordinates of location of the point at which the phase equality is determined, not the difference of coordinates between the start and end of the beam, i.e. between the source and receiver of signal, - this is important. We find the corroboration that in the left parts of (3), there are written down namely the coordinates of point at which we compare the phases straight in the relativistic derivations that are executed on the basis of (3). In these derivations (see any of listed references), in transition from the stipulation of equal phases to Lorentz transformation for the left part of (3), relativists used the formulas for transformation of coordinates, not directed sections. While applying to (3) the Lorentz transformation for directed sections, relativists would not yield the conventional expressions which they pass off as the exact solutions for Doppler problem.

In connection with the revealed features in the statement of problem, in order to analyse correctly Einsteinian approach to the solving, let us repeat the statement of problem after Pauli: "Consider a very far source of light   L which is at rest in the coordinate system K. The observer is in the system K'  which moves relatively K with the speed v in direction of positive values of coordinate x. Let the line connecting the light source and the observer makes the angle with the axis x in the frame K and, besides, the axis z is positioned normally to the plane determined by these two directions" [3, p. 36]. The graphic model that illustrates this statement of problem is shown in Fig. 1.

Fig. 1. The graphic model that illustrates in general case the relativistic statement of problem of Doppler shift

We see from the construction that after this statement of problem, the source and receiver can have arbitrary positions in their reference frames and this, of course, has to have a reflection neither on Einstein's stipulation (1), (2) nor on the Pauli equality (3). In accordance with Zisman, we are speaking of a real process which takes place at some point N ' at some instant of time, and we can choose any coordinates, they only have to be at rest as to the source and receiver accordingly. After the statement, the source S is at rest relatively the undotted frame K, and the receiver N ' - relatively the dotted moving frame K'.

At the same time, from the construction we see also a discrepancy against (3). Actually, as we already pointed, in the left and right parts of (3), the spatial delay phase is written as the module of radius-vector in the corresponding reference frame:

But in accordance with Fig. 1, these radius-vectors and  '  are determined in the following conventional way:

And while the inequality in (4) can be turned into the equality, if superimposing the source with the origin of the reference frame K, the inequality in (5) can be changed at no circumstances, as the source moves relatively the dotted reference frame in which the receiver is located. At the same time, the condition (3) has to be valid for any moment of time.