Mechanical Analogies for the Lorenz Gauge, Particles and Antiparticles
| Scientific Paper | |
|---|---|
| Title | Mechanical Analogies for the Lorenz Gauge, Particles and Antiparticles |
| Read in full | Link to paper |
| Author(s) | Valery P Dmitriyev |
| Keywords | electromagnetic fields, turbulent ideal fluid, Reynolds equations, Maxwell's electromagnetic equations |
| Published | 2000 |
| Journal | Apeiron |
| Volume | 7 |
| Number | 3-4 |
| No. of pages | 10 |
| Pages | 173-183 |
Read the full paper here
Abstract
An exact analogy of electromagnetic fields and particles can be found in mechanics of a turbulent ideal fluid with voids. The system is supposed to form a fine dispersion of voids in the fluid. This microscopically discontinuous medium is treated as a continuum. The turbulence is described in terms of the Reynolds stresses. Perturbations of the homogeneous isotropic turbulence are considered. For the high-energy low-pressure turbulence they are usually small. This entails the linearization of the Reynolds equations. The latter appear to be isomorphic to Maxwell's electromagnetic equations. The Lorenz gauge expresses the slight effective compressibility of the medium. A particle can be viewed as a cavity in the medium. A respective antiparticle is modeled with an agglomerate of the medium's material. Microscopically, these correspond to some nonlinear vortex formations in the "vortex sponge" which are of the cyclone and anticyclone type.
Overview
Valery P. Dmitriyev (Lomonosov University, Moscow) develops a mechanical substratum model of electromagnetism and of matter, published in Apeiron in 2000. The medium is an inviscid fluid in a state of developed turbulence, containing a fine volume dispersion of empty space — microscopically a "vortex sponge" of hollow vortex tubes threading the fluid in all directions, treated for calculation as a continuum with variable volume density V(x,t). Physical fields are perturbations of the background turbulence; particles are discontinuities of the medium.
The paper is a refinement of Dmitriyev's earlier incompressible-substratum model. Its distinctive claim is a correspondence between the gauge conditions of electrodynamics and the kinematics of the medium: the Coulomb gauge corresponds to an incompressible substratum, while the Lorenz gauge expresses a slight effective compressibility, arising because empty space dispersed through a microscopically incompressible fluid can be redistributed. A second claim follows from the same feature: because the density can deviate both above and below its background value, the model can represent particles and antiparticles symmetrically — as a cavity and as an agglomerate of the medium's material, "cyclone" and "anticyclone." Dmitriyev is careful to call the whole thing a mesoscopic description and states plainly that the self-organisation and stability of such a medium cannot be derived.
The argument
The turbulent substratum
Following the Reynolds scheme, velocity and pressure are split into mean and pulsation parts, u = ⟨u⟩ + u′ and p = ⟨p⟩ + p′. The ground state is homogeneous and isotropic:
- V0 = const, ⟨u⟩0 = 0, ⟨p⟩0 = const, ⟨u′iu′k⟩0 = c2δik,
so that c is fixed by the intensity of the background turbulence. The regime of interest is "low-pressure, high-energy": p0/V0 « c2. Averaging the Euler equation gives the first Reynolds equation, and multiplying by u′l and symmetrising gives the second, an infinite chain of moment equations. Integration of the first for isotropic incompressible turbulence yields what Dmitriyev calls a Bernoulli-like equation of state, V⟨u′1u′1⟩ + p = V0c2 + p0.
Maxwell's equations from linearised Reynolds equations
Small perturbations about the background satisfy δp/V0 « c2, δ⟨u′u′⟩ « c2, δ⟨u⟩ « c, which allows the two Reynolds equations to be linearised. Writing π = p/V0, the first becomes ∂tδ⟨ui⟩ + ∂kδ⟨u′iu′k⟩ + ∂iδ⟨π⟩ = 0. Differentiating the second gives a wave-type equation for δ⟨u⟩.
The dictionary is then imposed, with κ "an arbitrary constant":
- A = κcδ⟨u⟩, φ = κδ⟨π⟩, Ei = κ∂kδ⟨u′iu′k⟩, ji = κgi/4π.
Under this dictionary the two linearised Reynolds equations take the form of the two inhomogeneous Maxwell equations in potential form,
- (1/c)∂A/∂t + ∇φ + E = 0, (1/c)∂E/∂t − ∇×∇×A + 4πj = 0.
Dmitriyev notes that for a plane electromagnetic wave in an incompressible substratum the turbulence energy density is unperturbed, δ⟨u′1u′1⟩ = 0 — the wave carries a redistribution of Reynolds stress, not a change of total intensity.
The Lorenz gauge
The gauge condition is derived from mass balance rather than from the dynamics. Averaging the continuity equation gives ∂tV + ∇·(V⟨u⟩) = 0. Dmitriyev then writes δp = β2δV for the density-wave speed β and supposes that the turbulence-perturbation wave and the density wave propagate together, so that β = c and δp = c2δV. He remarks that thermodynamically this reads as the ideal-gas relation with c2 = kT. Linearising the continuity equation and substituting gives ∂tδπ + c2∇·δ⟨u⟩ = 0, which under the dictionary is exactly
- (1/c)∂φ/∂t + ∇·A = 0,
the Lorenz gauge. He draws a physical consequence: if the Lorenz gauge holds, an electrostatic field is accompanied by a slight variation of substratum density, and so "the scattering of a neutral particle by the electrostatic field should be expected." The density-perturbation wave is offered as a model of the photon.
Cavitons: proton, electron, neutron
An empty bubble in the incompressible fluid cannot fill with vapour, so equilibrium is reached instead by perturbing the turbulence at its wall. Outside the core, the perturbation falls off as
- δ⟨u′1u′1⟩ = π0R/|x − x′|,
a Coulomb form. This is Dmitriyev's model of the proton and its electrostatic field. The electron is the opposite object — "an islet of the quiescent fluid" — generating δ⟨u′1u′1⟩ = −c2re/|x − x′|.
The perturbation energy δU = ½∫Vδ⟨u′iu′i⟩ d3x is infinite for a Coulomb field. Requiring that the proton's positive divergence and the electron's negative one cancel gives the relation
- π0R = c2re,
so that in the low-pressure regime π0 « c2 the electron core radius is far smaller than the proton's. The two infinities do not cancel exactly; the finite remainder is identified with the energy of the neutrino. The neutron is a non-equilibrium cavity, and the bookkeeping is set by the observed decay n → p + e− + ν̅.
Particles and antiparticles
Antiparticles are obtained by mirroring the density and energy profiles about their asymptotes: the antiproton is an inclusion of lowered-energy fluid. Dmitriyev checks the scheme against annihilation, particle + antiparticle → photons (+ neutrinos and antineutrinos). The perturbation energies are exactly opposite, so δU+ + δU− = 0; the electromagnetic energies are equal and positive, ε+ = ε− > 0, summing to 2ε, "this finite quantity corresponds to photons"; and the density deviations integrate to zero, so with mass defined as m = ∫fδV d3x with f− = −f+, the masses cancel too.
The localised electron violates the linearisation condition, so a delocalised version is considered, split into N = c2/π0 "splinters," each with the same core radius but a field N times weaker. Dmitriyev observes that structurally the positron resembles his proton and the electron his antiproton; he takes the sharp density jump at the nucleon core boundary as an indication of internal structure and its absence in the electron as indicating "the absence" of internal structure.
The vortex sponge
The closing section gives the microscopic picture, credited historically to John Bernoulli Jr.: a random heap of hollow, randomly oriented straight vortex tubes, whose mean filament length per unit volume L sets the discreteness of particles and charges. A particle is a closed vortex formation enclosing empty space — a loop on a vortex filament for the neutron at rest. A torsional (helical or kink) wave on a filament models the electromagnetic wave; an axisymmetric, area-varying wave along a tube is proposed as a model of the gravitational wave. Dmitriyev cites Kelly's derivation of vacuum electromagnetics from ideal-fluid properties and Friedwardt Winterberg's Planck-aether model as related work, and opens the section by conceding that neither the self-organisation nor the stability of the system can be derived.
Assessment
This is careful continuum mechanics, not hand-waving. The Reynolds decomposition, the moment hierarchy and the linearisation are all standard and correctly executed, and Dmitriyev is scrupulous about flagging which steps are suppositions. He also gets a point of nomenclature right that most textbooks get wrong: the gauge condition (1/c)∂φ/∂t + ∇·A = 0 is due to Ludvig Lorenz, not H. A. Lorentz, and the paper spells it accordingly.
The central observation is genuinely elegant and, as far as it goes, exact. The Coulomb gauge ∇·A = 0 has the same form as the incompressibility condition ∇·u = 0, and the Lorenz gauge has the same form as a linearised continuity equation. Reading the choice of gauge as a statement about the medium rather than as a bookkeeping convenience is a real insight, and it delivers a testable-sounding consequence — that an electrostatic field should scatter a neutral particle, because it is accompanied by a density variation.
There are, however, three kinds of difficulty. The first concerns what has actually been derived. The dictionary produces the two inhomogeneous Maxwell equations in potential form; the two homogeneous ones are then identities of the potential representation and are not independent results, so the "exact analogy" is narrower than the abstract suggests. More awkwardly, the electric field is defined as Ei = κ∂kδ⟨u′iu′k⟩ — a divergence of a symmetric second-rank tensor with six independent components, mapped onto a three-component vector. No argument is given that the remaining components decouple or are unobservable. And κ is stated to be "an arbitrary constant," which means the correspondence is structural only: nothing in the model fixes the value of the elementary charge, of the fine-structure constant, or of any other electromagnetic quantity.
The second concerns the step that produces the title result. The Lorenz gauge follows only after β = c is assumed — the density-wave speed set equal to the turbulence-perturbation speed. Dmitriyev calls this "the supposition," and it is doing all the work: with any other β the continuity equation yields a gauge-like condition with the wrong coefficient. The paper's headline correspondence is thus asserted at the one point where it might have been derived from the mechanics.
The third concerns the particle models. That a Coulomb-form 1/r field emerges from a cavity's boundary condition is attractive, but the associated energy diverges, exactly as the classical self-energy of a point charge does, and the divergence is handled by cancelling the proton's +∞ against the electron's −∞. The difference of two divergent integrals is not defined without a regulator, and the regulator chosen determines what the "finite remainder" is; assigning that remainder to the neutrino gives it no computable value and no way to be wrong. The relation π0R = c2re is a stipulation of that cancellation, not a prediction. To the model's credit, the inequality it yields — an electron core far smaller than the proton's — does point the right way: the proton charge radius is measured at about 0.84 fm while the electron shows no structure down to 10−18 m.
Against measurement, the model is silent where it most needs to speak. It offers no route to the proton-to-electron mass ratio of 1836.15267, none to the neutron–proton mass difference of 1.293 MeV that permits free-neutron decay with a lifetime of about 879 s — the very reaction used as a constraint in section 7 — and, most seriously, no place for Spin. A cavity in a fluid has no obvious spin-½ structure, and the electron's g-factor, 2.002319304362, is known to twelve significant figures and is the single sharpest test any model of the electron faces. Nor is there any account of what actually causes the decay reaction the model invokes: the weak interaction appears only as an energy-balance constraint. The composite structure of the nucleon in terms of quarks — established by deep inelastic scattering — is gestured at as a "sharp jump of the density" rather than modelled.
Finally, an inviscid substratum with a wave speed set by its own background turbulence intensity reintroduces a preferred rest frame, and the paper does not address how Lorentz invariance is recovered or how the Michelson–Morley experiment null result and modern rotating-resonator bounds on light-speed anisotropy — below 10−17 — are to be reconciled with it. That omission is common to the whole family of mechanical-aether models, and it is the standing obstacle they have to clear before their structural analogies can become physics. Within its own declared scope — a mesoscopic analogy, with stability and self-organisation explicitly not derived — the paper is honest and internally consistent, and its gauge-as-kinematics observation deserves to be better known.