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The present paper discusses the relativistic Doppler effect and tries to found misunderstandings in the present state of the Special theory of relativity. The author's conclusion that he found some “blue shift” which contradicts with time dilation is wrong.

The weakest feature of the paper is that although the formulas, presented by authors, are in general correct, but they do not support the conclusions the author extract from them, and mistake is hidden in the interpretation.

Let's focus on the plane waves.

In general, the transverse Doppler effect, as it is studied in the available literature, means that an observer (let's call him the 1st observer), that receive an electromagnetic wave from a distant source, moving relative to the
*…show more content…*

In case of author considerations, they are measured by different observers, situated at different reference frames. If one wants to use 2nd observer to find the Doppler effect, for 2nd observer, his β=β₂=0, and his α'=α, formulas (1)-(2) will give ν'=ν₂'=ν. So there is no frequency shift for the 2nd observer. (as it should be for him, being in rest with the source).

For the 1st observer, substituting the β and α'=π/2 will give correct result: ν'=ν/γ, where γ=1/(sqrt(1-v²/c²)).

One should not mix this two cases, as the author does.

Let's look to the author's arguments, why one should use α=π/2 rather than α'=π/2 to define transverse Doppler effect. The author calls α as the proper angle and α' as apparent angle, in the analogy with the term "proper" length. This analogy seems to be not applicable. The "proper" length is the length of an item, measured in rest with this item. The term "apparent" length is for the length of an item, measured by another observer, moving relative to the item.

For the angle between two items, it is not obvious, relative to what part of the system the observer should be in rest, in order to measure the angle to call the angle as "proper" angle.

The author's idea, that using α=π/2 in order to compare with classical Doppler effect, also does not hold water, as in the classical theory

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