SELF

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S.B. Karavashkin and O.N. Karavashkina

3. Classical and relativistic solutions of Doppler effect in comparison of their approaches

To compare, write general Doppler solution in the relativistic formalism [2, p. 36, formula (15)]:

Comparing (11) with (9), we see that generally formulas are mutually irrelevant both in the form of mathematical regularity and in the amount of parameters on which the frequency shift depends. In classical formula, the shift depends not only on the angle and ratio of observer's speed to that of light, but also on the aimed distance H and absolute speed of body v; this considerably changes the physical meaning of the Doppler effect. So we will have not only to compare solutions but to understand, which of them describes the effect more accurately.

In Fig. 3 we show the regularities of signal frequency received by the moving observer, with respect to the angle , calculated on the basis of relativistic formalism after (11) (blue curve) and classical formalism (10) (lilac curve).

Fig. 4. Frequency of signal received by the moving observer, against the angle , calculated on the basis of relativistic formalism (11) (blue curve) and classical formalism (10) (lilac curve). Parameters: H = 1 km; c = 300 000 km/s; v/c = 0,5; = 1 MHz

We see from the plot that Doppler shift predicted by relativistic conception is much higher than the values predicted by the classical conception. And at = / 2, which, as relativists think, corresponds to the transverse Doppler effect, both curves pass the point ' =  = 1 MHz and are located at different sides of it. In other words, the relativistic conception predicts positive Doppler effect, and that classical - negative. This is the basic point, as we cannot reduce it to some similar values in transition to non-relativistic speed of observer's motion.

And, as we told above, the classical solution depends on many parameters, not only on ratio of observer's speed to that of light. In Fig. 5 we show the regularities of frequency of signal received by observer against the angle , at different aimed distances H.

Fig. 4. The frequency of signal received by moving observer against the angle , at different aimed distances H. Parameters: c = 300 000 km/s; v/c = 0,5; = 1 MHz

The plots show, the transverse Doppler effect in the view of classical formalism is revealed at small aimed distances and grows with their fall. But the effect always gives negative values of frequency shift.

It is important to mark here that as relativists state the problem, with lessening mutual speed of observer and source, all effects predicted by their conception have to be reduced to those classical. Should the predictions of classical conception were different only in sign, relativists could operate, neglecting the quadratic terms at small speeds. But factually the point is much deeper, as the sign of effect cannot be negated by disregarding of quadratic terms. This clearly speaks of erroneous premises on which the relativistic conception has been formulated, and below it will find additional basic corroboration.

To deepen the comparison, consider particular cases of longitudinal and transverse Doppler effect.