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2

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

Thus, we see that, using immediately the stipulation (3), relativists compare not the vectors and angles of which they say. To compare the angles which correspond to the light beam from the stationary source to the moving receiver, the graphical scheme has to be another. It is shown in Fig. 2.

Fig. 2. The proper graphical model which in general case has to illustrate the relativistic statement of problem of Doppler shift

In the presented model

and

When choosing convenient reference frames K   and K' , whose coordinate origin coincides accordingly with the source and receiver, the expressions (6) and (7) will be some simplified, as we can equalise x_S , y_S , x'_N ,  y'_N to zero. With it these expressions will take the following form:

and namely these expressions, not right parts of (4) and (5), have to be involved in (3). Then (3) will be transformed into

Here we have to draw our attention that x'_S and  y'_S depend on the local time of dotted frame K' and vary in direction opposite to the frame motion, and x_N and y_N  depend on the local time of undotted frame K and change their direction with the observer's motion together with the dotted frame.

Note also that the yielded stipulation (10) in no case will lead us to the conventional relativistic regularity of Doppler shift, as the conventional solution strongly corresponds to (3), and by virtue of one-valued solution of systems of equations, cannot satisfy any other condition. We can easily prove this last by a straight substitution of the solution [3, p. 36]

(where  = v/c) into (3).

Thus, we see that into the very statement of problem - or rather, into the condition of solving the problem of Doppler shift - relativists have put a gross mistake, which leads to outwardly like but wrong solution. Especially note, even with the introduced improvements, (10) also will not fully reflect the phenomenology of frequency shift in the mutually moving source and receiver, as in it a whole number of important points still has not been involved. First of all, relativistic transformations can only supplement the existing effect of frequency transformation, on whose basis is grounded the variation in phase of received signal at permanently varying distance between the source and receiver, and these variations in distance are registered in both reference frames. And secondly, transformation of frequency of signal reception relates by its properties to the measurement of time intervals, which cannot be done simultaneously; it requires at least two sequential events. So in Doppler effect calculation we have to proceed not from the equal phases in different reference frames at some moment of time and some particular point of space, but from the change of time intervals when the distance between the source and receiver increased, out of noting the relativistic transform of metric. But in presence of above solution and with the experimental corroboration of relativistic transformations just in the form on which relativists insist, they have to note this transformation of metric as a secondary effect superimposed on Doppler effect.