**This is the first article devoted to entirely new modeling of the universe, modeling, which is not based on the equations of general relativity. This article, along with several articles to follow, forms the basis for an unconventional approach to problems of cosmology.**

**Jozef Gelbard**

**The dual nature of gravity.**©

**Contents**

**Introduction**

**1.**

**Is there a gravitational mass defect?**The new definition of

gravitational
mass. Postulate of the existence of an absolutely
elementary

being.

**2.**

**Gravitational mass defect of a system of two bodies.**

**3.**

**The possibility of the existence of gravitational repulsion.**

**4.**

**Gravitational interaction of material points.**

**Modification of Newton’s law.**

**The phenomenon of collision in**

another
perspective. The first conclusions.

**5.**

**The potential energy of interaction of two material points**

**after taking into account the deficit of gravitational mass.**

**6. Gravitational potential of a system of two bodies.**

The potential of mass. The overall
potential.

**Appendix: Energy contained in the gravitational field.**

**Introduction**

This article presents an unconventional approach
to matters related to the effects of gravity. It also shows research
perspectives that haven’t been yet considered. This article precedes a series
of three articles concerning gravity in Planck-scale and cosmological
consequences of the proposed model. All four of them constitute one integrated whole.
They also form the physical basis for unconventional cosmology. But we will
come to that later.

My work
is based on the Newton’s theory of gravity, but I also take into account the
contributions of the special theory of relativity. I am particularly interested
in gravitational interaction at a very close range. Today the key role in this field
of research is played by the quantum field theory, though it does not consider
gravity, which is a serious drawback of this theory. The question refers to systems with a very high concentration of matter. In this regard, the general
theory of relativity has not
been tested, and what we "know" (black holes) is rather an extrapolation of what we know about dynamic
systems and those accessible to direct examination, which primarily include gravitational
lensing and, obviously, the perihelion of Mercury, as well as, possibly, recently
discovered gravitational waves. Very condensed matter, it is rather the domain
of quantum mechanics. The problem is that gravity is beyond quantum mechanics. And
yet, the black hole with the most mysterious material interior are already
factual reality (...) so the hope for solving the problem of that unknown matter
contained inside them may lie in quantum gravity which we are very eager to
find although so far it hides beyond the horizon.

My work aims at fulfilling the hope of solving at
least this problem. For instance by considering an option according
to which the matter beyond the gravitational horizon is the most ordinary
matter. I think that we should actually follow that direction.

Inconsistency existing
between the (currently applying) deterministic theory of gravity and the indeterministic
quantum mechanics (which takes place in sets of condensed matter) prompts to attempt
the "third way" that may not be looking for a compromise also may not
necessarily lead in the direction of quantum gravity in today's understanding
of the issues. It turns out that the Newton’s gravity (including the return to forces),
together with the findings of the special theory of relativity, bring quite
promising results. This is illustrated in the hereby essay along with three
works that follow it.

It is commonly
believed that the description of gravity in accordance with the procedures of
the general theory of relativity is more general, so it does not make sense to
apply the Newtonian approach. But here the question is not about forces, but about
space-time where gravity is the degree of its curvature. So the difference is
important, and what rises here my intuitive objection is the ontologisation of
space geometry which is not a material being. Do we really need "to throw
away" Newton’s approach as unreal ("since simply space")? I am
against it, if only for this reason that the remaining factors operate, however,
as forces. [Is the gravitation cast in some other mould? It
is after all a kind of inconsistency.] And besides, it's important, Newton’s
laws of motion are valid for all types of interactions. Has anyone wondered
about it? It is also worth noting that all the particles, without exception,
interact gravitationally but in other interactions some participate, some don’t.
Newton's law of gravity, in conjunction with the principles of dynamics,
suggests even that this interaction is primary, fundamental. This claim is immeasurably
strengthened by the fact of mass and energy rather an extrapolation of what we know about dynamic
systems and those accessible to direct examination, which primarily include gravitational
lensing and, obviously, the perihelion of Mercury, as well as, possibly, recently
discovered gravitational waves. Very condensed matter, it is rather the domain
of quantum mechanics. The problem is that gravity is beyond quantum mechanics. And
yet, the black hole with the most mysterious material interior are already
factual reality (...) so the hope for solving the problem of that unknown matter
contained inside them may lie in quantum gravity which we are very eager to
find although so far it hides beyond the horizon.

My work aims at fulfilling the hope of solving at
least this problem. For instance by considering an option according
to which the matter beyond the gravitational horizon is the most ordinary
matter. I think that we should actually follow that direction.

Inconsistency existing
between the (currently applying) deterministic theory of gravity and the indeterministic
quantum mechanics (which takes place in sets of condensed matter) prompts to attempt
the "third way" that may not be looking for a compromise also may not
necessarily lead in the direction of quantum gravity in today's understanding
of the issues. It turns out that the Newton’s gravity (including the return to forces),
together with the findings of the special theory of relativity, bring quite
promising results. This is illustrated in the hereby essay along with three
works that follow it.

It is commonly
believed that the description of gravity in accordance with the procedures of
the general theory of relativity is more general, so it does not make sense to
apply the Newtonian approach. But here the question is not about forces, but about
space-time where gravity is the degree of its curvature. So the difference is
important, and what rises here my intuitive objection is the ontologisation of
space geometry which is not a material being. Do we really need "to throw
away" Newton’s approach as unreal ("since simply space")? I am
against it, if only for this reason that the remaining factors operate, however,
as forces. [Is the gravitation cast in some other mould? It
is after all a kind of inconsistency.] And besides, it's important, Newton’s
laws of motion are valid for all types of interactions. Has anyone wondered
about it? It is also worth noting that all the particles, without exception,
interact gravitationally but in other interactions some participate, some don’t.
Newton's law of gravity, in conjunction with the principles of dynamics,
suggests even that this interaction is primary, fundamental. This claim is immeasurably
strengthened by the fact of mass and energy equivalence.
Indeed, mass determines the existence of gravitational field. That’s the path
worth exploring.

Let us add that Newton’s gravity theory
leads in most cases to the same results as GTR. So if the same result can be obtained
either by using forces (in a Newtonian way), or by treating gravity as a
curvature of space (with no forces), then the space-time method does not bear
any ontological characteristics. It is only a formal procedure. And for this
reason the ontologisation of space arouses doubts. So if GTR envisages results which
cannot be predicted by Newton’s theory, it is a sign that it is rather GTR that
should be modified. It only remains to check out whether this modification
leads to the (exactly) same results as GTR. I am convinced that the answer is yes,
even if the modification which I propose does not quite meet this
expectation... And if we don’t get the same results, then perhaps it is the modified
Newtonian approach that gives better description of nature. Well, this is the
unplanned arrogance on my part. Intuition tells me that we will come to the
fully convergent results but by using much simpler mathematics. In this wish of
mine I omit the GTR interpretation declaring as a fact the existence of
gravitational time dilation, which would be an ontological effect, not just resulting
from observation or procedure. I absolutely disagree with this ontology. The
argument against the existence of gravitational time dilation will be contained
in the essay dealing with black holes while this article is dedicated to other
issues.

**1.**

**Is there a gravitational mass defect?**

Warning:

Attention!
I must warn the more sensitive readers that this section contains content not
acceptable to many, especially those for whom physics is their daily bread.
They will be surprised. So I invite you to indicate any error in substance. I would
like to inform that taking up the issue by relying on the general theory of
relativity is not correct, because by the assumption, I do not refer in my
arguments to this theory. I advise you to follow the path of logic and not be
misled by habits of thought formed in times of learning and teaching
activities. Routine closes even the most open minds.

In nuclear physics there is a concept of mass defect.
Does mass defect exists also in gravity? I think so, although in the framework
of the general theory of relativity this matter is not considered. To lift a
body one has to invest some energy. Energy is equivalent to mass. Does the mass
of the Earth-body system increases? No, if the body is lifted at the expense of
some internal energy of the system. And if we have a closed system of two
bodies, and their relative freedom of movement is determined by their mutual gravitational
interaction? Also in this simplest case, the total energy of the system does
not depend on time. However, if we consider only the potential energy of
gravitational interaction of bodies (thus disregarding the kinetic energy,
and in general, all other kinds of energy existing in the given system), we
will notice that when the distance between them increases, so does the value of
the potential energy. This increase is equivalent to mass:

This mass was lacking
when the distance between the bodies was smaller. So we can consider this mass as
the (relative) deficit of the gravitational mass of the system. Further on there
will be given the strict (quantitative) definition of an absolute mass defect
of a gravitational system. Description of interaction on the basis of the
general theory of relativity, as mentioned above, does not include such a thing.
Is this right? Actually, it does not have to, thanks to a different approach,
and the results, as we shall see, are basically similar. So why go another way?
Because it’s simpler? Yes, but more important is the fact that my approach, in
the extreme case does not lead to a singularity that is alien to the real
nature. We'll see about that. For this reason, with regard to gravity (in the
traditional approach) the renormalization treatment fails in calculations
within the framework of quantum field theory. Although in these calculation it
does not matter "thanks to" the weakness of gravity in subatomic systems,
but overall this is a kind of deficit of the theory, and not, as some think,
the nature of things ("Gravity is something completely different for it is
only the curvature of space"). And if a body falls, then is the excess
energy emitted by some radiation? This hasn’t been somehow noticed in relation
to bodies. Wasn’t this noticed because of the "weakness" of gravity? And
what kind of radiation? Electromagnetic? From what bunch? Some gravitons? Or are
we just going astray.

In this context there is an urgent demand
for a new definition of gravitational mass, new in that it takes into account
the energy of gravitational binding. To describe the thing quantitatively let
define the concept of "gravitational mass" differently than it’s been
done so far.

**Gravitational mass is the mass of a system of bodies**, closely related to the fact of their mutual interaction, which applies in the same way to both bodies. Example: The gravitational mass of the Earth-Sun pair. No separate masses of each of these bodies.*At this point we should distinguish between the inert mass of individual bodies - elements, and the gravitational mass of the system. It would seem that in this context the supposition of equality of gravitational mass and inertial mass is becoming less relevant. However, if we consider that each and every body is a gravitational system, including the particles of the micro-world, it turns out that the mass of each body*

*is, in fact, the gravitational mass of that system. Therefore, what has been until now "the postulate" of equality between gravitational and inertial masses, in the context of our deliberations becomes more persuasive, becomes the unquestionable truth, conclusion – not a guess based on intuition. Equality of the inertial and gravitational mass constitute the conceptual basis of the general theory of relativity. As we can see, the new definition of the gravitational mass sanctions equivalence of both types of masses. On the other hand, phenomenologically, on our scale, the mass of bodies as objects of condensed matter (or material points), can be considered as an inert mass, because what’s important is their movement and not the gravitational field which they represent.*

So
it is an obvious fact that the mass of

__each__separate body is the mass of some system. Indeed, each body, even subatomic particles, is built of the smaller elements. Can this division proceed indefinitely? Definitely not! This could be attested by the fact of differentiation of particles, the multitude of their kinds, and also that there is a possibility of their systematization (standard model).
Let us
notice that the possibility of systematization of chemical elements according
to their characteristics (including periodicity) is consistent with the
existence of an atom as a basic component of a chemical element. This discovery
was made by John Dalton (1809), when he observed the constant quantitative
relationship of elements in chemical compounds.

This
leads us to the possibility of the existence of an elementary being common to
all the particles, which is the basic structural element of matter. This would
be an ultimate indivisible being. That is better than an endless abyss, even if
it is only an aesthetic requirement. So I postulate the existence of an

In atomic and subatomic systems gravity it very week.
But this doesn’t mean that it doesn’t exist. Some data suggest that at much
deeper level gravity is very strong. Provided of course, that there are some
entities of linear sizes much smaller than the distances characteristic of the
systems created by nuclear interactions. I think that they exist, that they are
parts of the structure of subatomic particles, that their (gravitational) interactions
determine the specific structures of these particles and the interactions existing
in their world, and through them, also in our world. So coming down, still deeper,
we come to
the end (not the endless singularity) in Planck scale. Not only that. I think
that gravity is the basis for other types of interactions. Let me repeat an
observation I made in the introduction, that all the particles, without
exception, interact gravitationally. Other interactions do not include all the particles.
For example, leptons do not participate in strong interactions. Gravity is
universal. This argument substantiates the thesis that it forms a base for
other interactions. This, I think, sounds rather convincing. Einstein felt that
a hundred years ago and for this reason stubbornly sought the unification of
all the interactions - under the rule of gravity.
**absolutely elementary being**. The mass of each particle, and in particular macroscopic body is therefore the gravitational mass of a system. The only exception is the elementary being itself. In the next work devoted to the so-called elsymons this being is defined and described.
Today physics attempts to reach the Planck
scale. It’s nothing new nowadays. But questions and thoughts remain. Here are
some of them. In our environment, gravity is much weaker than electromagnetism.
Yet it is the gravity that we feel, although we need for it this whole great
Globe. An electromagnetism? Bodies of our surrounding are electrically neutral.
Why gravity dominates among the celestial bodies? Is that because gravity is
only attraction? After all, we do not feel bodies’ electromagnetism, because in
this case the forces of attraction and repulsion compensate each other through
the duality of these interactions. Is this a sign that gravity is not dual,
that there cannot be gravitational repulsion? So why in the
"nano"-scale already close to Planck scale, gravity is to be very
strong, and above, among the atoms, it simply doesn’t exist in any measurements?
[A clue that at that level it is very strong is that the Planck mass is very
large in comparison with the masses of elementary particles and Planck length
very small.]

*This very weakness can be taken as another clue, this indiscernibility of gravity in the world of subatomic particles, as if a niche, and also the existence of the phenomenon of reflection in the world of particles, also electrically neutral. It occurs within an immeasurably small distance ("zero"), well below the range of nuclear forces, and in particular at relativistic velocities. This implies the existence of very large forces. Are these forces of an electrostatic or nuclear nature? One can doubt it. They're too big. And let’s not forget, these are the forces of repulsion.*
Thus, in
spite of all the current assumptions, the existence of gravitational duality
cannot be excluded. But I do not think it’s about some gravitational positive
and negative charges (as it is in electricity). By the way, the existence of
two types of electric charges testifies beyond doubt to the structural
complexity of electromagnetism. The aforementioned absolutely elementary being (if
it exists - in any case this possibility derives from our deliberations), it a
being one of its kind, just as the only invariant speed
(independent of the reference system) is the speed of light. The same can be
said about gravity, even the one which is (hypothetically) dual. I will elaborate
further on this topic - in this and in subsequent works.

#
2. Gravitational mass defect of a system of two bodies

Our
goal is to construct a mathematical definition of mass defect of a system of
two bodies (two material points). Of course, we are referring here to the
influence of gravity. Let an isolated system consists of two material points, while
their invariant mass in the traditional understanding - due to the fact of
their invariability. (they are
not, for example, collapsing star) - does not alter. These simple assumptions
will lead to a clear, unambiguous definition of mass defect. To define mass
defect, we need to notice that it is equal zero if the maximal potential energy
of the system equals zero. (In other conditions the potential energy is
negative.) This occurs when the distance between our material points tends to
infinity. In other words, there is no contact between the bodies. Or is it only
in this case?

When the distance between two points is equal r, the
potential energy of the system is equal to:
The increase in the potential energy of the system when the distance
between the points tends to infinity is equal to:

The mass of the system increases in an equal measure, that is:

This mass was lacking when the distance was r. So we have a mass defect
at this point. Taking into account equations (2) and (3) we have:

Of course, Δm is the (absolute) mass defect of a system
of two bodies, corresponding to the mutual distance between points equal to r. So
there we are.

**Examples of calculations.**

1. Calculate the
mass defect of two material points, both of which have same masses equal to 1
kg, and the distance between them is equal to 1 m

Solution:

Using equation (4) we
get:

It's very little. No wonder that this effect is undetectable, the more so
since it was not even expected. But the value:

is already
a measurable quantity. So one can be tempted to experimentally verify this
prediction. If someone wants to, I can provide moral support.

2. What is the mass
defect of the Earth-Sun system?

Solution:

Data:

Let us treat these bodies as
material points (spherical shapes and a correspondingly large distance). We
get:

Is
that a lot? It is the mass of a cube with edges of approx. 23 km, density of 5
g/cm^3 (close to the density of the earth). The energy equivalent to this mass would
remove Earth out of the solar system. No wonder, after all, mass defect is
equal to the binding energy.

On this occasion, we can perform a fancy calculation.
Let the binding energy (equal to the mass defect) be equal to some kinetic
energy. At what initial speed Earth would have to radially move away from the
Sun to free itself? Naturally, at the escape speed. Is that what we’ll get?

Indeed.

3. What is the average mass defect of the Mercury-Sun
system?

It turns out that it amounts to:

It wasn’t by accident that
I chose this planet. We know that earlier noticed "too large" shift of
this planet’s perihelion prompted scientist to test the general theory of
relativity (with positive results). As we know, according to this theory,
gravity is treated in geometrical terms – it causes the curvature of space. It
would be interesting to see what we would get, if we approach the issue
differently, according to the Newton’s theory, of course taking into account
the deficit of the gravitational mass of the system. Maybe we would get the
same result? Here's another possibility to check (falsifiability). If it turned
out that indeed..., it would mean that the curvature of space as such is not a
physical fact, and space is

__not__an autonomous entity. GTR would be a brilliant patent, a calculation procedure of great practical importance. Incidentally, the same could be said about quantum mechanics. This, however, does not result directly from our current deliberations. It is better to live aside the cognitive perspectives of all this, even more so because one of the possibilities is a fatal mistake (of course mine). But let’s calm down and remember that the systems’ mass defects resulting from our calculations are, after all, the values of gravitational potential energy, expressed in units of mass with an opposite sign (plus). "So what's new here, it's simply a trivial fun game with equations?" Well, the novelty consists in the fact that we have defined gravitational mass differently, that we are considering systems, not separate bodies. This, as we shall see, will lead to a modification of Newton's law and even to surprising results, namely, it can be assumed that in the systems of astronomical scale, mass defect can manifest itself by specific observable effects. And this makes the "theory" falsifiable. It would seem: a small formal beauty treatment (as the surgeons say).
"So what's new here, it's simply a trivial fun game
with equations?" Even if that’s the way you take it, dear reader,
please note that the sense of it, after taking into account the new definition
of the gravitational mass, is somewhat deeper. Further on you will come to
appreciate the interesting consequences of such an approach.

It can be
assumed that in the systems of astronomical scale, mass defect can manifest
itself with specific observable effects. And this makes the "theory"
falsifiable.

But let’s
not get ahead of the facts.

#
3. The possibility
of the existence of gravitational repulsion.

If the
distance between two bodies decreases, the gravitational mass defect of the
system increases. So let's solve the following problem:

Calculate the distance between identical
material points if the mass defect of their system is equal to the mass of one
of them.

Solution:

Assuming as proposed
that:
Let’s remind ourselves that the gravitational radius of the material
point of mass m is:

As you can see the resulting distance is equal half (!) of the
gravitational (Schwarzschild) radius of one of the elements.

**In this situation, the mass of the system is equal to the mass of one of the components. One can be even tempted to provide a new definition of the gravitational radius:****Gravitational radius is the double distance between two identical material points, at which the gravitational mass of the system is equal to the mass of one of these points.**As one can see, it's something completely new.
It is worth noting
that the above proposal does not apply to bodies of our surroundings. And yet
that’s what shapes our intuition. The radius of gravity of planet Jupiter, the
largest in the solar system, does not reach 3m. However, in relation to the massive
stars, and, of course, to galactic nuclei, the matter becomes important. We'll
see about that later.

And if we bring our material points even
closer to each other? It is easy to show (I leave it to the reader) that the
gravitational mass of the system can be also equal to zero - if these material
points will be even closer to each other. In this case, the distance is equal
to a quarter of the radius of gravity. Such a system is gravitationally saturated. As if it didn’t exist. False vacuum?

And even
closer? Then... the mass of the system should be negative. The system should
repel any foreign body. And our material points? Would they continue to attract
each other? Think about it. You’ll get an answer in a moment.

You may ask: Why is it that the atomic
nucleus, having certain size, does not collapse under the action of nuclear
forces? Answer: "Because the strong interaction includes also repulsion at
a shorter distance" – does not satisfy, neither that it is about fermions
and the Pauli exclusion principle: "Where does this exclusion really come
from?" But maybe it is gravitational repulsion that prevents the collapse
of the nucleus. And if this is true in relation to the atomic nucleus, then it
must be also true in relation to the whole matter.

However, it is no longer a system of two
material points. It is something much more complex, it already concerns the
structure of particles, or maybe even unification of interactions: nuclear and
gravitational ... The structure of particles, it is not the subject that I would
like to dwell upon at this stage. It is a little bit too early for that. But already
in the next article there will be a surprise.

And if the masses
of our points are not equal to each other? This case we considered above in an
exemplary calculation. Can the gravitational mass of the system be zeroed also
in this case? Let us consider in general the system of material points with
different masses, but on the assumption that the mass of the system comes to
zero. In this case we have:

is the reduced mass of the system.

And if we get them even closer? [We are
coming back to a system of two identical material points] Above we drew the
conclusion that the gravitational mass of the system becomes negative. Let's
think. The intensity of the gravitational field of this system (as a vector) is
directed outward. So, the repulsive force acts upon every "foreign" body.
The outward force also acts on the elements of the system, hence they also repel
each other. Further on I will present the facts confirming this conclusion. So we have

**gravitational repulsion**between material points. Then how is it with the nucleons forming atomic nucleus? How about reflection during collisions of elementary particles, especially those with great energy and uncharged?
Let's continue our discussion. First of
all, let us note that the resultant mass of a system of two bodies of two identical
material points can be expressed by the formula:

The force of repulsion should rise grow
rapidly with diminishing distance [

*very similar to electrostatic repulsion, causing a collision of**bodies*]. That repulsion prevents unlimited collapse toward singularity. Is Pauli exclusion principle a manifestation of this absolute limit? This is yet another new trail.
Thus our considerations lead to the
conclusion about the possibility of the existence of

I think that at this point one needs to mention the
so-called **duality of gravitational interaction**. But this is not envisaged by the General Theory of Relativity, I would even say that it rejects such a possibility. If after all this is true (which of course requires experimental verification), then the description of super-dense matter has much to gain.**asymptotic freedom**, envisioned already in 1973 (F. Wilczek, D. Gross, H. D. Politzer; Nobel 2004) [Today, when describing interactions at the quark level there is talk about the so-called quarks colour imprisonment.] This finding may be of great importance, also in the context of our discussion. It is possible that its explanation may lie in dual gravity. If successful, then ... oh, anything but.

I get the impression (privately)
that thanks to dual gravity we will close the gap between standard model and the
one based on gravity. Unfortunately, there is little chance that it would
happened, because it is inconsistent with established paradigms currently in
force. The Wilczek and his colleagues’ discovery made forty years ago concerns
bonds between quarks. The force of attraction between them

__decreases__as they approach each other and tends to zero at the point of contact. "And this contact, what it is?" you may ask. It's simply, still closer, repulsion at a much shorter distance. This can be compared to a stretched, and then compressed spring. One can expect that with further approach, the very quickly growing repulsive force will bring the approaching bodies to a stop. That's exactly how dual gravity works. "The imprisonment of colour" fits this approach. Perhaps this is the way towards unification of gravity with the strong forces. So there it is, my arrogant hypothesis.
It is
well known that at a greater range, strong interactions - forces of attraction
between nucleons - are the greater, the smaller the distance between them. With
this in mind, together with the existence of deeper asymptotic freedom, one
could suppose that the closer the elements of the system (nucleons) approach
each other, the force of attraction should reach its maximum and then decrease
upon further compression, now as quark systems (which are nucleons, and generally
hadrons). It is described roughly above. The discovery of asymptotic freedom
was a major surprise for the scientists. Perhaps it will be no less surprising
(for the reader) that this will become apparent in an illustrative manner later
in our deliberations devoted, of course, to dual gravity. Yes, gravity. And
here gluons are not even needed. So does it lead to the unification of these
two types of interactions? That's what I think, or actually suppose. For now, somehow,
this is not thought about. Is that right? I think it is because there is no
foothold for the time being.

What can be the consequences, or rather predictions
based on this conclusion? First of all, the gravitational collapse of a star or
a galactic centre is limited by repulsion, naturally, within an
appropriately close distance. Singularity cannot happen, although this does not
preclude the existence of the objects beyond the gravitational horizon. This
may in particular apply to the nuclei of galaxies, and maybe to the exceptionally massive stars (if at the same
time they are relatively stable). The point is that the average density
of an object enclosed by the gravitational horizon depends on its mass. It is
inversely proportional to its square. The mass of the nucleus of the galaxy may
be as much as billion times larger than the mass of the sun. The matter in the
nucleus of a galaxy has therefore the characteristics of matter known to us. Average
density of this matter may be, for example, approximately equal to the density
of water. Even the matter of a star which due a collapse got enclosed within
the gravitational horizon, does not differ from the matter known to us, in any
case, it is describable, even if we cannot expect any communication from those
places. Can’t we, really? But do we have to touch everything like a
toddler? Its density does not generally exceed the density of nuclear matter. And
by the way, in this context, also the gravitational time "dilatation"
becomes doubtful, and that for basic, not quantitative reasons. There will be
more about it too.

**4.**

**Gravitational interaction of material points. Modification of Newton’s law**

Our primarily aim is to derive the formula for the
modified law of gravity. In the new formula we will take into account the
gravitational mass defect of a system of two identical material points.
Gravitational mass of the system, that is, its resultant mass, is expressed by
the formula:

where: M – the actual mass of a
material point. Gravitational unit mass of the system is, obviously, equal to
half of the resultant gravitational mass. Thus, the force of the interaction
can be written in the following form:

This
expression can be simplified by taking into account the formula for the
gravitational Schwarzschild radius:
We get:

This corresponds to mass defect equal
to 2M, which immediately brings to mind the idea that at still smaller mutual distance
the elements of the system should repel each other. The force should be
negative. It is not just about aesthetics. The existence of repulsion is proven
by facts. The nucleus does not collapse upon itself, it can’t be even squeezed
any more. And as a matter of fact nucleons are not material points since they
are already complex objects - like a wall one cannot break through, one cannot
permeate. The reason is the extremely strong repulsion and within extremely
short, simply negligible distance.

*To illustrate the matter, in the world of our perception, collision-repulsion (the wall) is of course of electrostatic character. Range (braking distance) of this collision is, however, much wider than in the case of collisions of subatomic particles. It is at the level of an atom. Our considerations apply to a much shorter braking distance.*
The existence of repulsion is also demonstrated by
the phenomenon of particle collisions, not necessarily electrically charged, like
for example, collision of neutrons. I noticed it earlier. The collision does
not have to come down to just electrostatic repulsion. Gravity is more
universal. So as not to get distracted, let us focus on the bouncy, frontal collision.
During such a collision particles do not disappear, the principle of
conservation of energy, of momentum and angular momentum are fulfilled - in
each experiment (also the angular momentum, since the actual particles are not just
material points). In our perception the phenomenon of collisions is something obvious, accepted
without too much reflection (How? Why?). And yet this thing in itself is
exceptionally interesting because we are talking here about very strong
repulsion occurring over a very short distance and during a very short time.
This means the existence of extremely large forces occurring, in particular, in
the world of subatomic particles. Here is an illustrative example. An average
repulsive force between two neutrons colliding head-on within a range of, let's
say, 10^-20 m (acceptable), at a relative speed of several thousand km/s (that's
not such a high speed in the world of particles), amounts to more or less one
million newtons [N]. This in relation to a particle with the mass of 10^-27 kg,
is very much. Very much even at the scale of our perception. By the way, we see
that the forces, in this case the forces of repulsion occurring in the world of
subatomic particles are enormous. Let us
point out another fact. Generally subatomic particles move very fast. These little
things often fly at speeds close to the speed of light. Although we are talking
here about relative motion, in relation to ourselves all of them move at
enormous speeds, and that in spite of the fact that relativity of motion means
the possibility of a zero speed. How to explain it? And actually where this
speed comes from? They had to come to that speed as a result of very strong
interactions, rather at a very short-range, interactions which in our scale of
sizes do not occur. It's hard to even talk about any restfulness of these
particles (in relation to ourselves). In this regard they resemble photons to
some extent.

This
high speed may be a relic of the past when these particles were formed (and
that includes electromagnetic and nuclear interactions), when the Universe in
its present form was taking shape. Even before the atoms and molecules. It was
a natural movement due to the very high ambient temperature. When matter was
very concentrated, to the extent that also repulsion influenced the course of
events. I dare say that it was primarily gravitational repulsion which
manifested itself in collisions. Free path between collisions was very short. Collisions
constituted the integral part of the whole process of creation of we call the Universe.

*Let’s note that slowing down a particle moving at nearly the speed of light, and over distance equal to almost zero (after all, it’s about distances smaller than R/4), requires gigantic, simply inconceivable forces, far larger than electrostatic repulsion.*The world of particles is a different, extraordinary world escaping our imagination. It is also the world of gravity at its source. We will deal with it in the next article.
So it is a well-grounded
view that (we go back to our formula (10)) when the distance between our two
points is even smaller, namely: r <R/4, there should be repulsion. If there
was no repulsion but attraction, then, according to the formula (10), with progressively
smaller distance the force (of attraction) should have grown to infinity for a
distance equal to zero. But that belies the facts. In such a case everything would
immediately end in gravitational collapse. How could have ever come into being?
– quite a legitimate question. We wouldn’t simply exist. Conclusion: there is a
gravitational repulsion. Thus our formula (10) should be supplemented with a
factor taking into account the possibility of repulsion. There it is:

Which leads us to the final form of the modified Newton's law of
universal gravitation for a system of two material points:

To simplify this expression we can express (variable) distance between
our points as: r = xR (x> 0). So we get:

Now let us again use the Schwarzschild radius formula. We obtain a
relatively simple function of the variable x:

*As one can see, this formula does not contain mass. It is universal. This is a very important outcome, which could be a confirmation of the chosen path.*

The derivative of this function has zeros at the points:

At point x

_{1}we have maximum, while x_{2}is the point of inflection (after taking into account the coefficient G). The function has a vertical asymptote: F → - ∞ when x → 0. Here is a graph of our function:*[It is significant that the solutions of this equation (zero points and extremes) are so clear. This is very important. It's probably a sign that*

*we are on the right track. If the solution were, for example, irrational numbers, or even worse, if there was no clear solution, or no solution at all, it would be a sign that we went in the wrong direction. After all, we expect that nature is a kind of ideal which all its descriptions try to attain, even those that differ significantly among themselves. Yes, it's very important that we have obtained just this results]*

It would be interesting to see what is the peak
force (maximum in the graph) in newtons. To get some result we substitute x =
1/2 to the formula (12). And that’s what we get:

A very interesting solution. Maximum force does not depend on the mass
of interacting bodies. They may be two dust motes or two stars (if you do not
take into account their size). What is the numerical value of this force? It is
easy to calculate: 3–10^43N. That's quite a lot. For
comparison, two stars as large as the Sun, when the distance between their centres
is equal to one million kilometres, attracts each other with a force equal to
26.7–10^31N (excluding the mass defect).
This is much less. Thus, the gravitational interaction at the source is not all
that weak.

Is this the absolutely
biggest force? Results of certain considerations justify an affirmative answer,
although still larger will be the maximum repulsive force. There will be also
about that. There is therefore a borderline force, the topmost force, yes, as
there is the maximal borderline speed. As you can see, we have come to
interesting things from both, the physical and philosophical points of view.

###
The first
conclusions

The same formula (for the maximal force) can be
also reached by going down to the Planck scale. Some quite wrongly speak even about
the "Planck’s force". However, we came to the same equation through
general consideration of a system of two material points. This formula can also
be obtained from the formula for the gravitational field intensity at a point
on the sphere of gravitational horizon (Schwarzschild radius). It would be a
force at which interact with each other two identical black holes, which
centres would be at the distance of their gravitational radius, something
rather hard to find. With the material points it makes more sense. This formula
can be also derived from the general theory of relativity (!). But here, in
this work, this formula has been reached through the modification of Newton’s
law by assuming the existence of
gravitational mass defect. It's very meaningful.

Thus, if by modifying Newton’s law,
and by using the general theory of relativity one can obtain the same result, then:

1. The curvature of space does not
have to be an ontological fact. It is rather a particular (peculiar) feature of
the method, of the investigative procedure.

2. This proves the validity
(certainly not incompatibility) of the description based on a new definition of
the gravitational mass as the mass of a system, while taking into account the
gravitational mass defect (which directly leads to this, and no other
modification of the Newton’s law).

3. Modification of Newton’s law (such
and no other) is conceptually generally correct (not necessarily from the point
of view of a particular theory), if only because it generates anticipations -
it is falsifiable.

Taking into account the findings already presented in this
work (not even counting the overall results that I have come to and published
in other works), may necessitate certain changes of interpretation of both, the
quantum field theory and the theory of gravity, and this may get us closer to a
unified field theory.

**5. The potential energy of interaction of two material points after taking into account the gravitational mass defect**

We will base our considerations on Newton’s model of gravitational
interaction of two material points. However, we’ll take into account the
gravitational mass defect which becomes apparent at appropriately small
distances. For the record, it is not detectable in the scale of our perception.
This does not mean that it does not exist. We use the formulas:

The first one is the formula for the potential energy of interaction of
bodies

**M**and**m**, at a distance**r**from each other. For simplicity having no impact on the essence of things, I assume that we have a gravitational system of two identical material points. We shall calculate the potential energy of their interaction, taking into account the existence of a gravitational mass defect. We will use designations made earlier. So we get:

and: r = Rx (as we have already done in
calculating the magnitude of force, we get:

while for x → ∞ the potential
energy tends to zero, and for x → 0, the potential energy tends to +∞ (after taking
into account Γ = - 1 for x<1/4). So we have the potential energy at the
derived points:

In the first case there is the minimum,
and in the second we have inflection point. Here is the chart:

At
first, while exploring the matter, I felt rather tense. The result in the form
of, for instance, an irrational number would be even a legitimate reason to
reject the concept, let alone the lack of solutions (negative discriminant).
Continued research in this direction would make no sense. I expected a clear result,
as in the case of force. I found the result satisfying. By the way, let’s note
that (accidentally?) the minimum value of the potential energy is the third
power of 2/3. Is it third because we operate in three-dimensional space? Something
for the reader to pounder about.

**6. Gravitational potential of a system of two bodies**

Let us
recall that the gravitational potential at a point located at the distance r
from a material point of mass m, is given by the following formula:

Graphically,
the gravitational potential around a material point can be presented in the form of

a well with a characteristic shape (see
figure above). You can see here that the potential is ever-greater (negative),
the closer we approach the point source of the field, and tends to minus
infinity in the center (at the point). It reminds of singularity. In fact,
there is no body (point) isolated in an absolute way, and the material point is
only idealization, having no counterpart in Nature. Or maybe there is an absolutely
elemental being... then of course you absolutely cannot talk about singularity. It is a
notable thing for anyone who wants to describe the real Nature. "Funnel"
in our figure has in fact its bottom, as all material (and field) smallness has
its absolute limit. I think so, and I am not alone. Here singularity does not
exist, even if the field equations lead to it. Apparently, somewhere deep down
their relevance to real Nature breaks down. In a similar vein I will continue
my deliberations in the following articles. We also remember that the assumption
of the existence of an absolutely elementary being explains the equality of
gravitational and inertial mass. I pointed out this equality at the beginning
of the article. This equality is related to the principle of equivalence
postulated by Einstein. We already talked about it. I
would add that this equality applies to compact bodies. The new definition of
the gravitational mass slightly changes the approach.

And
what’s the case when we are dealing with a system of two (or more) material
points? In addition, very dense? When the material points
forming the system are
sufficiently close to each

other, the potential distribution is different (see
figure). This is of course an idealization, just as an idealization is a
material point. What’s important is that at the center of the system’s mass the
potential well is not, of course, infinitely deep. What is the
value of potential at that point? It's a natural question. We will deal with
this problem while taking, however, into consideration mass defect of the
system which, as we remember, depends on the distance between the points. This
should have a significant impact on the potential’s magnitude. One can think
that as the distance becomes smaller, the effect of mass defect on the value of
potential at this point becomes greater. We will express it mathematically. Let
us suppose that two material points of the same mass form the segment AB = r,
as shown below. We will calculate the ("classical") potential in the middle of the segment AB, at point O.
Potential is a scalar quantity, so the potentials of the field from each of its
sources simply add up.

We express it as follows:

As we shall see we’ll get the same result to
express potential on the surface of gravitational horizon of a particular
object. We will obtain the identical result to express the potential of the
Universe, and at its every point, not just on the line of horizon. That is what
makes the Universe different from any system. Indirectly it confirms the thesis
that the Universe is Everything, is the unity and uniqueness (does not make
part of any system). We’ll pursue these deliberations, especially in articles
devoted to the oscillations of the Universe. By the way, the convergence of
these results may confirm the validity of the chosen path. In addition, due to
the presence of constant c in the resulting expression, we are once again approaching
the unification of two interactions: gravitational and electromagnetic. It can
be thought that these interactions are the most elementary. But this
unification occurs in an appropriately small scale. [The fact is that even
quarks are charged.] So we have to move even
deeper. I believe that there also the laws of nature as we know them remain
valid.

###
Appendix

To
deepen our comprehension of the issue, let’s for a moment move away from the
mainstream discussion. Let’s calculate the field’s potential at a point separated
by Schwarzschild radius from a single material point. Here is the calculation:

Let us return to a system of two
material points. They are so close to each other, that the system’s mass defect
becomes significant. For this reason, as the system tightens, its mass gets
smaller, and the gravitational field around it weakens. You can also say that another
field, "antigravity" field increases. This field has its own
potential. It is the

It should be located exactly **potential of mass defect**:__in the middle of the system’s mass__, since it represents the system, and not any of its components. I propose to define it as follows:
It is, as one can see, positive
potential. Of course,

**r**is the distance between our material points. This definition applies, as you can see, to the specific case of equal masses of both material points, hence in the denominator we have half the distance between them. Obviously, this formula makes no sense in the case of a single body. When determining field’s potential in the middle of the system’s mass, depending on the distance between the material points, we should therefore also consider the potential of mass defect. In cases discussed below, due to a very small distance between points (in the order of the gravitational radius) - for a relatively distant outside observer - the center of mass represents the system.
Therefore

**the****overall potential**can be expressed as follows:
We will calculate it for three representative cases (at the centre of system’s
mass):

1.
(of one of the points). In this case: Δm = 1/2m. This can be easily demonstrated. Formula for mass defect will suffice. Want
to try? You’ve got my invitation. We calculate the two component potentials (formulas
(16) and (19)):

Thus:

This means the existence of the force
of attraction between the elements of the system (at least for now). On the
outside the system itself exerts the force of attraction.

In this case:
. So we get:

Also in this case, the system exerts the force of attraction which is, surprisingly,
even stronger (more condensed matter). We already know that this distance
corresponds to a maximum force of mutual attraction. Everything fits.

It is easy
to show that this time Δm = 2m , so we get:

And here we come to a breakthrough. The system becomes gravitationally
closed, simply disappears, and in case of our material points further
approaching each other, it "reappears" in all its repulsive form.
Repulsive blackness. There is something to detect here (though it is a black
hole in quasi-classical understanding). A black hole, but it does not attract,
it repels. "Dark energy"? So be it.

Rather than deal with special cases,
we could have immediately found the appropriate function and explored it. The
educational factors took precedence. But it is not too late. So let us present
our function:

Let’s substitute:

So finally we get:

An interested reader may
explore this function on his/her own (for a high school student - after the
reform of education – quite a challenge). Is this the
graph you obtained? :

We see here that after overstepping
the extreme value (
), with
material points getting still closer to each other, the potential passes
through the zero point and further on, as positive, rapidly increases. Perhaps
it is actually thanks to repulsion that nucleons do not overlap, do not
penetrate each other. If someone says, "the reason is that the nucleons
are fermions, so in relation to them applies Pauli exclusion principle," I
would reply: a) True, but in physics one should not conclude matters with
"little rules"; b) Pauli exclusion principle is an
"external" manifestation, it is one of the aspects of reality at a
much deeper level, in the world of truly elementary structures. In light of our
findings we may even think that it is an expression of our discovery:
gravitational repulsion. Pauli exclusion principle indicates the direction of
further research, continued exploration of the essence of things, not being however
neither final resolution, nor the philosopher's stone. Let’s add that according
to widespread opinion, gravitational collapse (towards singular black hole)
does not "consider" Pauli exclusion principle, as if it were too weak
to resist the pressure of gravity. Something is not right here. Does this mean
that this is just another potential barrier having nothing to do with gravity? Does this exclusion concern only subatomic systems and
detectable particles, or it is only an expression of quantization? And if that’s
the case then what is it in fact, this quantization? Nothing to do with
gravity? So what in fact causes this exclusion? What does it express, which
basic principles of nature? I
think that now we can decide (or, put it otherwise: make a cautious hypothesis)
that this exclusion is the consequence of the existence of gravitational
repulsion. Simple, right? But is it true? A singularity? That’s something we
won’t probably live to see. So let us treat repulsion as
an essential feature of nature. We will explore the matter there where it all
begins, in the world of planckons which
we’ll be dealing with in the following articles. And coming back to the Pauli
exclusion, we have to admit quantum mechanics is truly genial.

###
Appendix

So far we dealt only with a system
of two bodies. One can immediately ask: What are the consequences in relation
to more complex systems? I won’t undertake this task, because it's boring, and
in addition, if it turned out that I generally go astray (as it is currently though
without checking, without even reading my works), then a lot of effort would
end in a bin. But it is worth considering, though not the problem of three
bodies. Let the investigated system be (a conventional) object shrinking only
under the influence of gravity, theoretically from an infinitely large size (that
is from a dimension of an astronomical scale). When it collapses, its
gravitational mass defect gets bigger (in accordance with the definition
provided at the beginning of this article), which can be presented formally as
an increase of negative mass. As the object contracts further, this mass increasingly
compensates the positive mass. As a result the gravitational mass of the system
is reduced to zero. When does it happen? By analogy with the system of two
bodies, it would happen when the gravitational mass of the object is equal to
half of the initial mass (the mass of one of the two points if they constitute
a system). It can be put
otherwise. The mass of a shrinking system progressively
decreases. The increase the negative mass of the object and the simultaneous decrease
of its positive mass "meet" in the middle, at half of the invariant
energy, where balance equals zero (this does not mean half-way of the original
radius. Rather much deeper, taking into account the energy equivalent to mass).
At this point the gravitational field around the object disappears. It follows
that the summary energy contained in the gravitational field surrounding given
object is equal to half of its invariant mass (with a minus sign):

Let us remember this result. We will come back to it. [And by the way,
would it be possible to come to such a conclusion (if not general determination)
on the basis of the traditional definition of gravitational mass as the mass of
one body expressed by the well-known Newton’s formula? The calculation of the
energy contained in the gravitational field would be itself a formidable problem. One
would have to consider energy density of a field with non-zero gradient at a
given point, which in the simplest case would be the radial field around an isolated
material point, and then sum it up. Maybe someone will want to play with it.]

What are the dimensions of that
object at this moment in comparison with the initial dimension, if it is a gas ball
contracting in a natural way towards its centre? This is something for those
interested in the subject. In their study they will certainly take into account
the thermodynamic aspect associated with the dissipation of energy and an
increase of object’s temperature as it collapses upon itself. For ourselves this
does not matter, because contrary to what is today automatically thought,
change of gravitational energy into "thermal energy" is not all that
obvious. Dissipation of gravitational energy does not occur. The existence of this whole thermodynamics we owe to
phase change, which occurred at the end of the accelerated initial expansion
which was the first phase of Big Bang (Not inflation. I called this process
Urela – ultra-relativistic acceleration). There will be a lot about this in the
subsequent articles, particularly when we come to Big Bang. During this phase
change, part of the kinetic energy of the primary expansion dissipated. All of this
thermodynamics is a relic of the fact that this phase transition caused chaos
among particles of our worlds which were just forming at that time. Only
then the matter has gained a new parameter - the temperature, at the time the
highest in history. This chaos is also the cause of fluctuation and
heterogeneity of matter, thanks to which first appeared the stars, then the
galaxies. All this will be discussed in the subsequent articles. Gravity and
thermodynamic parameters of matter, are altogether different things. These
things should not be mixed - yet this is done automatically, simply as a matter
of course.

For us it is important that in this thermodynamic mess
gravity does not take part. If we remove internal energy, which is a relic of
origins, what determines the temperature of our object, and in addition liquidate
chaos and restore order, then we’ll obtain an object with absolute zero
temperature, and the motion of matter, neatly ordered (as one), towards the
centre . Of course, such an object does not radiate as it collapses.

So actually in our deliberations the thermodynamic
problem should not exist due to the fact that we consider only the gravitational
mass of the system, and not its total mass-energy. Gravitational mass is a
function of the location (mutual distance, concentration of matter), and the
temperature of the object does not matter.

Looking at it in a wider perspective, it can be
stated that absolute zero is not an intrinsically unattainable limit as the
speed of light. It expresses only the state of perfect order, the existence of
which cannot exclude any (basic) law of nature. Unreachability of absolute zero
is a secondary consequence of the disorder, which appeared at a very early
stage of the Big Bang and there's no turning back. This took a lot of energy.
For this reason, there is thermodynamics. This is a secondary effect, though
absolutely universal, a relic of the phase transition, which began this, and no
other development of the Universe. In fact, we exist thanks to chaos. In my philosophical
"credo" in the preface to the first of two books published in 2011 I stated
that there are no absolute limits in nature. If anything, they constitute only
axes of specified symmetries. Incidentally, this how we can (or rather should) treat
the invariant c, and absolute zero is an altogether different story.
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