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### 5.1.2 Spacetime density

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 rx rx = rxm, because it is galileian projection of the radius rm into the frame X of the falling spacetime structure) from Fig.10, we obtain:
(103)

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.
Respecting equation (5b), we may write:
(104)

where
rtmx
stands for a radius in units of a chosen frame, determining the orbital surface of a common orbit,
rlmx
stands for a radius in units of a chosen frame, determining the distance between the singularity point and the common orbit,
stands for a relativistic space density in a direction perpendicular to a speed direction, and,
stands for a relativistic space density in a direction of the spaceflow speed.
Substituting the space density at an angle and 0 acc. to equation (31) for the relativistic density in equation (104), we obtain:
(105)

where
vm/x
is defined by equation ((102)), and
cg
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:
(106)

(107)

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 (107).

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):

(107a)

where
rtm
stands for the frame intrinsic radius determining the frame intrinsic surface of a common orbit, and,
rlm
stands for the frame intrinsic radius, determining the frame intrinsic distance between singularity point and a common orbit.

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