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Taking into consideration paragraphs 3.3 and 4.3 we are now able to find out, that we may transform the distance and speed from the primed reference frame M to the reference frame N, moving together with the frame M at the same speed with regard to the unprimed frame, but unlike with it, having the same spacetime density as the unprimed reference frame. These transformed quantities may be dealt with in the unprimed reference frame in accordance with well-known galileian transformation. In this paragraph the reference frame N is taken as an unprimed frame and the transformed quantities Lnm and nm (see equations (5b) and (5c)) as galileian increments to the distance and speed in frame N.

The light propagation in common case, in unprimed and primed reference frames is shown on Fig 3 (below). The primed reference frame M moves at a speed vm/n with respect to the unprimed frame N. At time t=0 the starting points of the frames N and M are identical. At that moment the light beam is iniciated from the starting point (no matter whether from N- or M-point) at an angle n in anticlockwise direction from the vector of the speed vm/n. The light beam travels at the same angles n= m in both frames,when measured by the instruments breathing with the spacetime. If the leading of the light beam photons was strong enough, we could make it to pull two silk threads, one from starting point N and another from starting point M. Let the threads are of the same quality, breathing with spacetime and marked by the scale of length.

Fig 3 The length transformation

The observer from the frame N would see the dragging photon at the point X after time tn has elapsed. He would measure at that moment (by the instruments breathing with spacetime)
• the length of the distance NX amounting to the value ctn (product of the speed of light and the time elapsed )
• the distance NX directly red on the scale of silk thread, amounting to the same value ctn
• the angle n between speed vector vm/n and the silk thread NX
• the length of the silk thread MX of the value

where
• the length of the distance NM, amounting to vm/n*tn
• The angle nm between vm/n and the thread MX, different from n (higher in this case) in general,
• And he could calculate the speed of light photon travelling in M-frame (that of course equals to the value c in M-frame), transformed to N-frame, as
The observer from the frame M would see the thread MX at an angle m= n with respect to vm/n. This is why the dragging photon is situated in point P for the M-observer. In fact, points P and X are identical and M-observer does not see angle nm. He would measure at the same moment (by the instruments breathing with spacetime)
• the length of the silk thread MX = MP of the value Lmn= Lnm* s/m/n=ct*t/m/n=ct
• the distance MX = MP, directly red on the scale of silk thread, amounting to the same value
• the angle between vm/n and thread MX = MP of the value m = n, as mentioned above.
Looking back to starting point N, he would see
• the distance NM of the length vm/ntn*s/m/n (vm/ntn*s/m/n=Lmn - the distance travelled by the point N with regard to frame M).
Remark : s/m/n here is the spacetime density in direction MN.
• the distance NX = NP of the length ctn*s/m/n ( ctn*s/m/n=Lmn - the distance ctn transformed from N to M reference frame)
In general the speeds
• vm/n , expressing the relative motion of the primed system with respect to the unprimed system,
• and vx/m , expressing the speed of the object inside of the primed reference frame,
give the resultant speed vr/x/n , defined as galileian vector superposition of the speeds vm/n and vx/m / m/n/ ,
where
(9)

is the spacetime density defined as ratio of the space to time density in the frame M at the angle m in anticlockwise direction from the vector of the speed vm/n. In N frame the angle nm corresponds to the angle m.
We can simply derive from Fig 3 that
(10)

Respecting the triangle NMX in Fig 3 , we may write :

and eliminating time, we obtain
(11)

and hence
(12)

Now substituting for the angle from equation (10) in equation (12), we obtain:
(12a)

The equations (12) and (12a) say that the spacetime density of the primed system spacetime continuum depends on the angle. We may state that the spacetime of the primed reference frames is anisotropic for the observer from another reference frame. The equations (14) and (14a) give the following solutions for the four basic directions :
a/
(13a)

(13b)
b/
(13c)

(13d)
c/
>
(13e)

(13f)

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