Strain wave structure of electron and positron

Abstract

By treating physical space as an elastic continuum, we show that elementary particles and their fields can be represented by stress–strain wave packets in the elastic space continuum. Dynamic equilibrium equations of elasticity reduce to vector wave equations involving displacement vectors in this continuum. We derive the equivalence between displacement vector U and the magnetic vector potential A to show that electromagnetic fields are manifestations of stress–strain fields in this continuum. The structure of the electron is modeled on a spherically symmetrical strain wave solution of the vector wave equilibrium equation. The solution consists of a central standing strain wave core approximately 2 fm in radius, surrounded by a radially decaying field of phase waves propagating outward for the positron and inwards for the electron. Approximately 37.3% of the energy of the electron is contained in its wave field and the remaining in the central standing wave core. We also derive the Coulomb interaction between two electrons and verify Coulomb’s law of electrostatics. The intrinsic electrostatic field, intrinsic spin, and magnetic field effects of the electron are derived from its strain wave structure. We also verify the Biot–Savart law for motion-induced magnetic fields and indicate the origin of De Broglie matter waves.

1 Introduction

1.1 Representation of physical space as an elastic continuum

French physicist Louis Malus discovered the phenomena of light polarization as early as 1808. To explain the phenomenon of polarization, light waves were required to be transverse, as demonstrated by Thomas Young and A.J. Fresnel. In 1821, Fresnel, to explain the polarization of light, proposed a model of ether as an elastic medium, which could transmit transverse waves. This luminiferous ether theory inspired much subsequent research in the 19th century, leading to numerous dynamic theories of elastic solid ether proposed and developed by prominent scientists such as George Gabriel Stokes, Augustin-Louis Cauchy, George Green, Bernhard Riemann, and William Thomson [1]. Cauchy contributed significantly to the theory of elasticity and continuum mechanics; he was a key figure in developing the elastic continuum theory of the luminiferous ether, which posited that light propagated as an elastic shear wave. Regrettably, Cauchy’s elastic ether model was ultimately abandoned by later researchers because it contradicted our general perception of the motion of matter particles through an elastic medium.

With the introduction of Maxwell’s field equations in 1864, which explained a wide range of electromagnetic phenomena, a mechanical worldview gradually provided way to an electromagnetic one. Consequently, earlier concepts such as the luminiferous ether or an elastic solid ether were replaced by the idea of an electromagnetic ether—more precisely, the electromagnetic field. Through Maxwell’s development of the electromagnetic theory of light, the electromagnetic field was granted an independent status, capable of existing on its own, much like matter.

However, the independent status of the electromagnetic field was not enough in itself: the dimensional properties of permittivity ‘ε₀’ and permeability ‘μ₀’ had to be ascribed to empty space—that is, “nothingness”. The propagation of independent electromagnetic field waves at velocity c through empty space depended on the characteristic parameters ascribed by ε₀ and μ₀. By associating the characteristic parameters ε₀ and μ₀ with an “elastic continuum” pervading the entire space, we could view electromagnetic waves as stress–strain waves propagating through this continuum. Accordingly, the transportation of energy across physical space could be viewed as a propagation process of a specific type of stress–strain wave through the elastic space continuum.

It is difficult to visualize the transportation of matter as some sort of propagation process through the elastic space continuum, even though the equivalence between mass and energy has been well established. At the subatomic scale, all primary constituents of a body—electrons, protons, and neutrons—are known to cover only approximately 10^–12% of the entire volume of that material body. These subatomic or “elementary” particles, occupying such a tiny fraction of space, are generally identified by their energy and interaction characteristics. Hence, these particles may be conceived as packets of energy entrapped in a characteristic wave formation in the space continuum. Therefore, the transportation of clusters of such particles can be viewed as a “propagation” process through the continuum. It seems likely that all the EM phenomena and all transportation processes involving entrapped energy packets occur in the elastic space continuum with the characteristic properties of an elasticity constant 1/ε₀ and inertial constant μ₀.

1.2 Elasticity equations in the elastic space continuum

1.2.1 Stress and strain tensors in the elastic continuum

Displacement vector U at every point P, as a function of position coordinates and time, constitutes a displacement vector field in the elastic continuum. The displacement vector from points P to P′ in an orthogonal coordinate system represented by its base vectors a_i is given by Equation 1where uⁱ are the contravariant components of vector U as functions of the coordinates of P and time t. Displacement vector field U(xⁱ,t) is said to be well defined over a region of space when every component of it is finite and continuous within that region. Specifically, displacement vector field U may be a periodic function of coordinates xⁱ and t within the field region and must be 0 at the boundaries and outside this region. Obtaining specific solutions for displacement vector field U(xⁱ,t) under specified initial and boundary conditions is the main objective in the analysis of any deformation in the elastic space continuum.

In a certain region of space, displacement vector field U(xⁱ,t) represents an infinitesimal deformation field in that region. The infinitesimal deformation at any point P(xⁱ,t) is best quantified through the components of strain tensor S, defined by the covariant derivatives of the displacement vector field with respect to spatial coordinates x^j as

The first partial derivatives ∂uⁱ/∂t of uⁱ with respect to time constitute local velocity field vⁱ. The second partial derivatives ∂²uⁱ/∂t² constitute the local acceleration field. Therefore, inertial body force density component Fⁱ will be represented by the -μ_0.∂²uⁱ/∂t² term. Kinetic energy density W_k in the region under dynamic strain will be represented by (μ₀/2). [(v¹)²+(v²)²+(v³)²]. Since μ₀ε₀ = 1/c², kinetic energy density W_k can be written as

However, strain energy density W_s contained in the strain field (2) is proportional to the sum of squares of the spatial strain components and is given by

Comparing Equations 3, 4, we may consider (1/c) times the ∂uⁱ/∂t component as a kinetic or temporal strain component. Therefore, three components of the displacement vector uⁱ (x¹,x²,x³,t) will produce nine spatial strain components and three kinetic or temporal strain components, which are functions of the space and time coordinates.

The state of stress at any point P(x¹, x², x³) of the elastic continuum under infinitesimal strain is represented by stress tensor T. For an infinitesimal rectangular plane area σ_+j perpendicular to the x^j coordinate direction, the net force per unit area T_j acting on σ_+j is given by Equation 5,Here, τⁱ_j represents the components of stress tensor T, which are functions of space coordinates (x¹, x², x³) of point P and time t. Stress components τⁱ_j are + ve when corresponding components of force act in the direction of the +ve Xⁱ axis and the normal surface is also along the + ve X^j axis. When the normal surface is along the −ve X^j axis, positive values of τⁱ_j are associated with oppositely directed forces. Therefore, for an infinitesimal rectangular-faced volume element δV = δx¹. δx². δx³, with point P(x¹, x², x³) as its center and faces parallel to coordinate planes, the stress components τⁱ_j will correspond to forces in opposite directions at the opposite ends of the volume element.

1.2.2 Dynamic equilibrium conditions and modified Hooke’s law

Under the static equilibrium of a material body, the vanishing of resultant moments, to preclude rigid body motions, is ensured by the symmetry of stress and strain components. However, in an elastic continuum there are neither rigid body motions nor static equilibrium. Hence, in the elastic space continuum, the condition of symmetry of stress and strain components is not applicable. The dynamic equilibrium equations for an elastic continuum can be shown to reduce to a set of three partial differential equations as in [2]:

or, in tensor notation,

Here, body force component −Fⁱ is associated with inertial force component μ₀.∂²uⁱ/∂t², and μ₀ is the inertial constant for the elastic continuum. Therefore, equilibrium Equation 7 can be written as

In orthogonal curvilinear coordinates with metric tensor components g^ij and g_ij, the standard dynamic equilibrium equations for the elastic continuum take the form

Lamé constants λ and μ are used to describe the correlation between stress and strain components in the theory of elasticity. While constant μ relates the corresponding stress and strain components, constant λ relates a normal stress component in one direction to the strain components in transverse directions. Therefore, the magnitude of λ is basically linked to the material’s atomic structure; for the elastic space continuum, with no internal granularity, it reduces to 0. This leads to a zero magnitude of Poisson’s ratio and a corresponding simplification of Hooke’s law of elasticity. Ultimately, this simplification leads to a common velocity of propagation of longitudinal and transverse strain waves, which is consistent with physical observations.

Since stress and strain tensor components are not required to be symmetrical under dynamic equilibrium conditions, with zero magnitude of Poisson’s ratio, Hooke’s law of elasticity will modify to a simple form as

In electrical units, dimensions of (1/ε₀) are Nm²/Coul². However, the dimensions of the elastic constant (1/ε₀) are required to be N/m² in mechanical units. Therefore, to ensure the dimensional compatibility of electrical and mechanical units for the elastic constant, we must assign the dimension m² to the Coulomb unit of charge. However, the ratio e²/ε₀ = 2.9x10⁻²⁷ Nm² is free of electrical units, and in all force and energy computations, e and ε₀ often appear as the e²/ε₀ ratio, which remains constant in both units [2].

Substituting stress–strain relation (9) in the dynamic equilibrium Equations 8A, 8B, we derive the corresponding vector wave equation in terms of displacement components uⁱ as

or

Vector wave Equation 10B is the standard equilibrium equation of elasticity in the elastic space continuum, which can be transformed to any coordinate system.

1.3 Representation of electromagnetic fields as stress–strain fields

In free space, electrical field E and magnetic field strength B are constituents of the electromagnetic field and satisfy Maxwell’s field equations. Magnetic vector potential A, having the dimensions of volt second per meter or newton per ampere, serves as a common link between fields E and B. Time-varying magnetic vector potential A can be induced in free space by the oscillations of Hertzian dipoles [3].

A Hertzian, or an infinitesimal, dipole is a tiny segment of a current-carrying conductor in which an oscillating current produces electromagnetic radiation. This oscillating current creates time-varying magnetic vector potential A in the surrounding space. This potential is used to derive the time-varying electrical and magnetic fields that constitute the electromagnetic wave radiating from the dipole. For infinitesimal length l of a Hertzian dipole, the magnitude of vector potential A(r,t) can be approximately given bywhere I is the effective current (dq/dt) due to the oscillating charge, ω is angular frequency, and k = ω/c is the wave number. Therefore, when vector A is induced by electronic oscillations, its intensity will be proportional to eω × µ₀.

Electric and magnetic fields can be represented by dynamic stresses and strains in the elastic space continuum with physical properties of ε₀, μ₀, and c. Considering a time-dependent displacement vector U in the elastic continuum that satisfies the dynamic equilibrium equation or vector wave Equation 10B, a solution of Equation 10B for U that satisfies the essential boundary conditions will represent a transverse wave field if ∇.U = 0. Displacement vector field U can now be identified with conventional electric and magnetic fields E, B, and A in free space through the following relations:

The displacement vector field U satisfies all electromagnetic field equations that are satisfied by A, E, and B in free space. Hence, displacement vector field U may be considered a crucial link between the stress strain field in the elastic space continuum and the conventional electromagnetic fields in free space. The stress–strain fields in a finite bounded region of physical space constitute a structure of electromagnetic fields and elementary particles [2]. In a finite region of strained space continuum, if the solutions of Equation 10B satisfy the boundary and stability conditions, then that region of space will exhibit an observably distinct entity. Such distinct regions of the elastic space continuum with finite and stable total energy content can be called “strain bubbles” or “elementary particles.”

2 Strain bubble representing electron structure

2.1 Spherically symmetric strain bubbles

Of all the known elementary particles, the electron was the first to be detected and is the most actively studied and researched. As a part of our understanding of the electron, we have already determined its charge, mass, spin, magnetic moment, and various interaction characteristics. Apparently, everything that is worth knowing about electrons is already known. Nevertheless, most scientists still regard electrons as point charge, point mass, and structureless elementary particles. Many new models of their structure have been recently proposed based on the current theoretical paradigm, but we are still not in a position to know or visualize their shape, size, and inner structural details.

Some models, such as the Kerr–Newman electron, are primarily mathematical models that attempt to unify concepts from general relativity, quantum mechanics, Higgs field, and string theory, but they do not make any physical sense [4]. [5] introduced a Zitterbewegung electron model based on geometric interpretations of the key equations of Planck, Schrödinger, and Bohm. The electron is modeled as a massless current ring rotating at the speed of light [5]. [6]’s Helical Solenoid Model proposes a relationship between the electron’s charge and magnetic flux, suggesting that the electron is fundamentally linked to both electric and magnetic properties. [7] proposes a Zitterbewegung model of the electron which shows that it is the charge and not the mass that undergoes rapid spatial oscillation and that there are measurable consequences of this charge zitterbewegung.

The quantum electrodynamics (QED) electron model assumes the physical existence of a quantum vacuum, electron field, and photon field in addition to the physical existence of Maxwell’s electromagnetic field supported by empty space, with physical properties of permittivity ε₀ and permeability μ₀. In QED, the electron model consists of a “bare electron,” which is a half-spin point particle obtained as a solution of Dirac’s equation and is surrounded by a quantum field of virtual photons obtained from the solution of the QED Lagrangian. In contrast, we have developed a structure for the electron and its electrostatic wave field by analyzing some of spherically symmetric solutions of equilibrium equations of elasticity in the elastic space continuum.

The equilibrium equations of elasticity (10) can be expressed in spherical polar coordinate system yⁱ with coordinates (r, θ, ϕ) and corresponding physical components of the displacement vector written as u^r, u^θ, and u^ϕ.

The non-zero metric tensor components g_ij and g^ij for this coordinate system are

The corresponding Christoffel symbols of the second kind Γⁱ_jk are given by

The physical components u^r, u^θ, and u^ϕ of displacement vector U are related to the corresponding contravariant components u¹, u², and u³ through Equation 15 as

The covariant derivative of uⁱ is given by

The elements of strain tensor Sⁱ_j can be computed from Equations 2, 16, 18A by taking covariant space and time derivatives of displacement components uⁱ for spatial and temporal strain terms. The mixed tensor components Sⁱ_j can be converted to the corresponding physical components by using the relationwhere g^ij are the usual metric tensor components for the spherical polar coordinate system. The physical spatial strain components thus obtained are listed below:

The corresponding physical components of temporal strain are given by

The vector wave Equation 10B given in tensor notation can now be rewritten in terms of physical displacement vector components (u^r, u^θ, u^ϕ) as follows:

The wave Equations 20A–20C constitute a set of three simultaneous partial differential equations that involve displacement vector components u^r, u^θ, and u^ϕ. Unlike wave equations in a conventional Cartesian coordinate system, these equations in spherical polar coordinates may be considered “mutually coupled” in the sense that none of these equations can be solved independently of one another. The general solution of these mutually coupled equations is very intricate, and we have to seek some simple alternatives. The electron structure will now be shown as a special solution of wave Equations 20A–20C. One of the lowest order solutions of these equations is obtained when we restrict u^θ = 0, u^ϕ to be independent of ϕ coordinate and u^r to be independent of both θ and ϕ coordinates. The resulting equations reduce to

Equations 21A, 21B can be solved for displacement vector components u^r and u^ϕ by the separation of variables, respectively. Substituting R.T for u^r in Equation 21A and R.ϴ.T for u^ϕ in Equation 21B and then dividing both equations by R.T and R.ϴ.T, respectively, we obtain

and

Solving Equations 22A, 22B after substituting x = kr, we obtain two solutions for T as cos (kct) and sin (kct), one solution for ϴ as sin(θ), and two solutions for R as j₁(x) and y₁(x), where j₁(x) and y₁(x) are the spherical Bessel functions of the first and second kind, respectively. Some Zitterbewegung-based models have been proposed in which charge is assumed to undergo rapid spatial oscillation [7]. The proposed spherically symmetrical strain wave solution is also expected to provide Ztterbewegung-type spatial oscillations in the strain field structure of the electron.

2.2 Spherical strain wave solution for the electron structure

2.2.1 Standing wave core

The oscillating wave type solution in terms of displacement vector components u^r and u^ϕ of Equations 21A, 21B, 22A, 22B, with a spherically symmetric boundary surface, is given below in terms of spherical Bessel function j₁(x) for the electron core. The corresponding solution for the positron core will be obtained by changing the sign of displacement component u^r.

where

From Equations 10A, 10B, 11, 12, we can deduce that integration constant U₀ is proportional to (ε₀c) × (eωµ₀) or U₀ = bek, with b as a dimensionless component of the integration constant and e as the magnitude of electron charge in coulombs. The core is bounded by one of the nodes of the spherical Bessel function.

Equations 23A, 23B represent the standing strain-wave-type oscillations within the core region. Another similar solution which has a singularity at the origin and is hence not admissible for the electron/positron core is given by the spherical Bessel function y₁(x):where and is bounded inside by the first node with b₁ ≤ , where (b₁) = 0 and b₁ = 2.798386

2.2.2 Radial phase wave field

However, if the above two solutions (Equations 23A, 23B, 24A, 24B) are combined, we obtain an oscillating core given by Equations 23A, 23B for 0 ≤ x ≤ b₁ and a propagating radial phase wave solution for the electron field for x ≥ b₁, given by

where ψ₋ = x + kct, and

This strain wave field consisting of phase waves propagating inwards from infinity to the core boundary at the speed of light c represents the electrostatic field of an electron. Another similar solution consists of phase waves propagating outwards from the core boundary to infinity at the speed of light c representing the electrostatic field of a positron. Changing the sign of the oscillating core radial displacement components given by Equations 23A, 23B yields an outward propagating radial phase wave solution for the positron type electrostatic field for x ≥ b₁ given by

where ψ₊ = x − kct, and

2.3 Strain energy in the electron structure

2.3.1 Core and field strain components

The strain energy of the core and strain wave field can now be computed from the corresponding strain components. Here, wave number k is of the order of 10¹⁵ m^-1. Solutions u^r and u^ϕ are in phase quadrature in the core and in the field. The physical strain components of strain tensor S and corresponding strain energy density W can therefore be directly computed from the sum of squares of strain components as per Equations 3, 4, 19A, 19B:

Since, in the strain wave field, surfaces of constant phase propagate outwards or inwards at the speed of light c without any associated transport of strain energy, we may term this wave field as a “phase wave field”. However, there is a special feature in this field: in the standing or oscillating wave solutions, the temporal and spatial strain components are in quadrature and independently store strain energy; however, in the phase wave field, the propagating phase waves are not associated with any transportation of strain energy. Hence, the energy density in the phase wave field will be governed only by the maximum amplitude of these phase waves or, more precisely, by their RMS values. The physical strain components for the electron core, computed from relations (19) and (23), are listed below in terms of functions j₁(x) and their derivatives j₁'(x):where

The strain components for the positron core will be the same as given above but with opposite signs. In the strain wave field of the electron represented by Equations 25A, 25B, the strain components can be computed on similar lines as for the core. However, there is one major difference: since the phase waves associated with sin(ψ₋) and cos(ψ₋) do not transport energy, we can say that there is no strain energy associated with the sin(ψ₋) and cos(ψ₋) terms. Therefore, we can treat ψ₋ as a constant parameter under differentiation for strain computations in the electron wave field.

We can associate inward propagating phase waves, represented by sin(ψ₋) and cos(ψ₋) terms, with negative charge and outward propagating phase waves, represented by sin (ψ₊) and cos (ψ₊) terms, with positive charge. The alternating currents flowing through power transmission lines are often modeled by their equivalent root mean squared (RMS) valued currents flowing in the direction of wave propagation [3, 8]. Similarly, for field energy computations, we can model the field displacement vector components for electrons (Equations 25A, 25B) by their RMS values with a negative sign, and those for the positron (Equations 26A, 26B) by their RMS values with a positive sign. Accordingly, we shall associate the sign of field displacement components for electrons and positrons with their charge or the direction of phase wave propagation and not with the orientation of their polar axes in all field interaction computations.

The amplitude of these phase waves at any point will remain constant with time but will keep decreasing with radial distance. Hence, the physical strain components for the electron wave field, computed from relations (19) and (25) by treating ψ₋ constant, are listed below in terms of functions j₁(x, ψ₋) and y₁(x, ψ₋) and their derivatives j₁'(x, ψ₋) and y₁'(x, ψ₋):

where and

2.3.2 Core and field strain energy

Strain energy density W, calculated by Equations 27–29, is given below as W_cs for core spatial strain energy, W_ct for core kinetic or temporal strain energy, and W_f for field strain energy:

In W_c above, the coefficients of the cos² (κct) and sin² (κct) terms are not exactly equal, which indicates slight fluctuations of energy density within the core. To compute the total energy in the core, we must integrate the coefficients of cos² (κct) and sin² (κct) terms over the whole core volume, which extends up to x = b₁. Writing the coefficient of cos² (κct) as W_cs1 and coefficient of sin² (κct) as W_cs2 in Equation 30A, we obtain

Similarly,

Integrating these coefficients over the entire volume of the core and using numerical integration for the spherical Bessel functions, we obtain

Similarly,

and

Therefore, using Equations 32A, 32B, 32C, 33A, 33B, 33C, 33D, we can sum up total spatial strain energy E_cs and total kinetic or temporal strain energy E_ct in the electron core, bounded by 0 < x < b₁, as a function of oscillation time t:

We can similarly sum up total electron core strain energy as

As per Equation 35A, total strain energy E_c in the core shows slight oscillations of approximately 4.6% (Figure 1) from the mean. The displacement components at the core boundary will also slightly oscillate during the time period, which may be considered driving the wave field oscillations. The mean value of total strain energy in the electron core is given by

FIGURE 1

In Equations 34A, 34B: kc represents the angular frequency of standing strain wave oscillations in the electron core; E_cs represents spatial strain energy content in the core region, the magnitude of which keeps oscillating with time (Figure 1); E_ct represents kinetic strain energy content in the core, which also keep oscillating with time. It is notable that the sum of spatial and kinetic strain energies is fairly stable during the oscillation time period, accounting for the structural stability of the electron. The time period of strain wave oscillations in the core works out to approximately 15 yocto-seconds (Figure 1). It is quite astonishing to note that during this extremely small oscillation period of 15 ys, approximately 82 keV of strain energy oscillates between spatial and kinetic strain energy components in the core.

Next, we compute total strain energy in the wave field by integrating W_f in Equation 31 over the field region (x ≥ b₁) while treating the wave parameter ψ₋ as constant.

Segregating the coefficients of cos² (ψ₋) and sin² (ψ₋) in W_f and taking their root mean square values, we obtain

Therefore, the total strain energy contained in the electron core and field structure is

It is clear from Equations 36, 37 that approximately 37.34% of the total mass energy of the electron is contained in its strain wave field and the remaining 62.66% of its mass energy is contained in the central standing wave core.

3 Coulomb interaction between two electrons

3.1 Qualitative description of Coulomb interaction

In classical electrodynamics, the Coulomb interaction arises because each electron produces an electrostatic field, and the other electron feels a force due to this field. The repulsive force between two electrons is given directly by Coulomb’s law:

In quantum electrodynamics (QED), the electrostatic Coulomb force (Equation 38) between two charged particles, such as two electrons, is not an “action at a distance” force in the classical sense but rather an effective interaction mediated by the exchange of virtual photons. The two electrons interact by continually exchanging virtual photons, which are believed to be temporary fluctuations in the electromagnetic field. The interaction is not really instantaneous; the field disturbance travels at speed c. The static Coulomb potential emerges as the time-independent approximation of the full retarded interaction.

In the proposed strain wave structure of electrons presented above, the radial phase wave fields take the place of virtual photons and provide the mechanism of the Coulomb interaction. The radially diminishing strain energy density W_f in this wave field is given by Equation 31. This energy density, associated with inward or outward propagating phase waves of radially decaying amplitude, may be viewed as momentum density waves representing the electrostatic fields of the electron and positron. When two electrons or two positrons are brought closer with their centers a finite distance apart, their wave fields will become superposed and the strain energy density in the intervening region of space will increase, leading to repulsion between them. When a positron and an electron are brought closer, their wave fields will become superposed and the strain energy density in the intervening region of space will decrease, leading to attraction between them. The actual mechanism of the superposition of strain wave fields in the overlapped region of space is a little more complex due to the inherent complexity of interaction between propagating phase wave fields of interacting particles. To simplify somewhat, we have modeled the effective displacement components as positive for the outward propagating phase waves and negative for the inward propagating phase waves and neglected the oscillatory effect of sin(ψ) and cos(ψ) terms in the effective displacement components.

To compute the Coulomb interaction energy between two electrons, we must superpose the effective strain tensor components Sⁱ_j of their respective wave fields in a common coordinate system and then compute the total energy of their combined fields. This combined field energy E_sup may be more or less than the sum of their isolated field energies E₁ and E₂. The difference [E_int = E_sup−(E₁+E₂)] is termed “interaction energy” E_int. For two electrons, the respective strain components add up, and since the energy density is proportional to the sum of squares of strain components, their total field energy will be more than the sum of their separate field energies. Conventionally, the interaction energy is termed “+ ve” in this case. Similarly, for interaction between an electron and a positron, the combined field energy will reduce, and the interaction energy will be termed “−ve”. The negative interaction energy implies that due to the superposition of fields, part of the initial total field energy of the system of interacting charges is released by the system and may be transformed to some other form [9]. The computation of the interaction energy of two electrons involves the transformation of effective displacement or field strain components from one coordinate system to another to effect superposition in a common coordinate system.

Let us assume that a spherical polar coordinate system (r,θ, ϕ) with its origin at point O is the common coordinate system in which the physical strain components of the two electrons are to be superposed to compute their interaction energy. Let S^α_β(1) be the physical strain components of first electron and S^α_β(2) be the physical strain components of second electron as transformed into this common coordinate system. The indices α and β stand for r, θ, and ϕ. Thence, in accordance with Equation 27, the strain energy densities of electrons W₁ and W₂ and the energy density of their combined superposed field W_sup, are computed as

Therefore, the interaction energy density W_int will be given by Equations 39, 40

It is notable that the interaction energy density between two charge particles is given by the sum of the product of their identical strain components in a common coordinate system. The total interaction energy can now be computed by taking the volume integral of Equation 41 over the entire overlap region of space.

3.2 Superposed strain field interaction between two electrons

We now compute the strain field interaction of two electrons located at points O and A, with collinear axes and centers separated by distance R along axis OZ (Figure 2). Let any space point P refer to two spherical polar coordinate systems, one yⁱ(r,θ,ϕ) centered at point O and the other xⁱ(ρ,δ,ϕ) centered at point A. For point P, the following relations hold between these coordinate parameters:

FIGURE 2

For coordinate transformation between the yⁱ(r,θ,ϕ) and xⁱ (ρ,δ,ϕ) coordinate systems, the Jacobean matrices of their partial derivatives obtained from above relations are

and

From Equation 29, the far field effective displacement vector components in yⁱ(r,θ,ϕ) coordinate system at point P, due to the first electron located at point O, are written as

and

The corresponding effective strain components at point P, due to the first electron, are given by

Similarly, the far field effective displacement vector components in the xⁱ(ρ,δ,ϕ) coordinate system at point P, due to the second electron located at point A, are written as

In accordance with Equation 17, the contravariant components of the displacement vector, corresponding to the physical components given by Equation 48A, are given by

These contravariant components of the displacement vector in the xⁱ(ρ,δ,ϕ) coordinate system can be transformed to the yⁱ (r,θ,ϕ) coordinate system using the Jacobean matrix derivatives of Equations 44A, 44B. After transformation, the contravariant components can again be converted back to their physical components in the yⁱ(r,θ,ϕ) coordinate system and are given below:

From these physical components of the displacement vector in the yⁱ(r,θ,ϕ) coordinate system, we can compute the strain components using Equation 19A:

3.3 Strain field interaction energy computation

Equations 47, 50 represent the wave field effective strain components at point P due to two electrons located at points O and A and referred to as common spherical polar coordinate system yⁱ(r,θ,ϕ). From these strain components, we can compute the interaction energy density using Equation 41, where we must first compute the sum of the products of identical strain components from both electrons:

Equations 51A, 51B, 51C, 52A, 52B, 52C, 52D contain product terms of all direct and shear strain components produced by components u^r and u^ϕ of the effective displacement vectors of both electrons. If we reverse the orientation of the spin axis or polar axis of the electron, then the sign of components u^r and u^ϕ will remain unchanged since they have been modeled on the direction of propagation of the field phase waves. However, if we replace one of the electrons with a positron, then due to the opposite propagation of their field phase waves, the effective displacement components and the corresponding field strain components of the two interacting particles will have opposite signs. Hence, the products of their strain components will change sign, and the total interaction energy will become negative.

If we let W_int1 be the interaction energy density from the product terms of Equations 51A, 51B, 51C and W_int2 be the interaction energy density from the product terms of Equations 52A, 52B, 52C, 52D, we obtain from Equation 41:

By substituting y = r/R in Equations 53A, 53B to facilitate their volume integration for computing total Coulomb interaction energy E_int, ρ²/R² will be replaced with (y²+1-2ycos(θ)). The volume integration of W_int1 and W_int2 will yield

Similarly,

The total interaction energy of two electrons separated by distance R in an axial direction will be given by the sum of E_int1 and E_int2:

With 1/R dependence of total interaction energy between two electrons, the Coulomb interaction law of electrostatics stands proven. Since contributions of both displacement components u^r and u^ϕ are equal, we may assume that this interaction energy will be insensitive to the relative orientations of the two electrons/positrons [9]. Therefore, comparing the interaction energy of Equation 55 with that given by the general Coulomb interaction law for two electrons separated by distance R, we obtain

This is an important result. Substituting this value of dimensionless constant b from Equation 56 in Equation 37 and comparing the total strain energy of the electron with its known mass energy (m_e.c²), we obtain wave number k and the corresponding angular frequency ω of the strain wave oscillations [10]. Other physical parameters of the electron, such as oscillation frequency f, oscillation time period τ, and core radius R₀, can then be computed thus:

This yields k = 1.349033 x 10¹⁵, angular frequency ω = 4.0443 × 10²³ rad/s, oscillation frequency f = 6.4367 × 10²² Hz, time period τ = 1.5536 x 10⁻²³ s = 15.5 ys, and electron core radius R₀ = b₁/k = 2.074x10⁻¹⁵ m = 2.07 fm.

In this analysis of strain wave structure of electron- and positron-type charge particles, we have restricted the central core region to the first zero of the spherical Bessel function y₁(x) in order to minimize the total strain energy and ensure the overall stability of the strain wave structure. It is, however, theoretically possible to extend the core boundary from the femtometer region to the picometer region in order to pack a large amount of strain energy in the core, even at the cost of the overall stability of the strain wave structure. It appears that a well-known, short-lived elementary particle, the muon, may belong to this type of strain wave structure with an enlarged central core region.

4 Electromagnetic characteristics of the electron structure

4.1 Intrinsic spin characteristics

It is now evident from Equations 23A, 23B that displacement vector component u^r in the electron core oscillates in the radial direction at angular frequency ω, given by the cos (ωt) term. Displacement vector component u^ϕ in the electron core oscillates in the azimuthal direction, given by the sin (ωt) term. The combined effect of these two orthogonal oscillations is that at any point in the core, displacement vector U, consisting of orthogonal components u^r and u^ϕ, will continuously rotate in the transverse plane at the angular frequency ω = 4 x10²³ rad/s. Angular velocity vector Ω representing this angular rotation of displacement vector U will be pointing as normal to the transverse or azimuthal plane—in the direction parallel to the polar axis given by θ = 0. This spinning action of displacement vector U can be associated with the intrinsic spin of the electron.

Similarly, it is evident from Equations 25A, 25B that vector components u^r and u^ϕ in the phase wave field are in phase quadrature, producing the rotational effect for displacement vector U in the electron field. At any point in the principal transverse plane (θ = π/2), the magnitude of resultant displacement vector U in the wave field remains constant while “spinning” with constant angular velocity ω. The direction of this spin angular velocity vector is parallel to the polar axis and remains constant with time, though this displacement vector does appear to propagate along the phase waves even while spinning. The phenomenon of this intrinsic spin is a very unique feature of electron/positron structure, which characterizes the orientation of the particle in various interactions.

Since vector field U produces the corresponding strain field, the intrinsic spinning motion of vector U will also be accompanied by the spinning motion of the associated strain field about the intrinsic spin axis. With this spinning motion of the strain field, the whirling motion of the associated strain energy will produce intrinsic strain energy vortices throughout the electron structure. These strain energy vortices will give rise to the angular momentum of the electron about its spin axis.

4.2 Intrinsic electric field

The radial phase wave field of the electron defined by Equations 25A, 25B, 29 constitutes the intrinsic electrostatic field of the electron. Furthermore, as seen from Equations 36, 37, approximately 37.3% of the total mass energy of the electron is contained in this electrostatic wave field. The radial phase wave field consists of three direct strain components and four shear strain components. Whereas the shear strain components contribute to the intrinsic spin and magnetic field effects of the electron, the direct strain components contribute to its electrostatic electric field. Out of the three direct strain components, radial direct strain component S^r_r, with its inward propagating ψ₋ phase waves, represents the electric field of the electron. Inserting the value of proportionality constant U₀ = bek into Equation 29, far from the core region, the radial direct strain component S^r_r can be written as

The term cos(ψ₋), representing a phase wave propagating radially inwards, can be replaced in interaction computations by radial unit vector in effective displacement and strain components. The corresponding stress components associated with the radial direct strain given by Equation 58A can be obtained using Hooke’s law (Equation 9):where b is a dimensionless constant, e is the magnitude of electron charge, and r is the radial distance from the center of the electron. However, electrical field intensity E in the electrostatic field of an electron, as derived from the Coulomb interaction, is normally given by

Therefore, it is obvious that radial direct stress T^r_r of Equation 58B represents electrical field intensity E in the electrostatic field of the electron, even though two other direct stress components also contribute to the overall electrostatic field strength. The overall stress–strain wave field of Equations 25A, 25B, 26A, 26B, 29 represents the electron’s intrinsic electrostatic field with inward propagating ψ₋ waves and that of positron with outward propagating ψ₊ waves. The radial lines associated with the radially inward or outward propagating phase waves represent the electric field lines of the electron and positron, respectively. The Coulomb interaction between two electron/positron particles is governed by the superposition interaction between their intrinsic strain wave fields, as demonstrated in Section 3 above, but in simplified form it can be said to be the interaction of the electrical fields of the particles.

Practically, however, electric field intensity E given in Equation 58C is derived as the gradient of potential energy function ϕ(r) representing the interaction energy per unit charge. For a single isolated electron, potential function ϕ(r) does not represent any entity that physically exists anywhere in the field of the electron because potential energy is essentially the Coulomb interaction energy between two charges. Nevertheless, field intensity E, as a gradient of the potential function, is known to physically exist and define the electrostatic field even for an isolated electron. The energy density of electrostatic field W_f is also known to depend on electric field E thus:

However, as shown in Section 3, strain energy density W_f in the electrostatic wave field of an electron is given by Equations 29, 31. Replacing sin(ψ) and cos(ψ) terms with their RMS values, W_f in the far field region is given by

From Equations 59, 60, we obtain an equivalent electric field E that represents the field energy density of a positron/electron as given by Equation 61

4.3 Intrinsic magnetic field effects

As shown above, the intrinsic spin of electrons (as well as positrons) is associated with the intrinsic rotation of the displacement vector U in the transverse plane of the particle. Even though vector components oscillate at high frequency ω, the rotation of U is caused by their phase quadrature as represented by the cos (ωt) and sin (ωt) terms associated with u^r and u^ϕ, respectively. Essentially, azimuthal displacement vector component u^ϕ is responsible for the intrinsic spin and the rotation of vector U in the transverse plane. As such, the shear strain field associated with displacement vector component u^ϕ will contribute to ∇xU and, hence, the intrinsic magnetic field perpendicular to the transverse plane, in accordance with Equation 14. In essence, the shear strain field associated with the rotation of vector U will produce a sort of torsional strain around the intrinsic spin vector, which is perceived as a magnetic field. While the rotation of displacement vector U creates the intrinsic spin and intrinsic angular momentum effects, the corresponding torsional effect of associated shear strain components creates the intrinsic magnetic field.

Of the strain components listed in Equations 28, 29 for the core and radial phase wave fields of the electron, components S^ϕ_r, S^r_ϕ, and S^ϕ_t are associated with the intrinsic spin of the electron and are at their maximum in the principal transverse plane. In accordance with Equation 14, the intrinsic magnetic field perpendicular to the principal transverse plane—for the core (B_c) and the wave field (B_f)—are given by the difference (S^ϕ_r − S^r_ϕ) and S^ϕ_t strain components, as per following relations:

and

Numerical computations using Equation 62A show that the average value of the magnetic field in the core is of the order of 2 x 10¹² Tesla and is parallel to the polar axis or intrinsic spin axis of the electron. Similar numerical computations using Equation 62B show that the magnetic field in the radial wave field of the electron is parallel to the intrinsic spin axis and radially propagates as a phase wave, with maximum intensity of approximately 1.8 x 10¹¹ Tesla at the core boundary and decaying in amplitude with the cube of the radial distance. With this intrinsic magnetic field, when the electron is placed in an external magnetic field B_ex, with spin axis aligned to the external field, the combined magnetic field energy will increase when B_c and B_ex are parallel and decrease when they are anti-parallel. Currently, these magnetic field effects of the electron are empirically modeled on the similar magnetic field effect of a tiny current loop and are hence represented by an equivalent magnetic moment.

4.4 Magnetic field induced with uniform motion of electron and positron

We now consider radial component u^r of the displacement vector for electrons or positrons (Equations 25A, 26A). The cos(ψ) terms in these radial components represent surfaces of a constant phase propagating inwards or outwards at speed c. Consider a large number of surfaces of constant phase which are separated by a radial distance of one wavelength λ or phase difference of 2π. For a static particle, the radial electric field lines are always normal to these constant phase surfaces. However, when the particle is moving with uniform velocity v, the transverse radial lines from the instantaneous position of the particle will no longer remain normal to these constant phase surfaces. We illustrate this effect by considering a positron moving with uniform velocity v along a positive Z-axis. From Equation 26A, u^r is given by

Letting this particle be moving along the Z-axis with velocity v, we consider the position of this particle at ten instants of time, t₁, t₂, t₃, … t₁₀, separated by equal intervals of time, dt. In a plane passing through uniform velocity vector v, the surfaces of the constant phase generated at these instants will form circles with their centers shifted along the Z-axis from A to B (Figure 3). During this time period, radial field line BD will appear to be deformed or dragged behind to BC with the drag angle given by tan⁻¹ (v/c). That is, the displacement vector at point D in the transverse plane will transform to the sum of old radial vector u^r corresponding to direction AC plus the additional motion induced displacement vector u^z corresponding to direction CD or AB. This will amount to an induced velocity-dependent displacement component u^z parallel to velocity vector v as u^r. (v/c).

FIGURE 3

This velocity-dependent component of the displacement vector (Equation 64) may be treated as the motion-induced displacement vector. This will produce a magnetic field through the curl of the displacement vector (Equation 14). For simplicity, we consider the transverse plane passing through B and perpendicular to the velocity vector. To analyze the induced magnetic field in this plane, we use a cylindrical coordinate system (ρ, φ, z), where ρ is the radius and φ is the azimuthal angle in the transverse plane. Treating ψ as a constant under differentiation for strain computations (Section 2.3.1) and taking the RMS value for the cos(ψ) term, the magnitude of the magnetic field induced by u^z in the transverse plane is given by

The direction of this induced magnetic field B (Equation 65) at any point in the transverse plane passing through the instantaneous position of the particle will be perpendicular to the plane containing velocity vector v and position vector ρ as given by the curl of U. Hence, motion-induced magnetic field B will be tangential to the concentric circles and form clockwise circular loops around the direction of motion in accordance with the right-hand rule with respect to the velocity vector. This verifies the Biot–Savart law for magnetic fields induced by the uniform motion of electron or positron-type charge particles.

4.5 Matter waves induced by the uniform motion of electrons and positrons

As discussed in Section 4.4, we again consider a positron moving with uniform velocity v along a positive Z-axis. Considering the position of this particle at equal time intervals dt, the surfaces of constant phase generated at these instants will form circles in a plane passing through the Z-axis, with their centers shifted along the Z-axis (Figure 3). Along the Z-axis, ahead of the moving positron, separation between two adjacent circles is given by (c-v)dt. That is, the radial phase wave field will compress in the forward direction by a factor of (1-v/c). Hence, the number of constant phase wave fronts per unit length will increase by a factor of (1 + v/c) in the forward direction. This indicates that the modified wave number k' will be given by

Therefore, displacement vector component u^r of Equation 63 is modified along the forward direction (+Z-axis) as

Splitting the cosine term into two parts, with z-vt = z' and z' > R₀,

Here, the terms (b.e/z'). cos (k (z-ct)) and (b.e/z'). sin (k (z-ct)) represent the original phase waves in the electrostatic wave field, which serve as carrier waves. The terms cos (k (v/c) (z-ct)) and sin (k (v/c) (z-ct)) represent the amplitude modulation terms, which modulate the carrier phase waves. Wavelength λ_m of the modulation wave is given by

Since Equation 57 informs us that k is proportional to the total energy and mass of the electron/positron particles, term kv in Equation 68 will be proportional to momentum p of the particle. Hence, Equation 68 indicates that the wavelength of the modulation wave is inversely proportional to the momentum of the particle. As such, we can qualitatively conclude that the motion-induced modulation waves over the inherent phase waves of the electrostatic field represent the well-known De Broglie matter waves.

Equations 66, 67A, 67B, 68 represent a highly simplified treatment of motion-induced modulation waves superposed over a radially-decaying electrostatic phase wave field of electron/positron particles. For detailed treatment of motion-induced effects, we must begin from some modified form of the equilibrium Equations 10A, 10B, like the Klein Gordon equation, which we have not attempted here. Moreover, it is relevant to note that motion-induced deformation of the electrostatic field will lead to an increase in the strain energy of the intrinsic strain wave field of electrons and positrons. All kinetic energy of the motion of the electron will possibly be stored in its deformed or retarded strain wave field. With the deceleration of an electron in motion, the reduction in its kinetic energy of motion will be associated with radiation from its deformed strain wave field.

5 Summary and conclusion

In this study, we have shown that a physical space with the characteristic properties of permittivity ε₀ and permeability μ₀ that supports the propagation of transverse electromagnetic waves can be represented as an elastic space continuum with an elasticity constant 1/ε₀ and inertial constant μ₀. We have also shown that all electromagnetic phenomena, energy entrapping, and transportation processes that appear to occur in empty space can be perceived as occurring in the elastic space continuum. The elementary matter particles can be represented as packets of energy entrapped in a characteristic wave formation in the elastic space continuum.

In the elastic space continuum under stress–strain deformation, only dynamic equilibrium can exist. Analysis of the elastic space continuum under dynamic equilibrium conditions shows that stress and strain tensor components need not be constrained to be symmetric. As such, Hooke’s law of elasticity reduces to a simple form, and the dynamic equilibrium equations of elasticity lead to the standard wave equation in displacement vector components. We have shown an equivalence between magnetic vector potential A and displacement vector U in the elastic space continuum. Through this equivalence, all electromagnetic fields can be interpreted as stress–strain fields in the elastic space continuum.

In Section 2, we developed a spherically symmetrical strain wave solution for electron/positron-type charge particles. This is the lowest order solution of vector wave equation in spherical polar coordinates in terms of displacement vector components u^r, u^θ, and u^ϕ. The solution consists of a central standing strain wave core of approximately 2 fm radius surrounded by a radially decaying field of propagating phase waves. These phase waves are found to propagate outward at the speed of light for the positron and propagate inwards at the speed of light for the electron. These phase waves do not transport any strain energy but characterize the interaction properties of electrons and positrons by their characteristic direction of propagation. Approximately 37.3% of the total mass energy of the electron is contained in its wave field, and the remaining 62.7% of its mass energy is contained in the central standing wave core.

In Section 3, we derived the Coulomb interaction between two electrons or positrons by taking effective displacement vector field components, positive for outward propagating phase waves and negative for inward propagating phase waves. The interaction energy density between two electrons was first computed by superposing the effective strain tensor components Sⁱ_j of their respective wave fields in a common coordinate system and then computing the total interaction energy of their combined fields. This interaction energy is found to be inversely proportional to the separation distance between the centers of two interacting electrons, thus verifying Coulomb’s law of electrostatics. Comparing the total strain energy of electrons with their known mass energy, we obtain an oscillation frequency of displacement and strain components as approximately 6.4 × 10²² Hz and corresponding oscillation time period of approximately 1.5 × 10⁻²³ s or approximately 15 ys. Electron core radius R₀ is found to be approximately 2.0 × 10⁻¹⁵ m or 2 fm.

Since the oscillating displacement vector components u^r and u^ϕ are in time quadrature, total displacement vector U appears to continuously rotate or spin at all points of the core and the field of the electron. This spinning motion of displacement vector U around an axis parallel to the polar axis constitutes the intrinsic spin of the electron, and the polar axis constitutes its spin axis. The intrinsic spinning motion of vector U is accompanied by the spinning motion of the associated strain field, giving rise to intrinsic strain energy vortices throughout the electron structure. These strain energy vortices produce the angular momentum effects of the electron about its spin axis.

The radial stress component in the phase wave field constitutes the electrical field of electrons, and the radial lines representing the direction of propagation of phase waves represent the electric field lines of electrons and positrons. The intrinsic magnetic field B associated with the intrinsic electron spin is given by Equations 62A, 62B; it is found to be of the order of 10¹² Tesla in the core region and 10¹¹ Tesla in the near field region. In Section 4.4, we also computed the magnetic field induced by the uniform motion of electron- or positron-type particles and verified the Biot–Savart law for induced magnetic fields. In Section 4.5, we showed that motion-induced modulation waves, with wave length proportional to the inverse of momentum, are superposed over a radially decaying phase wave field. This can be viewed as the origin of De Broglie matter waves as modulation waves superposed over characteristic phase waves of the electrostatic field.

References

• 1.

  ChristovCI (2009). On the nonlinear continuum mechanics of space and the notion of luminiferous medium. arXiv:0804.4253v3. 10.48550/arXiv.0804.4253

  • CrossRef
  • Google Scholar

• 2.

  SandhuGSDhindsaIS. Strain-wave structure of a photon and the electromagnetic field. ResearchGate (2025). 10.13140/RG.2.2.23770.81602

  • CrossRef
  • Google Scholar

• 3.

  DeclanT. Wave functions for the electron and positron. Appl Phys Res (2022) 14(1). 10.5539/apr.v14n1p59

  • CrossRef
  • Google Scholar

• 4.

  BurinskiiA. Kerr–newman electron as spinning soliton. arXiv:1410.2888v1 (2014). 10.48550/arXiv.1410.2888

  • CrossRef
  • Google Scholar

• 5.

  VassalloG. Charge clusters, low energy nuclear reactions and electron structure. J Condensed Mat Nucl Sci (2025) 39(2025):220–40. 10.70923/001c.134007

  • CrossRef
  • Google Scholar

• 6.

  ConsaO. Helical solenoid model of the electron. Prog Phys (2018) 14(2):80–9. Available online at: https://fs.unm.edu/PiP/PiP-2018-02.pdf (Accessed June 13, 2025).

  • Google Scholar

• 7.

  DavisBS. Zitterbewegung and the charge of an electron. arXiv:2006.16003 (2020). 10.48550/arXiv.2006.16003

  • CrossRef
  • Google Scholar

• 8.

  DeclanT. Charged particle attraction and repulsion explained: a detailed mechanism based on particle wave-functions. Appl Phys Res (2022) 14(No. 2):2022. 10.5539/apr.v14n2p38

  • CrossRef
  • Google Scholar

• 9.

  SandhuGS. Dynamic electron orbits in atomic hydrogen. J Mod Phys (2023) 14:1497–525. 10.4236/jmp.2023.1411087

  • CrossRef
  • Google Scholar

• 10.

  FleuryMJJRousselleO. Critical review of zitterbewegung electron models. Symmetry (2025) 17:360. 10.3390/sym17030360

  • CrossRef
  • Google Scholar