5.1.5 Acceleration of the spacetime structure falling into a singularity
(Gravitational acceleration)
Standing (means: not moving with respect to a singularity point) in a common distance
r_{lmx} from a singularity, we can detect, in our own frame, that the spacetime structure
is falling into a singularity at a speed v_{x/m} (frame intrinsic spaceflow
speed), and is beeing just accelerated at an acceleration g (frame intrinsic
acceleration of the spaceflow speed).
Again, standing in a common frame (in a distance r_{lmx} from singularity), and
observing the spacetime structure flowing through our own frame, we could detect (if, for example, the quantum of the
spacetime structure was dyed) by the length instrument calibrated on our frame (frame of the observer),
and the clock calibrated on the chosen frame (frame corresponding to the distance r_{lm}
from the singularity point),
that the spacetime structure is falling into a singularity at a speed v_{zx/m},
following the equation (114),or, (115)
different than the frame intrinsic speed according to equation (110).
This speed may be considered as a real con-spacetime speed of the spacetime structure
through a common orbital frame (in its own units), respecting changes of the space and
time densities relating to a chosen reference frame.
The acceleration of the motion of the spacetime structure in frames of the Z_{ct} zone
(contemporary with the chosen frame) may be defined as a derivation of the speed v_{zx/m}
by time of the chosen frame:
(127)
where
g_{zx/m}
stands for the Z_{ct} spaceflow acceleration.
The speed v_{zx/m} is defined by equation (114).
To calculate the derivation dv_{zx/m}/dr_{lmx}, however, we would have to know
the derivation of the speed of light , c_{g}, by the radius. For the time being we do not know it. The derivation dv_{zx/m}/dr_{lmx} can be resolved in a following way:
in case of a constant speed of light c_{g}:
(128)
in case the speed of light c_{g} varies along with a radius (space density):
(129)
The derivation dr_{lmx}/dt_{o} may be expressed:
(130)
Solving the equations (114), (127), (128), (129) and (130), we have:
in case of a constant speed of light c_{g}:
(131)
in case the speed of light c_{g} varies along with a radius (space density):
(132)
The three spaceflow speeds we have been dealing with in chapter 5.1.3, may be expressed as the speed of
the space flow
that has passed the same distance l_{m} expressed in the same frame intrinsic units, within the same
frame intrinsic time interval t_{m}, but expressed as:
galilean projection of the time interval t_{m} into Z_{ct} frame, giving
time interval
for the speed v_{zx/m},
frame intrinsic time interval
t_{m} for the speed v_{x/m},
galilean projection of the time interval t_{m} into Z_{cr} frame, giving
time interval
for the speed v_{ox/m}
That is,
(133)
In a similar way as the acceleration g_{zx/m} was defined, we may define:
g_{x/m}
as the frame M intrinsic spaceflow acceleration, and,
g_{ox/m}
as the Z_{cr} spaceflow acceleration (the acceleration of the space flow, expressed
in length and time units of a chosen frame).
We came to conclusion, in chapter 5.1.1, that the speed v_{m/x} does not change with the radius.
Respecting the equation (110), we can say, that
the frame M intrinsic space flow gravitational acceleration in case of a constant speed of light will be:
(133a)
the frame M intrinsic space flow gravitational acceleration, in case the speed of light varies with the radius,
will depend on the derivation dc_{g}/dr_{lmx}
Respecting the definition formula of the Z_{cr} spaceflow acceleration,
(135)
and the derivation (from equation (134))
(136)
we receive, by substitution from eq. (134) and (136) to equation (135), the equation defining the
Z_{cr} spaceflow acceleration:
(137)
Summarizing the results of the chapter, we can say:
Measuring the acceleration of the space flow by means of the meter calibrated on the spacetime structure belonging
to a respective point (radius), and by means of the clock calibrated on a spacetime structure of the chosen frame (like
of the frame on Earth surface), we obtain the results corresponding to equation (131) or (132). This acceleration acts
in direction of the space flow (accelerating it).
Measuring the acceleration of the space flow by means of both, the meter and clock, calibrated on spacetime structure
belonging to a respective point (radius), we would come to conclusion, that the space flow moves without acceleration,
or, with acceleration (if any) due to change of the speed of light with the radius. Note: The spaceflow acceleration due to change of the speed of light might become significant in gravitational
fields of the big cosmic bodies (like stars) or in gravitational fields of the bodies of a very high mass density
(like nucleons).
Measuring the acceleration of the space flow by means both meter and clock calibrated on the chosen frame, we would
find out that the acceleration is constant (acc. to equation (137)), but acting in opposite to the direction of the space flow
(slowing down the spaceflow speed).
Note: The acceleration of the body moving in gravitational field will be described in subsequent chapters.