Gravity due to Space Flow

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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 10³³ 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*10⁸ m/s and the data from Tab1, we are able to calculate that even for the gravitational object of the mass of 2*10³³ 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:

	(184)

Applying the equation (108) for the speeds:

v_zn/m: 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:

	(191)

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