1st Revision : August 8th 2003 (the extent of theme not changed) The subject of revision :
The more thorough explanation of the theme
Modification of the indexing
At that time already, when the Special relativity was acknowledged, admitting the time dilation and length contraction,
even long before, when the theories admitting various space geometries (or the space curvature) were accepted by scientists,
the fact that the spacetime continuum has its own structure was accepted. We can hardly imagine the curvatured spacetime
without the idea that the things deformed in a curvatured spacetime are firmly bound up with it.
The imagination of the spacetime structure makes possible to define the Unit flow of the space, as the number of the spacetime
cells that has passed through the unit area perpendicular to the vector of the speed, within a time unit. Because the number of
the spacetime cells is proportional to the volume, it is :
(29)
We have :
(30)
Determining the space flow in the frame M with regard to frame N, the "contemporary" speed
_{mn} = v*_{m/n} has to be
used instead of v , to obtain speed proportional to the spacetime density. The speed
_{mn} defines the number of spacetime cells of the M-frame spacetime
structure, that has passed in direction of its speed through a perpendicular reference area, within time unit of N-frame.
The change of the space (or, spacetime) density however brings the change of the frequency of the spacetime structure. Applying
the definition of the time density from chapter 3.2, we can easily derive that the frequency of the
spacetime structure must be directly proportional to the time (or spacetime) density. The higher (lower) frequency must be
understood as the sign of the higher (lower) energy of the spacetime cell. From this point of view we may consider one spacetime
cell as n spacetime quanta if the time density (frequency) became n-times higher (lower). This is why now the spaceflow
speed (marked (v)_{tn}) has to be considered as _{t/m/n} times
higher (lower), and we have :
(31)
where speed (v)_{tn}
defines the distance in units of M-frame, travelled within time unit of N-frame, if the distance was defined as the lined up
series of the spacetime quanta.
See Fig7a :
Fig7a The speed of the space flow
Thus
(32)
(32a)
(32b)
where
dQ_{mn}
stands for the differential of the space flow cells in M-frame, recalculated for the time unit of the reference (N) frame, and,
dQ_{emn}
stands for the differential of the space flow quanta in M-frame, recalculated for the time unit of the reference (N) frame
Space flow into a mass point
Admitting possibility of the existence of the space flow brings us to idea of the spacetime flowing from all directions to a mass
point, giving chance to explain the mass point as a spacetime structure of a very high spacetime density, and, to explain the
acceleration due to gravity as an acceleration due to change of the speed of the space falling into a mass point. The question
where from and to is the spacetime travelling on its way to a mass point cannot be ignored in general, but may be disregarded
in this work.
At an angle = the spaceflow speed v is orientated
in direction with the speed v_{m/n} at which mass point is moving with respect to N-frame. The equation
(12a) must be modified for this configuration:
Substituting for _{s/m/n} to equation (32) from (33), we obtain:
(34)
(34a)
The area differential on the spherical area of the radius r (see Fig7b):
Fig7b
Thus
(35)
(35a)
where ln stands for natural logarithm.
We can simply derive that Q_{mn} = Q_{m} = Q_{n} = 4r^{2}v
in equation (35) represents the frame own space flow (reaches the same value in all frames).
Thus
(36)
In Special relativity the following transformation equation for the mass was derived:
(37)
where M is mass of an object as measured by an observer moving at a speed v with respect to the object, and
m, called rest mass, is the object's mass as measured in its own frame. This transformation equation is considered as
verified, because it was confirmed by many laboratory experiments, especially on synchrotrons and synchrocyclotrons.
We can see that curve spaceflow-speed of the mass-point according to equation (36) is very similar to the curve mass-speed
according to equation (37). This is why we may say that quantity of a mass is proportional to quantity of a spacetime flowing
into it. The equation (36) therefore may be modified to express also mass- speed dependence:
(38)
(39)
The derived formula (38) gives the following results in comparison with formula (37) accepted by Special relativity:
v_{m/n} / c
Mass acc. to equation (37)
Mass acc. to equation (38)
Ratio m_{2} / m_{1
}
m_{1}
m_{2}
0,01
1,000033m_{o}
1,00005m_{o}
1,000017
0,05
1,000835m_{o}
1,001252m_{o}
1,000417
0,2
1,013663m_{o}
1,020621m_{o}
1,006864
0,5
1,098612m_{o}
1,154701m_{o}
1,051054
0,8
1,373265m_{o}
1,66667m_{o}
1,213652
0,95
1,9281m_{o}
3,2056m_{o}
1,660916
0,99
2,67338m_{o}
7,088812m_{o}
2,651323
0,9999
4,95221m_{o}
70,7124m_{o}
14,27896
In common case, when object X of the mass m moves in frame M at a speed v_{x/m} , and frame
M at a speed v_{m/n} in frame N, using equation (38) we may write
For mass of the object X, moving at a resultant speed v_{x/n}, as detected from frame N:
(40)
For mass of the object X, moving at a speedv_{m/n} in N frame, as detected from N frame:
(41)
For mass of the object X, moving at a speed v_{x/m} in M frame, as detected from N frame ( it means
the mass we detect moving in M frame at a speed v_{x/m}/
_{m/n} ) :
(42)
Solving relations between equations (40), (41) and (42), we obtain the following formulas:
(43)
(44)
(45)
The mass m of the object X, moving in M-frame at a speed v_{x/m} reaches value M_{xm}
when detected from M-frame: