Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). https://dx.doi.org/10.5281/zenodo.845357

S. G. Fedosin

Sviazeva Str. 22-79, Perm, 614088, Perm region, Russian Federation

e-mail: intelli@list.ru

ABSTRACT

The formula is derived for the electric force inside a uniformly charged spherical body, as well as for the Coulomb force between the charged bodies from the standpoint of the model of the vacuum field with charged particles. The parameters of the fluxes of charged particles are estimated, including the energy density, energy flux and cross section of interaction with the charged matter. The energy density of gravitons in the Le Sage’s gravitation model is expressed in terms of the strong gravitational constant. The charge to mass ratio is determined for the charged particles that make up photons and the charged component of the gravitational field. These particles are identified as praons, while the praon level of matter is considered a lower level relative to the nucleon level of matter. The analysis of the main problems of the Le Sage’s model shows that these problems can be eliminated in the modernized model.

Keywords: vacuum field; graviton field; electric force; praons; infinite nesting of matter.

1. INTRODUCTION

The similarity of Maxwell equations for the electromagnetic field, on the one hand, and the Heaviside equations for the gravitational field in the Lorentz-invariant theory of gravitation [1,2], on the other hand, as well as the similarity of formulas for the Coulomb force and the Newton force implies a large probability that the same physical mechanism is responsible for that. For example, as it was shown in [3], gravity may be due to the action of electromagnetic micro quanta with a wavelength equal to the Planck length.

Earlier in [2] and [4], we derived the formula for the Newton's law of universal gravitation and the expression of the gravitational constant in terms of the graviton field parameters, using the modernized Le Sage's theory of gravitation. In addition, in [5] we found the expression for the body mass as the function of luminosity of the gravitons interacting with the body, as well as the expression for the strength of the gravitational field inside the body.

Now we intend to derive the formula for the Coulomb force between the charged bodies and to specify the parameters of the vacuum field, consisting of the graviton field and the field of charged particles. In the modernized Le Sage's theory of gravitation the all-permeating fluxes of the vacuum field particles consist of neutrinos, photons and charged particles, the properties of which are similar to high-energy cosmic rays. The presence of charged particles in the dynamic vacuum field allows us to describe the electrostatic forces and as a result to justify the electromagnetic phenomena.

2. THE INTERACTION PICTURE

To understand the electric interaction of the bodies at a distance from each other, consider Figure 1 which shows the motion of small charged particles of the vacuum field near the two bodies, one of which is neutral and the other is positively charged. As can be seen, both positive and negative particles act symmetrically on the positively charged body, which does not result in emerging of any additional force in comparison with the force of gravitation. The same applies to the second neutral body.

Figure 2 a) shows that the positive particles push the negatively charged body to the left, and Figure 2 b) shows that the negative particles push the positively charged body to the right (when the smallest particles pass through the body similarly to gravitons, they transfer their momentum to them). Consequently, both bodies will be attracted to each other.

Figure 3 shows the lines of motion of the negative particles of the vacuum field near two positively charged bodies. Both bodies attract the negative particles and obtain an additional momentum from them, which leads to repulsion of bodies. The motion of the positive particles of the vacuum field in Figure 3 is not shown. It is assumed that they are repelled from the bodies and therefore their interaction with them is weak.

For two negatively charged bodies the interaction is similar to the one shown in Figure 3, only it is necessary to replace the signs of all charges. This results in the repulsion of similarly charged bodies. The described above picture can be found in [6]. The common in all the Figures is the fact that depending on the sign of the charge of two bodies the number of charged particles falling on the body changes so that after calculating the momentum transferred from these particles the electric force with required direction emerges. Thus, we reduce the interaction between the charges at a distance to the interaction by means of the charged particles of the vacuum field.

3. THE COULOMB FORCE

To determine the expression for the electric force we use the approach applied in [4-5]. Let’s assume that the fluence rate of the charged particles of the vacuum field is defined by idealized spherical distribution of the following form:

.                                                          (1)

According to (1) we suggest that some detector per unit time  measures the charged particles of the vacuum field in the amount  that fall on the detector from the solid angle  per unit surface area  perpendicularly to this surface.

We will assume that inside the matter of each charged body an exponential change in the number of charged particles of the vacuum field takes place, as the flux of these particles travels some path  in this matter:

,                       ,                   (2)

where  is the cross section of interaction of the moving charged particles with the matter,  is the concentration of charges associated with the matter.

Denoting the positive elementary charge by , for the absolute values of charges and the area  of ball segments in Figure 4 we obtain the following:

,                 ,                .     (3)

The detector is located at point 0 in the middle between the two segments. For it, each segment is seen at the same solid angle  at the distance , while the transverse areas of the segments are the same and equal . It means that before we apply further arguments for the two large bodies, we should cut these bodies into segments and then calculate the total electric force between all the possible pairs of segments by means of vector summation of particular forces.

Let us first consider the case when the charge  is positive and the charge  is negative. Comparison with Figure 2 shows that interaction leads to attraction due to absorbing and scattering of charged particles falling on the charges and passing through them. As a first approximation we can assume that the main contribution is made by the flux of negatively charged particles falling on the charge  from the left and the flux of positively charged particles falling on the charge  from the right.

Decrease of the flux of charged particles on the left side after passing the first segment in Figure 4 according to (2) depends on the thickness of this segment and on the concentration of charge:

.

After that the flux of charged particles passes through the second segment with further decrease of the flux:

.

We will denote the average momentum of a charged particle of the vacuum field with , and further we will assume that in case of interaction of such a particle with the charged matter the change in the momentum of the particle is approximately equal to . This is possible if the charged particle is stopped by the matter or is reflected by the electromagnetic field sideways so that the change in the momentum vector has the same order of magnitude as the particle momentum vector itself.

Then the force acting on the second segment from the left, taking into account (1) is equal to:

.

Decrease of the flux of charged particles, passing through the second segment from the right side, and the force from this side are, respectively:

,        .

For the force of electrical action on the second segment we find a symmetrical expression, which is equal by its absolute value to the force of electrical action on the first segment:

.

Expanding the exponents in the linear approximation by the rule , taking into account (3), we obtain for the force of attraction between two oppositely charged segments the following:

,         .     (4)

In (4) the force  is directed oppositely to the vector  of the distance from the first segment to the second segment, since the charge  is negative.

According to the Coulomb's law, the formula for the electric force between two charged bodies is as follows:

.                                                           (5)

Comparing the values of the forces in (4) and (5), we arrive at the expression for the vacuum permittivity in terms of the parameters of charged particles fluxes in case of idealized spherical distribution:

.                                                       (6)

The vacuum permittivity in (6) depends on the cross section  of interaction of charged particles fluxes with the matter, on the average momentum of one charged particle , on the fluence rate  and on the elementary charge .

From the expression for the force we determine the electric field strength of one charge at the place of the second charge:

.                                                      (7)

We will assume now that the charge  in Figure 4 is positive like the charge . This situation corresponds to Figure 3, from which it follows that after the passing the charge  the flux of charged particles effectively increases before falling on the charge . For the flux of particles moving from the charge  and falling on the charge  the situation is symmetric. In order to take into account the effect of increasing of the flux of charged particles, we will introduce an additional coefficient . Then the flux of charged particles from the left side after passing the first segment in Figure 4, taking into account (2), changes to the value:

.

When passing through the second segment the flux decreases:

.

The force acting on the second segment from the left, taking into account (1), is equal to:

.

For the flux of charged particles passing through the second segment from the right and the force from this side we obtain, respectively:

,        .

For the force of electrical action on the second segment we obtain:

.

In this expression, we will expand the exponent and use (3):

.                        (8)

The repulsion force (8) after changing of the sign of the charge  must be equal by its magnitude to the attraction force in (4). For this the following condition must hold: . There is a way to prove this relation. To do this, we should consider the situation in Figure 3, estimate the fluxes of charged particles from all sides and their interaction with the charged bodies, so that we could determine how much these fluxes increase when falling on the bodies as compared to the situation in Figure 1. We will return to this issue again in Section 6.

In Figure (1) we see that if one of the bodies has no charge, then the charged particles of the vacuum field do not interact with this body electrically. They pass through it almost freely, except for the gravitational action. As a result, between the charged and uncharged bodies there will be only the force of gravitational attraction.

4. THE ELECTRIC FIELD STRENGTH INSIDE THE BALL

In order to estimate the field inside a uniform ball it is more convenient to proceed from spherical distribution (1) to cubic distribution in the form of a mixed derivative for the flux of charged particles of the vacuum field directed in one way:

,                                                             (9)

where the fluence rate  indicates the number of charged particles , that during time  fell on the area  of one of the cube faces, limiting the volume under consideration, which is perpendicular to the flux.

Figure 5 shows the section of a uniform charged ball with a radius , inside which there is a small test body in form of a ball with a radius .

The fluxes of charged particles of the vacuum field move along the paths 1, 2, 3, as well as other paths, passing the section of the small ball, which is at a distance  from the center of the large ball. If we replace the small ball with the cube of the same size, then in case of idealized cubic distribution it is enough to consider the vertical fluxes along the path 2. The fluxes of charged particles passing through the other faces of the small cube will be symmetrical and will not influence the electric force. This means that with this approach we will take into account the fluxes along inclined paths 1 and 3 not directly, but indirectly. All these fluxes in case of vector summation will give the force, acting on the small ball and should be added to the force, calculated for path 2.

Let the volume of the small ball be equal to the volume of some cube. Then for the volume of a cube with an edge  and for the absolute value of charge  of this cube we obtain the relations:

,                              ,                      (10)

where  is the concentration of charge in the small ball.

Distribution (9) replaces the actual distribution of the charged particles of the vacuum field in space with the idealized cubic distribution, when only six fluxes of charged particles fall on the given cubic volume perpendicularly to the faces of the cube.

By analogy with (2) we can write the dependence of the fluence rate of the charged particles of the vacuum field on the distance traveled in the matter:

,                              .            (11)

Let us first assume the charge  of the small ball in Figure 5 as negative and the charge of the large ball as positive.

The flux of charged particles falling from above travels the path  in the large ball with the concentration of charge  in its matter, and reaches the small cube, with which we replaced the small ball. According to (11) at this point the fluence rate decreases to the value:

.

Then the flux passes through the small cube with concentration of charge  and decreases again:

.

The force from this flux of charged particles is proportional to the square of the face of the small cube and to the number of charged particles, which transferred their momentum per time unit to the cube matter:

.             (12)

On the lower side of the large ball the flux of charged particles first passes the path  to a small cube and then passes through the cube:

,                  .

The force acting on the small cube from this side equals:

.             (13)

The total force is the difference between the forces (12) and (13):

Since exponents in this expression are small enough, the exponents can be expanded in the small parameter by the rule: . With this in mind, we obtain:

.

In this expression, we will take into account that the charge density of the large ball is given by the formula: , and will use (10):

,                      .

The force  acts on the small ball with the negative charge  in Figure 5 so that the force is directed toward the center of the large ball and oppositely to the radius vector  from the center of the large ball to the small ball. By definition, the electric field strength is the ratio of the force, acting on the test body, to the charge of the test body. Then the vector of the electric field strength inside the large ball will be:

.                                               (14)

In electrostatics, the vector of the electric field strength inside a uniform charged ball is determined by the formula:

.                                                           (15)

From comparison of (14) and (15) we find the expression of the vacuum permittivity in terms of the parameters of charged particles fluxes in the cubic distribution approximation:

.                                                        (16)

The difference between the used cubic (9) and spherical (1) distributions leads to the fact that the formulas for vacuum permittivity (16) and (6) differ by a numerical factor.

If the small ball in Figure 5 has not a negative charge but a positive charge , then its interaction with the charge of the large ball should be considered in view of Figure 3 for the interaction of two positive charges. It means that it is necessary to introduce an additional coefficient  in order to take into account the effect of increasing of the flux of charged particles.

As a result, the fluence rates  and  and the force (12) from the flux of charged particles falling from above on the small cube, by which we replaced the small ball, will change and be equal to:

,         ,

.           (17)

Similarly, at the lower side of the large ball for the fluence rate and the force, instead of (13), we have:

,                  ,

.          (18)

The total force equals the difference between the forces (18) and (17):

Expanding the exponents by the rule: , we find:

.

Let us assume that the charge density of the large ball is given by the formula: , and for the coefficient  the relation holds: , which was found in the previous section. Then, with regard to (10), we obtain:

.

The force  is directed radially from the center of the large ball, and the expression for this force after dividing by the charge  leads to the electric field strength (14).

5. THE PARAMETERS OF THE FLUXES OF CHARGED PARTICLES OF THE VACUUM FIELD

We will estimate the energy density for cubic distribution of charged particles fluxes of vacuum field in space. Suppose there is a cube with an edge , into which charged particles fly from six sides perpendicularly to the faces of the cube. The speed of charged particles is assumed to be equal to the speed of light, so that in the time  the cube will be completely filled. In view of distribution (9) the number of charged particles in the cube will be: . If the energy of one charged particle is ,  then with the help of (16) for the energy density of charged particles of vacuum field we find:

.                                           (19)

Now we will use the spherical distribution (1) to estimate the energy density of charged particles of vacuum field. An empty sphere with radius  can be filled with charged particles in the time , if the graviton fluxes are directed radially and correspond to the full solid angle . The number of charged particles inside the sphere will equal . Multiplying this number by the energy of one charged particle and dividing by the sphere’s volume we can find the energy density. In view of (6) and the condition , we obtain:

.                                        (20)

The energy density (20) with spherical distribution is 3/2 times greater than with cubic distribution (19), which emphasizes that our estimates are approximate due to the use of two idealized distributions.

Earlier in [5] we have applied the concept of the graviton field to calculate the Newton’s gravitational force between two bodies and the gravitational constant. This allowed us to estimate the energy density of the graviton field for cubic distribution and the rate of the energy flux of the graviton field in one direction:

J/m3 ,                             (21)

W/m2 ,

here  is the average energy of one graviton, is the average momentum of one graviton,  is the number of gravitons falling per unit time on unit area from one of the six spatial directions in cubic distribution, ,  is the gravitational constant,  is the mass of one nucleon of the matter, m2 is the cross section of interaction of gravitons and the matter.

The energy density  in (21) is associated with the gravitational constant  and with gravitation at the level of nucleons. Similarly, the energy density of the charged particles of the vacuum field  in (19) is associated with the electromagnetic action of the field on each elementary charge  of the matter.

Further we will need the similarity coefficients, with the help of which in the theory of infinite nesting of matter [2], [6] we will calculate the physical quantities inherent in each particular level of matter. As the typical parameters of a neutron star we will take the mass equal to 1.35 Solar mass or kg and the stellar radius equal to  km.

Dividing the mass of the neutron star by the proton mass , we find the coefficient of similarity in mass: . Similarly, we calculate the coefficient of similarity in size as the ratio of the stellar radius to the proton radius: ,

here the quantity m in the self-consistent model of the proton [7] was used. We can estimate the minimum possible radius of the neutron star based on the ratio of the star volume to the total volume of all the nucleons in the stars: ,   km. A star with the radius of 12 km exceeds this limit, there are some gaps between the nucleons and the nucleons remain to be independent particles.

The coefficient of similarity in speed equals the ratio of the characteristic speeds of the matter inside the star and the proton, respectively. For the star the characteristic speed  is calculated from the energy equality from the standpoint of the general principle of equivalence of mass and energy, generalized with respect to the absolute value of the total energy to any space objects:

,                          m/s.

Similarly, we find for the proton the equality of the characteristic speed of its matter and the speed of light:

m/s,

while m3·kg-1·s-2 is the strong gravitational constant, calculated from the equality of electric and gravitational forces in the hydrogen atom,  is the vacuum permittivity,  is the electron mass and according to [7] for the proton . Hence, the coefficient of similarity in speed is equal to: .

As it was shown in [2], the ratio of the absolute value of strong gravitation energy density to the electromagnetic energy density of the proton is equal to the ratio of the proton mass to the electron mass . Indeed, for the energy of the fields and their ratios, in view of the definition of the strong gravitational constant , we have: ,  ,  .

We believe that the same ratio exists for the energy densities of graviton field and charged particles of the vacuum field, which allows us to estimate the energy density of charged particles of the vacuum field:

J/m3.                                            (22)

Let us substitute (22) into (19), using the value of  from (21), and take into account the proximity of the proton mass and the average mass of a nucleon , as well as the definition of the strong gravitational constant in the form . This gives an estimate of the cross section of interaction of the charged particles of the vacuum field with the charged matter:

m2 .         (23)

This cross section has a value that almost exactly coincides with the geometrical cross section of a nucleon and significantly exceeds the cross section m2 of interaction of gravitons with the matter. In order to find the significant difference between  and , we will express  from (22), use  from (19) and take into account the definition of :

.                                        (24)

From comparison of (24) and (21), provided that , it follows that if in (21) we pass from the cross section  to the cross section , then at the same time it is necessary to substitute the gravitational constant  with the strong gravitational constant . In (24) the energy density  of the graviton field at the level of nucleons is fully expressed in terms of the parameters of the nucleon level of matter. Similarly, in (19) the energy density  of the charged particles of the vacuum field is expressed in terms of the parameters of the nucleon level of matter. In this case, both in (19) and (24) the same cross section  of interaction of the vacuum field particles with the matter consisting of nucleons is used. Note that in [4] it was found that the cross-section  of the interaction between the gravitons and the matter of nucleons must be equal by the order of magnitude to the cross-section of the proton.

By analogy with (24) for the graviton field at the stellar level we can write:

.

If in this expression we shall consider the following relations in accordance with the dimensional analysis, coefficients of similarity and (24):

,        ,        ,

then we obtain the relation  J/m3, in which the energy density of graviton field at the stellar level , needed to keep the matter in neutron stars, linked to the energy density . Since the energy density  is required for the integrity of the nucleons in the field of strong gravitation, then .

In view of (16), (19), (22) and the relation , for the rate of the energy flux of charged particles of the vacuum field in one direction we find:

W/m2 .                              (25)

Due to the fact that the above-mentioned energy density  of charged particles of the vacuum field is less than the energy density  of graviton field in (21), the rate of the energy flux of charged particles of the vacuum field  is less than the rate the energy flux of the graviton field .

6. THE ESTIMATES OF FORCES AND ENERGIES

In [2] and [6] the assumption is made that some neutron stars magnetars can have a positive electric charge of up to  C, where  is the elementary electric charge and the similarity coefficients are used in accordance with the dimensional analysis.

The proton electric energy on the surface of the charged magnetar will reach J  or eV. The corresponding electric force will be equal to  N. It is assumed that it is precisely the electrical energy in the magnetar field that is the energy source of high energy cosmic rays.

For the absolute value of the gravitational energy of the proton on the surface of the magnetar similarly we have:  J.

This energy and the gravitational force, associated with it, are clearly not enough to keep the proton, on which the repulsive force is acting from the entire charge of the magnetar. However the magnetar looks like a huge atomic nucleus consisting of a number of closely-spaced nucleons. Between nucleons there is strong interaction, which holds them together. In the gravitational model of strong interaction [6] the idea of strong gravitation is used to describe the strong interaction. The nucleons in the atomic nuclei are attracted to each other by strong gravitation and repel from each other by means of the torsion field, which arises from the rapid rotation of the nucleons. According to the Lorentz-invariant theory of gravitation [1-2], the torsion field arises similarly to the magnetic field in electromagnetism, and in the general theory of relativity it corresponds to the gravitomagnetic field. The balance of attractive and repulsive forces, arising from strong gravitation, can be responsible for the integrity of the atomic nuclei, as well as for the integrity of the charged neutron star.

We did the estimates of forces and energies in the atomic nuclei in [6]. For example, the nickel nucleus  consists of  nucleons, among which there are 28 protons and 34 neutrons. The mass of this nucleus is kg, and the radius is obtained from experiments on the scattering of electrons by the formula: m, where m. Based on these data we will estimate the force, acting from the nucleus on the proton located on the nucleus surface, with the help of strong gravitation: N.

The surface of the magnetar as a neutron star apparently consists of the nuclei of such elements as iron, nickel and heavier nuclei, since their binding energy per nucleon is maximum. If the proton was near one of these nuclei on the magnetar surface, the force  would keep the proton, acting against the force of electrical repulsion N from the magnetar charge. But the concentration of nuclei on the stellar surface is such that the proton on the average will be located somewhere between the nuclei at a distance  from them.

To keep the proton the condition  must hold, which implies that m. For a cube with the edge , at the corners of which there are 8 nuclei , and the proton is in the center of the cube, the matter  density is equal to kg/m3. The matter density on the magnetar surface must exceed this value, so that the condition of stability with respect to electric forces is satisfied. On the other hand, the estimates in [8] of the matter density in the crust of the neutron star imply that at a density of  kg/m3 and more the nuclei  begin to decay. Consequently, heavier nuclei must prevail in the magnetar crust, in particular, a typical nucleus according to [8] is . From these calculations it follows that the magnetar charge is almost the maximum charge that the star can have without loss of its integrity. And the main contribution into the stability of a star is made by not ordinary but strong gravitation, acting at the level of atomic nuclei.

With the help of the similarity coefficients we can calculate the mass, radius and charge of the praon – the particle, which relates to the proton, as the proton relates to the magnetar: kg, m, C. If the praon is located at the surface of the proton, its electrical energy and gravitational energy in the strong gravitational field will be equal: J, J.

The ratio of these energies is the same as the ratio of the electric energy of the proton at the surface of the magnetar to the gravitational energy of this proton in the gravitational field of the magnetar. In the substantial model of the proton and neutron, presented in [6], it is assumed that the nucleons consist of neutral and charged praons, just as neutron stars consist of nucleons. In addition, by analogy with the composition of cosmic rays, consisting mainly of relativistic protons, we can assume that the charged component of the vacuum field can consist of praons accelerated by positively charged atomic nuclei up to high energies.

At the present time cosmic rays are registered with energies up to eV or 9.6 J per nucleon [9]. Assuming that this is the energy of the accelerated proton, we will divide it by the coefficient of similarity in energy and will find the corresponding energy of the praon: J. Equating this energy to the energy  of a charged particle of the vacuum field, we can estimate the concentration of these charged particles as the concentration of relativistically moving praons. In view of (19) and (22) we obtain:

m-3 .

Multiplying this concentration of charged particles by the charge of one praon  and the speed of light, we can estimate the density of the current in the vacuum in one direction, which arises from the flux of positively charged praons in one direction at cubic distribution:

A/m2 .

Beside the current density , we should expect another similar current density  in the same direction, which arises from the flux of negatively charged praons. This should ensure a certain degree of vacuum electroneutrality and existence of electrical forces of repulsion and attraction.

Now we will consider the question of neutron star’s matter permeability for gravitons and charged particles of the vacuum field, respectively. The fluence rates from a unit solid angle similarly to (2) have the form:

,                 .

If the neutron star has a radius of 12 km and a mass of 1.35 solar masses, then the average concentration of nucleons will equal m-3. The average concentration of the positive charge in the magnetar is m-3. Assuming that km, for the exponents in view of (21) and (23) we find: , . It follows that if we put three neutron stars in the way of the flux of gravitons, the flux will reduce approximately by a factor of , where is the base of the natural logarithm. But for the flux of charged particles of the vacuum field in order to reduce it noticeably we need to put in a line about 140 magnetars.

This difference in fluxes allows us to explain the saturation effect of the specific binding energy, when the nuclear binding energy per nucleon, depending on the number of nucleons in nuclei, first increases, reaching a maximum of 8.79 MeV per nucleon for the nucleus , and then begins to decrease. For light nuclei the increase in the specific energy agrees well with the increase of the specific gravitational energy of the nucleus in the strong gravitational field, when the energy increases in direct proportion to the square of mass and in inverse proportion to the radius of the nucleus. The saturation effect comes into play in the range of 17 to 23 nucleons, forming the nucleus. Besides, adding a new nucleon to the nucleus increases the energy not proportionally to the square of mass, but to a lesser extent. This is due to the fact that gravitons of strong gravitation cannot permeate the nucleus with a lot of nucleons, as is evident from the exponent. Each new nucleon is simply pressed to the nucleus from the outside by the strong gravitation, until for the large nuclei this force reaches the maximum, conditioned by the pressure of the graviton flux. However, the charged particles of the vacuum field in these conditions have almost 50 times larger path length, and therefore the positive electrical energy of the nucleus’ protons further decreases the negative gravitational energy of the nucleus, making the main contribution into the observed decrease in the specific binding energy of massive nuclei.

Earlier in [4] we estimated the maximum force between two stellar objects:

N,

where  for the case of uniform density of each object, and it is assumed that the graviton fluxes are fully retained by these objects, which are located close to each other.

A similar expression for the maximum force at the nucleon level of matter, after replacing the gravitational constant by the strong gravitational constant, in view of the coefficient of similarity in speed  has the form:

N.

We should note that the corresponding ratio of the gravitational energy and the force between two protons to their electrostatic energy and force is equal to the ratio of the proton mass to the electron mass. Indeed, for the forces and their ratios in view of the definition of the strong gravitational constant , we have: ,  , .

We can explain this by the fact that in the expression for  the exponent for the flux of charged particles of the vacuum field in the magnetar and hence in the proton is less than the corresponding exponent for the flux of gravitons in the expression for . The gravitons are retained in the proton matter more than the charged particles of the vacuum field, and therefore the gravitational force is greater than the electric force.

After passing from dense and charged objects such as magnetars and protons to the bodies surrounding us the situation with the ratio of forces is changing. The gravitational force decreases rapidly with decreasing of the mass of bodies, and we can hardly influence it. However, by changing the charges of bodies we can change their electrical interaction, so that the electric force can be many times greater than the gravitational force between these bodies. This can be seen from the ratio of the electric and gravitational forces for two identical bodies with the mass  and charge , which is proportional to the squared charge: .

Let us take for example two iron balls with the radius cm each. With the density of iron 7874 kg/m3 it gives the mass of each ball of approximately 4.1 kg. For the equality of the gravitational and electrical forces it is enough to charge the balls up to C, so that the potential of each ball reaches the value V. Let us estimate the electrical energy of the praon, flying near the ball, taking into account that above we estimated the charge of the praon with the value C: J. On the other hand, the energy of a praon, regarded as a relativistic particle similar by its properties to cosmic rays, has been found above in the form: J. Comparison of these two energies allows us to make the following conclusions. Firstly, even weakly charged bodies, which interact at the level of low gravitational force, can influence the motion of praons near them and deflect them aside. This substantiates the pattern of motion of the charged particles of the vacuum field near the charged bodies in Figures 1-3 and our calculations of the electric force. Secondly, if we decrease the charges and increase the sizes of bodies, there can be deviations from the Coulomb law. However, these deviations should be distinguished from the gravitational force, which in this case becomes greater than the electric force.

The last conclusion can be specified as follows. In order to find the deviations from the Coulomb law, it is desirable that the condition of small potentials is satisfied V. To reduce the dependence on the gravitational force, there are the following conditions  or . Hence for the corresponding electrical potential of one ball, we have: V or  kg/m.

For the iron balls it gives cm,  kg. Another complication in the experiments for finding deviations from the Coulomb law occurs due to the fact that in conductive bodies the uncompensated charges are located in the thin layer on the bodies’ surface, with a thickness of the order of 1 or 2 atomic layers. Free electrons easily go out of the equilibrium position in the external electric field, either repelling or being attracted to the source of the external field, thereby changing their concentration on the body. Due to this, in two interacting charged metal balls additional electrical forces appear, which are usually calculated by the method of images.

7. INTERACTION OF THE BODY’S CHARGE WITH THE VACUUM FIELD

The Coulomb law, due to the presence of charged particles in the vacuum field, can be explained with the help of Le Sage’s model. However, not only the fluxes of charged particles influence the interaction of charged bodies, but the charges of bodies themselves influence the fluxes of charged particles around the bodies. One example of this influence is deflection of the charged particles from their trajectories, as it was described in the previous section. In addition, each charged body achieves a certain balance of energy and momentum during interaction with the vacuum field.

Let us consider the energy density of the charged particles of the vacuum field inside the charged body and near it. Suppose there is a body in the form of a cube with an edge . The number of charged particles  per unit time through a unit area during particles’ motion in the matter decreases according to formula (11). During time  six fluxes of charged particles from each side will pass inside the cube through the faces with the area  and will change up to the value:

,                     ,

where  is the number of charged particles that passed through the cube.

If charged particles flew through the same empty volume, the number of charged particles coming out would be . Consequently, the number of charged particles, which interacted with the matter of charged body during time , equals:

.

As it was shown in [5], almost all the energy of the graviton field, which interacts with the matter, is re-emitted back to the graviton field, without heating the bodies significantly. This also applies to the fluxes of charged particles the vacuum field, that transfer their momentum to the matter with return of the energy back to the vacuum field.

Let us estimate in view of (19) the energy density of those charged particles that interact with the bodies’ matter:

.                                                  (26)

From (26) we will calculate the luminosity of charged particles of a body in the form of a cube, multiplying  by the volume  and dividing by the time . Expressing the charge concentration in terms of the charge, in view of (19) we have:

,                 .             (27)

From (27) it follows that the luminosity  of the charged particles, understood as the luminosity of those charged particles fluxes that interacted with the charged matter of body and gave their momentum to it, is proportional to body charge . This means that the body charge can be expressed in terms of the parameters of the charged particles fluxes interacting with the body.

In (27) there is a product  equal to the number of uncompensated elementary charges in the body under consideration. Then the charged particles luminosity per one elementary charge, in view of (19), (22-23) will equal:

W.                         (28)

The ratio of the luminosity  to the average energy of a charged particle  gives the number of charged particles that interact with one uncompensated elementary charge of matter per unit time and gave their momentum to it. According to (28), this number of charged particles is equal to the product , while the cross section  characterizes the effective area of elementary charge’s interaction with charged particles, and the coefficient 6 is associated with the six sides of cubic distribution of charged particles fluxes  in (11).

Expression (27) can be given a different meaning, if we assume that the area of the cube face is connected with the cross section  by the following relation: , where  is some numerical coefficient,  is the number of uncompensated elementary charges in the cube. Then under the condition  (27) can be rewritten as follows:

.

This relation shows that the emission rate is proportional with accuracy to a coefficient  to the electric energy of the charged body, derived from the body in the time  of passing the body characteristic size by the charged particles.

We note one more aspect concerning the interaction between the electromagnetic and gravitational fields. The concept of the general field [10] shows that the vector fields, including the electromagnetic and gravitational fields, are the components of one general field. And in case if the theorem of equipartition of the energy is satisfied, the equations of particular fields no longer depend on each other and are similar in form to the Maxwell equations. If the fields interact with each other, then in the Hamiltonian it is manifested in the terms with the field energy, where the cross-terms with the products of different field strengths appear. This is possible, for example, in non-stationary processes in the systems that have not reached equilibrium. From the viewpoint of the vacuum field, it means that in stationary conditions the gravitons and charged particles of the vacuum field interact with the matter relatively independently, creating gravitational and electromagnetic forces. If there is no equilibrium in the system, then the kinetic energy of matter and the energies of some fields are transformed into the energy of other fields, and the exchange of energies between gravitons and charged particles in the vacuum field is possible as well. This leads to the cross-terms in the system’s energy.

8. PHOTONS AND PRAONS

In this section we will try to specify which particles can be responsible for electromagnetic phenomena. The charged particles of the vacuum field not only lead to the electric forces in the Coulomb law, but should be part of the photons, i.e. the electromagnetic quanta emitted by atoms. Let us consider, for example, a photon with the wavelength m and the angular frequency s-1, arising in the hydrogen atom in the transition of an electron from the second to the first level in the Lyman series. The probability of this transition equals  s-1 [11], which gives the average lifetime of an electron at the second level s, as a measure of duration of photon emission during the transition. In quantum mechanics [12] there is a formula for the oscillator’s oscillations decay time in  times, where  is the base of the natural logarithm, with the help of which we obtain the following estimate:

s.

where  is the vacuum permeability.

The duration of photon emission can be calculated directly within the Bohr model of a hydrogen atom. In this model, the electric force between a proton and an electron acts as a centripetal force in the electron’s rotation around the nucleus in the form of a proton. In this rotation, the electron must emit an electromagnetic wave, since it is constantly accelerated towards the nucleus. The formula for the charge emission rate during its rotation is well known, which allows us to relate the electron velocity and the effective force acting on the electron from emission. Moment of this force decreases the angular momentum of the electron, leading to a decrease in the radius of rotation. Hence we can derive the dependence of the radius on the time [6]. From this dependence we find the duration of photon emission as the time of transition of an electron from the second to the first level of energy. Given that the average radius of the electron rotation on the second level equals , and the average radius of the electron rotation on the first level is the Bohr radius , we have the following:

s.                                      (29)

For the instantaneous power of electromagnetic emission we obtain the formula:

.

This implies a strong dependence of the emission rate on the current radius  of the electron rotation, which is inversely proportional to the fourth power of this radius. It turns out that the main photon energy is emitted when the electron approaches the lower energy level.

Knowing the emission duration we can find the length of the photon . To calculate the volume of the photon we also need its midsection. In the first approximation, we assume that the mean radius of the photon equals , which is equal to . We note that in the substantial model of electron [6], it is considered as a thin disk that has on the main energy level the inner radius  and the outer edge , and the Bohr radius  is obtained as a certain characteristic radius of the disk and the average radius of the electron rotation. On the second level, the outer edge of the electron disk is greater than the average radius of the electron rotation  on this level. With this in mind, the volume of the photon will equal: .

Further on we will use a simplified model of photon from [2], [13], according to which the photon consists of charged particles, the rotation of which around the photon’s axis creates the angular momentum of the photon. In addition, inside the photon as well as in the electromagnetic wave there must be mutually-perpendicular periodically varying electric and magnetic fields. Electromagnetic energy of the photon consists of the equal electric and magnetic components, and for the total energy density we can write: , since in the wave . The electric field strength  inside the photon will be characterized by the amplitude . The field inside the photon oscillates, varying from zero to the peak value, so for the average density of the electromagnetic energy of the photon, we assume that . We also assume that the photon energy is equally divided between the mechanical energy of the charged particles and the electromagnetic energy. The photon energy  is proportional to the Dirac constant  and the angular frequency . Dividing the photon energy by the photon volume, we obtain the energy density, which can be equated to the doubled density of electromagnetic energy inside the photon:

,                               .                   (30)

Substituting in (30) the photon angular frequency s-1, the duration of the photon emission  from (29) and the photon radius , we estimate the amplitude of the electric field strength inside the photon: V/m. For comparison, the proton creates at the Bohr radius the electric field strength V/m.

From the mechanical point of view we can consider in a simplified way the photon as a long thin cylinder, rotating with the angular frequency . If inside the cylinder there are  particles, each of which has a relativistic mass , then in case of uniform distribution of particles the angular momentum of the cylinder must be equal to the Dirac constant, as it is supposed for all photons:

.                                                         (31)

From (31) it follows that the mechanical energy of the particles’ rotation, calculated as half the product of the angular momentum  and the angular velocity of rotation, is equal to the half of the photon energy: . The other half of the photon energy must be the electromagnetic energy, which was taken into account in (30). Since the angular momentum of the electron in the atom is quantized and proportional to , from (31) it follows that the total relativistic mass  of the charged particles rotating inside the photon must be of the order of the electron mass, in order to ensure the angular momentum  of the photon. However, the mass  is only a small part of the mass of the entire flux of charged particles of the vacuum field, that pass through the electron disk per time of the photon emission  from (29). The total relativistic mass of particles of the entire flux per time  is expressed by the product of the energy flux rate (25), the time  and the area of the electron disk , and then dividing by the square of the speed of light in order to pass from the energy to the mass:  kg, which is much greater than the electron mass.

Let us consider the motion of some charged particle inside the photon, located on the radius . This particle rotates at a certain velocity  around the axis of the photon, and besides it moves at the speed of light, as well as the photon, in the direction of its propagation. For the particle’s period of rotation we can write:

,                                .                (32)

In this model of a photon, there is a relationship between the centripetal force, required for the particle’s rotation, and the electric force, exerted on the particle with the charge  and the mass . In view of (32) we have:

.                                                    (33)

Let us express from (30)  and substitute it in (33) in order to determine the ratio  for the charged particles inside the photon. In view of (29) for , as well as the assumed relation  and the value of the photon angular frequency s-1, we find:

C/kg.                              (34)

For the level of stars, the charge to mass ratio should be the highest for the charged magnetar, as a neutron star with the mass kg that, according to our assumption, bears the electric charge C. This gives: C/kg. At the level of atoms, the same is true for the proton, for which C/kg. What does the relation (34) give to us? From this relation it follows that we must refer to a lower level of matter, that is, the praon level of matter. For the charged praon at rest, the charge to mass ratio, in view of the results of Section 6, is: C/kg.

Now we will take into account that in (34) the mass of the charged particle is the relativistic mass, i.e. the ratio of the particle’s energy to the square of the speed of light. This mass can be written as: , where  is the Lorentz factor for the particle, moving almost at the speed of light. Substituting the mass  in (34) and using the value  for the praon, we can determine the Lorentz factor: .

Earlier in Section 6, we referred to the fact that the protons in cosmic rays reach the energy eV, while the rest energy of the proton is eV. Consequently, for the most energetic cosmic-ray protons the Lorentz factor is as follows: .

We see that the Lorentz factors for praons and protons are close enough to each other. All this means that the photon is a tightly bound flux of praons, the energy of which is maximum and corresponds to the energy of cosmic rays at the nucleon level of matter. Besides, praons are related to protons, just as protons are related to a charged neutron star a magnetar. From photon’s neutrality it follows that it must consist both of positively and negatively charged praons.

9. THE ANALYSIS OF CRITICISM OF THE LE SAGE’S THEORY

If we assume that the charged particles of the vacuum field are relativistic praons, then according to (28) the power of energy of the fluxes of praons interacting with each uncompensated elementary charge of some charged body is equal to  W. In [5] we found that the power of energy of the fluxes of gravitons, creating the ordinary gravitation, calculated per body’s nucleon is equal to:

W.                                                  (35)

At the nucleon scale level of matter we must move from ordinary gravitation to strong gravitation, which holds the matter of nucleons. As it follows from (21-24), for this the cross-section of the interaction of gravitons  in (35) should be replaced by the interaction cross-section :

W.                                                  (36)

Then it follows from (22) that ,

that is the ratio of the emission power of the fluxes of gravitons of strong gravitation calculated per body’s nucleon to the emission power of the fluxes of charged praons calculated per each body’s uncompensated elementary charge is equal to the ratio of the proton mass to the electron mass.

According to (35), every second one nucleon of a gravitationally bound body should re-emit the energy  J of the graviton fluxes incident on this nucleon so that ordinary gravitation would be possible. If there were no such re-emission, then we could estimate from the equation  the temperature, to which each nucleon and any body of nucleons could be heated from the fluxes of gravitons per second:  K. If we substitute as  the energies, resulting from (28) and (36), then the corresponding temperatures would be even higher.

It is the problem of heating of bodies, which is the main problem of Le Sage’s theory, both for the fluxes of gravitons leading to gravitation and for the fluxes of charged praons creating electromagnetic interaction at the nucleon level of matter. Actually, since the fluxes of gravitons and charged particles must transfer some part of their momentum to the matter to give rise to gravitational and electromagnetic forces, it also seems that some part of the energy of these fluxes should turn into the kinetic energy of motion of the matter and thus heat it to high temperatures, which is not observed. Many researchers of Le Sage’s theory, including Thomson [14] and Poincaré [15], based on the arguments of classical mechanics and the relationship between energy and momentum, pointed to this and other circumstances, considering them an obstacle to the theory’s acceptance.

Now we raise the following question: is such a mechanism possible, when the fluxes of smallest relativistic particles transfer some momentum to the body’s matter, but at the same time they almost completely conserve their energy and are re-emitted into the surrounding space without heating the body significantly? To answer this question, let us remember that there are fields known in physics that do not perform work on the particles and do not change their energy. This is the magnetic field, as well as the torsion field in the covariant theory of gravitation [6], known as the gravitomagnetic field in the general theory of relativity. A fast charged particle, passing through the region of space with the magnetic field, is deflected from the original direction of motion by the Lorentz force, in which case the amplitude of the particle’s momentum and its energy do not change. Despite this, the pressure force from the particle is exerted on the source of the magnetic field. This happens because the momentum like any vector can change both in magnitude and in direction, and any change in the momentum is associated with the corresponding force.

The analogue of nucleons at the level of stars is a neutron star, and the fluxes of praons correspond to cosmic rays. The cosmic rays, passing close to a neutron star, will interact with the strong magnetic field of the star and be deflected by it. Obviously, if the flux of cosmic rays on one side of the star is stronger than on the others, then the stronger flux will start to shift the star due to the magnetic pressure. The same effect takes place also due to the torsion field, which is especially strong in rapidly rotating neutron stars and it interacts even with neutral fast-moving particles, since it does not act on the moving charge but on the momentum of particles.

The fluxes of praons moving in the matter are influenced not only by the magnetic fields of nucleons, but also by the electric fields of uncompensated charges of individual protons and electrons. These fields also deflect the fluxes of praons without significant changes in the energy of praons, which is a consequence of potentiality of the electric field. Actually, if the fluxes of positively charged praons fly towards the proton, they are first decelerated by the electric field of the proton and decrease their energy, and then when they fly past the proton, they start accelerating from the proton under the influence of the same field and increase their energy up to the previous level. The gravitational force acts on the gravitons in a similar way.

With the help of the described mechanism the fluxes of gravitons and praons can create the gravitational and electromagnetic interactions in the matter of bodies without heating these bodies significantly. Passing through the matter the fluxes of gravitons and praons, consisting of a large number of tiny particles, act simultaneously on the nucleons, electrons and atomic nuclei, compressing them in the direction of the gradient of the corresponding flux and creating the gravitational and electromagnetic acceleration. The fact that the interaction cross-section  characterizes both the electromagnetic interaction of the fluxes of praons with nucleons and the strong gravitation from the fluxes of gravitons at the level of nucleons, and is equal by the order of magnitude to the cross-section of the nucleon, suggests that the interaction forces can actually emerge near the surface of nucleons. Here, the electric and gravitational fields, the magnetic field and the torsion field of nucleons reach the maximum and can effectively interact with the fluxes of praons and gravitons. As gravitons, leading to strong gravitation, in [5] we suggested graons as the particles that make up praons just as praons make up nucleons or as nucleons make up a neutron star. For graons in order to become gravitons they must be accelerated up to relativistic energies in the processes near the surface of praons.

Let us now consider the second problem of the Le Sage’s theory, according to which during motion excess pressure of gravitons and charged particles in front should emerge (drag effect), proportional to the velocity of the bodies’ motion. As a result of resistance to the bodies’ motion from the fluxes of gravitons, long-term rotation of planets around the Sun would not be possible and the principle of free inertial motion in the absence of forces would not hold true. Using cubical distribution of the charged particles of the vacuum field (9) and the fluence rate (11) of these particles, for the change of fluence and for the force, acting on a fixed charged cube with an edge  from one side, we find the following:

,

.                   (37)

The quantity  in (37) is equal by the order of magnitude to the relativistic momentum of one charged praon. While the cube is fixed in an isotropic reference frame, the fluxes of praons pass symmetrically through the cube from all sides and it remains motionless. Now suppose the cube is moving along the axis  from left to right so that one cube’s face is perpendicular to the axis . In case of the cube’s motion, the praons coming from its right side would have increased energy , besides the fluence amplitude  would increase due to the increased rate of occurrence of the praons incident on the cube. Since praons are ultrarelativistic particles, their momentum is proportional to the energy: , besides . These relations follow from the Lorentz transformations applied to the momentum and the rate of occurrence of particles.

On the other hand, we can expect that the cross-section of the praons’ interaction with the matter should be directly proportional to the square of the de Broglie wavelength, and inversely proportional to the square of the energy of praons: .

This dependence of cross-section in the quantum theory of elastic scattering is typical of ultrarelativistic photons – the greater their energy is, the weaker they interact with each other [16-17]. At the same time, we assume that photons are composed of praons, and the interaction of praons with the electromagnetic field of nucleons is a special case of photon’s interaction on virtual photons.

If ,  and  depend on the energy of praons, as described above, then the force in (37) remains unchanged both for a fixed and a moving cube. In (37) there is also a product , which is equal to the number of uncompensated charges inside the cube and is independent of the cube’s motion. The force  in (37) is the force, acting on the cube from one side, it is opposed by a similar force from the opposite side of the cube. Thus, the cube can move by inertia and the decelerating force from the fluxes of charged particles of the vacuum field, proportional to the velocity of motion, does not arise. This approach to the problem of pressure on the moving bodies from the fluxes of gravitons has been previously used in [2].

The problem of aberration in the Le Sage’s theory is illustrated by an example, in which in the motion of two gravitationally bound bodies near each other it seems that in view of the limited velocity of the gravitons’ motion, a certain additional force takes place. Indeed, while the gravitons moving from one body reach the second body, it will move in its orbit from the position that is dictated by the Newton’s theory of gravitation for instantaneous gravitation. As a result, the gravitons will reach the second body at some other angle to the orbit, which gives an additional force component. This problem has been considered in [4] for the case, when two bodies are moving synchronously in the direction perpendicular to the line connecting the bodies. In this case it was shown that the problem of aberration of the gravitational force disappears, if we apply the relations of special theory of relativity to gravitons, which take into account that the velocity of ultrarelativistic particles is not infinite and is almost equal to the speed of light. In both cases, for fixed and moving bodies, gravitons reach these bodies at the same angle with respect to the axes of the proper coordinate system.

The hypothetical problem of gravitational shielding in Le Sage’s theory suggests that if we place between two bodies the third body, it will lead to a more noticeable change in the forces between the bodies, than in case of the Newton's law of gravitation for three bodies. The measurements of the possible Moon’s shielding of the Sun’s gravitational influence on the Earth during Solar eclipses do not find any deviation from the theory within the limits of measurement error [18]. According to [4], this situation is due to the smallness of the cross-section  of gravitons’ interaction with the matter. This allows us to expand the exponents in the expressions for the forces into binomials with sufficiently high accuracy and ensures the principle of superposition of gravitational forces for several bodies. A noticeable deviation occurs only for such dense objects as white dwarfs and especially for neutron stars. According to (23), the cross-section  of praons’ interaction with the charged matter is 20 orders of magnitude greater than the cross-section  of gravitons’ interaction with the matter. However, the concentration  of uncompensated charges in the matter is usually many orders of magnitude less than the concentration  of nucleons. This allows us to expand the exponents in the expressions for the forces just as in the case of electromagnetic interaction of bodies by means of charged particles of the vacuum field, and thus to substantiate the principle of superposition of forces.

10. CONCLUSION

Based on the assumption that the electric force appears due to the action of the fluxes of charged particles that exist in the vacuum field, we derived an expression for the electric field strengths inside the ball (14) and outside it (7). These expressions are in good agreement with the formulas for the field strengths in electrostatics. From the field strengths we can easily proceed to the scalar potentials of the electric field, since the strength is up to a sign determined as the potential gradient.

Once we find the electric scalar potential, then with the help of a special procedure [19] we can find the 4-potential, the stress-energy tensor of the electromagnetic field, the electromagnetic field equations, the electromagnetic force, as well as the contribution of the electromagnetic field into the equation for the metric. This means that the electromagnetic field theory both in the flat Minkowski space and in the curved spacetime is fully proved at the substantial level through the charged particles fluxes of vacuum field. And the dependence of metric on the electromagnetic field potential allows us to take into account the influence of the inhomogeneous charged particles fluxes on the results of space-time experiments, based as a rule on the use of electromagnetic waves and devices.

In (19) and (22) we made an estimate of the energy density of the charged component of the vacuum  field, in (23) we presented the cross section of charged particles’ interaction with the matter, in (25) we estimated the rate of the energy flux of the charged particles in one direction. Based on the principles of the theory of infinite nesting of matter, the densest objects at each level of matter are assumed as the sources of the charged particles of vacuum field neutron stars and magnetars, nucleons and atoms, praons as the components that make up nucleons, etc. These objects emit neutrinos, photons and high-energy cosmic rays that can make contribution to the vacuum field at all levels of matter.

In the formula (27) we expressed the body charge in terms of the emission rate of those fluxes of charged particles of the vacuum field, which interacted with the body’s matter and transferred their momentum to it. Due to this interaction, the contribution was made by the charged component of the vacuum field into the mass as the measure of body’s inertia. The inertia of the body is manifested in its acceleration, when the balance changes between the falling on the body and outgoing energy fluxes of the vacuum field. We can distinguish in the vacuum field three components, one of which with the energy density  is associated with the strong gravitation and the rest energy of particles, determines the integrity of nucleons and atomic nuclei, and is mainly responsible for the inertia of bodies. Another component with the energy density  is responsible for the ordinary gravitation, and the third component in the form of charged particles with the energy density  leads to electromagnetism. The last two components make their own contribution to the mass of bodies.

We will also note the difference in how the origin of the electrical force is understood. In our approach, the fluxes of charged particles of the vacuum field are the source of electrical force, they exist as a necessary complement to the matter in the form of elementary particles and the bodies composed of them, are involved in the processes of gravitational clustering of the scattered matter, and are generated by the emission from the densest objects, such as praons, nucleons and neutron stars. It is precisely the fluxes of charged particles of the vacuum field that are the cause of the so-called displacement currents in the vacuum, which are proportional to the rate of change of the electric field with the time. Here, an example is the chargeable capacitor, between the plates of which there is a magnetic field, despite the absence of the electron current in the capacitor.

In electrostatics, the electric force is not explained. In quantum electrodynamics by means of selecting the Lagrangian of the field’s interaction with the matter the formula is derived that resembles the formula for the electric energy of the interaction between two charges in electrostatics [20]. As interpretation the pattern is suggested, in which the charged bodies exchange virtual photons with each other, which leads to the electrical interaction. Besides, here the uncertainty principle is used, limiting the lifetime of virtual photons. Due to virtuality, the photons are attributed very exotic properties, including the possibility of energy negativity or the presence of the momentum without energy. The photons’ energy is considered to be proportional to the Planck constant, and therefore the possibility of existence of photons and particles, belonging to the lower levels of matter and with another Planck constant, is not considered. The obvious disadvantage of this approach is the difficulty to explain the origin of virtual particles as such and their unique properties.

If we consider the fluxes of charged particles in the vacuum field as the source of the electric forces, it becomes possible to consider their scattering in the process of quantum transitions in atoms. In [6] the substantial model of electron in the form of a disk is considered, in which the charged matter rotates differentially, and ensures the magnetic moment of the electron. In addition, the electron spin is explained as the result of the shift of the disk’s center relative to the nucleus and rotation of this center in addition to the matter rotation in the electron cloud. If the electron transits into the quantum state with lower energy, it emits a photon, which carries with it the angular momentum that is proportional to the Dirac constant. In this process, the scattering of charged particles of the vacuum field on the electron disk, taking into account the action of the magnetic and electric fields in the wave zone, leads to the formation of a photon as an object preserving its structure for a long time.

In Section 8, we studied the model of the photon, emitted in atomic transition in the hydrogen atom. Associating the photon parameters and its structure with the parameters of the emitter – the charged electron disk, we managed to determine the charge to mass ratio for the particles that make up the photon. As a result, it turned out that photons consist of praons of very high energies, comparable to the energies that cosmic rays would have if these rays emerged at the nucleon level of matter near the protons. These relativistic praons must form the basis of the charged particles of the vacuum field. Indeed, in the interaction of praons of the vacuum field with the electron in atomic transition, the twisting of praons takes place under action of the fields along the axis of the electron disk, and the appearing photon carries away the excess angular momentum of the electron from the atom. Meanwhile, part of praons of the vacuum field is part of the photon, so that the speed of the photon actually is the speed of praons in the fluxes of particles of the vacuum field.

In contrast to the chaotic motion of the praons in the vacuum field, the praons in the photon are rigidly bound to each other by both electromagnetic and gravitational forces. The situation here is similar to the situation with the nucleons, which only in special circumstances can form extremely stable formations – the atomic nuclei. According to the gravitational model of strong interaction [6], the nucleons in atomic nuclei are attracted to each other by strong gravitation and repel each other by means of the torsion field, arising from the rapid rotation of the nucleons. In order to form the nucleus, the nucleons must interact with each other only in a strictly defined orientation of the spins and magnetic moments and must have sufficient initial energy that allows rotating the nucleons up to the desired rotation speed by means of gravitational induction. The praons in the photon can interact with each other in a similar way. We can even calculate the gravitational constant for the praon level of matter with the help of the coefficients of similarity from Section 5 and the strong gravitational constant m3·kg-1·s-2 in the following way, using the theory of dimensions and SPФ symmetry, according to [2]: m3·kg-1·s-2 .

In the gravitational field with this large gravitational constant, the praons of the photon can form sufficiently rigid structure, so that the photon could fly large cosmic distances without decaying.

In Section 5, for the ratio of the absolute value of energy in the field of strong gravitation to the energy of electric field of the proton we found: . Аналогичное равенство следует и для праона, для чего необходимо постоянную сильной гравитации  заменить на постоянную гравитации для праонного уровня материи  и подставить массу и заряд праона из раздела 6:  .

Concurrent consideration of the evolution of objects at different levels of matter, such as the level of praons, nucleons and neutron stars, allows us to draw conclusions not only as to the origin of gravitational and electromagnetic forces. For example, if for a neutron star with the mass  Solar mass and the stellar radius km we calculate the average binding energy per nucleon, we will obtain J or 47 MeV per nucleon,

which is greater than the binding energy of atomic nuclei. Taking into account that neutron stars are born in supernova explosions, when the explosion energy is carried away by neutrinos and emission, and is converted into the kinetic energy of the discharged shell, a significant part of the binding energy is emitted from the star and transferred into the environment. In [13], we estimated that 61% of all praons are part of nucleons, and the rest 39% form new particles – nuons (which are structurally the analogues of white dwarfs at the level of elementary particles) or exist separately. The same proportion remains at the level of stars: 61% of all nucleons over time will be part of neutron stars, and the rest of nucleons remain either as a gas or as the matter of white dwarfs. Nuons as the analogues of white dwarfs, due to their significant presence in cosmic space, can ensure the red shift effect in the spectra of distant galaxies, explain the background radiation and the dark matter, etc.

Consequently, the concentration of free protons in the visible Universe must be of the same order as the averaged over the entire space concentration of nucleons in stars, that is of the order of concentration of baryons m-3, according to the findings of the Lambda-Cold Dark Model [21]. With this in mind, the product of the concentration of baryons and the binding energy of a neutron star in the calculation per nucleon will give us the estimate of the maximum energy density of emission in cosmic space: J/m3. Indeed, the energy density in the relic radiation equals J/m3, and the energy density in the stellar radiation, magnetic fields and cosmic rays is of the same order of magnitude, as well as the kinetic energy of the motion of gas particles. The sum of these energy densities does not exceed the maximum energy density .

In conclusion, we will estimate the length of free path of the charged particles of the vacuum field in the cosmic space, taking as the charge concentration in a first approximation the value  of the elementary charge per cubic meter, which is equal to the average concentration of baryons in the Universe. This approach gives only the minimum value of the free path length, since on the average the matter in the Universe is neutral, and  must reflects the average concentration of the total charge of the Universe. From the ratio  at a given concentration of charges and the value  according to (23), we find the free path length of charged particles: m. This value is 3 orders of magnitude greater than the visible size of the Universe, which is estimated by the value of 14 billion parsecs or m. Consequently, the charged particles can easily reach our Universe from a distance, where they can be produced in a concentration sufficient to meet the required energy density. We do not support the model of the Big Bang, which limits the lifetime of the Universe to the value of 13.8 billion years, explaining in a different way the phenomena associated with this model [13]. Then the charged particles of the vacuum field can have enough time to get into our Universe from the outside and reach here the equilibrium concentration with the value m-3.

In the last section, we considered the main objections to the Le Sage’s theory and presented our arguments supporting this theory. This was possible due to taking into account the special theory of relativity and the peculiarities of interaction of gravitons and charged particles of the vacuum field with the matter. In our opinion, the modernized Le Sage’s model most naturally explains the origin of gravitational and electromagnetic forces, and the theory of infinite nesting of matter explains the evolutionary origin of gravitons and charged particles of the vacuum field. In addition, we can assume that the role of gravitons of ordinary gravitation is played by charged praons, and the role of gravitons of strong gravitation is played by charged graons. For neutral bodies the action of fluxes of these particles leads to gravitational forces, while in charged bodies there are also electromagnetic forces.

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