Mass, Energy, Space
| Scientific Paper | |
|---|---|
| Title | Mass, Energy, Space |
| Read in full | Link to paper |
| Author(s) | Henrik Vilhelm Broberg |
| Keywords | Mass, Energy, Space, Singularity, Vacuum, Force, String |
| Published | 1991 |
| Journal | Apeiron |
| Volume | 1 |
| Number | 9-10 |
| No. of pages | 39 |
| Pages | 161-195 |
Read the full paper here
Abstract
This paper deals with the conceptual origin of particle mass and its relation to energy and space. It has been impossible, along the way, to avoid the question of the unification of the forces, and some results have been achieved in this area as well.
Overview
Henrik Broberg's long Apeiron paper grew out of a year (1982–83) at the Laboratoire d'Astrophysique Théorique in Paris, spent on a European Space Agency grant working with Jean-Claude Pecker "in the unorthodox surrounding of Pecker, Vigier and other followers of de Broglie, who did not take the 'standard model' for granted — least of all 'Big Bang' cosmology." The paper's organising idea is a single number. Broberg posits that a photon exchanges energy with the vacuum on every cycle, and that the exchange per cycle is exactly hH — the Planck constant times the Hubble constant. From this he constructs a photon–vacuum cross-section proportional to photon energy, a "quantum volume" of vacuum, and a universal surface-to-mass parameter A ≈ 0.7 m2/kg. He then attempts to derive the rest masses of the electron, muon, pions, proton and neutron from A, the electron charge and the vacuum permeability alone.
The departure from the standard account is on two fronts at once. Cosmologically, the cosmological redshift is not expansion but a per-cycle energy loss to the vacuum — a tired-light mechanism — and the Big Bang singularity is replaced by a possible steady state. Microphysically, particles are not point-like quanta of quantum field theory but singularities: rotating closed strings whose surface-to-mass ratio is fixed by A, wrapped just inside their own Schwarzschild-like event horizons. Broberg invokes Einstein's 1919 suggestion "that the gravitational constant ... could have another value in the system of a particle than in the space outside the particle," and takes it literally: in his scheme Newton's G is not a constant but a function of the mass of the system it applies to.
The argument
The photon–vacuum cross-section and the constant A
Hubble's law implies a photon losing energy in proportion to its own energy over an interval of time, hence an exponential law and a constant loss per cycle of hH. Broberg observes that this would be "the smallest amount of energy observed to participate in any physical process," corresponding to a wavelength of order the scale factor of the universe, "and it is, therefore, tempting to think of this as a minimal energy quantum." Defining a cross-section σγ = AE/c2 and a quantum volume Vq = Ah/c, he finds that applying the cross-section to the minimal quantum hH gives approximately the square of the Planck radius, σq ≈ 2π2Rp2, which holds when H−1 = 15 billion years. Rearranged, this is
- A = 4πG/Hc ≈ 0.7 m2/kg.
He immediately draws the consequence: "the latter expression gives G as a function of H, if these two properties are variables and the others true constants." If H is read as an inverse scale factor, it becomes "a ground state resonance frequency of the Universal space," and universes of different sizes would have different gravitational parameters. Broberg notes that a surface-to-mass ratio of about 1 m2/kg recurs across scales — for a nucleon, for the large-scale mass content of the universe, and in the surface characterising the Hubble redshift.
Strings instead of points
The electron's point-particle treatment is rejected on the familiar ground that the self-energy of a charged sphere diverges as the radius goes to zero; truncating Coulomb's law at the Planck radius gives a confined electrical energy about 1020 times the electron's mass energy, "hardly possible." Broberg's alternative is to distribute the charge as N dots along a fibre of length L. The dot–dot force is independent of N, and the sum over all dots converges as Σ1/n2, so the fibre tension is finite; a string of about 10−15 m carries an energy of order the electron rest mass. He is careful about the status of this: "This little exercise is hardly an exhaustive explanation of the electron mass — that is not the point. It does, however, show the possibility offered by a string-like geometry." He distances himself explicitly from super-string theory, which "ha[s] become increasingly complicated and now require[s] many more than the normal four dimensions."
A rotating string relates surface to mass through A. Two limits appear as singularities: as the rotational velocity goes to zero the radius must shrink to zero for the string to complete its revolution in the cycle time; as it approaches c the periphery vanishes by Lorentz contraction. The two limits differ by a factor of two in surface, which Broberg reads as a massless spin-1 wave being able to convert into an integer-spin system of double surface, which in turn collapses into two half-integer-spin particles — a mesonic intermediate and then baryons, or directly an electron–positron pair.
The electron as a toroidal coil
The electron is modelled as a thin toroid carrying a current from a charge moving in a spiral of N turns. Working out the stored magnetic energy and combining with the surface–mass relation gives, after eliminating the radius,
- me = μ02Q4/(4π3A3) ≈ 9.10 × 10−31 kg (0.511 MeV),
with string radius RS ≈ 2.25 × 10−16 m and a Schwarzschild-like system radius Rg = 4πRS ≈ 2.826 × 10−15 m. The electric energy on the loops falls off as 1/N and disappears for large N, leaving the mass entirely magnetic. Setting g = Ac4/(2πM) makes the gravitational and electrical interaction amplitudes inside the electron identical, and solving that equality returns the electron mass. Broberg concludes that "in the electron system, the strong force, the gravitational force and the electric force therefore appear to be unified," the weak force not being a force at all but "the result of decaying unstable particle systems." The associated surface force comes out at about 83 kN at R = 10−15 m, which he notes "agrees with what is assumed for the quarks by present day theories."
He also identifies a four-term ladder of radii each separated by the inverse fine structure constant: the electron event horizon, the Compton wavelength, the inner de Broglie orbit of the Bohr hydrogen model, and the Rydberg wavelength (this last needing an extra factor of two).
A microwave background without a Big Bang
If protons radiate to the vacuum at the same rate at which photons are redshifted by Hubble's law, and if the black-body surface is a sphere of the electron Compton wavelength, Broberg computes a temperature of 2.9 K — "in accordance with the observed background radiation." He suggests the background "emanates mainly from the hydrogen clouds in space."
The particle masses
The remaining masses follow from the electron geometry with simple integer or geometric factors. The neutron comes from six string components paired along three axes, giving m = 2πh2/(Ac3) ≈ 1.68 × 10−27 kg (940 MeV). Using the periphery of the electron system as a wavelength gives half the charged pion mass, so two such waves give 2.489 × 10−28 kg (139.5 MeV); the neutral pion comes out at 2.407 × 10−28 kg (135.0 MeV); a generalised pion at 137 MeV, between the two. The proton is three charged strings each of charge Q/3 and twisted twice, giving 1.673 × 10−27 kg (938 MeV) — or, as a shortcut, 139.5 × 3 × 95/(4·6·63) = 938.2 MeV. The muon, "a particle without a quark substructure ... an aborted effort to create an electron," comes to 1.883 × 10−28 kg (105.55 MeV). Broberg reports three- to four-significant-figure agreement in each case.
An appendix generalises this into a single hadron mass formula, E(N) = 86 MeV × N2/3, where N counts the waves enveloping the singularity: N = 2 gives the pion, N = 16 the η, N = 36 the nucleon group, with the meson spectrum generated by quantum-oscillator excitations on the π and η ground states.
The paper closes with speculations offered as such: inertial mass as a local vacuum-pressure effect requiring no Mach's principle; gravitons as "holes of negative energy" ejected into the vacuum when a particle absorbs a quantum; and the suggestion that the Sun's output is vacuum energy absorbed by its particles rather than fusion, since "the underground facilities for neutrino detection have failed, after more than 20 years, to detect any neutrinos released from the assumed fusion process."
Assessment
The genuinely interesting core of this paper is the numerical coincidence at its centre. That 4πG/Hc should come out at the same order as the surface-to-mass ratio of a nucleon, and that feeding that number through a magnetostatic toroid calculation should return the electron mass to three figures, is not nothing — it belongs to the same family of large-number relations that Eddington and Dirac took seriously, and Broberg is right that a relation tying G to H would make gravity a cosmological rather than a local quantity. The paper is also unusually honest about its own status: the fibre calculation is offered as an illustration, not a proof; the neutron mass formula is described as "an oversimplification"; and the final section is explicitly labelled speculation. The string-tension resolution of the electron self-energy divergence is a real idea, arrived at independently of and more simply than the super-string programme.
The difficulties are severe and mostly arise from the same source: A is fitted, not derived. Broberg sets H−1 = 15 billion years to make the cross-section equal the Planck area, then uses the resulting A to compute the electron mass. The electron mass therefore depends on the Hubble constant, and since A enters the electron mass as A−3, a 10% change in H shifts the predicted mass by about 33%. The value of H is not known to that precision even now, and was known far less well in 1991; the agreement to three figures is thus a consequence of choosing H, not a test of the theory. The same objection applies with more force to the hadron masses, where a free integer N is assigned per particle after the fact (N = 2, 16, 36) and no independent rule selects which integers correspond to real particles.
The variable-G claim conflicts directly with measurement. If G ∝ H and H is an inverse age, then Ġ/G ≈ H ≈ 7 × 10−11 per year. Lunar laser ranging to the Apollo retroreflectors constrains |Ġ/G| to below about 10−13 per year — some three orders of magnitude tighter — and Big Bang nucleosynthesis and binary pulsar timing give comparable limits. Broberg's alternative reading, that G varies with the mass of the system rather than with time, fares no better: laboratory Cavendish measurements, planetary ephemerides and galactic dynamics all use the same G across some forty orders of magnitude in system mass, and a 1040 enhancement of g at nuclear scales would be visible in precision atomic spectroscopy, where it is not seen.
The cosmological half of the paper faces the standard difficulty of tired-light models, which Broberg does not address. If the redshift is a per-cycle energy loss, distant light curves should not be stretched — yet the light curves of Type Ia supernovae are observed to be broadened by exactly (1 + z), a time dilation independently confirmed in their spectral evolution. Nor does a scattering-like interaction with the vacuum obviously preserve image sharpness at cosmological distance. The derivation of a 2.9 K background from proton radiation is arithmetically suggestive but the wrong shape of argument: the microwave background is a black-body spectrum to better than one part in 104, with an angular power spectrum showing acoustic peaks; a superposition of emission from hydrogen clouds at varied temperatures and redshifts does not produce a Planck curve of that fidelity, and the paper offers no account of the anisotropies at all.
One factual claim has since been decided against the paper. Solar neutrinos were detected before 1991 — by the Homestake chlorine experiment from the late 1960s and by Kamiokande in 1987 — at roughly a third of the predicted rate; the deficit, not the absence, was the puzzle, and it was resolved in 2001–02 by SNO's demonstration that the missing electron neutrinos had oscillated into other flavours, with the total flux matching the solar-fusion prediction. Broberg's suggestion that the Sun's energy "does not come from fusion at all" rested on a premise that was already inaccurate when written and is now closed.
Read as a derivation of the particle spectrum, then, the paper does not succeed: too many of its numbers are inputs. Read as an exploration of what follows if the vacuum is a physically active medium exchanging hH with every photon cycle, and if particle mass is a surface rather than a point property, it is a serious and mathematically careful piece of work whose central coincidence has not, to this article's knowledge, been explained away.