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8.2. Formulas for aberration and Doppler effect derived from new transformations

One of possible solutions of the wave equation (8.3) in the inertial reference frame B is a plane wave

, (8.23)

where is an amplitude of the electromagnetic wave;  Ф' is the electromagnetic wave phase, which can be written in the form

(8.24)

w ' is an angular frequaency of electromagnetic oscillations mesured by a device, which is at rest relatively the oscillations source in the primed reference frame; a', b', d'  are direction cosines of a perpendicular to the wave front in the inertial reference frame B (in the primed reference frame).

The electromagnetic wave phase can not depend upon choise of a reference frame (see p. 247 in [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977). That is why the phase (8.24) must be the invariant of the new coordinate and time transformation. Having applied transformation (6.9) (because the source of this electromagnetic wave is at rest in the inertial reference frame B) to the expression (8.24), for the phase Ф in the inertial reference frame A we shall have the expression

.(8.25)

where

.(8.26)

..(8.27)

(8.28)

.(8.29)

Expressions (8.26) and (8.27) we can rewrite in the form

..(8.30)

..(8.31)

where q ' is an angle (in the inertial reference frame B) between a line connecting the source of electromagnetic oscillations with an observer and  the vector of observer's velocity in the inertial reference frame B (the observer is at rest in the inertial reference frame A); q is an angle (in the inertial reference frame A) between a line connecting electromagnetic oscillations source with the observer and  the vector of source velocity in the inertial reference frame A (the source is at rest in the inertial referencccce frame B).

From the expression (8.31), which is a formula of abberation law, we can find the value

(8.32)

and substitute it into the formular (8.30). We shall obtain

(8.33)

The expression (8.33) is a formula for Doppler effect for angular frequency.

As the electromagnetic wave is emitted by a source, which is at rest in the inertial reference frame B, this wave propagates in the inertial reference frame A at a speed, which is determined by the expression c_u = c_o (1 + u²/c_o²)^1/2 (2.1), having in the reference frame A the angular frequency   w. Then the wavelength of electromagnetic oscillations in the reference frame A can be determined using the expression

(8.34)

Substituting the formula (8.33) into the expression (8.34), we shall obtain

(8.35)

where is the wavelength of electromagnetic oscillations in the inertial reference frame B (the wavelength of oscillations emitted by an immovable source).

The expression (8.35) is the formula of Doppler effect for the wavelength.

At from formulas (8.33) and (8.35) we shall obtain

(8.36)

(8.37)

From formulas (8.36) and (8.37) it follows, that in the new space-time theory the transversal Doppler effect for angular frequency is absent, but for the wavelength in the new space-time theory the transversal Doppler effect results in the red shift (like in the special relativity theory).

At from formulas (8.33) and (8.35) we can obtain

(8.38)

(8.39)

From formulas (8.38) and (8.39) it follows, that in case, if a source of electromagnetic oscillations is moving towards an observer,   the frequency of oscillations received by the observer increases, and their wavelength decreases.

At from formulas (8.33) and(8.35) we can obtain

 (8.40)

 (8.41)

From formulas (8.40) and (8.41) it follows, that in case, if the source of electromagnetic oscillations moves from the observer, the frequency of oscillations received by the observer decreases and their wavelength increases.

In the formulas (8.26)...(8.41) we can see a parameter

.  (8.42)

When the speed u changes from zero to infinity this parameter in formula (8.42) changes within limits from zero to unity. That is why the dependence of wavelength for electromagnetic oscillations received by an observer upon source movement speed, which can be written in the new space-time theory in the form of equations (8.35), (8.37), (8.39) and (8.41), qualitatively coincides with analogous dependence from the special relativity theory.

But the dependence of  frequency of electromagnetic oscillations received by an observer upon the speed of these oscillations source  in the new space-time theory essentially differs from the analogous dependence from the special theory of relativity. This can be explained by the fact, that in the special relativity theory (SRT) instead of formula (8.30) we have the formula

..(8.43)

where , instead of formula (8.33) we have the formula

.(8.44)

instead of  formula (8.36) we have

...(8.45)

instead of  formula  (8.38) we have

....(8.46)

and instead of  formula  (8.40) we have

. . (8.47)

For the transversal Doppler effect the formula (8.36) from the new space-time theory differs from the formula (8.45) from the SRT not only quantitatively, but qualitatively too. According to the formula (8.36) from the new space-time theory the transversal Doppler effect for the frequency is absent. But according to the formula (8.45) from the SRT the transversal Doppler effect for the frequency results in the red shift (in decrease of frequency).

The formula (8.40) also essentially differs from the SRT formula (8.47). Indeed, according to the SRT formula (8.47), if the source is moving away from an observer and source speed increases, the frequency of oscillations received by an observer tends to zero. But according to the formula (8.40) from the new space-time theory in case of retreating source speed increase the  received frequency tends to a value of 0.5 w _o   and can not become less than 0.5 w _o .

The experiment [57]. Champeney D. C., Moon P. B. Absence of Doppler shift for gamma ray source and detector moving on the same circular orbit. - Proc. Phys. Soc. , 1961, v. 77,  pp. 350 -352  can be considered as experimental confirmation of formula (8.36) from the new space-time theory, i.e. as experimantal confirmation of absence of transversal Doppler effect for the frequency.

At present time it is not clear whether the formula (8.40) is confirmed by experiments, or not. Because in astronomic observations the Doppler effect is superimposed on the effect of light quanta "extension", which results not only in the increase of wavelength (see formula (5.15)), but in the decrease of angular frequency of electromagnetic oscillations. Taking into account the effect of light quanta "extension" and Doppler effect  in astronomic observations of objects, which are at a distance of D from an observer on the Earth and which are moving to or from the the Earth, the angular frequency and the wavelength of electromagnetic oscillations received by an observer on the Earth will be determined by expressions

(8.48)

(8.49)

Taking into account the effect of light quanta "compression" and Doppler effect, these formulas take the form

 (8.50)