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 L_{nm} 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 v_{m/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 v_{m/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
t_{n} has elapsed. He would measure at that moment (by the instruments
breathing with spacetime)
the length of the distance NX amounting to the value ct_{n} (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 ct_{n}
the angle _{n} between speed vector v_{m/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 v_{m/n}*t_{n}
The angle _{nm} between v_{m/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 v_{m/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 L_{mn}= L_{nm}*
_{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 v_{m/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 v_{m/n}t_{n}*_{s/m/n}
(v_{m/n}t_{n}*_{s/m/n}=L_{mn} - 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 ct_{n}*_{s/m/n} (
ct_{n}*_{s/m/n}=L_{mn} - the distance
ct_{n} transformed from N to M reference frame)
In general the speeds
v_{m/n} , expressing the relative motion of the primed system with respect to the unprimed system,
and v_{x/m} , expressing the speed of the object inside of the primed reference frame,
give the resultant speed v_{r/x/n} , defined as galileian vector superposition of the speeds
v_{m/n} and v_{x/m} / _{m/n/
nm} ,
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 v_{m/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 :