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 r_{lmx}, and,
In the chosen orbital frame at the radius r_{x}
Fig 15 The spacetime structure in gravitational field
where represents
C_{m}
the real spacetime structure cell in the common frame,
C_{mo}
the galileian projection of the cell C_{o} from the chosen into the common reference
frame,
C_{o}
the real spacetime structure cell in the chosen frame,
C_{om}
the galileian projection of the cell C_{m} from the common into the chosen frame
D_{o}
Diameter of the cell in the chosen frame, and,
D_{m}
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 r_{x}, the object revolving at the radius
r_{lmx} around the gravitational object, we will measure (by the meter breathing with the
spacetime structure) the circumference length l_{o} within the angle alpha:
(205)
the same, as the observer from the orbital frame at the radius r_{lmx} would measure
(l_{m}=l_{o}) 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
l_{m}):
(206)
and, the galilean projection of the frame intrinsic circumference length travelled by the
revolving object in the common orbital frame at the radius r_{lmx} 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
dl_{m} within the frame intrinsic time differential dt_{m}. Respecting the
linear geometry, we may write for the galilean projection dt_{o} of the time differential
dt_{m} 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
r_{x}/r_{lmx} times lower (higher) than it is in linear geometry, the respective galilean
projection of the time differential dt'_{o} of the time differential dt_{m}
into a chosen frame must be r_{x}/r_{lmx} times lower (higher) :
(207a)
Thus, we can derive for the Z_{ct} 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 (v_{zocr}), that maintains him in a circle, without vertical motion. We can easily derive from the equations (207b) and (207c) that the Z_{ct} orbital speed for the motion in a circle will be :
in linear geometry
(209)
in spherical geometry
(209a)
where T_{o} indicates period of rotation.
Considering the equation (209a) and Fig15 we certainly must come to a conclusion, that speed v'_{zo} = 2*PI*r_{lmx} / T_{o} can be explained by two ways :
The length of the trajectory l passed by the revolving object on the circle of the radius r_{lmx} is r_{lmx} / r_{x} times greater (in units of the chosen reference frame) than the length l_{o} of the trace, marked by the line connecting the revolving object with the centre of gravity, on a circle of the radius r_{x} (the chosen reference frame). Both paths (l and l_{o}) are passed within the same time interval.
The real frame intrinsic length of the trajectory passe by the object however is l_{m}=l*r_{x}/r_{lmx} and the respective Z_{ct} speed would reach
v'_{zo}=2*PI*r_{x}/T_{o} in a line geometry. The motion however takes place in a spherical geometry. That is why the Z_{ct} speed of the revolving object v_{zo}=2*PI*r_{lmx}/T_{o} must be taken into consideration.
Thus, if we put the common circumference of the circle of the radius r_{x} (the circle from the chosen frame of space density = 1) close to the circle of a common radius r_{lmx}, 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 r_{lmx}=r_{k} and at the orbital Z_{ct} speed v_{zo}=v_{zok} (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 r_{x}), within it the object passes the same path, must be inversely proportional to the radius r_{lmx}, because the period of rotation of the revolving object, providing that its vertical shift was stopped at a radius r_{lmx}, would be r_{k}/r_{lmx}
times smaller than such period of rotation at the radius r_{k}. Consequently must be :