To examine the mass distribution under the surface of the gravitational object, let us have a look on the relativistic
phenomenon (how much the total mass exceeds the rest mass due to its motion with respect to the spacetime structure
falling into mass). According to equation (38) we
can express the relativistic mass :
(516)
where
m
stands for the relativistic mass
m_{o}
stands for the rest mass
Designating
(517)
as the relativity factor, and applying the equations (170) and
(180), we can derive :
(518)
We can derive from the equation (516) that even at the relativity factor k_{r} = 0,1 it will be
m = 1,00335 m_{o}.
Designating k_{r} = 0,1 as sufficient to apply the equation (516) without the relativistic
modification, that is in form
(519)
we can derive from equation (516)
(520)
where
m_{k}
stands for maximum gravitational mass (evenly spread under the respective radius), at which the relativity effect causing
mass increase may be neglected, and the equation (519) may be considered as sufficiently precise.
The equation (520) can be modified using the average mass density instead of mass;
(520a)
The curve "average mass density vs radius" acc. to equation (520a) is shown on Fig20, for k_{r}=1,005.
The maximum mass m_{k} under radius r_{lmx}
that may be considered as not influenced by relativistic mass increase.
Fig20 shows that, to derive the equation of the mass distribution, the relativistic mass increase due to the motion of the mass
inside gravitational objects with respect to the falling spacetime structure need not be taken into consideration, since even at the
objects of the radius 2*10^{11} m (higher than the distance Sun - Earth) the average mass density
about 6000 kg/m^{3} (the mass density occurring at planets) is allowable at the relativity factor
k_{r} < 1,005. This relativistic mass increas however plays the significant role for the cosmic objects
created of the mass of high mass density, like the neutron stars.
Another factor interacting the space flow streaming into a mass, that is also the mass quantity, is the space (spacetime) density, that may
influence the mass by the similar way like we derived it in chapter 4.8.
The space density inside the gravitational object may be significantly influenced by the mass distribution
(depends on mass and radius).
However now we are speaking about the space density inside gravitational field of the mass, that is located under the respective
radius, not about the space density inside atomic nucleus, nucleon or similar mass particle. In chapter
4.8, respecting the postulate of the constancy of the speed of light, we have derived that
the relative speed, with respect to the chosen frame, induces the variation of the space density inside the gravitational field of
the mass particle, giving rise to the mass increase (relativity effect). In chapter 5.3.4.2
however, we have derived that any gravitational object is able to influence the space density inside the mass particle in case
only, if the object reaches, by its own gravitational field, in position of the mass particle, the space density higher than it is
induced by the mass particle itself. This effect may occur in case only, when the gravitational field of the influencing object
is extremely intensive, since the space density on the proton's surface reaches the value about
and we can derive that, to reach such high space density in proton, the object of the Earth mass (concentrated in a very small
volume) should be situated in a distance from the proton not longer than
This is why, deriving the mass distribution in gravitational object, we need not take into consideration the relativistic mass increase
(decrease) due to increase (decrease) of the space density in the gravitational field inside the gravitational object.
Ignoring both the relativistic mass increase due to speed, and the relativistic increase (decrease) due tu change of the
space density, the mass distribution under surface of the gravitational object may be considered
depending on the mass density dustribution, and on this feature of the space density
that is responsible for the mass distribution Note:observing from the chosen reference frame, like from the earth surface, the constant
cube vulume, not breathing with the space, we would have to find out that the quantity of mass in this cube differs depending on the
position of the cube inside gravitational object, in consequence of the change of the space density.
Between radia r_{lmx} and
r_{lmx}+dr_{lmx} the mass differential dm may be expected :
(521)
where
stands for the frame intrinsic space density (measured by the instruments breathing with spacetime structure in a
respective position inside gravitational object)
In general, we can tell, that under the common radius r_{lmx} the mass m is situated,
to which the gravitational radius r_{x} is related according to equation
(169):
The gravitational space density :
(522)
Solving the equations (521) and (522) we receive the common equation of the mass distribution vs radius inside gravitational
objects :
At = o =const
At constant frame intrinsic space density the differential equation (523) gives the solution :
(524)
where
(525)
K is common constant.
At the boundary condition :
at r_{lmx} = R_{o} (the outer radius of the gravitational object) :
m = M_{o} (the total mass of the gravitational object)
- the equation (524) becomes :
(526)
Substituting for m from equation (526) to equation (522), we recwive the equation defining the gravitational
space density :
(527)
Applying the equation (526) we can express the average mass density under respective radius :
(528)
The examination of the equations (526), (527) and (528) leads to the following conclusions :
The gravitational object can be created of the mass of any frame intrinsic mass density. The object of the mass
M_{o} may be created of the low (high) mass density, since the low (high) mass density is
compensated by high (low)space density to reach the requested average mass density. Note
The decision whether the object would be able to exist, made of any frame intrinsic mass density, depends on
conditions of stability of the object (like gravitational acceleration,pressure, themperature and many others), and
is not discussed in this chapter.
At k = 3, or, acc. to equation (525) at the frame intrinsic mass density
(529)
The mass distribution becomes directly proportional to the third power of the radius. See Fig21.
Fig21
The gravitational space density becomes constant, not depending on radius :
(530)
The average mass density under radius becomes constant, not depending on the radius :
(531)
At k < 3, or, acc. to equation (525) at the frame intrinsic mass density
(532)
The mass becomes directly proportional to the power lower than 3 of the radius.See Fig22
Fig22
The space density becomes increasing with the decreasing radius. See Fig23.
Fig23
The average mass density under radius becomes increasing with the decreasing radius. See Fig24.
Fig24
At k > 3, or, acc. to equation (525) at the frame intrinsic mass density
(533)
The mass becomes directly proportional to the power higher than 3 of the radius.See Fig25
Fig25
The space density becomes decreasing with the decreasing radius. See Fig26.
Fig26
The average mass density under radius becomes decreasing with the decreasing radius. See Fig27.
Fig27