4.4 The spacetime density.

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 imagenm (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 imagen in anticlockwise direction from the vector of the speed vm/n. The light beam travels at the same angles imagen= imagem 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 observer from the frame M would see the thread MX at an angle imagem= imagen 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 imagenm. He would measure at the same moment (by the instruments breathing with spacetime)
Looking back to starting point N, he would see In general the speeds give the resultant speed vr/x/n , defined as galileian vector superposition of the speeds vm/n and vx/m / imagem/n/ imagenm ,
image (9)

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

Respecting the triangle NMX in Fig 3 , we may write :
and eliminating time, we obtain
image (11)

and hence
image (12)

Now substituting for the angle from equation (10) in equation (12), we obtain:
image (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 :
image (13a)

image (13b)
image (13c)

image (13d)
image (13e)

image (13f)