5.3.2.2 Speed and acceleration of the free object in a spaceflow line
We can judge from Tab1 that for gravitational objects up to
10^{33} kg due to small value of the ratio v_{m/x}/c the relativistic factors
may be neglected for practical calculations (perhaps except of the calculations of the long-term gravitational
influence).
Differentiating the equations (179a) and
(179b), we have :
(181)
(181a)
Substituting from equations (180) and (181) to equation
(171b), and neglecting the relativistic factors, we receive:
(181b)
where
g_{zn/m-}
stands for the Z_{ct} acceleration of the object on its downward motion
And, substituting in a similar way from equations (180a) and (181a) to equation
(171b), we receive :
(181c)
where
g_{zn/m+}
stands for the Z_{ct} acceleration of the object on its upward motion
Now, designating :
(181d)
we can derive :
(181e)
(181f)
Applying c=2,997*10^{8} m/s and the data from Tab1,
we are able to calculate that even for the gravitational object of the mass of 2*10^{33} kg the ratios
g_{2+/1} and g_{2-/1} do not exceed the value of 5,1 %. This is
why we can say, that for practical calculations (except of the calculations of the long-term gravitational influence)
the equations (181b) and (181c) may be modified, and the Z_{ct} object acceleration
defined as :
(182)
Neglecting the relativistic factors and applying the equation (182), we can rewrite the equation
(174):
(183)
Resolving the differential equation (183) for the boundary condition
v_{zn/m}=v_{zn/mk} at r_{lmx}=r_{lmxk},
we receive the common formula for the Z_{ct} speed of the object moving in a spaceflow
line:
the object's Z_{ct} speed (speed of the object in length unit of the space density
that the object is just passing through, within time unit of the chosen frame),
v_{n/m}
the object's frame intrinsic speed (speed of the object in length and time units of the space density that
the object is just passing through), and,
v_{on/m}
the object's Z_{cr} speed (object,s frame intrinsic speed, expressed in length and
time units of the chosen frame),
we can write:
(185)
(185a)
Since, neglecting the relativity factor,
(185b)
substituting from the equations (184) and (185b) to the equation (185) we obtain:
(186)
where
v_{n/mk}
stands for the object's frame intrinsic speed, corresponding to its Z_{ct} speed
v_{zn/mk}, at the radius r_{lmxk} (the boundary condition).
(186a)
and, substituting from equations (186) and (185b) to equation (185a), we obtain:
(187)
where
v_{on/mk}
stands for the object's Z_{cr} speed, corresponding to its frame intrinsic speed
v_{n/mk}, and to its Z_{ct} speed v_{zn/mk},
at the radius r_{lmxk} (the boundary condition).
(187a)
The equations (186) and (187) make possible to derive the respective frame intrinsic and Z_{cr}
acceleration, applying the following equations:
(188)
(188a)
Resolving the equations (186) and (188) we obtain :
(189)
Remark 1 :
Once the object has reached the frame intrinsic speed v_{n/mk}=-v_{m/x} at any radius
r_{lmxk} it moves in its following course without its relative motion with respect to the spacetime
structure. Therefore
it must be g_{n/m}=g_{x/m}=0. We can see, that in this case the equation (189)
gives g_{n/m}=0. This result is in accordance with the equation
(133a).
Once the object has reached the frame intrinsic speed v_{n/mk}=v_{m/x}
(in a drection out of the gravitational field), its frame intrinsic acceleration again becomes
g_{n/m}=0 (see equation 189), and the speed
v_{n/m}=v_{m/x} will not come back below the limit v_{m/x}
without being braked by the outward force (see equation (186). The speed v_{m/x} therefore
represents the frame intrinsic escaping speed from the gravitational field.
Resolving the equations (187) and (188a) we have :
(190)
Remark 2 :
The object moving at the frame intrinsic speed v_{n/mk}=v_{m/x} (without motion
with respect to the spacetime structure), that is, at the Z_{cr} speed
(190a)
does not appear to move without acceleration in Z_{cr} frame. In fact in this frame it appears to
be braked, since in this case the equation (190) gives the acceleration
(190b)
This result is in accordance with the equation (137).
Remark 3 :
Resolving the equation (190) for g_{on/m}=0, we receive:
(190c)
This equation defines radius r_{lmx} at which the Z_{cr} acceleration,
or braking of the object in free fall reaches zero, that is, the radius at which the object's Z_{cr}
accelelation or braking in course of its free fall has been reduced to zero and the object now begins to be braked
or accelerated.
Remark 4 :
Substituting for v_{n/mk}=0 to equation (189), we receive:
(190d)
The equation (190d) says, that the frame intrinsic acceleration of the object not moving with respect to M-frames
(or, with respect to the gravitational object) is constant at r_{lmx} = r_{lmxk}
(that is at the moment when v_{n/m} = 0), and then becomes decreasing with decreasing radius
(since v_{n/m} goes up).
Remark 5 :
Substituting for v_{on/mk}=0 to equation (190), we receive the equation
(190e)
defining the course of the Z_{cr} acceleration of the object along with radius
r_{lmxk}, the speed of that has become zero
at the radius r_{lmxk} (at the radius r_{lmxk} the object has not been moving
with respect to M-frames).
The examples of the object free fall acceleration calculated in accordance to the equations (182), (188) and (188a)
are shown on Fig13 and Fig14.
Fig13
Fig14
Resolving the equation (186) for v_{n/m}=0, we can derive the radius r_{c}
at which the object culminates if its motion has begun at the radius r_{lmxk} and at the speed
v_{n/mk} of a positive value: