On the Kinetic Origin of Mass
| Scientific Paper | |
|---|---|
| Title | On the Kinetic Origin of Mass |
| Read in full | Link to paper |
| Author(s) | Henrik Vilhelm Broberg |
| Keywords | singularities, Rotating rings, cosmic background radiation |
| Published | 1993 |
| Journal | Apeiron |
| Volume | 1 |
| Number | 15 |
| No. of pages | 11 |
| Pages | 10-20 |
Read the full paper here
Abstract
Rotating rings (or toroidal plasma filaments) are associated with singularities in space-time in two limits: when the rotation velocity approaches c and zero, respectively. The resulting geometry incorporates a constant relationship between surface and enegy for the kinetic energy of the rotation, as well as the rest-masses of the rings. This relation provides the missing link that unifies the electric and gravitational forces. The geometry generates configurations similar to vortex rings in hydrodynamics, and leads to a model of elementary particle masses. An interpretation of the thermodynamics of de Broglie is given in which the cosmic background radiation and other cases of radiation in space are identified as due to energy exchanges between particles and the vacuum.
Overview
Broberg builds elementary particles out of rotating charged rings in a subquantum medium, in the toroidal ring tradition he shares with Bergman, Wesley and Carroll, and grafts onto it a thermodynamic reading of de Broglie's "hidden thermostat". The organising idea is a proposed new constant of nature, A, the ratio of interaction surface to energy for any self-contained quantum oscillator. Broberg treats A as fixed for all such systems — "a stability constraint, comparable to a radiationless condition" — and takes the surface to be the equivalent of the probability density of the Schrödinger equation, comparable to energy confinement in a Schwarzschild singularity or to bag confinement of quarks.
From this single constant, together with μ0, e, h and c, he claims to obtain the masses of the electron, muon, both pions, the neutron and the proton "to four significant figures"; the fine structure constant as a ring-counting parameter; a Coulomb-form gravitational law inside the electron system that is offered as "the missing link that unifies the electric and gravitational forces"; a value for the Hubble constant; and a temperature for the cosmic background radiation of about 2.8 K. He is explicit that the model is built in four dimensions, that "Newton's 'constant' is not at all the only constant of its kind", and that his approach rejects the arbitrary normalisation used, for example, for the Higgs field.
The argument
The surface-energy constant
For a thin ring of rest mass m0 rotating at vr, the swept surface is φ = π(vrτ/2π)2 and the kinetic energy m0(γr − 1) in mass terms; their ratio is defined as A. Two singular limits follow for a fixed period τ: as vr → 0, Am0 → 2π(cτ/2π)2; as vr → c, Amk = π(cτ/2π)2. The two differ by a factor of two, and it is this factor that Broberg identifies with the internal spin factor s = ½.
A translating ring traces a spiral, generating a tube of N loops with vr2 + vL2 = c2, the spiral carrying charge while the rings carry mass. The photon is the vr → 0 case, modelled as a sphere Lorentz-contracted to a flat disk, and its surface relation defines the quantum volume V0 = Ah/c, taken as fundamental for all quantum oscillators, with Am·cTc = V0 in general.
The electron and the value of A
Taking the toroid's stored magnetic energy E = ½μ0(iN/L)2V and eliminating the radius, Broberg obtains
m = (1/4π)·{[μ0e2]2/(A(1 + 1/4N)2)}1/3
and identifies it with the electron mass, "made up entirely of magnetic energy" when N is large — the electric potential energy of the loops falling off as 1/√N and being cancelled by the magnetic tension. Matching to me ≈ 9.108 × 10−31 kg gives A ≈ 0.70 m2/kg ≈ 1.25 × 10−26 cm2/MeV, and he states that "the above numerical value for A will be used in the following for all particles and fields". The associated radius is R ≈ 2.25 × 10−16 m and the tube length L ≈ 1.413 × 10−15 m.
Particle masses
Each particle is a different loop count and folding of the same geometry.
- Muon — the electron geometry reduced to one of the four surface components, with a single loop, so that both electric and magnetic fields contribute: mμ = (h/2c)·[(1 + ¼)/Aμ0e2]1/3/√5 = 1.883 × 10−28 kg (105.55 MeV).
- Charged pion — two quanta of Compton time equal to the circumference of the Schwarzschild-like singularity of the electron system: mπ± = h/(πc(Aμ0e2)1/3) = 2.489 × 10−28 kg (139.5 MeV).
- Neutral pion — a tube whose length and circumference are both cτ: mπ0 = (2h2/Ac2)1/3 = 2.407 × 10−28 kg (135.0 MeV).
- Neutron — three pairs of interacting rings, A divided by six as "a surface manifold of sixth order": mn = 6(πh2/Ac2)1/3 = 1.6785 × 10−27 kg (941.716 MeV); deducting four electron masses gives 939.67 MeV.
- Proton — three pion-like quark structures each carrying e/3: mp = 1.672 × 10−27 kg (938.07 MeV).
Force unification and the fine structure constant
Defining Ge = Ac2/Re as the analogue of Newton's constant inside the electron system and inserting the model's own expressions gives R2F = c2μ0e2/4π when N → ∞, so that "the 'gravitational' or attraction force law in the electron system is of equivalent form and magnitude to Coulomb's Law". Broberg writes the two laws in one form, F = Q1Q2/4πε0R2 ≡ Ge(n1me)(n2me)/R2, and generalises to an energy density ρg = ρ0(Φ1/D2)(Φ2/D2) in which the normalised particle surfaces are read as a probability that a force quantum interacts with both. The Schwarzschild counterpart of the electron system, 2Geme/c2, works out to 2L = μ0e2/4πme, the classical electron radius.
The fine structure constant then emerges as a ratio of ring geometry: α = r/2R·(1 + 1/4N)1/2 = 4πL/λC = λC/2πRB = 4πRB/λRyd, and also α = N−1/2(1 + 1/4N)1/2, so that α "is a parameter relating a number of consecutive rings in the electron-system to each other".
Hubble constant, graviton and the 2.8 K background
Requiring a disk of Planck radius to sweep one quantum volume gives cT = Ac2/2πG = 1.5 × 1026 m (16 × 109 light years), "in good agreement with the accepted value of the so-called Hubble radius", whence H = 2πG/Ac ≈ 2.00 × 10−18 s−1 and a Hubble time of 1.6 × 1010 years. The corresponding minimal quantum — a candidate graviton — has mass hH/c2 ≈ 1.47 × 10−68 kg, which "has a positive or a negative value, depending on whether the Planck radius is imaginary or not", a negative-mass "hole" in the vacuum transferring negative momentum and hence attraction. The corresponding universal mass field is M0 = π(cT0)2/A ≈ 1053 kg, compared with 1011 galaxies of 1011 solar-mass stars.
Finally the "vacuum thermostat": with vacuum density ρ0 = 2πG/A2c2, an energy inflow dme/dt = cρ0Ame through the absorbing surface, and an absorbing-to-radiating surface ratio Φabs/Φrad = α2/4, the Stefan-Boltzmann law gives T ≈ 2.8 K. Broberg suggests the background could equally arise from spontaneous creation and absorption of electron-positron pairs in a Dirac ether, that a photon meeting a vacuum "hole" loses one elementary quantum — so cosmological redshift should be enhanced along lines of sight through galaxies — and that stellar radiation partly comes from the same vacuum exchange, which would explain the low solar neutrino rate.
Assessment
Broberg's arithmetic is correct throughout, and the check is worth setting out because it also shows where the content lies. Recomputing from his own formulae with modern constants: μ0e2 = 3.2257 × 10−44; the electron formula with A = 0.70 gives 9.082 × 10−31 kg; R = 2.249 × 10−16 m and L = 1.4132 × 10−15 m, matching his 2.25 × 10−16 and 1.413 × 10−15. The muon expression returns 1.8836 × 10−28 kg = 105.66 MeV against a measured 105.658; the charged pion 2.489 × 10−28 kg = 139.63 MeV against 139.570; the neutral pion 135.06 against 134.977; the neutron 942.0 MeV, which after his four-electron subtraction is 939.97 against 939.565; the proton 938.60 against 938.272. cT = 1.500 × 1026 m, H = 1.998 × 10−18 s−1, graviton mass 1.473 × 10−68 kg = 8.26 × 10−39 MeV, M0 = 1.01 × 1053 kg, and the background temperature 2.786 K — every printed figure reproduces. This is careful work, and the mass agreements at the few-parts-per-thousand level across six particles are not nothing.
What the check also shows is that the load is carried by A, and that A is not new.
A is the classical electron radius in disguise. Solving the electron formula for A gives A = μ02e4/(4πme)3, or equivalently A = re2/4πme with re = μ0e2/4πme, and numerically 0.69369 m2/kg. It is a rearrangement of e, μ0 and me, containing no information those three do not already carry. Broberg's derived quantities are correspondingly familiar: R = re/4π, L = re/2, and his "Schwarzschild radius of the electron system" 2L is exactly re, which the paper itself states.
The unification is an algebraic identity in which A cancels. Substituting Ge = Ac2/Re together with his own Re and me into Geme2 gives c2μ0e2/4π, and since μ0c2 = 1/ε0 that is e2/4πε0 exactly — the two agree to every decimal place, as they must, because A drops out of the product. Nothing is being unified: Ge was defined so that Geme2 would equal the Coulomb constant. Newton's G plays no part in this equality; it enters the paper only later, in the Hubble and temperature relations. Similarly, α = re/λ̄C = 4πL/λC is the textbook relation between the classical radius and the reduced Compton wavelength, restated in ring language rather than derived.
The "four significant figures" comes from A, not from the geometry. The model's real, A-independent predictions are dimensionless mass ratios, and these can be extracted in closed form. Using α = μ0ce2/2h, the muon formula reduces to mμ/me = (π/α)(5/4)1/3/√5 = 207.40, against the measured 206.768. The charged pion reduces to the strikingly simple mπ±/me = 2/α = 274.07, against 273.132. The proton gives 6(11.25)1/3/α = 1840.9 against 1836.15, the neutral pion 4π(2α2)−1/3 = 265.4 against 264.14, the neutron 24π(π/4α2)1/3 = 1849.9 against 1838.7. Every one is high, and all but the neutron are high by very nearly the same 0.26 to 0.34 per cent — exactly the shift produced by using A = 0.70 rather than the electron-exact 0.69369, since all masses scale as A−1/3. In other words, Broberg rounds A up by 0.9 per cent, which brings the muon, pion, proton and neutron into four-figure agreement in MeV at the cost of putting the electron — the system from which A was obtained — 0.28 per cent low. The four-figure agreement is a property of the rounding, not of the model. The honest statement of the result is that the ring geometry predicts mπ± = 2me/α and mμ = (π/α)(5/4)1/3me/√5, each accurate to about three parts in a thousand. That is interesting; it is not four figures. The neutron additionally needs an unexplained subtraction of four electron masses, introduced only because the geometry overshoots by 2.4 MeV.
The cosmological relations are Dirac-type coincidences. Because A = re2/4πme, the Hubble relation is H = 8π2Gme/re2c — a combination of G, me, e and c with nothing cosmological in it, in the family of large-number coincidences that runs from Eddington and Dirac to Weinberg's relation. It yields 61.7 km/s/Mpc, about ten per cent below the Planck and SH0ES determinations of 67-73, which is genuinely close; but such combinations are numerous, and matching one of them is weak evidence. The 2.8 K result deserves the same caution for a specific reason: T enters as a fourth root, so agreement to two per cent in temperature corresponds to agreement only to about eight per cent in the underlying quantity ρ0Φabs/Φrad, and the factor α2/4 in that ratio is fixed by a choice of which surface radiates. The measured value is 2.7255 K.
Three conflicts with measurement. First, the claim that the low solar neutrino rate is explained by radiation from vacuum exchange has since been removed as a problem: the Sudbury Neutrino Observatory measured the total all-flavour solar neutrino flux by the neutral-current channel in 2002 and found it equal to the standard solar model prediction, the deficit being flavour oscillation. Second, the prediction that cosmological redshift "should be enhanced when electromagnetic waves pass through areas of space with strong gravitational interactions, such as in galaxies" is testable and is not seen; redshifts of galaxies and of the quasars behind them track distance, not intervening mass. Third, the background radiation cannot be a superposition of vacuum-exchange emission from scattered sources: COBE's FIRAS instrument found it to be a blackbody at 2.7255 K to about 50 parts per million, and no distributed emission process reaches that.
Broberg is more candid than most about the status of his work, and this should be recorded. He writes that "this analysis is mainly an empirical comparison between geometric concepts and particle energies, and consequently does not conform in detail with the quark model", that the mass formulae "must be seen as approximations" because the one-way vibration velocity has been set to c, and that the charge and spin relations of the quanta have not been analysed. Read as what he says it is — an empirical geometric numerology in the toroidal ring tradition, one that does produce two clean dimensionless relations for the pion and muon masses in terms of α alone — it is a more interesting paper than its abstract's claim to have found "the missing link that unifies the electric and gravitational forces", which the algebra does not support.