V.5 No 1

3

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

At the same time we cannot think that relativists did not see the defects in conventional Einstein's stipulations. One of the main defects - inconsistency of angles of light beams and radius-vectors - to a definite extent tried to improve M. Born, writing the phase equality for a plane problem as [2, p. 293]

To obtain this expression, Born considerably changed the meaning of light beam description in the more classical approach. He formulated the statement of problem so: "Suppose, the relative speed of frames S and S 'is directed with the coinciding axes x and x', but the direction of light wave propagation is already perpendicular to these axes, - say, parallel to the axis y'.. But it would be also inadmissible to think the normal to the front of light wave in the system S  to be parallel to the axis y.

The distance from the start of the packet of light waves at the moment t'₀ up to its end at the moment t'₁, when observing in the frame S ' , is equal to y'₁ - y'₀ ; in the frame S  it will be not simply y₁ - y₀  but will depend also on x₁ - x₀ , - say, will be equal to a(x₁ - x₀) + b(y₁ - y₀) " [2, p. 292- 293].

But comparing (12) with (10), we see that in (12) he did not take into account the main feature of model: in the moving reference frame he had to note the changing location of the source, and in the stationary reference frame - the changing location of receiver. By the statement of problem, the observer in moving reference frame does not see the trajectory of beam radiated by the stationary source, as we usually see the trajectory of bodies due to the fact that the information of its motion arrives much faster than the body shifts. In this problem the observer permanently receives the signals, continuing its motion at the same time. Thus, in order to arrive to the perpendicular at the moment of time  t₁, the observer has at the moment t₀ to be not at the perpendicular and to receive not that beam of light which he will receive at the moment  t₁. The classical formalism, on whose basis we made our calculation in the main paper, notes these factors, but Born's (12) undoubtedly does not. So it is basically incorrect to compare (12) with (10) and the more with the calculation made in frames of classical formalism, as in (12), in the changed appearance, all mistakes of Einstein's conditions remain. We would also mark, in the relativistic approach to the solution that reduces everything to Lorentz transform, and noting that, as we said above, the very expression (10) does not note the dynamic of processes, it also cannot bring a correct solution. But this feature of (10) is additionally worsened in M. Born's derivation by an ungrounded excluding some components from (12) and remained mess in the angles of radius-vectors and vector of light beam. In this way Born yielded a clear mathematical ersatz and further could not cope with it without new clearly unfounded redefining. Actually, if  x₀ , y₀ , z₀   determine the source location at the moment t₀ and x₁ , y₁ , z₁ determine the receiver location at the moment  t₁, and these locations are different by the statement of problem, the directing cosines a and b will be different at these moments, too, and the left part will have basically other appearance. As we know from analytical geometry, at the initial moment t₀

at the moment  t₁

and in (12) we have

In change of differences of projections in (15), the beam direction changes automatically, but not in the direction necessary to follow the shift of receiver, but in a fully abstracted direction, and the parameters a and b will not retain their constancy in time, which we would have to note in transformations.

Thus, the left part of (12) will not describe the trajectory of moving source. The same in the right part of (12). In this expression Born has not only excluded the x-component but made constant the directing cosine of the angle of y-component, retaining the very component and in this way ignoring the mutual motion of the source and receiver. Well, even if y'₁ is fixed in the moving reference frame, y'₀ relates to the source and will not be constant in time. The presence of y-component says only that the observer is located not in the origin of the moving reference frame. With it, as Born says, the directed cosine in the moving reference frame will be invariable in time and equal to 1, while in the undotted frame, if retained statement of problem and appearance of (12), and noting (15), this angle will vary in time. As we see, Born's interpretation of Einstein's method corroborates, it is absolutely undetermined, - just what we said above.

And Born did not analyse his clearly 'invented', not modelled equation, he advanced to Lorentz transformation.

"If we use the formula

and put in it

expressing the fact that the experiment is true at the fixed point in the frame S ' , the calculation becomes more complicated, as in this case we have to know the value a " [2, p. 293]. As we see, the conclusion which we made, analysing the basic equation (12), is corroborated. Born refused to place the observer into the origin of moving reference frame, but referred not to his real reason that would disable him to yield a 'like' expression for Doppler effect; he confined himself to a first excuse which came to mind, while the directing cosines can be clearly determined from (15). And Born retained the y-component, just because it forms an important equation in the general system of equations leading to the relativistic solution.

However, the excuse does not solve the problem; then Born undertakes a 'cardinal step': "But if we use the formula

putting in it

we yield without trouble

(in this case Born means = (1 - (v/c)²)^1/2  - Authors)" [2, p. 293]. Well, what do mean the conditions (19) for the modelling equation (12)? They mean, from the view of stationary reference frame K , the light beam has been received by the observer in K'   immediately at the moment when it was radiated. In this case the between-events time interval would be zero, i.e.

With it the modelling equation (12) will be also simplified and will take the form

not (20).