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### 5.3.3 Common free motion in gravitational field

#### 5.3.3.1 Spherical and linear geometry

We can see the relations between the size of the cells of the spacetime structure on Fig15 in two orbital frames:
• In a common orbital frame at the radius rlmx, and,
• In the chosen orbital frame at the radius rx
Fig 15 The spacetime structure in gravitational field

where represents
Cm
the real spacetime structure cell in the common frame,
Cmo
the galileian projection of the cell Co from the chosen into the common reference frame,
Co
the real spacetime structure cell in the chosen frame,
Com
the galileian projection of the cell Cm from the common into the chosen frame
Do
Diameter of the cell in the chosen frame, and,
Dm
Diameter of the cell in the common frame, measured by meter not breathing with spacetime structure and calibrated on the chosen frame.
It must be:
(204)

Observing, from the chosen orbital frame at the radius rx, the object revolving at the radius rlmx around the gravitational object, we will measure (by the meter breathing with the spacetime structure) the circumference length lo within the angle alpha:
(205)

the same, as the observer from the orbital frame at the radius rlmx would measure (lm=lo) in his own frame within the same angle. This feature appears to be principal for the comprehension of the difference between the spherical and linear geometry, since in linear geometry the observer in the chosen frame will measure the con-space circumference length (galilean projection to lm):
(206)

and, the galilean projection of the frame intrinsic circumference length travelled by the revolving object in the common orbital frame at the radius rlmx into the chosen orbital frame will be:

The length interval in the chosen frame, however, corresponding to one turn of the revolving object in the common frame, equals to the value:

The object in orbital motion (perpendicular to the spaceflow speed) is passing the frame intrinsic length differential dlm within the frame intrinsic time differential dtm. Respecting the linear geometry, we may write for the galilean projection dto of the time differential dtm into the chosen reference frame :
(207)

In a spherical geometry however, since now the galilean projection of the length interval into a chosen frame is rx/rlmx times lower (higher) than it is in linear geometry, the respective galilean projection of the time differential dt'o of the time differential dtm into a chosen frame must be rx/rlmx times lower (higher) :
(207a)

Thus, we can derive for the Zct speed (speed in frame intrinsic length units of the common frame and time units of the chosen frame):
a/ in linear geometry
(207b)
b/ in spherical geometry
(207c)
The equations 207 - 207a (and, subsequently also 207b - 207c) seems to be in conflict with one another, since of course, the time differential of the common orbital motion must be uniquely determined as the one value (hence the real geometry is just one). In fact however, this contradiction is apparent only. The equation (207) has been derived for case only of the zero orbital speed, because to derive it, it was presupposed the gravitational space density same as for the motion in a space-flow line :
(208)

The equation (207a) has been derived for case only when the revolving object reaches such orbital speed (vzocr), that maintains him in a circle, without vertical motion. We can easily derive from the equations (207b) and (207c) that the Zct orbital speed for the motion in a circle will be :
in linear geometry
(209)

in spherical geometry
(209a)

where To indicates period of rotation.
Considering the equation (209a) and Fig15 we certainly must come to a conclusion, that speed v'zo = 2*PI*rlmx / To can be explained by two ways :
• The length of the trajectory l passed by the revolving object on the circle of the radius rlmx is rlmx / rx times greater (in units of the chosen reference frame) than the length lo of the trace, marked by the line connecting the revolving object with the centre of gravity, on a circle of the radius rx (the chosen reference frame). Both paths (l and lo) are passed within the same time interval.
The real frame intrinsic length of the trajectory passe by the object however is lm=l*rx/rlmx and the respective Zct speed would reach v'zo=2*PI*rx/To in a line geometry. The motion however takes place in a spherical geometry. That is why the Zct speed of the revolving object vzo=2*PI*rlmx/To must be taken into consideration.
• Thus, if we put the common circumference of the circle of the radius rx (the circle from the chosen frame of space density = 1) close to the circle of a common radius rlmx, we will have to discover that these two lines are created of the spacetime structure of the same space density (that is of the space density in a linear geometry). Let us designate this space density as the apparent gravitational space density for the calculations concerning orbital motion. For the motion in a circle (on condition that vertical speed = 0) it will then be :
(209b)

and,
(209c)
Starting common motion of the object in a gravitational field at the radius rlmx=rk and at the orbital Zct speed vzo=vzok (see Fig 15a), assuming of course some vertical displacement of the object after certain time interval, we can expect the following scenario :
• The frame intrinsic speed of the object in an apparent gravitational space density = 1 must stay constant.
• The time interval measured in a chosen reference frame (at the radius rx), within it the object passes the same path, must be inversely proportional to the radius rlmx, because the period of rotation of the revolving object, providing that its vertical shift was stopped at a radius rlmx, would be rk/rlmx times smaller than such period of rotation at the radius rk. Consequently must be :
(210)

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