l'··· the length of the abscissa breathing with space in the primed frame l··· the length of the abscissa breathing with space in the unprimed frame
or, in general, l_{m} = l_{n}.
(2)
{l_{m}}_{n}
stands for the length of the abscissa in a frame "m", measured by the ruler not breathing with the space and
calibrated for the frame "n".
stands for the density of the space. This quantity is applied instead of the "space curvature",frequently used to
indicate the spacetime change. It means the number of the spacetime cells, or the wave packets, divided by the volume,
they occupy. When measured by the instruments breathing with the space, the density would stay the same, and it is
. We can take this density. For our
considerations the third root of the ratio of the "m" to "n" systems space densities is used, when measured
by the instruments not breathing with the space and calibrated for "n" system , designated
as .
Remark: The other relevant explanation of the space density or space curvature would not abolish the equation (2).Time events breathing with spacetime
The equation (3) says that flow of time is the same in different spacetime structures when measured by instruments breathing
with spacetime structures.In other words, any physical quantity in course of the same physical phenomenon, taking place in
different spacetime structures, reaches the same value at the same time, measured by the instrument breathing with spacetime
structure. However when time is measured by instrument not breathing with spacetime, the time intervals at which the physical
quantity mentioned reaches the same values, are different (are not contemporary). t'··· time in the primed frame t'··· time in the unprimed frame
or, in general, t_{m} = t_{n}.
(4)
{t_{m}}_{n}
··· time elapsed in the frame "m", measured by the clock not breathing with the time (or spacetime) and calibrated
for the "n" frame.
··· the time density. It means the number of the spacetime oscillations in the frame "m" within the time unit that has
elapsed in the frame "n", divided by number of the spacetime oscillations fixed to define the time unit.
Remark: The other relevant explanation of the time density would not abolish the equation (4).
Taking the time unit equal to one oscillation , we have :
when time (number of oscillations) is measured by the clock breathing with time,
does not equal 1 in general, when detecting the number of the spacetime oscillations
in frame "m" within the time unit measured by the clock not breathing with time and calibrated for "n" frame time unit.
For our considerations the ratio of the "m" to "n" systems time densities is used, when measured by the instruments
not breathing with time and calibrated for "n" system , designated as .
It seems, the space and time in two of more spacetime structures occurring in the same place, appear to be not breathing
with the other spacetime structures, in a similar way as an abscissa not breathing with space is defined in paragraphs 2.1
and 2.2. The following illustration shows that the length of abscissa lm of the system M , not breathing with the space of the
system N can be measured as the length L_{nm} in the system N (by the instrument from system N,
breathing with space), and that similarly the abscissa l_{n} can be measured as
L_{mn} in the system M. In the same way the time intervals T_{nm} and
T_{mn} can be determined.
To discuss more thoroughly the con-spacetime events, imagine the motion of the objects in two the following reference
frame systems (see Fig2 ) :
Reference frame system M, moving at a constant speed v_{m/o} with respect to a primed system,
And the frame system N, of the same space and time density as there are in an unprimed system, but moving at the same
speed v_{n/o} = v_{m/o} together with M system.
Let the object M move in M-frame at a speed v_{m} = 0,5 m/s. The motion covers the distance
l_{m} = 2m at time interval 4s. The same distance l_{n} might be travelled
at time interval t_{n} = 4s by another object, N, moving in N-frame at the same speed
v_{n} = 0,5 m/s. The two object mentioned however are not found on the same place at the same time,
because
and therefore, looking from frame N, we can find :
object N on the distance 2m at time t_{n} = 4s
and object M on the distance 1m at timet_{m} = 4s
Looking for the object M in N-frame, we have to take into consideration that
at time interval t_{m} = 4s in M-frame, the time interval only
has elapsed in N-frame , and therefore
(5a)
where T_{nm} (T_{mn}) is the time interval in frame N (M) , corresponding to time
interval in frame M (N).
and, that the distance lm =2m in M-frame is detected as
Fig 2 Motion of one object in two reference frames
in N-frame, and
(5b)
where L_{nm} (L_{mn}) is the distance in frame N (M), corresponding to the
distance in frame M (N).
Now , starting from time interval T_{mn} = 4s in M-frame we can see that object has travelled the
distance
in M-frame, and the same object is found in frame N :
at the distance
and at time
Taking:
where means speed in frame N, corresponding to the speed
in frame M(it may be used in frame N for superposition, as a galileian increment)
we have
and, in general,
(5c)
The equations (5a), (5b) and (5c) represent the transformation equations between M and N reference frames. The frame N,
moving at the same speed with frame M, but having the same spacetime density as the unprimed frame, is imaginary only.
But it will become real, if we put it into unprimed system, and take quantities L_{nm},
T_{nm}, and as suitable for galileian transformation in N-frame.