Electron, Universe, and the Large Numbers Between

Scientific Paper
Title	Electron, Universe, and the Large Numbers Between
Read in full	Link to paper
Author(s)	Manfred Geilhaupt
Keywords	Electron, Reimann, Dirac, Planck, Gravity
Published	2008
No. of pages	24

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Abstract

We show how to calculate mass and charge of the single free electron and present a simplified model of the particle having finite size and shape from internal dynamics. The full internal structure reveals Reimann's differential geometry. The model readily accounts for the electron's duality nature as well as other particle characteristics (i.e., spin, mass, charge, and magnetic moment). The model goes further by identifying a new Large Integer Number (1022) liken to but more basic than Dirac's ?Large Number?. We show how this new Large Number might be explained by theory rather than by numerology or mere coincidence. The electron model and the new Large Number predict an ultra-low quantum of mass (the massquant) radiated/absorbed by the electron's internal and external dynamics. Furthermore, the model reveals an elegant symmetry between the micro and the macro universe and gives also a solution for quantum gravity due to many body electrons. At least there is an important application in techniques due to energy management and energy storage in the hydrogen atom's electron.

Overview

The paper is a joint work by Manfred Geilhaupt (Hochschule Niederrhein, Germany) and John Wilcoxen (USA). Its starting point is a complaint Geilhaupt attributes to Einstein and repeats throughout: "Eine Theorie, die Ladung und Masse a priori setzt ist unvollständig" — a theory that puts in the charge and mass of the electron by hand is incomplete. In both quantum mechanics and in Boris Unrau's mass-operator extension of the Klein-Gordon equation, which the authors discuss in the introduction, "the rest mass of the electron remains an input parameter." Their aim is a theory whose only inputs are h, c and G — through the Planck mass and Planck charge — and from which the electron's rest mass and elementary charge come out.

The proposed mechanism is thermodynamic. The authors treat the internal dynamics of a single free electron with its centre of mass at rest as a "quasi internal Carnot cycle", so that the Second Law applies to a single particle: the rest mass cannot be constant in time, because the cycle must pay an "energy tribute" back to what they call, after de Broglie, the subquantique milieu. Formally the theory is described as Quantum ThermoDynamics (QTD) combined with general relativity, Maxwell's electrodynamics and quantum operators, under a single extra hypothesis: "Only the laws of nature will not change in space or time", i.e. the laws do not depend on the space-curvature parameter λ.

From this the paper derives an extended electron structure, a value for the fine structure constant, a "large integer number" N = 1×10²² from which Dirac's large number follows as N²/24, an ultra-light "mass-quant" of about 5.2×10⁻⁵¹ kg said to be exchanged as gravitational radiation, and a set of scaling relations connecting the electron to the observable universe. The four claims the authors themselves put "in short" at the head of the paper are: the local scalar and vector functions of the free electron are time dependent; the electron has an internal time-dependent structure; the electron decays; and charge is related to rest mass and space changes.

The theory

The simplified electron model

The model is built from an action radius r_G read off the electron's spin, m·c·r_G = ½ℏ, together with a periodic action time t_G from Hamilton's total action h = m·c²·t_G. The same t_G serves both the circular and the linear half-quantum of action, "because internal mass actions simply need to be completed at the same time."

The resulting picture (Figure 1) is emphatically not a point particle but "a spinning and oscillating 3-dimensional energy-volume due to internal action defining its rest energy." There is an inner oscillating nucleus of radius R_G, an outward-stretched mass shell of radius r_G, and outside that a spinning charge "skin" at radius ½g_sr_G, with g_s the electron g-factor; the electrodynamics of the spinning skin gives the magnetic moment. The authors report that the gravitational-electromagnetic mass tensor works out to T_GE = m'c²/(2π³r_G) = 8.178×10¹⁶ J m⁻³, a form they read as "indicat[ing] a torus shape of the restmass". Wave-particle duality is then not a duality of description but of parts: the electron "appears to be nearly a point mass due to nucleus... oscillation acting together with the outward shell", while oscillation and spin together supply the matter wave of wavelength 2πr_G.

Several relations are listed as consequences of the model:

internal mass rotational velocity V_r = c/2
internal mass vibration velocity V_o = cα
velocity of light c = 4π(r_G/t_G)
classical electron radius r_e = (2α)r_G
Bohr radius r_B = (2/α)r_G
Bohr orbit velocity v_B = cα = (c/2)(2α)

The last of these, relating the electron's internal motion to its bound orbit in hydrogen, leads the authors to suggest that "this electron model applies to the bound as well as the free electron", with (2α) appearing "as a coupling constant between the electron and proton of the hydrogen atom's action."

Equation of motion and the two differential equations

Sections 3.1 and 3.2 rehearse, at length and with long German quotations from Einstein's Grundzüge der Relativitätstheorie, the passage from pre-relativistic to relativistic to general-relativistic equations of motion. The authors' point of departure is that in all of these the rest mass is treated as constant: "Here m(t) is new aspect concerning elementary particles like the electron which Einstein did not discuss in his book."

They note that Einstein explicitly excluded any dependence of a body's properties on its history, on the grounds that otherwise there could be no sharp spectral lines. Their reply is that once an internal Carnot cycle is assumed, the rest mass cannot be constant, because the decay emission is not exactly balanced by regenerative absorption — and it is not, "because gravity is only attractive." They accept that this makes the theory testable: "The experiments should find the change of mass state conditions on a sub-microscopic and on a large astronomical time scale... This, of course, will be the central test to verify this theory!"

Setting the external force and the connection terms to zero for a free electron at rest, and combining the Coulomb force from Maxwell's electrodynamics with the Newtonian force recovered from general relativity, they obtain two coupled differential equations for the internal radial motion r(t) with m = m(r(t),t). A damped oscillatory solution is proposed,

f(t) = f₀·e^−a(t)·N·cos{ω₀(t − t₀) + δ₀},

with a(t) = (t − t₀)/τ the accumulated heat loss per cycle divided by E₀, and τ a finite lifespan.

Results: mass, charge and the fine structure constant

The stated outputs are:

rest mass m = (1/N)√(παh'c/(24·24G))
charge e = √(2h'cαε₀)
fine structure constant, zero order, α₀ = (3/4)[1 − a(t)]² = 1/137.112

The decay parameter a(t) is set equal to ln(3), which the authors tie to the dimensionality of space: "We assume that the 3-dimensional space is a GR restriction and the only convenient one applied to nature." With a relativistic correction factor β = 1/√(1 − V_i²/c²) using an internal velocity V_i = αc, this improves to α₁ = 1/137.031, and with a further metric factor X₄₄ = 1/g₄₄ = 1.00001796 it reaches α = 1/137.035998. The authors are candid that this is not a closed derivation: "we can not present a complete theory calculating all these parameters (α, N, and τ), especially N, uniquely from a closed theory", and they use the CODATA value of α for the subsequent numerical work. Section 3.3's overall claim is that the set of equations matches CODATA values "to within 8 digits."

The large number N and the mass-quant

From the ratio of the Coulomb to the Newtonian force of the electron's internal action the authors obtain f_C/f_G = N²/24, and setting this equal to the textbook external two-electron force ratio 4.166×10⁴² gives N = 1×10²². The point they draw from this is that Dirac's number is recovered as N²/24 = 4.1666×10⁴², but "the large number N is derived from internal action of a single electron, whereas Dirac's is from external interaction of two electrons" — which they take to suggest "that perhaps Coulomb and Newton forces come from the same physics in micro and macro space."

Because the rest mass goes as 1/N, a change ΔN implies a quantum of mass Δm. This mass-quant m_Q is given as roughly 5.224×10⁻⁵¹ kg (about 2.93×10⁻¹⁵ eV), with an ultra-low internal rotational frequency ω_Q ≈ 4.45 Hz and an associated radiated power

P = m_Qc²ω_Q ≈ 2×10⁻³³ watts.

The authors note immediately how far this is below detection: the strain amplitude is about 10⁻⁴³, against a best gravity-wave detector sensitivity of about 10⁻²³.

Quantum gravity from the Second Law

Section 4.3 proposes that the mass-quant is the gravitational interaction. In Figure 3 an electron absorbing one mass-quant emits two by stimulated emission in the forward direction, producing a backward momentum; because gravity is only attractive there is no symmetric reverse process. "So quantum gravity of two or more electrons is a caused by the II Law (energy tribute)!" Gravity, on this picture, "is due to a certain internal thermo-dynamic friction or due to stimulated emission of mass quants into forward direction. Gravity is the back action."

The authors then reach the same 2×10⁻³³ W figure by two further routes. Applying J. P. Ostriker's formula for the gravitational radiation power of a spinning macroscopic object, P = εG(mℓ²ω₀³)²/c⁵, to their spinning torus electron gives the same order. So does an estimate for the hydrogen atom built from the proton and Bohr rotational frequencies. They summarise the coincidence as

P_electron = P_proton = P_mass-quant = P_{Hubble mass} = P_hydrogen = 2×10⁻³³ watts

and conclude that "the structure of the hydrogen atom is very much determined by the gravitational as well as the electromagnetic forces of individual particles making up the atom" — indeed, that "from gravitational radiation equations we obtain the electromagnetic properties."

Large numbers and cosmic symmetry

Section 5 reviews the large-number tradition — Weyl's 4×10⁴² ratio of the electron's electrostatic to gravitational field, Eddington's coincidences, Dirac's proposal that G varies with cosmic age (which the authors note "lost favor when new evidence was found in support of the constancy of G over time"), Jordan's 10⁶⁰, Weinberg's pion-mass relation, and the anthropic response, which "may have merit, but does not make a definitive statement about the physical laws."

They then list a series of relations in which N, the number 24 and α recur: cosmic tension over internal electron force, the mass and radius of the observable universe over the electron's, the ratio of the electron's internal rotational frequency to the Hubble constant, the Planck-to-electron mass ratio m_Planck/m_e = √N/(24α), the typical-galaxy-to-solar mass ratio 10¹¹ ≈ √N, and several Planck-era-to-today ratios equal to (2N³α/24π²) ≈ 8.06×10⁶⁰.

Using H₀ = 71 km/s/Mpc from WMAP they compute a universe radius R_u = c/H₀ ≈ 1.3×10²⁶ m, a universe mass from Mach's relation M_u = c²R_u/G ≈ 1.76×10⁵³ kg, and a minimum "Hubble mass" m_H ≈ 2.7×10⁻⁶⁹ kg. The model's own values differ from these by a factor Ω₁ = √(24/2πα) ≈ 90.08, which would require H₀ ≈ 79 km/s/Mpc, "which is outside current parameters. Why the difference? We do not have an answer to this question but we do find it most interesting that the difference is related to two of the three recurring numbers." The neatest relation offered is the geometric mean

m_e = m_Q²/m_H,

placing the electron mass exactly between the mass-quant and the Hubble mass.

Hydrogen sub-levels as an energy source

The conclusions add a practical conjecture. If N/2 and/or N/3 are possible, energy sub-levels should exist in the hydrogen atom below the ground state, with E ∝ (1/n²)(1/N); Figure 6 shows an N/3, n = 1 level at −40.83 eV, a 30.83 nm transition from the −13.6 eV ground state. The authors say a special-relativistic mass correction, assuming the internal velocity equals the first Bohr orbit velocity, shifts this to 30.39 nm — the wavelength reported by Randell Mills and P. Ray in extreme-ultraviolet spectroscopy of catalysed hydrogen plasma. They also cite a private communication about a "thermal energy device" claimed to produce 1.5 kW excess power. Their claim is that "1kg Hydrogen atoms might replace 45000 liters of oil and more."

Assessment

The paper's motivating idea is a serious and old one, and it is stated cleanly: a theory that inserts the electron's mass and charge as parameters has not explained them, and it is legitimate to ask for a scheme in which they emerge. The specific move — applying the Second Law to a single particle's internal cycle, so that rest mass acquires a time dependence and a finite lifespan — is unusual and at least has the virtue of being framed as testable; the authors say so themselves and identify secular mass drift as "the central test." The extended, spinning, oscillating structure with a charge skin displaced from the mass shell is a concrete geometrical picture that does at least reproduce the g-factor placement of the magnetic moment, and the relations r_e = 2αr_G, r_B = (2/α)r_G put the classical radius, the Compton-scale action radius and the Bohr radius in a single geometric ladder. The identification of N²/24 with Dirac's number, whatever one makes of it, is a genuinely tidy arithmetical observation.

Against that, much of the paper's structure is arithmetic rather than derivation. The authors admit that N cannot be obtained from the theory — it is fixed by setting N²/24 equal to the measured force ratio, which makes the subsequent recovery of Dirac's number from N circular rather than predictive. Similarly α is derived only to zeroth order (1/137.112, wrong in the third digit); the improvement to 1/137.035998 comes from two successive correction factors, one of which is a metric coefficient g₄₄ chosen to close the remaining gap, and after this the CODATA value of α is used as input for everything else. The identification a(t) = ln(3) is asserted from the dimensionality of space rather than derived. The claim of agreement "to within 8 digits" therefore describes a fit, not a prediction. The large-number section is frank that no complete theory of the numbers is on offer, and the Ω₁ ≈ 90 discrepancy in the cosmological quantities is left explicitly unexplained — it would require a Hubble constant the authors themselves say is outside the observational range.

Two points conflict with well-established results. First, the paper's central physical claim — that the electron continuously loses rest mass through the internal Carnot cycle, so that "the electron decays" — is in tension with the extraordinary precision with which electron mass, charge and the fine structure constant are measured to be stable. Laboratory and astrophysical searches for time variation in α have set limits far tighter than any decay of the kind described would allow, and the electron itself has no measured lifetime; the authors do not quantify their predicted decay rate against these bounds, which is the one number a reader most wants. Second, the hydrogen sub-level proposal places bound states below the ground state of hydrogen, which contradicts the ordinary Schrödinger and Dirac spectra confirmed to many digits by spectroscopy; the supporting evidence adduced is the Mills 30.4 nm work, which has not been reproduced by independent groups, plus an unpublished private claim of excess power. Readers should note that the 30.39 nm figure is not predicted cleanly either — it requires a relativistic correction applied after the fact to shift 30.83 nm onto the reported line.

Finally, the paper is hard to read as a piece of exposition. Long untranslated German passages from Einstein occupy several pages, and the promised derivations are repeatedly deferred to papers "in preparation" or to a "separate paper." The authors invite readers to reconstruct the fine structure result themselves from equations (3.22), (3.23) and (3.25), which is not a substitute for showing it.