SELF

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S.B. Karavashkin and O.N. Karavashkina

2. Analysis of the Hubble law on the basis of longitudinal Doppler effect

To substantiate the statement that the corollaries following from the additional supposition of Doppler nature of red shift are contradictive, let us first prove the following. When supposing that the red shift in the form (1) is conditioned by the velocities of source and observer, we in this way suppose quite specific regularity of motion of source and observer as to some selected point of space at which their velocity is zero.

To prove it, consider the kinematic diagram shown in Fig. 2.

Suppose, the point O is some centre around which both the observer A and source B move. Let the law of their motion be

where H₀ = 100 km/pc^.s;   r_A ,  r_B and  r_AB   are measured in megaparsecs;  v_A  and v_B   are measured in km/s.

Consider the formed triangle OAB . Noting that usually in such problems only longitudinal Doppler effect is taken into account, determine the projections of velocities onto the side AB :

Since from this triangle it follows that

The velocity of motion of the source and observer relatively each other will be

where

Consider separately the amplitude and phase parts of (7). If we substitute (3) to V  in (8), we will yield

as it follows from the construction in Fig. 2. Now if we substitute (3) into (8) for the phase  , we will yield

Given from the sine theorem

we yield

Whereat, (7) will take the following form:

It is consistent with (2) and in this way proves our statement that the specific regularity of velocities of the source and observer with respect to the point O , on one hand, and our guess that the Hubble law is conditioned by the mutual velocity of the source and observer, on the other, correlate.