5.3.2 Free motion of the object in a spaceflow line Light speed in a spaceflow line

To explain and solve the motion of an object in a direction of the space flow we may use the relativistic factor derived for the space flow, since it was derived for this direction. However to resolve the motion of an object in the opposite direction we have to define the downward direction as a chosen relativistic frame, and express the upward motion as its the galilean projection into it. Designating
vzn/m acc. to equation (108)
as the Zct downward object speed,
as the Zct upward object speed,
as the frame intrinsic downward object speed, and,
as the frame intrinsic upward object speed,
we have:
equation (171)

The equation (171) shows that to express the upward quantities, the space density
equation (171a)

has to be applied instead of the space density as defined by equation (107) for the downward quantities.

On this condition, and, being aware of the fact that in the equation (132) defining the Zct downward spaceflow acceleration, the quantity cg express, in fact, the absolute value of the speed of light, we can derive, by the same way as in the chapter (5.1.5) the common equation for the Zct acceleration of the free object in the spaceflow line :

equation (171b)

To express the gzn/m-, the negative value of the light speed cg and, to express the gzn/m+, the positive value of the light speed cg has to be substituted to the equation (171b),
In a common orbital reference frame we may write:
equation (172)

stands for the differential of the Zct speed of the common object moving in the spaceflow line( the speed of an object in a common orbital frame, in length units of the common orbital frame, within time unit of the chosen frame)
stands for the frame time differential in a chosen orbital frame.
We may substitute for gx/m from equation (132), and for dto
equation (172a)

stands for the frame intrinsic length differential
Applying equation (160a), and taking drlm=dLlm, we may write:
equation (173)

Now, substituting to equation (172) for:
gzn/m from equation (172),
dto from equation (172a),
drlm from equation (173),:
we receive :
equation (174)

The equation (174) may be used to find out the speed of light cg dependence along with the radius rlmx. To do that, we have to take:
Solving the equation (174) by substitution mentioned, we receive:
equation (175)

Integration of the differential equation (175) is not simple.However, accepting the condition
equation ,
we may ignore this expression, and, in a such way we obtain :
equation (176)

The solution of the equation (176) for cg < differs from the solution for cg > at the same radius rlmx. This is why we may expect, that on Earth surface, as well, the downward value of the speed of light differs from its upward value. We may assume that this feature is caused due to different relativistic spacetime deformation in the upward direction. In fact, the space density (see equation (107) ) for the upward motion should be modified :
equation (176a)

and, consequently, the equation (107a) should be modified for the upward frame intrinsic radius :
equation (176b)

Thus, looking for the boundary conditions to solve the equation (176), we may accept the following procedure:
Accepting c g< =c at rlmx = rlmo ,
equation (176c)

stands for the frame intrinsic radius in the space density on Earth surface,
the speed cg> = c will be reached at the radius rlmxo1 :
equation (176d)

Thus we obtain the solutions :
> for the downward direction (cg < 0 ) :
equation (177)

equation (177a)

for the upward direction (cg > 0 ):
equation (178)

equation (178a)
Applying the equations (169) and (170) to calculate the basic parameters of the gravitational fields, rx and vm/x, we can analyse the equations (177), (177a), (178) and (178a), and derive simplified equations for : which are exact enough for gravitational objects of the mass up to 1032 kg :
equation (179)

equation (179a)

equation (179b)

The basic parameters of the gravitational fields of some chosen objects, and their respective constants k and k 1< , are shown in the following table:
Object Mass (kg) rx (m) vm/x (m/s) k (m/s) k1 (m/s) k /k1
Proton 1,672*10-27 4,17211*10-11 7,31334*10-14 10035,3 8,95037*1012 0,11212*10-8
Earth 5,997*1024 6,37928*106 1,11823*104 10035,5 8,95121*1012 0,11211*10-8
Sun 1,989*1030 4,42068*108 7,74905*105 10048,7 9,00861*1012 0,11154*10-8
Common 2*1033 4,42881*109 7,76331*106 10205,7 1,27507*1013 0,08004*10-8
We can easily derive, that for the objects like planets and smaller, the equations (179) and (179a) may be simplified once more:
equation (180)

equation (180a)