An Equivalence for Both Coulombian and Gravitational Interactions

The main purpose of this article is to demonstrate that electromagnetic and gravitational phenomenology are two different expressions of the same interaction that we can call “universal interaction”. In order to reach the aim, it is therefore necessary to imagine the matter in a slightly different way, as well as equivalent, to what has been done until today by the literature everywhere accepted. Even if differently imagined, this proposed structure of matter cannot and must not escape from reflecting the measurements and phenomenologies widely experimented in laboratories all over the world. In the proposed model the proton radius; the electron mass; the Avogadro constant; the existence and the mass of neutron and the existence of neutrino are theoretically derived. The main consequence is therefore a more general rewriting of Newton’s law of universal gravitation. A definitive value for the universal gravitational constant is proposed.


Introduction
The present paper, which is based on an application of the philosophical concept of Occam's razor to the modern questions of physics, aims to investigate and determine a logical truth in the field of modern physics. The determination of one of the possible logical truths to be studied imposes as an immediate consequence of its scientific verifiability in the sense of physical-mathematical truths.
More generally, we will build an algebraic logical model through which we will be able to express simultaneously two physical truths that are still far from being In an original article by D.M. Snyder [1], the author succeeds in merging in a more general unifying consideration the curved space-time of general relativity to Lorentz's transformations of special relativity. One of the key points of the proposed observations lies in the intuition that the "[...] spacetime curvature may be developed by considering an accelerating reference frame or pattern of such reference frames as a pattern of local Lorentz frames moving at slightly different uniform, translational velocities relative to one another [...]". Quoting the main observations of the author further "[...] As noted, the derivation of the spacetime curvature associated with an accelerating reference frame may be based on the notion that an accelerating reference frame is composed of a pattern of local Lorentz frames (essentially, local, inertial reference frames) [...]". " [...] If the spacetime continuum is locally Lorentzian, and the velocity of light is therefore locally invariant and finite, then the locally invariant and finite velocity of light provides a connection between the fundamental forces of gravitation and electromagnetism [...]". Starting from these considerations, the proposed work focuses on defining as the founding stone of the equivalence between gravitational and electromagnetic forces the existence of an elementary mechanism that C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology acts simultaneously as a source of field, clock and controlled microsystem that defines within itself the invariance of the speed of light.
The effective existence of an equivalence for both gravitational and coulombian interactions seems also consistent with the well known phenomenon of the gravito-magnetism [2].

The Elementary "On Board" Clock
 Let's define "primary matter" an agglomeration of any size constituted by at least one entity (a brick) primitive and indivisible (quantized) that we will call PAN.  . Let's refer to this as an "elementary clock'' (see Figure 1).
 In order for this clock to remain in rotation we need to hypothesize the existence of a momentum exchange between the two PANONE that we will call generically ex p and about which we will say nothing in this paper.
The speed of the ideal point belonging to the surface circumscribing the PANONE is c, verifying in this case the condition: If stimulated to move with speed any v parallel to p ω this object must not violate the fundamental principles namely: 1) The speed of the extreme point of the surface surrounding the PANONE must remain c.
2) The angular momentum must be preserved.  The first of the two conditions imposes as a direct consequence that: It might be argued that the created elementary clock needs infinite energy to be rotated around the central axis. But this is not the case because of the definitions. We can therefore generalize by saying that all the implications of special relativity are valid [4].
The surface underlying the generic PANONE due to the length contraction could be seen in a freeze image of the clock in Figure 2.
The transversal remains a circle, the longitudinal section (see Figure 2) is following:

Presumption of Behaviour at the Boundary m ∂
Let we define the surface The frontier surface m ∂ transforms (see Figure 3) each element of the primary matter existing in space-time into a luminal entity i e of a special space { } 2 R ct × and vice versa. For the entity that moves at luminal speed, the temporal coordinate becomes infinite and the longitudinal spatial coordinate becomes immeasurable. An observer placed in an inertial frame of reference external to the entity sees the spatial longitudinal dimension and the temporal dimension of the entity merge into a single spatial coordinate of the type ct. We can therefore assert that the frontier of primary matter behaves as a geometric function that applies a transformation of the space-time.

The Elementary Far-Field Interactions
Let us make the following considerations:  We define "elementary interaction" as the interaction between two agglomerates of primary matter with dimensions considerably larger than a PANONE.
In this case, two protons respectively with mass p m each is inscribed in a surface m ∂ of radius p R placed at a distance 2 p r R  (far field) interacting in a philosophically empty space.
 The internal mechanics of the clocks of the primary matter determines a periodic and isotropic expulsion of entities (see Figure 4) moving with speed c with a frequency of 2π  Each entity has a non-zero momentum equal to 0 p  ;  The mass where r is the radial versor outgoing from the emitting primary matter body.
The variation in the momentum due to collisions intercepted by the body Equation (9) has the size of a force.

The Number p ℵ of Emitted Entities
Because of the invariance of charge [4] [5] and being present in the last relation C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology 9 exclusively constants, we are obliged to deduce that the energy of entities is inextricably linked to the energy state of the PANONEs of primary matter, only in this case it's is possible to correlate the energy of each emitted psudo-entity to the Lorentz factor of the emitting body, then the entity energy has to be wrote as It is therefore of fundamental importance to determine the number of Panones p ℵ constituting the peel of the primary matter and which are affected by the geometric transformation of the boundary m ∂ . We will hypothetize (until proven otherwise) the primary matter to be spherical with radius In the simplest image of an isotropic emission with spherical symmetry we consider to be exactly 0 2λ the thickness of the "zest" more external than the sphere of primary matter (Panones diameter).
The p ℵ number can simply be calculated using the d V definition of the volume occupied by only one Panone: The number of issued entities is an even number.

An Interpretation of the Fine Structure Constant and Planck's Constant
For each of the agglomerations of primary matter from which the luminal entities protrude it is possible to define a surface S m = ∂ that encompasses its material extension and calculate the flow of this volumic energy through it as follows: We observe that the flow of a volumic radiating energy through a surface external to its material distribution is the origin of a force field that we will assume, in the following, to be exactly the coulombian force field. We will not express this relation in local form as it is not possible to consider the source as punctiform.
From this observation it transpires that the sources of the field are describable as a volumic radiating energy, i.e. there is a substantial equality between mass C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology and energy, which cannot be separated from the propagation of energy waves. In the following part of the paper we will deal exactly with these topics.
We can then impose the 9 as following: From this last relation it emerges that the charge would not seem to be an invariant quantity but this is not possible [4] [6]. We can therefore imagine that each emitted entity corresponds to a material PANONE inside to the surface m ∂ (therefore subject to the laws of the SR).
It therefore makes sense to believe that this phenomenon, which combines electromagnetism and electroweak force, originates from an unfathomable phenomenon of dispersion of the outer peel of the material from which it originates.
Using the Sommerfeld notation for the coulombian interaction [7] and because of Equation (14) we derive a new expression of Planck's constant: This last equation shows that the relativistic mass of the emitting body and the momentum 0 p of the emitted entities linked each other and proportional to p γ . In the above relation we just used the symbol  in order to remark that the hypothesis is the boundary behavior. Then we can now write: The Planck constant as well as in Equation (16) alone is sufficient to express the entire behaviour of the classic fundamental interactions interpreted by this model. Equation (16) report alone could be defined as the universal relation of electrostatic interactions in far field. In the near field, careful calculations must be made to evaluate the non-linearities.

Verification of the Model through Determination of Proton Charge Radius
The determination of the proton charge radius is one of the open questions of modern physics [8].  [14] and Laser spectroscopy of atomichydrogen measurements using QED theory. In a very recent as well as beautiful article by Miller, the author discusses how it is not possible to define a spherical shape for the charge radius of the proton even if one can identify its precise spatial density [15].
The here proposed work of investigation, through different tools and breaking with traditional and modern frameworks, starting from definitions and premises completely different from the theoretical ones used by Miller, aims to trace a new path of research which tries to answer to a subset of opened questions that have not yet been answered. The following theoretical prediction is reflected in the current experimental results. We will therefore allow ourselves to continue to use the concept of "frontier of matter" as defined above, reserving for ourselves, in the following, the right to define it further.
Directly from Equation (15) By multiplying and dividing by 2 3 and because of the definition Equation (2) we can write: where p m is the mass of primary matter and p R is its radius. We can therefore make explicit the radius of primary matter taken under examination as follows: C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology The radius expression thus calculated is fully consistent also with the CODATA [16] values for the proton radius with a standard relative uncertainity of 6 × 10 −3 . It has to be observed that the calculated value coincides exactly with the values of the proton charge radius calculated with ep-scattering and spectroscopy.

The Gravitational Equivalence: From Relativistic Mass-Energy Relation to Avogadro's Constant and Electron Mass
In this section of the proposed research the principal hypothesis is made. The history of humanity is evolved by successive advances and new concepts are constructed on the foundations of the old ones. In the following, we make a very strong historical-anthropological assumption: humans refer to all that appears different by different names. And for each of the different things that he sees and experiences he creates models. Humans write their models in the logical-mathematical language: the only one they know, so that they can understand and represent them in the subspace of their perceptions. The external reality is therefore not knowable in its entirety, but exclusively observable within the subspace of human perceptions in which it is projected.
In this specific case we will consider that gravitational and electromagnetic effects are two different manifestations of the same universal interaction evolved historically both independently. If the two models are equivalent then there must exist a constant that correlates them, which we will call language constant.
The correlation constant between two equivalent models of which the observables are completely different (masses and charges) by the will of the ancient observers cannot and must not be any number. The language constant needs to C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology be a special number, that is unitary, and its units of measurement must reflect in themselves the characteristics of logic and language.
By using the proton radius relation 18 we can obtain the following mass-energy relation: where 0 p E is the rest energy of a proton with rest mass p m . Directly from 21 The member to the left of the last equation (less than R 2 ) represents the expression of the coulombian force using the Sommerfeld relations. The one to the right is certainly the Newtonian interaction. We can write both equations then in vector form as follows: where we use N n to discriminate it from P n and separate the respectively gravitational and electromagnetic cases. We can now define where G is the Newtonian constant of gravitation and ξ is an appropriate constant measured in [m/s•mol]. Using the definition 24 in Equation (22) we are able to calculate the following quantity: It's simple to verify that: where A N is the Avogadro constant. We can calculate now ξ as following: this ratio has to be proportional to the amount of matter that deviates from that of the proton, then we can write: where we called x m the mass of the unknown particle. Then the unknown is following: The following Table 1 shows the values of the most important and accredited G measurements.
Removing the maximum and minimum values from Table 1 the average value We note that the mass not introduced in the a priori model is also linked to an electrical balance. An imbalance of charges in the universe would compromise the entire physical-chemical balance and, at the same time, falsify the whole of physics as it is known today. Therefore we suggest to adopt universally the following value of the universal gravitation constant: The proposed value with a relative standard deviation of 2.25 × 10 −6 in relation to mean G appears to be strongly compatible with all laboratory measurements mentioned here.
We can furthermore estimate the exact value of the Avogadro constant: ( )

Conclusions
Once a language has been defined (symbols and syntax), the relation between space and time (speed) of the observed entity (Panone, mol −1 ) is an invariant.
The invariance of the speed of light in all the reference frames then appears only as an exclusive consequence of the invariance of the characteristics of language.
In order to understand the world in a logical form, it must be possible to describe it in a language made up of symbols and rules. In the case of physics, the

The Peculiarity of the Matter Structure
We have discovered that the structure of matter (see Figure 6) is made up of elementary particles that agglomerate with each other, following a similar pattern.
Then we can solve (see Table 2) completely our system of unknown.
We found new definitions for many natural constants (see Table 3).    The invariance of the language constant and the mechanism of the behaviour of the entity on the boundary can be erected to first principles that can be observed experimentally. We could define them as "fundamental behaviors" or "primordial behaviors" of entities. We have therefore demonstrated that the principle of mass conservation, the principle of energy conservation, the prin- consequence of what has been said is that there is no need for axioms. We only need the properties of the space in which we define the mechanisms of frontier that do not necessarily have to be expressed in a Euclidean geometry.

Reshaping the Newton's Law of Universal Gravitation
We could finally extend the Newtonian law to the microscopic world and rewrite it as following:  the negative signum is to be used for antimatter;  the positive signum is to be used for matter.
It appears that between matter and antimatter the gravitational force is repulsive. Then we can write e more general law of gravitation: where 2 1 r r − is the vector distance between the interacting masses.

Progressive Entities with Negative Momentum
When we say that "opposite charges attract each other" it is precisely because the mass of one of them is negative so the body of negative mass hit by the entity regresses in the opposite direction.
When we say "two bodies of the same charge sign reject each other" means that the entities of the electric field carry with them the information about the primary matter which generated them, so that they have a negative momentum if the mass which generated them is negative. It is also observed that the Lorentz dilatation in order to express itself requires that clocks are always oriented perpendicular to the motion of the primary matter. Then we can write the momentum of a entity or of the entity as following: is the scalar product between the speed versor of a general emitting body and the angolar momentum's versor of its clocks. r is the propagation speed versor of the entity. Then 0 p is positive for matter and negative for antimatter. Therefore, the mass of a Panone is directly related to its behaviour in terms of clock. If we define in this way the elementary momentum, its determination becomes ambiguous for bodies in absolute state of stillness. Here we implicitly mean that ˆi v and ˆp ω are only parallel or antiparallel.

From Gravitation Law to Neutron Composition and Neutrino Mass
Propagating the result of Equation (36) until its origin (Equation (19)) we obtain the expression of the radius of an agglomerate of primary matter globally neutral (null charge). But this is surely the neutron radius:

Simplified Solutions for the System
Following the system of simplified solutions using the neutron mass:

Final Comments about the Proposed Framework
The proposed theoretical framework shows the potential to be able to adopt a unified approach for the description of gravitational and electrostatic phenomena in far field until now considered irreconcilable.
All the experimental results currently achieved in the field of physics are preserved unaltered in the context of the proposed theoretical approach and no hypothesis made contradicts either classical theories or modern theories that remain alternatively valid. The illustrated theoretical system is able to demonstrate for the first time by purely theoretical paths the existence of natural structures of primary importance given as axiomatic by the currently accepted theories (existence and mass C. D'Errico Journal of High Energy Physics, Gravitation and Cosmology of the electron, of the neutron and of the neutrino) or known by exclusively experimental means (the Avogadro constant).
Nowadays the generic concept of mass and charge are still very elusive, despite their apparent simplicity they conceal considerable pitfalls and it is extremely easy to fall into banalities or tautologies. The discussions and decisions that have been expressed, for example, in resolution 1 of the 26th GFCM (2018) [36] are "revolutionary" and attempt to fill these kinds of gaps. In this specific case, it was thought to correlate the whole SI to Planck's constant, but to do this they were forced to precipitate and clumsily and dogmatically fix Avogadro's constant (I cite: "[...] The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.02214076 × 10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, N A , when expressed in the unit mol −1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles. [...]"). The theory that we propose here, instead, opens a huge and important door within the general conception of the study of physics by placing a fundamental emphasis precisely on these themes of great conceptual relevance and of general (and not special) interest.
The proposed model offers a streamlined numerical processing compared to the models currently in force, for example it is extremely simple to process problems to many bodies with a simple algebraic model of equations to differences by calculating the resultant k-th step of the momentum. The overall model is therefore not only simplified but also less axiomatic with regard to the transformations of mass into energy and vice versa.
The further, currently tortuous, explanations about the internal structures of matter can be considered as alternatives and equivalents.
In our case, instead, by giving geometric definitions to the primary matter, it is possible to give to the charge a material consistency and a physical space that allow to find a link between the two main phenomenologies.
A further point in favour of the proposed framework is that it weaves into a single concept most of the fundamental constants of physics, providing fundamental meanings and offering inextricable bonds that correlate them all at the same time.
To a careful eye it will not have missed that the modeling of the frontier of matter in §3. coincides with the definition of entanglement and that the elementary clock defined in Equation (2) leads directly to a form of Heisenberg's uncertainty principle. It is established that spontaneous emission, the entanglement and Heisenberg's uncertainty principle (elementary clock) are the basic principles from which the entire order of nature spreads out. In accordance with the definitions of "frontier of primary matter" phenomena such as "photonic upconversion" and phononon generation are a direct consequence.

C. D'Errico
From the model emerge also two further characteristics of matter absolutely not negligible. The first characteristic is that the weak interaction is a consequence of the universal interaction. The second characteristic is that the strong interaction is useless because it is described by the model itself.
A last and important step taken by this framework is the relevance of the language and how once a language is chosen for the description of a problem, the solutions become a formal logical expression developed within the language itself. The critical analysis that gave birth to this paper is therefore extremely philosophical: is it possible to unify equations that describe solutions to problems formalized by different people, in different epochs with different languages and formalizations? Is it necessary to choose a new, more universal language to describe all physics in a single all-inclusive system? Is the language itself able to reveal all the truths at the same time? Are the different solutions expressed in the individual languages all equivalent and illuminate different aspects of reality? Is it necessary to have a single theory to describe the whole of physics or is it necessary to have infinite theories, all of which are irreconcilable, which together give an all-encompassing description of the truth of the facts, that is, those which Wittgenstein defined as "Tatsache"?
The limit of the physical world is nothing more than its representation in the subspace of our imagination and its acceptance is only the "how" this neural representation is reproducible through physical experiments. Choosing a traditional or an alternative path turns out to be onerous only if we analyze the dichotomy in the time domain, but if the universe is timeless we must then have confidence in the visions that leave unchanged and incorporate the physical phenomena accepted and experienced. This has been taught by Galileo, Newton, Cauchy, Maxwell, Einstein, Heisenberg, Gödel, Wittgenstein and many others.
Further developments and contributions of the scientific community in this field could lead to the development of equivalent Maxwell equations for the gravitational field capable of describing an "electrogravitational" model capable of proving equivalent to the model of general relativity.
Proceeding through the contribution of the international community in this direction could certainly be thrilling.