SELF

36

S.B. Karavashkin and O.N. Karavashkina

Tending t_B   to zero, we will obtain

To pass from (23) to the frequency representation, remember that in the statement of problem we spoke of the radiation and reception of two sequential fronts of light. So for the observer A  and source B  the phase difference d is the same. Noting this, use the standard definition of frequency

For the non-relativistic case, (26) essentially simplifies:

If now we make more specific the regularity of velocities of the source and observer as to the point O  and substitute (3) into (27), we will obtain, accurate to the higher-order, as follows:

According to the basic construction in Fig. 2, in (28)

Substituting (29) into (28) and making simple transformation, we yield

Thus, using the regularity (r) in the form (3), we came again to the Hubble law, to its nonlinear form. "For far galaxies, the dependence between z  and  r  reveals the deviation from simple proportionality, so we can write

… After the modern data, q₀   lays between 1 and 3" [8, p. 511]. Comparing (30) and (31), we see that these expressions are equivalent. The only difference is, the quadratic addition in (30) has a strong anisotropy, and the sign before cos   determines it. In transition from  r_AB > r_A  to r_AB < r_A    the direction of anisotropy alternates. To show the value of this anisotropy, introduce to (40) the supposedly known for today parameters of the model. The value  r_A   can be estimated as 75 million light years = 10,75 megaparsecs. "The second-order galaxy (i.e., the system of assemblages consisting of more than 10 000 galaxies) has the linear size about 100 million light years, and its centre is at the distance about 35 million light years to the Virgo constellation. Our Galaxy, thus, is well nearer to the edge than to the centre of the Supergalaxy" [8, p. 512].

At the indicated value  r_A   the anisotropy in the near region conditioned by the quadratic term of (30) will be negligibly small, but at the distances  r_AB 1000   megaparsecs the anisotropic addition will amount  4*10 ^-4 , with the value of linear term equal to 0,2, i.e.  z = 0,2 4*10 ^-4 . This is inconsistent with the observations. According to the known data, the anisotropy is observed in the near region of Metagalaxy. "More detailed study reveals some visual anisotropy of the red shift, and the values H₀   in some directions differ in 1,5 times and more… Such anisotropy of the red shift is revealed mainly in the nearer regions of the Metagalaxy to which the most part of the observed red shift relates" [8, p. 511]. This means, the anisotropy is typical for the nearer, not for far regions of the Metagalaxy.