V.5 No 1

3

Supplement to the study of classical transverse Doppler effect in respond to received criticism

After you did your transformations and yielded your (12)

which you could yield well simpler by way of equalising the speed of source to zero, you are comparing this expression with standard general Einsteinian expression for Doppler effect which he gave in his basic work of 1905 (item 6 of the first part):

With this, your formulas take the following appearance:

<< "Obviously if

?this reduces to the previous >>

Well, do you know, what do you compare with what, and the main, under which conditions?

The question is not rhetorical, it is the matter of great principle. You have yielded your (12) (in this our post it is (1) ) as the expression for transverse Doppler effect - and are comparing it at the angle corresponding to longitudinal Doppler effect. Yes, it is just so, dear Tim. In the item 6, part 1 of his work of 1905, Einstein applied his formulas of relativistic transformation to find Doppler effect. He took another way than you do and compared the intensities of a plane light wave propagating in void relatively the undotted stationary and dotted moving reference frames. He yielded that if "the line 'source of light - observer' makes an angle with the observer's speed as to the coordinate system being at rest as to the source of light, the light frequency ' received by observer will be determined by the relation (2 of our post)". Similar and independent calculation by W. Pauli that brought him to the absolutely same result was grounded on the angle equal to Einstein's . Pauli defined this angle so: "Let the line connecting the light source and observer makes an angle with the axis x in the system K …" [W. Pauli. Relativity, part 1, item 6, p. 36 in Russian edition]. To visualise these angles equal, we are showing you in Fig. 4 both Einstein's and Pauli's schemes modelling their statements of problem.

Fig. 4

Thus, you can see, the angles are obviously equal. But it does not follow from this that the condition of transverse Doppler effect is implemented in the relativistic version at

Substituting this value to Einstein's formula, we yield

i.e., the expression which, as relativists think, has to describe the longitudinal Doppler effect, and with your value of angle (3 of our post), which has to describe the longitudinal Doppler effect, we yield

- just the expression which relativists, and you, take as the transverse Doppler effect. This means the accuracy up to the opposite. Naturally, in this correct consistence of solution to the angles, the formula for relativistic longitudinal effect will NEVER match that classical. We can easily reveal the inverse correspondence in effects and their relativistic predictions. It would be better if we do so after Pauli - as his derivation is more transparent, it can be followed easier, - but we can do it by Einstein, with the same result leading us to the fact of inverse correspondence. In this connection, the numbers which you give as so-called experimental corroboration are nothing else as a careless study, which is so much typical for relativists. Such manner of unscrupulous analysis of experimental schemes and methods to calculate the experiment by relativists has been clearly shown in our work.

And it is quite typical that you in your calculation have yielded the formula for longitudinal effect. As we showed you before, you did not account in your model the non-central motion of observer as to the source, and without it the problem is reduced to determination of just longitudinal effect, when at some initial instant the source and observer were located at one point, and then the observer began distancing from the source. But if you account the non-central motion, this will fully destroy your calculation.