Being situated in a chosen orbit (in a chosen frame, not moving with respect to a singularity point M,
and trying to measure radius rx of the spacetime structure passing through any
another common orbit ), we must find out that we can measure only the galileian projection of the
respective radius rx into our chosen orbit, because the instruments, we use, are
usually breathing with the spacetime structure. See Fig.10. Fig.10 Space density in gravitational field
Generalizing the relation between radia rmx and rxrx = rxm, because it is galileian projection of the radius
rm into the frame X of the falling spacetime structure)
from Fig.10, we obtain:
where stands for the space density due to geometry
of the spacetime structure in a gravitational field.
In common case the following circumstances have to be taken into consideration:
The spacetime density depends on direction in general. For our considerations a radius
suitable to calculate the orbit surface in equation (101) must be used, determined by the
space density in a direction perpendicular to a vector of speed vx/m.
To calculate the speed vx/m, the space and time densities in a speed
direction must be known.
The relativistic behavior of the spacetime structure must be implemented.
stands for the speed of light in a respective common orbital frame M (we do not know yet, whether the speed
of light stays constant, or varies along with the space density).
And, substituting from equation (105) to (104) we have:
The equation (107) defines the space density of the spacetime structure tn the gravitational field. The space density
is increased with the decreasing radius. Respecting the equation
(23), we can derive, that the spacetime (or time,)
density of the spacetime structure in the gravitational field equals to the square root of its space density acc. to equation
The space density of the own frame (frame intrinsic space density) equals to 1. This is why we can modify the
equation (106) to obtain the radia measured inside of the own frame (the frame intrinsic radia):
stands for the frame intrinsic radius determining the frame intrinsic surface of a common orbit, and,
stands for the frame intrinsic radius, determining the frame intrinsic distance between singularity point and
a common orbit.