Gravitational Space-Time Quantization for Charged Wormholes and the Diophantine Uncertainty Relation

This research work proceeds from the assumption, which was still considered by Einstein, that the quantization of gravity does not require additional external procedures: quantum phenomena can be a consequence of the properties of the universal gravitational interaction, which maps any physical field upon the space-time geometry. Therefore, an attempt is made in this research work to reduce the quantization of physical fields in GRT to the space-time quantization. Three reasons for quantum phenomena are considered: Partition of space-time into a set of unconnected Novikov’s R- and T-domains impenetrable for light paths; the set is generated by the invariance of Einstein’s equations with respect to dual mappings; The existence of electric charge quanta of wormholes, which geometrically describe elementary particles in GRT. This gives rise to a discrete spectrum of their physical and geometric parameters governed by Diophantine equations. It is shown that the fundamental constants (electric charge, rest masses of an electron and a proton) are interconnected arithmetically; The existence of the so-called Diophantine catastrophe, when fluctuations in the values of physical constants tending to zero lead to fluctuations in the number of electric charges and the number of nucleons at the wormhole throats, which tend to infinity, so that the product of the increments of these numbers by the increment of physical constants forms a relation equivalent to the uncertainty relation in quantum mechanics. This suggests that space-time cannot but fluctuate, and, moreover, its fluctuations are bounded from below, so that all processes become chaotic, and the observables become averaged over this chaos.


Statement of the Problem
A natural approach to the problem of quantizing gravity, to which Einstein (the creator of the general theory of relativity) inclined, is considered: quantum phenomena should be a consequence of the gravitational field properties. The logic is as follows: GRT (general theory of relativity) is a geometrizing model. Any physical field must be mapped onto the geometry of a curved space-time, the curvature of which, for its part, is equivalent to a universal gravitational field [1] 1 . It is enough to solve the Einstein equations linking physics with geometry, together with Maxwell's equations for an electromagnetic field, to express all physical quantities characterizing, say, electromagnetic interaction, through the metric of pseudo-Riemannian space.
Moreover, the arbitrary functions of the radial coordinate arising in this case in the Cauchy problem for the initial conditions (the first integrals) describe the geometric images of the sources of the electromagnetic field, i.e. electric charges [2], which were not present in the formulation of the problem and which, therefore, are formed by the gravitational field itself. That is, the sources of the field are formed by space-time itself. As it turned out, they have a non-trivial topology of pulsating wormholes with non-closing static throats [2]. We got the opportunity to investigate the internal structure of elementary particles by eliminating in this way the main singularity in the flat space of the special theory of relativity, i.e. the Coulomb divergence of a point charge field.
That is, GRT can really reduce the quantization of a physical field to the quantization of the multidimensional pseudo-Riemannian space-time. Quantization of space-time means the existence of two phenomena: discretization of 4-dimensional pseudo-Riemannian space, i.e. dividing it into a set of unrelated domains, and the impossibility of accurately determining its geometric characteristics, which acquire a fluctuating character and, by analogy with quantum mechanics, are linked by the gravitational uncertainty relation. This is what we should get.
The results of the work are presented in three sections and in Appendix 1, which provides a proof for the complete geometrization of an electromagnetic spherically symmetric field consisting of a free radial electric field and dust-like neutral matter in a synchronous accompanying frame of reference. According to the exact particular solution of the Einstein and Maxwell equations [2], it is a periodic in space and time chain of pulsating wormholes with static throats (holes, necks) of extreme dimensions geometrically, which are geometric images of electric charges formed by the electromagnetic (and hence gravitational) field in curved space-time ( Figure 1).
The second section describes a fundamental property of centrally symmetric Einstein's equations, i.e. their invariance in relation to dual mappings the mutual 1 Recall that this relativistic principle of equivalence is well known in Newtonian nonrelativistic physics and is very accurately carried out experimentally in the form of equality of the inertial mass of a body (an analogue of matter) and its gravitational mass (an analogue of a gravitational field) in Newton's second law, as a result of which all bodies in a vacuum being in a gravitation field falls with the same acceleration. Journal of Modern Physics Figure 1. One period of a wormhole with two electric charges ±Q-necks (throats) of radius R h that exit into two parallel static Reissner-Nordstrom vacuum worlds, in the state of maximum expansion (R max ) and in the state of maximum compression (R min ). The arrows show the direction of the radial electric field. replacement of the time coordinate with the spatial radial coordinate. This leads to the existence of one more exact particular solution of the Einstein-Maxwell equations, dual conjugated to the given one, and to the emergence of a discrete set of so-called Novikov's R-and T-domains [3], which are closed and unconnected with each other, and which boundaries "B" are impenetrable for light paths (being isotropic geodesic lines) [4]. This is the first effect of natural "self-quantization" of the gravitational field.
As it will be illustrated in this section, this leads to the fact that the flux of photons continuously emitted by the source will come to an observer located at another point in space in discrete portions, i.e. quanta. This is the second effect of "self-quantization" of the curved space-time.
It was shown in [2] that the gravitational-electromagnetic interaction is sufficient to construct the simplest classical model uniting an electron, proton and neutron, with eliminating the main contradictory prohibitions, and to calculate their radii of curvature at unclosed throats of wormholes that go out into two parallel vacuum spaces. The radii of the proton and neutron are in good agreement with their experimental values (for a proton, with an accuracy of 0.04%).
An electron, which in the flat space regarding to the special theory of relativity (SRT) would have to be single-point, i.e. structureless, also turns out to be a structural object in this model, having a finite radius of curvature of the throat much larger than that of a proton. Note that the very fact that the proton turned out to be not a single-point structured object is direct experimental evidence that the space-time in the microworld is also curved. That is, the gravitational inte-Journal of Modern Physics raction is universal.
In the third section of this research work, it will be shown that, according to this exact solution of the Einstein's and Maxwell's equations, the relationship between the physical and geometric parameters of particles at the throat is described by a Diophantine equation (an integer variable equation with rational coefficients). This requirement of the rationality of the coefficients inevitably leads to the fact that an arithmetic relationship arises between the rest mass, electric charge and the radius of the Gaussian curvature of the throat, which has the discrete spectra nature. That is, the physical characteristics (masses and charges) are quantized. nucleons at the throat is not less than a certain value, that is, it is limited from below. This means that fluctuations of such quantities cannot but exist; they will be maintained by the gravitating system itself, since as the increments of physical constants tend to zero, the increment in the number of charges and the number of nucleons at the throat would tend to infinity, which is impossible.

Dual Spaces. Discretization of Novikov's R-and T-Domains
Einstein's equations in GRT: in 4-dimensional pseudo-Riemannian space in the coordinate system in a set of arbitrary metrics that transform into each other, with a non-degenerate Jacobian in a class of metrics with a Lorentzian signature within the framework of centrally symmetric spaces with orthogonalized metric which transform into themselves at the following dual mapping: , with this mapping, the signature is preserved; the time coordinate surfaces "B" (boundary), on which the Jacobian (4) and the second-order determinant (7) degenerate: can be completely covered by a family of geodesic lines [5].
Thus, using the invariance of the GRT equations with respect to dual maps, we have a way to generate new exact partial solutions of the Einstein equations that describe space-time more fully.
Let's give a simple illustrative example. Let's take the well-known Schwarzschild solution for a gravitational field of a point mass m [1]: One can make sure quite simply by substituting solution (14) into Einstein's Equation (1) [1] that we have obtained a new solution missing for the domain g r r < , in which the radial coordinate r has turned into a time coordinate τ , and time τ has become a radial coordinate r (Figure 2, left side).
Part of a boundary hypersurface "B" is a surface in T-domain ( Figure 2, left side).
Since the time and the radial coordinate that create the "B" boundary are different characteristics, then R-and T-domains are not related in any way. Therefore, no signal can cross the "B" boundary from the R domain to the T domain and back.
We construct a family of light paths for this example, Substituting (15) in (13) and (14), we obtain the expressions for the coordinate velocity of light R l β " " and T l β " " which is tangential for isotropic geodesic lines: Figure 2. Parts of "R" and "T" Novikov regions in the Schwarzschild point mass field and light trajectories. In parentheses are the coordinate axes in the "T" area. "B" is the boundary between these regions-the event horizon, impenetrable for light trajectories. In the "T" region, the time coordinate τ and the radial coordinate r are reversed.  [9] have not yet been successful, both analytically and numerically due to mathematical difficulties. Therefore, we have depicted in Figure 3 only a qualitative picture of this family for semi-closed solutions, which clearly illustrates this phenomenon of curved space-time discretization.
If some observer resting in the "R"-domain continuously emits light rays, then observers in the "T"-domains will not see these signals at all, and observers at other points of the "R"-domains will see intermittent flux of light quanta. That is, the quantization of light turns out to be a property of curved space-time.
We also note that the integral gravitational action g S for each of these discrete domains of space-time is finite [1]: where  is a trace of the Ricci tensor,  where  is Planck's constant. Thus, the fundamental constant of quantum mechanics can also have a gravitational origin.
The results obtained in GRT allow us to take a fresh look at the event cone known in SRT (Figure 4(a)). This event cone consisting of two timelike do-

Discrete Diophantine Spectra of Wormhole Parameters and Relationship of Fundamental Constants
Consider the simplest centrally symmetric system consisting of a dusty substance (without pressure and temperature) and a free radial electric field (without sources) in an accompanying frame of reference. When integrating the Einstein and Maxwell equations in the Cauchy problem, three first integrals arise, being three arbitrary functions of the radial coordinate r, the setting of which is equivalent to setting the initial conditions for the dust energy density s ε , the radius of the 2-dimensional Gaussian curvature R and the rate of its change R ⋅ : the electric charge Q, the total energy of the gravitational field g ε and the function f characterizing the curvature of space inside the wormhole and at its throats, through which it can be extended into two parallel external Reissner-Nordström vacuum spaces [5].
The GRT equations can be integrated to the end in three cases when one of the first integrals becomes a constant [2] [10]. We will consider the simplest solution, in which the electric charge becomes a constant: where e is a fundamental charge, it is an irrational number for our world and N Q is an integer number of such charges.
If we take into account that we initially considered a free electric field without sources, then the appearance of a new physical constant ±Q during integration means that space-time itself has formed a source of the electromagnetic field, i.e. π is a gravitational radius proportional to the total gravitational energy ( ) is the critical radius proportional to the electric charge Q of the particle, which is always paired with its antiparticle, the throat of which can be extended into a parallel vacuum space; is the charge to the mass ratio of the inner world, Thus, we can say that e is a geometric quantum of electric charge, i.e. a topological characteristic of a given particle. Along with it, we can introduce a quantum of mass, the critical mass c m that differs from the charge only by a constant: and the quantum of length, which is a critical radius c r : where the classical (electromagnetic) and gravitational radius are compared in magnitude.
Taking into account (24), Equation (22) where e N is the number of electrons at the throat of the wormhole, taking into account that can be rewritten in the form of a Diophantine equation: where p N and Q N are integer variables; λ is a rational coefficient equal to: We will not solve arithmetic relations (31) and (32) here. We will make only a qualitative conclusion: first, the masses of the electron e m , quanton c m , and proton p m are related arithmetically, i.e. depend on each other. Second, they have discrete spectra of values, outside of which the conditions for the existence of a stationary throat are not satisfied. Therefore, the considered gravitational system (wormhole) will be nonequilibrium, which means that it will fluctuate.
This statement obtained within the framework of this gravitational model obviously can be verified experimentally. That is, the nature of quantum fluctuations is non-fulfilment of arithmetic equilibrium conditions.
We present below the procedure for solution of the Diophantine equation (30).
We represent the rational coefficient λ as the sum of an integer Λ and a proper Here is the simplest illustrative example of this arithmetic: This is a Diophantine catastrophe. We will discuss it below.
The same qualitative disturbances will be expected for the spectrum of the necessary throat radii:

Fluctuations of Fundamental Constants, Wormhole Parameters and Diophantine Uncertainty Relation
What is the cause of a Diophantine catastrophe? The fluctuations of physical constants (electric charge, rest masses of elementary particles) tending to zero cause an infinitely large required increment in the number of charges and the number of nucleons to ensure a stationary state of wormhole throats. Since such numbers are not found, the gravitational system begins to fluctuate.
The perturbed system is described by a perturbed Diophantine equation: in which the perturbed rational coefficient can be represented as: and we write the fluctuation δλ λ  itself in the form: Consequently, when the number of fundamental charges is bounded from above, the fluctuations of the physical constants are bounded from below. This means that the gravitational system of elementary particles cannot but fluctuate.

Conclusions
The invariance of Einstein's equations in GRT with respect to dual mappings is considered. First, this symmetry makes it possible to obtain new exact partial solutions of the Einstein equations that complement the obtained ones to a more complete description of space-time. Secondly, this symmetry leads to discretization of space-time, or its division into a set of unconnected Novikov's R-and T-domains impenetrable for light paths. This is the first effect of "self-quantization" of the gravitational field. This, in turn, leads to discretization of the flux of photons continuously emitted by an emitter in separate portions, or quanta. This is the second self-quantization effect of a curved space-time.
It is shown that the relationship between physical and geometric parameters at the wormhole throats, which are exact particular solutions of the Einstein-Maxwell equations, and which describe the internal structure of elementary charged particles, is described by a Diophantine equation in integer variables with a rational coefficient. This leads to the appearance of discrete spectra of physical and geometric parameters of wormholes. This is the third effect of "self-quantization" of the gravitational field.  However, using the permutation of the second partial derivatives and, substituting the partial derivatives 0 A′ and 0 A ⋅ from (A5) and (A6) in (A8), it is possible to obtain from (A9) the explicit required expression for the scalar potential 0 A in terms of the geometry: