5.3.2 Free motion of the object in a spaceflow line
5.3.2.1 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
as the frame intrinsic downward object speed, and,
v_{n/m+}
as the frame intrinsic upward object speed,
we have:
(171)
The equation (171) shows that to express the upward quantities, the space density
(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 Z_{ct} downward spaceflow acceleration, the quantity c_{g}
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 Z_{ct} acceleration of the
free object in the spaceflow line :
(171b)
To express the g_{zn/m-}, the negative value of the light speed c_{g}
and, to express the g_{zn/m+}, the positive value of the light speed c_{g} has to be
substituted to the equation (171b),
In a common orbital reference frame we may write:
(172)
where
dv_{zn/m}
stands for the differential of the Z_{ct} 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)
dt_{o}
stands for the frame time differential in a chosen orbital frame.
We may substitute for g_{x/m} from equation (132),
and for dt_{o}
(172a)
where
dr_{lm}
stands for the frame intrinsic length differential
Applying equation (160a), and taking
dr_{lm}=dL_{lm}, we may write:
(173)
Now, substituting to equation (172) for: g_{zn/m} from equation (172), dt_{o} from equation (172a), dr_{lm} from equation (173),:
we receive :
(174)
The equation (174) may be used to find out the speed of light c_{g} dependence along
with the radius r_{lmx}. To do that, we have to take:
Solving the equation (174) by substitution mentioned, we receive:
(175)
Integration of the differential equation (175) is not simple.However, accepting the condition
,
we may ignore this expression, and, in a such way we obtain :
(176)
The solution of the equation (176) for c_{g} < differs from the solution for
c_{g} > at the same radius r_{lmx}. 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 :
(176a)
and, consequently, the equation (107a) should be modified for the upward
frame intrinsic radius :
(176b)
Thus, looking for the boundary conditions to solve the equation (176), we may accept the following procedure:
Accepting c_{ g< } =c at r_{lmx} = r_{lmo},
where
(176c)
stands for the frame intrinsic radius in the space density on Earth surface,
the speed c_{g> } = c will be reached at the radius
r_{lmxo1 }:
(176d)
Thus we obtain the solutions : >
for the downward direction (c_{g} < 0 ) :
(177)
where
(177a)
for the upward direction (c_{g} > 0 ):
(178)
where
(178a)
Applying the equations (169) and (170)
to calculate the basic parameters of the gravitational fields, r_{x} and v_{m/x},
we can analyse the equations (177), (177a), (178) and (178a), and derive simplified equations for :
c_{ g < }..... the downward speed of light, and,
c_{ g > }..... the upward speed of light in the gravitational field,
which are exact enough for gravitational objects of the mass up to 10^{32} kg :
(179)
(179a)
and,
(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: Tab1
Object
Mass (kg)
r_{x} (m)
v_{m/x} (m/s)
k (m/s)
k_{1} (m/s)
k /k_{1}
Proton
1,672*10^{-27}
4,17211*10^{-11}
7,31334*10^{-14}
10035,3
8,95037*10^{12}
0,11212*10^{-8}
Earth
5,997*10^{24}
6,37928*10^{6}
1,11823*10^{4}
10035,5
8,95121*10^{12}
0,11211*10^{-8}
Sun
1,989*10^{30}
4,42068*10^{8}
7,74905*10^{5}
10048,7
9,00861*10^{12}
0,11154*10^{-8}
Common
2*10^{33}
4,42881*10^{9}
7,76331*10^{6}
10205,7
1,27507*10^{13}
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: