V.3 No 1

35

On the nature of red shift of Metagalaxy

Developing this proof, we can show that in the premise of Doppler effect, the relation between the motion of the source and observer as to some stationary centre O  and the Hubble law remains true if we take into consideration not only longitudinal but full Doppler effect. For it, we have to change a little Fig. 2 and to consider in more details the propagation of the wave front from the source B  to the observer A .

Let the front of the wave radiated by the source B    at the point B₁  reaches the observer A   at the point A₁ , and the front of wave radiated by the source at the point B₂  in the time interval t_B  is received by the observer in the time interval  t_B  at the point A₂   (see Fig. 3).

As we can see from Fig. 3, the difference between t_A  and  t_B   is determined by the time t  which is necessary to the beam to pass the difference of intervals between  A₂B₂   and A₁B₁ , i.e.

In its turn, in accordance with the statement of problem, the difference between t_A  and  t_B   is conditioned by the motion of source and observer as to the mutual reference point  O .

All the further proof is derived in frames of geometry, taking into account the smallness of  t_B . This enables us to think v_A and v_B constant during the studied time intervals. From  A₁A₂F

As the result we yield

After we substitute (15), (16) and (17) to (18) and take into account that in accordance with the construction and (14)

we will yield, accurate to the first order of smallness of the increments of  t_A ,  t_B   and t , the following: