Einstein's Theory of Relativity
versus
Classical
Mechanics
by Paul Marmet
Chapter Twelve
On the Formation of Pseudo Black Holes.
12.1 - Formation of a
Protostar.
In this chapter, we will consider what happens to a large volume of gas
when taking into account the gravitational field of each individual
atom. As an example, we use a nebula containing N atoms of hydrogen.
Due to Newton's universal law of gravitation, all these individual
electrically neutral particles attract each other. Consequently, each
atom slowly drifts toward the center of the system. The gas becomes
more and more compact as a function of time and the nebula occupies a
gradually smaller volume of space.
During the collapse of the nebula, the velocity of the particles
increases due to the increasing gravitational potential created by the
increasing concentration of matter. The density and the velocity of
individual atoms augment so that the temperature increases while the
radius of the volume of gas decreases. Consequently, the gas becomes
very hot. These high temperature and density produce a high pressure
that reduces the collapsing rate.
Due to Planck's law of radiation, the gas emits its thermal energy as
electromagnetic radiation to outer space. This phenomenon causes a
reduction of the internal temperature and pressure so that the star can
progress with further shrinking. These two processes go on
simultaneously as long as the star has enough mass to produce a
gravitational force sufficiently large to produce further shrinking.
The shrinking rate of the star depends on the rate of emission of
energy of the star through radiation. An equilibrium exists between the
atomic, molecular or nuclear forces which provoke emission of radiation
at high temperature and the gravitational forces.
In the above qualitative description, we consider that the number N of
hydrogen atoms does not change during the contraction of the nebula
into a star. However, a large amount of energy has to be emitted from
the star through radiation in order to get rid of the thermal energy.
One must take into account the principle of mass-energy conservation
requiring the mass of the star to decrease because of the radiation
emitted due to Planck's law of radiation. 12.2 - Mass-Energy Conservation in Clusters of Atoms.
In order to satisfy the principle of mass-energy conservation, let us
calculate quantitatively the amount of energy that must be emitted from
the protostar when it is transformed from a nebula to a high density
star. Let us start with an initial very large diffuse nebula. We will
calculate the change of gravitational energy when the nebula takes the
shape of a hollow sphere of radius R.
Let us calculate the gravitational energy when N hydrogen atoms coming
from the nebula have all reached the distance R from the center of
mass. When the first atoms reach that distance, the sphere is
infinitely thin. The potential energy met by each new individual atom
increases with the number of atoms (mass) that has already reached the
distance R. This process goes on until all atoms have formed a sphere
of radius R. We have then a spherical protostar.
In order to calculate the total internal gravitational potential of
such a star, let us use the building up principle and accumulate
individual hydrogen atoms, one by one. In the case of the Sun, the
number of hydrogen atoms needed is about 1.2×1057.
Each
individual atom is systematically brought from a large distance in
outer space to the location at a distance R from the center of the
stellar mass being formed. We consider the approximation of a hollow
sphere because we want to keep the potential constant inside the star.
The very first step in the formation of the star is to bring two
hydrogen atoms together at a distance R. At that distance, the atoms
have acquired gravitational energy E{1} due to the gravitational
potential between them. This gravitational energy is given by:
 |
12.1 |
where mH
is the mass of the hydrogen atom. The two particles remain trapped at a
distance R in this gravitational potential if the amount of
electromagnetic energy emitted is equal to E{1}. The equivalent loss of
mass to stabilize this interaction is equal to:
 |
12.2 |
Therefore, after stabilization by the emission of radiation, using
equations 12.1 and 12.2, we find that the remaining mass M{1} of the
pair of hydrogen atoms (at distance R) is:
 |
12.3 |
After the formation of the first pair of hydrogen atoms, let a new
hydrogen atom fall (at a distance R) into the gravitational field
produced by the new pair. The new hydrogen atom of mass mH
interacts at a distance R from the pair of mass M{1} previously formed
and described in equation 12.3. Using Newton's law, the gravitational
energy between the pair of hydrogen atoms with mass M{1} and the
individual hydrogen mH atom is:
 |
12.4 |
We might want to explain how the new hydrogen atom can be at an
effective distance R from the previous pair of atoms. The distance R
mentioned here means that the new atom is located at a distance R from
the previously formed pair so that the gravitational potential between
the new atom and the pair is equivalent to the potential that would
exist if the previously formed pair of atoms were close together and
the new atom were at a distance R from the pair. This description is
supported mathematically by a theorem (used in electrostatics) which
shows that the potential created at the surface of a spherical
distribution of charges is the same as if all the charges were located
at the center of the sphere. We will apply this same theorem here for
the case of the gravitational potential of particles approaching the
spherical distribution of matter forming the star.
In equation
12.4, the mass DM{2} lost after emitting
thermal energy is:
 |
12.5 |
The total
mass M{2} of the three hydrogen atoms is then:
| M{2} = M{1} + mH - DM{2} |
12.6 |
 |
12.7 |
Equations
12.7, 12.3 and 12.4 give:
 |
12.8 |
Of course, when a star is formed, the energy does not have to be
emitted immediately after the addition of each individual atom. When
particles are brought together, they form a hot gas in their
gravitational potential which cools down later by the emission of
radiation. There is no difference of energy if the radiation is emitted
immediately or later.
Repeating
the operation and adding a fourth hydrogen atom to the set of three
atoms gives:
 |
12.9 |
Equations
12.8 and 12.9 give:
 |
12.10 |
Adding
another hydrogen atom to the growing mass gives:
 |
12.11 |
Equations
12.10 and 12.11 give:
 |
12.12 |
Let us
define:
 |
12.13 |
Then:
| M{4} = 5mH - 10mH2 Z + 10mH3 Z2 - 5mH4 Z3 + mH5 Z4 |
12.14 |
Adding
another hydrogen atom gives:
| M{5} = 6mH - 15mH2 Z + 20mH3 Z2 - 15mH4 Z3 + 6mH5 Z4 - mH6 Z5 |
12.15 |
The seventh
hydrogen atom gives:
| M{6} = 7mH - 21mH2 Z + 35mH3 Z2 - 35mH4 Z3 + 21mH5 Z4 - 7mH6 Z5 + mH7 Z6 |
12.16 |
Going on
with more individual atoms but limiting our calculations to the fourth
power of mH gives:
| M{7} = 8mH - 28mH2 Z + 56mH3 Z2 - 70mH4 Z3 |
12.17 |
| M{8} = 9mH - 36mH2 Z + 84mH3 Z2 - 126mH4 Z3 |
12.18 |
| M{9} = 10mH - 45mH2 Z + 120mH3 Z2 - 210mH4 Z3 |
12.19 |
| M{10} = 11mH - 55mH2 Z + 165mH3 Z2 - 330mH4 Z3 |
12.20 |
| M{11} = 12mH - 66mH2 Z + 220mH3 Z2 - 495mH4 Z3. |
12.21 |
The
coefficients of the equations above can be generalized to give:
 |
12.22 |
For a star like
the Sun, the value of N is about 1057. Then for N>>1
equation 12.22 gives:
 |
12.23 |
which is
identical to:
|
|
12.24 |
 |
12.25 |
Let us
define:
Equation
12.25 becomes:
 |
12.27 |
This can be
written (N is so large that it can be approximated to ¥ ):
 |
12.28 |
We recall
that Y = NmH
is the total mass of the nebula that formed the star. This would be the
mass of the star if there were no energy (mass) lost through radiation
during the formation. M{N} is the final mass of the star made of N
hydrogen atoms after taking into account the thermal energy emitted as
explained above.
12.3 - Mass of a Star
versus the Amount of Matter Used for Its Formation.
Equation 12.28 gives the mass of the star as a function of the amount
of matter Y used to form it. Of course, when a larger amount of matter
falls into the gravitational potential, thermal energy is emitted and
the amount of mass lost into radiation increases. In these
calculations, the value of Z (from equation 12.13) is kept constant
when we study a star having a fixed radius R. Figure 12.1 shows the
final mass of the star (after temperature stabilization) as a function
of the total mass falling on it, using Z = 1 in equation 12.28.
We see on figure 12.1 and from equation 12.28, that for a very small
amount of hydrogen atoms, the total mass of the star is almost the same
as the mass of the atoms used before the formation. However, when the
number of atoms accumulated in the star becomes larger, the
gravitational potential acting on each newly added hydrogen atom
becomes increasingly important.
Figure 12.1
More energy is lost in thermal radiation after each new hydrogen atom
is added. Consequently, an increasing fraction of the new mass is lost
when the star becomes more massive.
Here is a numerical example obtained from equation 12.28. When the
total input of mass from the nebula is 0.01 (YZ = 0.01), independently
of the value of Z, about 99.5% of that mass remains in the star. For
one unit (YZ = 1.0) of input mass, the final mass is 63% of the initial
matter. When the input mass is ten units (YZ = 10.0), only 0.005% of
the new mass is added to the star. Finally, when the amount of matter
given by the nebula to form the star becomes much larger, the new mass
added to the star becomes almost completely transformed into energy due
to the gigantic gravitational potential. Therefore the mass of the star
no longer increases when the value of YZ gets very large (as shown on
figure 12.1).
12.4 - Mass of a Star
versus its Radius.
Within the limits explained above, let us now consider a different way
to build a star. Instead of increasing the amount of matter from outer
space while forming the star at a constant radius, we use a constant
number of hydrogen atoms from the nebula but all matter is contracted
into a star of radius R.
When the
star is initially very big, the gravitational potential at its surface
is negligible. A very large star appears almost like a concentrated
nebula without an intense gravitational potential. However, when the
radius gets smaller, the high density star generates a much higher
gravitational potential so the increase of temperature generates
radiation which causes a loss of mass-energy of the shrinking star.
Using equation 12.28, we can calculate the radius of the star formed
from a contracting nebula containing a constant number of atoms of
matter. During the decrease of the radius, the star is maintained at a
relatively low temperature (of a few tens of thousand degrees), due to
Planck's emission of radiation.
Figure 12.2
When the
total number of particles N (= Y/mH)
coming from the nebula is kept constant, Z(R) becomes the variable (see
equation 12.13). For Y = 1, let us calculate the residual mass of the
star as a function of its radius R. After temperature stabilization,
the relative mass of the star (with respect to the mass of the initial
nebula) as a function of the radius R is given by equations 12.13 and
12.28. This is illustrated on figure 12.2.
We see that when the radius of the nebula (or the star) decreases, the
star loses mass as electromagnetic radiation more and more rapidly.
12.5 - Maximum Mass of a
Star versus Its Radius.
Let us assume now that the mass available Y is so large that the
product YZ is always larger than 10. In that case, the value of the
bracket in equation 12.28 reaches a maximum of 1.0. Let us substitute
equation 12.13 in equation 12.28. This gives:
 |
12.29 |
Since the
maximum value of the bracket in equation 12.29 is 1.0, the maximum
value of M{N} as a function of R is:
 |
12.30 |
Equation 12.30 shows that the maximum mass of a star increases linearly
with its radius R. Above this limit, any mass falling freely on the
star reaches a kinetic energy equal to its mass so that the same amount
of radiation energy is freed and there is no net increase of mass of
the star. The incoming particle is totally transformed into radiation
which totally escapes from the star.
12.6 - Complete
Transformation of Mass into Energy.
There is another way to find the maximum mass of a star of radius R. We
have seen that the gravitational energy E(Pot) of a particle of mass m
at a distance R from the surface is given by:
 |
12.31 |
We know that independently of their masses, all particles reach the
same velocity when they fall from outer space to the surface of the
same star. During their fall, particles acquire kinetic energy. The
kinetic component of energy of a particle moving at velocity v is given
by (g-1)m in the equation:
where
 |
12.33 |
During the fall of a particle in the gravitational potential of a star,
no energy is coming from outside the system. Consequently, the total
energy of the falling particle remains constant during an unperturbed
fall.
This result
is different from
the inertial acceleration of a mass absorbing energy given by an
external independent source. Due to that external source of energy, the
total energy of the particle increases as given by equation 12.32.
However, when falling freely in a gravitational field, the kinetic
energy increases at the expense of the gravitational energy of the
particle.
Let us
consider a particle reaching the surface of a star (of maximum mass).
The velocity corresponds to g = 2 (v =
0.866c). Then the kinetic energy Ek is equal to the initial
mass at rest:
When the particle hits the surface of the star, the kinetic energy is
released and emitted toward outer space (either immediately as gamma
rays or later as thermal energy). When this happens, the loss of mass Dm
is equal to the mass of the particle m. At the surface of the star, the
kinetic energy of the particle is equal to the gravitational energy it
has lost. We have:
 |
12.35 |
Therefore,
in that limit case, the mass Mlim of
the star is:
 |
12.36 |
Consequently, any mass falling from outer space to the distance Rlim from the star of mass Mlim
will be totally annihilated into radiation. As expected, this result is
identical to equation 12.30. Consequently, when the surface of the star
is at such a deep gravitational potential, there is no possibility of
increasing the mass of the star any further. Finally, if a particle has
an initial velocity toward the star when entering the outer limits of
the gravitational field, more energy will be removed from the star
through radiation than the amount added by the particle. The mass of
the star then decreases since more mass escapes by radiation than the
amount of mass added by the particle.
Of course, near the surface of a star (which has a maximum mass), the
gravitational potential is enormous so that clocks run at a very slow
rate. Matter located in this extreme gravitational field will interact
according to the proper parameters existing at that location.
Consequently, the spectrum of the Planck radiation emitted from this
deep potential will be emitted according to the local clock which runs
very slowly. The spectrum will be displaced toward longer wavelengths
with respect to outer space where clocks run more rapidly as explained
in chapter one. However, after its emission from the location in the
deep gravitational potential, light will not be redshifted again while
traveling against the gravitational field as explained in chapters one
and ten.
If we
consider a particle reaching the ultimate potential at a distance Rlim
from the center of the star, there is no possibility for it to move
deeper inside that radius because there is nothing left of the
particle. It would be absurd to discuss the behavior of particles at or
inside that extreme radius since they no longer exist and all their
energy and mass have been transformed completely into radiation.
Comparison.
This relationship for the maximum mass of a star can be compared with
the Schwarzschild radius. Let us note that the Schwarzschild radius RS
has an incomprehensible meaning in our context. Just as for general
relativity, it is not compatible with the principle of mass-energy
conservation. It is given by the relationship:
 |
12.37 |
12.7 - Proper Values in
Extreme Gravitational Potentials.
Let us consider that an observer in outer space measures the distances
between the center of a star (having the maximum mass Mlim)
and different bodies stationary at different distances. Using his
proper units, the outer space observer can measure the distances
between the center of the star and the closest body existing around it
(which is near Rlim) up to the more
distant masses. However, the observers located on each of those bodies
will use their proper units to make their measurements of their own
distance from the center of the star. They must use these proper values
in order to apply correctly the well-known physical relationships. We
have seen that the absolute length of the meter is longer for an
observer located closer to the star. Consequently, when measuring the
same absolute radius, the number of proper meters will
be smaller for the observer close to the star than for the outer space
observer.
Using the
equations given in chapter four,
we see that when the distance from the star is large (in the Newtonian
limit), the number of proper meters measured by an outer
space observer is almost identical to the number obtained by an
observer not too close to the star. However, when the observer is close
to the extreme minimum radius Rlim, the use of the extremely dilated proper meter will give a number
of proper meters approaching zero (and not Rlim(o.s.)). For this reason, physical phenomena taking place
near location Rlim (using internal
proper values) appear very strange to an outer space observer.
Near that
location (Rlim),
the Bohr and nuclear radii get very large and the corresponding energy
inside particles becomes extremely small with respect to the external
mechanical forces. In outer space, we are used to see internal (atomic
and nuclear) forces of matter being much larger that the mechanical and
gravitational forces. Near a degenerate star, nuclear forces are much
weaker. This phenomenon favors reactions between particles.
Let us also recall that in the first chapters of this book, we were
calculating very small relativistic interactions (i.e. Mercury
precessing around the Sun). It was then enough to consider the first
order of a series expansion. However, when we consider bodies with
kinetic energy in a very deep gravitational potential, these
approximations are no longer accurate.
12.8 - Beyond the Extreme
Gravitational Potential.
Let us consider a star having a maximum mass and therefore surrounded
by an extreme potential. We have seen that when an hydrogen atom gets
closer to the surface of the star, its mass decreases when brought to
rest and its clock slows down in the same proportion. We have seen that
the same maximum gravitational potential can exist at the surface of
stars having different radii. When the nucleus of this star approaches
that extreme limit of gravitational potential, the number
of particles forming that star approaches infinity while the mass of
each atom approaches zero. The product of these two parameters
approaches a constant (for a given radius) as shown in equation 12.30.
Finally, extrapolating (to a smaller radius) beyond this extreme
potential, the mass of the falling hydrogen atom disappears at the same
time as the clock becomes infinitely slow and finally stops running at Rlim.
In
fact,
one
can
say indifferently that the clock has stopped running
or that the clock has disappeared and no longer exists. Therefore
clocks become infinitely slow at the same time as they disappear
completely out of existence. In physics, it is absurd to study matter
inside the critical radius Rlim.
12.9 - Formation of Matter
in a Deep Gravitational Potential versus the Formation of Matter and
Anti-Matter.
We have seen above that mass can be transformed into radiation in a
deep gravitational potential without requiring a reaction between
matter and anti-matter. In physics, there is another well-known
mechanism transforming mass into radiation: the annihilation of a
particle with its anti-particle. For example, we know that an electron
and a positron can be annihilated into radiation. As expected, the
corresponding inverse mechanism is also known from the interaction of
photons creating a pair of matter and anti-matter. It is important to
notice that the reaction of annihilation of matter with anti-matter is
extremely rapid so that matter formed at the same time (and at the same
location) can survive only during an extremely short time before being
annihilated. Particles and anti-particles destroy each other at a very
high rate. This system is quite unstable. Furthermore, since matter and
anti-matter are formed simultaneously at the same location, it is
ultimately improbable that they could separate out to form independent
galaxies. Consequently, another mechanism of formation of matter
without involving anti-matter is required to explain our universe if we
want to avoid ad hoc hypotheses.
12.9.1 - Inverse
Gravitational Mechanism.
We have seen in this chapter how matter falling in a deep gravitational
potential is finally transformed into radiation. This mechanism cannot
be maintained forever in the universe because all matter would be
transformed into radiation. We have explained above how the formation
of matter through the mechanism of matter and anti-matter cannot lead
to the formation of huge clusters of galaxies of matter in the universe
as we observe them. There must be an equilibrium between the formation
and the annihilation of matter in the universe. Mass-energy
conservation is not compatible with the creationist theory that claims
that the universe was formed from nothing ten or fifteen billion years
ago.
It is well
known in physics
that for every mechanism, an inverse mechanism exists. The simple
absorption of radiation by matter is to some extent an intermediate
mechanism of transformation of energy into mass without involving
anti-matter. However, in that case, atoms become more massive but no
new atoms are formed.
A simplistic
description of the inverse mechanism corresponding to the annihilation
of matter in a gravitational field is the following. Since radiation is
emitted when atoms hit a surface located in a deep gravitational
potential, we can foresee that energetic radiation hitting the surface
of the same star could generate particles with sufficient kinetic
energy so that they could reach the escape velocity vesc
( = 0.866c) of a star with extreme mass and be freed in outer space. Of
course, other mechanisms involving gravity can be suggested but are
beyond the discussion of the present book.
When matter falls into an extreme gravitational potential, it is
transformed into energy without involving a reaction between matter and
anti-matter. Consequently, the inverse reaction must equally correspond
to the formation of matter without the creation of anti-matter. We have
seen that a reaction generating matter plus anti-matter is not
acceptable to explain the origin of matter in the universe, because of
the extremely fast inverse reaction returning matter into radiation. We
see now that a mechanism using gravity can explain the transformation
of matter in the universe.
The
transformation of matter into radiation (and its inverse reaction) is
an extremely slow process since the time for a star to emit the thermal
energy during its formation depends on its size but generally takes at
least a few hundred million years. One can expect that the inverse
reaction transforming radiation into neutral particles can take a few
billion years before forming nebulae which later evolve into stars and
later into other bodies with a very deep gravitational potential. Such
mechanisms would finally form a complete cycle transforming matter into
radiation and vice versa. On the average this cycle would repeat itself
every ten or fifteen billion years. In such a case, after a full cycle,
the information about the exact previous structure of the universe
would be lost. From this mechanism, matter of the universe could be
recycled periodically. During that cycle, since there would be large
variations in the time taken by concentrations of masses to evolve, the
universe would always look more or less the same through time. The
possibility of such a mechanism becomes highly probable when taking
into account the red shift mechanism taking place in our universe as
demonstrated [1] in previous
papers.
12.10 - References.
[1] P. Marmet, A New
Non-Doppler Redshift, (Book), Physics Dept. Laval University,
Québec, Canada, 64p., 1981.
also:
P. Marmet, A New
Non-Doppler Redshift, Phys. Essays, 1, 24-32, 1988.
also:
P. Marmet, Redshift
of Spectral Lines in
the Sun's Chromosphere, IEEE, Transactions on Plasma Science:
Space and Cosmic Plasma 17, 238-243, 1989.
also:
P. Marmet and Grote Reber, Cosmic Matter and the Non-Expanding
Universe, IEEE, Transactions on Plasma Science, 17, 264-269,
1989.
also:
P. Marmet, Non-Doppler
Redshift
of
Some
Galactic
Objects, IEEE, Transactions on Plasma
Science, 18, 1, P. 56-60, 1990.
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