5.3.3.3 Influence of relativity factor on common free motion

Influence of relativity factor on basic parameters of gravitational field

All equations defining common free motion in a gravitational field has bee derived in previous chapters on condition that the influence of the relativity of the motion (expressed by the relativity factor) was ignored. The routine calculations of the parameters of free motion are sufficiently accurate even if the relativity factor is ignored. For many calculations however, especially for gravitational fields of the objects characterized by considerably greater mass than the mass of the planet Earth, the influence of the relativity factor must be taken into consideration.

The gravitational acceleration in Z_{ct} frame (length units in a frame of the observed object, time units in a chosen frame) is defined by the equation (138b). Applying the equation (180a), we have :

(235)

where

c_{g}

stands for speed of light in a spaceflow line,

c

stands for speed of light on Earth surface and,

r_{x}

stands for the gravitational frame intrinsic radius.

Substituting from the equation (235) to (138b), we receive :

(236)

where

g_{zx/m}

stands for the Z_{ct} gravitational acceleration

Let us designate factor p :

(237)

In a chosen reference frame (r_{lmx} = r_{x}) :

(238)

Factor p determines how the gravitational acceleration is influenced by the relativistic behaviour of the spacetime structure in a gravitational field. Unfortinately the factor p is not the constant, neither in a gravitational field of one specific gravitational object (like the Sun). Its size is affected not only by the mass of the gravitational object, but also by the distance from the centre of gravity (radius r_{lmx})
This fact could result in a very complicated equations describing common motion in a gravitational field. However for calculations of trajectories related to motion, taking place in a relatively narrow orbital zones, like move planets, the mean value of the factor p, taken from its values at the limit radia (r_{k}, r_{f}), may be considered as constant, with a sufficient accuracy.
It should be :

In the equation (162) we presupposed that K, expressing the coefficient of proporcionality between the space flow and mass, is the universal constant. Now we can see that due to influence of the relative motion of the space flow with regard to the speed of light, the parameter K cannot be considered as universal constant, but it varies with the mass of the gravitational object. Therefore the new ratio T_{g} between the gravitational frame intrinsic radius and the frame intrinsic space flow speed will be :

(242)

where

T_{g}

stands for the new constant, influenced by the relativity factor; this constant must be calculated for each gravitational body,

T^{'}_{g}

stands for the original constant calculated when the relativity factor was ignored; T^{'}_{g} = 570,48 seconds,

r_{x}

stands for the new gravitational frame intrinsic radius, influenced by the relativity factor,

r^{'}_{x}

stands for the original gravitational frame intrinsic radius, not influenced by the relativity factor,

v_{m/x}

stands for the new frame intrinsic space flow speed, influenced by the relativity factor and,

v^{'}_{m/x}

stands for the original frame intrinsic space flow speed, not influenced by the relativity factor.

stands for the charactaristic radius of gravitational field calculated from the circumference of any orbital circle (meaured in place of the respective orbit) with its centre identical with the centre of gravity and,

r_{lmx}

stands for the charactaristic radius of gravitational field, defined by the distance between any point inside gravitational field and the centre of gravity, on condition that the space between this point and the centre of gravity was created by spacetime structure of the same gravitational space density (not the apparent gravitational space density) as it is in the respective point.

Substituting for c_{g} from the equation (235) :

(248)

Since r_{tmx} is a radius calculated from the circumference length of the circle with its centre identical with the centre of gravity and, r_{lmx} represents the distance from the centre of gravity, the equality of these two radia should be expected. In a weaker gravitational fields they really may be considered equal. With greater mass of the gravitational body however the ratio r_{tmx} / r_{lmx} rises up. In a gravitational field of the Sun it has to be taken into consideration.
The growth of the ratio r_{tmx} / r_{lmx} in principle means that the space density in a horizontal direction of the gravitational field becomes lower than the space density in a vertical direction. Subsequently the lower horizontal space density brings, in fact, the equivalent reduction of the angle of the meridian circle in a gravitational field, below its primary value of 360 grades. Therefore reaching the angle in this deformed spacetime structure will be registered, in the non-deformed geometry (of the equal space density in both vertical and horizontal directions), at an angle greater than 360 grades. For our calculations of the Z_{cr} orbital speed in this deformed spacetime structure therefore, we shall use (as derived before) the spacetime deformation density (as the square root of the space deformation density):

(248a)

From this point of view the equations (226) and (226a), defining the Z_{cr} orbital speed have to be modified. The Z_{cr} speed v_{ood} (that is the speed expressed in length and time units of a chosen frame) for the deformed spacetime structure shall be used :