The current orthodox Physics assumes in general terms: 1-That background space of physical phenomena, it is a three-dimensional euclidian space. This means that physics space is considered a uniform and flat three-dimensional space in the one which to each position determined by a set of three coordinates corresponds a single physical point. 2-That electrons and positrons are charged pointlike particles. 3-That electromagnetic interaction and gravitation are two phenomena that are not related one each other at all.

In our geometric theory of fundamental entities these suppositions are revised and it is speculated: 1-That the structure of space of physical reality could be that of a "multiple space" in the one which to each position would correspond more than one " physical point". This space would be inspired in spatial sections of a Kerr complete spacetime. 2- That nude electrons and positrons are something like "miniblack holes" of Kerr. 3-That electromagnetism is a consequence of exchange, (by effect of the gravitation) of actually elementary particles that surround nude particles that are modelized by miniblack-holes.

The Kerr metrics corresponds to a family of exact solutions of the equations of Einstein for empty field that describes black holes with mass and with angular momentum. In the section 2 of this article are described the peculiarities of this solution given to know in 1963, that is to say much time after of development of quantum mechanics.

The Kerr metrics discovers a complex spacetime that can not be covered with a single coordinate system as the customarily used to cover flat spacetime. Since they are needed multiple coordinate systems to cover complete Kerr spacetime is said that it consists of multiple universes. In section 3 of this article various versions of complete Kerr spacetime are described and possible relationship to the structure of physical space and also to a model of nude electrons and positrons is remarked.

In our model of charged particles these are presented as miniblack holes of Kerr. They are suppousedly orbited for certain type of particles whose speed tend to c, (PF, see article 1). To this end possible paths of massless particles around Kerr black holes are described in section 4

Finally they are presented in section 5 of this article hypotheses that sustain to the geometric electron-positron models as well as a possible gravitational mechanism of electromagnetic interactions.

In 1963, R.P. Kerr presented an exact dependent of two parameters (m and t) solution to the Einsteins’s equations. Those parameters can be interpreted as the mass and as the specific angular momentum (a=S/m; S=spin) of a rotating black hole.

R.H. Boyer and R.W. Lindquist, [1] studied complete spacetime described by the metrics of Kerr. Also they expressed this metrics in some well adapted and suitable for an intuitive interpretation of gravitational field, coordinate system. These coordinates of Boyer and Lindquist, (BL), are nothing but spheroidal coordinates adapted to the curved space. Their relationship to the cartesian coordinates can be deduced from the following equations:

It can be noted that r=cte surfaces are ellipsoid and also that for a=0 BL coordinates coincide with the spherical ones.

The expression of the Kerr metric in BL coordinates is as follows:

Taking into account previous invariances and accepting possible existence of negative masses and negative universes, they could be considered four inequivalent situations defined by the same metric according to the signs of: {a,m,r} . These situations would be corresponding to the following sets of signs: 1 {+,+,+}; 2 {+,-,-}; 3 {-,+,+}; 4 {-,-,-}.

  To solutions: r=r₊ y r=r_-, correspond two singular closed surfaces that can be avoided with convenient coordinate changes. These "pseudosingular" surfaces are typical of black holes and are usually named horizons of the spacetime.

Fit to consider 3 special cases:

1 When a=0 metric (2) coincides with Swarzschild metric, the singularity is reduced to a point, and exists a single horizon located at r=2m.

2 When a=m the two horizons are superpossed at r=m.

Pseudosingularity reveal that with a single BL system can not be labeled all the events of a complete Kerr spacetime. The different possibilities of complete Kerr spacetimes that we consider interesting in order to constrct our model, are the object of the next section.

To deduce all the regions that exist in the complete Kerr spacetime and to obtain coordinate patches to cover these regions is the objective of analytical prolongation of the Kerr metric. Results and procedures of this analytical prolongation can be found in detail in references [1][2][3], therefore here we only expose a summary of the more relevant results.

Supposing a=m it does not exist region {1} since in this case r₊=r_-=m ; (see figure 1 and figure 2).

In the same way as is extended the region {0} the regions: {1} {2} {-1} {-2} can be extended obtaining new regions of same types which also can be extended. The final result consists of a succession of infinite regions as those described in the previous paragraph. These regions are usually represented by Penrose’s diagrams where is accomplished a kind of conformal compression of the spacetime in order to represent the farthest reaches of the spacetime. These spacetime diagrams preserve the property of rdrawing universe lines of photons inclined to 45 or 135 degrees. The horizont lines, (represented in gray ) indicate that horizonts are surfaces generated by two families of photonic paths that are usually named "principal null congruences".

In figure 1 (spanish version) diagrams corresponding to the case 0<to<m. are represented.

The figure [1.a] shows a complete Kerr spacetime diagram consisting of a indefinite repetition of the four square region {0}, {1*=1}, {0*}, {-1*=-1} and the four square region {2*}, {3*=3}, {2}, {1*=1} overlapped to the previous by the common region {1*=1}.

The square {0*} represents another positive external region that united to the late make a bridge which is similar to the Einstein-Rosen bridge of Schwarzschild metric.

Nothing prevents to speculate that in physical reality, a certain number of exterior universes such as those just described are actually superposed. In this model each point of space would be made of a duperposition of certain number of points each one of them belonging to a different universe. In the case of the minimal complete Kerr spacetime, (0<a<m) there would be four of these points: two of they belonging to positive leaves of the space and two of they belonging to negative leaves. The four leaves of the space would be disconnected except in the center of miniblack holes (particles) where they are connected through the interior regions.

In figures [2a] and [2b] are represented spacetime diagrams of Kerr holes having a=±m. Cited figures should be supposed limitless upward and downward. Versions of complete spacetime would be constituted then by these infinite positive and negative succession of universes that result of the successive analytical prolongation of the initial universe. Also they would be complete spacetimes that result of identifications of a universe with other subsequent or previous universes. In the figure these identifications has been supposed in order to form 2 versions of complete Kerr spacetime having four universes: 1.- {{0=8}, {2}, {4}, {6}} and 2.- {{0*=8*}, {2*}, {4*}, 6*}}. Each one of these versions would correspond to two spacetimes endowed with opposite values of angular momentum. We consider that each one of these two versions of complete spacetime could be suitable for spin up and spin down electron models respectively.

Finally it remains the case of a Kerr black hole with a>m, that correspond to what is designated with the name of "nude singularity". In this case they do not exist horizons and complete spacetime consists of only two universes, a positive one and a negative one, separated by a ring singularity. The spacetime diagram is represented it in figure [2c].

Until here the description of some versions of complete Kerr spacetime. All have in common the existence of negative and positive exterior universes. What is suggested by all they is that space of physics could be multiple. This mean that in a point of the space in reality would be superposed a multiplicity of points belonging to various positive and negative universes. On the other hand the existence of horizonts and innocuous singularities (that is to say non-destructive), suggests that central part of black holes would be an interconnection zone between different spaces. What is intended is to build a model where the particles would be mini Kerr black holes that locally connect these superposed spaces.

The reality of the versions of spacetime obtained by the various identification of universes has been rejected in the astrophysics realm for motives related to causality principle: In fact in these spacetimes can occur closed timelike universe lines that would suppose a violation of the cited principle, Carter (1968). Nevertheless there is no no test of the fact that the causality principle in its most restrictive form has to be taken into account in the area of elementary particle physics.

The issue that remains is to decide which among available versions is in principle best suited for construction of a geometric model of particles. We delay this topic until section 5.

The gravitational field of a Kerr black hole, encloses many surprises. To the already indicated in previous sections one must to add the repulsive character of the singularities at short distances while at greater distances they are attractive. Other interesting characteristic is the fact that in the gravitational field around singularities exist infinite possible closed orbits of null particles (particles whose speed is c). All this can be noticed by the study of the geodesics of the metric.

The orbits and paths of photons and neutrinos in the Kerr spacetime has been systematically investigated by many authors with the help of the first deduced by B. Carter (1968), [3] geodesic equations. Here they will be qualitatively described the features of the movement of null particles, (NP) approaching to a singularity of a Kerr hole from an exterior universe.

For L values greater than the corresponding to the exterior orbit, the NP will escape toward the exterior universe, but for smaller impact parameters to the corresponding to the interior orbit several things can occur:

If the impact parameter is zero or very next to zero values the NP will cross the ring singularity and entered the negative space, where will reach values of negative r and will feel a negative, that is to say repulsive, gravity. That will make the null particle be scattered from the singularity.

When the impact parameter is the adequate so that the particle is approximated to the singularity by the exterior part of the ring there is a balance between the attraction and the repulsion and as a result the NP can acquire circular orbits that are more stable that the considerate up until now. These stable orbits are interesting because they make the proximities of the singular ring a kind of null particles store.

Is yet possible others circular orbits but running in the negative space. They are those which acquire particles within a certain range of impact parameters corresponding to paths approaching the ring nearby it interior part. These orbits are called pendular since the path of the particle elapses in a ellipsoidal surface in the negative space bouncing between different points of the singularity.

With respect to the particles approaching to the black hole from negative space, only those going in axial or nearly axial paths, will get the positive universe. For other particles the singularity is repulsive and will make them bounce back toward the spatial infinite of the negative universe. Tend to be said that seen from a negative universe, the Kerr black hole is an antigravity source.

Until here a qualitative summary of possible null particle paths in the proximities of a Kerr singularity where is supposed that the particles do not disturb considerably to the gravitational field.

In order to imagine what would occur in a "mini complete Kerr spacetime" as that of our model there would be taking in accoun all mutiplicity of universes and singularities that it comprises and study all the possible closed universe lines. Also there would be interesting to consider and to study the case in the one which the orbiting particles provoke a considerable disturbance of the metric.

To give verisimilitude to a geometric model of electrons and positrons, as well as to speculate with a gravitational mechanism of electromagnetism, it is necessary to introduce some ideas or additional hypothesis:

Provided the mass be positive, both negative and positive values of the angular momentum do not present any interpretation problem (neither do the assignment of appropriate for them spacetimes). They supposedly correspond to the electron which would have signs: {m=+,a=+} and {m=+,a= -}, each one related to the two possible values of the angular momentum The corresponding spacetime diagrams for electrons are represented in figures [3a] and [3b].

The external field of a rotating negative mass would be multiple, likewise as that created by a rotating positive mass. The curvatures of the different leaves of the exterior space of a negative mass black hole would be equal to those of a positive mass black hole. This cause that both fields would be indistinguishables at certain distance of the horizons. Nevertheless the absorption or emission of actually elementary particles, would proceed in opposed way in a negative mass field than in a posive one. We recall that the particles "actually elementary" of our geometric model, the PF, are supposed local disturbances each of them evolving on their proper leaf of the space.

The spacetime diagrams related to these negative masses are presented in the figures [3c] and [3d]. Each one of they corresponds, as in the case of [3a] and [3b] figures, to each one of the two possible signs of the angular momentum.

We believe that the introduction of gravitational fields of negative mass provide a convenient symmetry to our model that not contradicts to the gravitation Einstein theory. Furthermore we believe that the idea is attractive for aesthetics and philosophical reasons

In order to complete the symmetrization of the gravity theory we consider the possibility of test particles of positive and negative mass. The convenience of this concept will be put of manifest in our based on gravitation explanation of electromagnetic interactions. The elemental particles of our geometric model, (the prephotons , PF) can have positive or negative (but infinitesimal) mass. The essential peculiarity of the negative mas particles is that during their movement the speed vector and the linear momentum vector have opposite senses. This property would characterize to these negative mass particles whose behavior on impacts with other particles would be the opposite to which one would expect on the impacts with positive mass particles.

With the cited symetrization of gravitational sources and test particles one can built models of electromagnetic atractions and repulsions that are based on the exchange of PF. This exchange will be between Kerr black holes of equal signs for the case of repulsions and of different sign in the event of attractions. The mechanism would be as follows:

The proximity of two orbited by PF Kerr miniblack holes corresponds in our geometric model to the proximity of two charged particles. The mutual disturbance of the spacetime due to this proximity would provoke that both black holes emit and absorb some PF of the residents in their respective stable orbits that surround to their singular rings. Concretely in the mechanism that we forward expose, (that it is not the sole imaginable), a signed + hole (modelizing to the electron) would emit double number of positive PF that of negative PF. In a symmetrical way a signed - hole (positrón) would emit double number of negative corpuscular objects that of positive corpuscular objects.

Figure [4] (spanish version) represent the attraction between a positron and an electron.

The ellipse on the left part, with a sign + in it center, corresponds to the electron. The ellipse on the right part, with a sign – in it the center corresponds to the positron. Concretely what these ellipses intend to represent are the stable orbits in the proximities of the singular ring of a spacetime of positive mass and of a spacetime of negative mass. Each one of these particles emits three PF that in the figure are represented with numbered circles that surround to the corresponding sign. The reaction of the charged particles to the emission or absorption of the PFs is represented by the arrows that indicate the reaction movements. It can be appreciated that this reaction is opposed to the customary when the signs of both the charged particles and the PFs are different.

The PF number 1 is a positive truly elementary particle emitted by the electron and elastic dispersed by the positron. The effect that produces on the charged particles is a displacement of both toward the left that we represent by the two arrows pointing out that direction.

PF 2 also is a positive mass elementary particle and it is emitted from the electron and absorbed by the positron. The effect of these processes is a displacement of both charged particles toward the left just as the produced by PF 1.

PF 3 is negative, imitted by the electron and crosses the singular ring of the positron. Its effect is only felt on the electron and consists of a displacement toward the right of this electron.

The PF number 4 is negative and it is emitted by the positron being dispersed by the electron. The net effect is opposed to that of PF 1: That is to say the displacement of the electron and the positron toward the right. This effect compensates to that produced by the emission of PF 1.

PF 5 is a mass negative one, that is emitted by the positron reaching a stable orbit around the singular ring of the electron. Its effect is equal to that produced by PF 4, that is to say a both particles displacement toward the right, that is to say that is a compensating opposed effect to that produced by PF 2.

PF 6 is a positive one emitted by the singular ring of the positron that crosses the singular ring of the electron. Its effect is a displacement of the positron toward the left.

In short: the effects of PFs 1 and 2 are compensated with those of 4 and 5. The net effect is produced by PF 3 and consist on a displacement of the electron toward the right and a displacement of the positron toward the left. That is to say an attraction of these two particles.

To obtain the conservation of the number of PF in the orbits around each ring, it is necessary that two of the residents in the unstable orbits PF (the fotonic spheres) of each particle goes to the respective ring orbits. This transfer are represented in the figures by the corresponding signed circumferences. In the referred transfers no net displacement of the particles are produced since it deal with internal momentum transfer.

The electromagnetic repulsion mechanism is represented in the figure 4b. In that we represent two electrons. As will be supposed the repulsion mechanism between two positrons would be similar to that of the electrons we explain next:

PF 1 is a positive one, emitted by the electron on the left and dispersed by that on the right. The effect consists of some displacement on opposite sense of both particles that is to say tending to separate them.

PF 2 is a negative one, emitted by the electron on the left and absorbed by that on the right. The effect would be a both particles approximation that would compensate the separation due to the PF 1.

PF 3 is a positive one, emitted by the electron on the left that crosses the singular ring of the other electron. Its effect would be only felt by the electron on the left and consists of a displacement of this electron toward the left.

The net effect of PF 1, 2 and 3 would be this displacement toward the left.

In a symmetrical way the net effect of PF 4, 5 and 6 would be a displacement of the electron on the right toward the right. The total net effect of these two displacement would be a repulsion of the two electrons.

References:

[1] Boyer, R.H., Lindquist, R.W. (1967). J. Math. Phys., 8,265.

[2] Carter, B. (1966). Phys. Rev., 141,1242.

[3] Carter, B. (1968). Phys. Rev., 174,1559.