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 deltabig.gif (843 bytes)tB  is received by the observer in the time interval  deltabig.gif (843 bytes)tB  at the point A2   (see Fig. 3).

 

fig3.gif (7301 bytes)

 

As we can see from Fig. 3, the difference between deltabig.gif (843 bytes)tA  and  deltabig.gif (843 bytes)tB   is determined by the time delta.gif (843 bytes)t  which is necessary to the beam to pass the difference of intervals between  A2B2   and A1B1 , i.e.

(14)

In its turn, in accordance with the statement of problem, the difference between deltabig.gif (843 bytes)tA  and  deltabig.gif (843 bytes)tB   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  deltabig.gif (843 bytes)tB . This enables us to think vA and vB constant during the studied time intervals. From  deltabig.gif (843 bytes)A1A2F

Image1600.gif (1453 bytes) (15)
from  deltabig.gif (843 bytes)B1B2E

(16)

As the result we yield

(17)
and from deltabig.gif (843 bytes)A2KB2   

(18)

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

(19)

we will yield, accurate to the first order of smallness of the increments of  deltabig.gif (843 bytes)tAdeltabig.gif (843 bytes)tB   and delta.gif (843 bytes)t , the following:

(20)
or, taking into account (14),

(21)
whence

(22)

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