Weber-type Laws of Action-at-a-Distance in Modern Physics
| Scientific Paper | |
|---|---|
| Title | Weber-type Laws of Action-at-a-Distance in Modern Physics |
| Read in full | Link to paper |
| Author(s) | Thomas E Phipps |
| Keywords | current elements, Ampere?s law, Lorentz (Biot-Savart) law |
| Published | 1990 |
| Journal | Apeiron |
| Volume | 1 |
| Number | 8 |
| No. of pages | 18 |
| Pages | 18-35 |
Read the full paper here
Abstract
Recent data indicate that the law of action between electric current elements proposed by Ampere is notably superior to the Lorentz (Biot-Savart) law in its ability to describe laboratory observations of currents flowing in single circuits. Ampere's law conforms to Newton's third law and thus cannot be covariantly expressed. Since all field theories of retarded action violate Newton's third law in describing nonstatic situations, it appears that the observational evidence in question weighs against all field theories as applied to the description of force actions. A reexamination of force instant action-at-a-distance modes of description is therefore indicated. We investigate here the possible revival of such a formulation proposed by W. Weber before 1850. The virtues of this approach are (a) mathematical simplicity, (b) rigorous conformity to Newton's third law, and (c) agreement with Ampere's law of action between current elements — hence with the observations just mentioned. Two different "modernizations" of Weber's approach are examined, dependent on whether energy or force methods are viewed as more fundamental in mechanics. Implications for plasma physics are touched upon.
Overview
This is an early Apeiron paper by Thomas E. Phipps, Jr., and it is the theoretical companion to the experimental work on longitudinal Ampère forces that he and Peter Graneau were pursuing at the time. Its starting point is a nineteenth-century embarrassment that Phipps says has been "so smoothly assimilated" into twentieth-century physics as to go unremarked: Maxwell's field equations do not by themselves give the mutual actions of charges, and must be supplemented by a separate intercharge force law. Since fields are not observable but charges and forces are, he asks whether the field can be dispensed with entirely for the description of force — keeping it, if at all, only for radiation.
The paper's departure from the mainstream is not incremental. Phipps accepts the timelike invariant of relativistic kinematics, c2dt2 − dr2 = c2dτ2, as empirically confirmed by muon-lifetime experiments (he cites Bailey et al., 1977), but only as a statement about happenings on a single particle's worldline. What he rejects outright is spacetime symmetry: the spacelike invariant obtained by changing the algebraic sign, and everything that follows from it — length contraction, the Minkowski representation, the "elsewhere," and above all the relativity of simultaneity. In their place he puts a kinematics in which the spacelike invariant is simply Euclidean length, with an operational distant simultaneity established by his "V* transport" clock-synchronization method. That restores an invariant "now," which is what a genuine action-at-a-distance force law requires.
The argument
Why Newton's third law is the crux
Phipps' empirical lever is Newton's third law of action and reaction, understood strictly as equality, oppositeness, collinearity and instantaneousness. Nobody, he writes, has ever gone into a laboratory and observed a violation; it is still relied on "on alternate Tuesdays to refute perpetual motion schemes," yet the rest of the week relativists have "very little use" for it. The doctrines that displaced it — universal covariance and causal retardation at speed c — were founded solely on far-zone (radiation) evidence and then extended to near-zone force actions, an extension he says has "no greater weight of empiricism behind it today than did Ptolemaic doctrine in its time."
Two lines of evidence are marshalled against retarded force. The first is the Ampère longitudinal force, repeatedly observed by Graneau and others at magnitudes sufficient to explode wires and buckle railgun rails, and by Phipps and Phipps at currents and frequencies low enough to exclude heating or induction. Ampère's law of action between current elements obeys Newton's third law and therefore cannot be covariantly expressed; its existence "breaks" spacetime symmetry, and, Phipps notes acidly, this has been known since Ampère's own day. The second is gravitation: no gravitational aberration has ever been detected, and Laplace concluded from celestial mechanics that any retardation of gravity's action would require a propagation speed of not less than 108c. Mach's Principle, he adds, cannot even be coherently stated without distant simultaneity.
Weber's law
Wilhelm Weber (1804–90) is presented as "the first and (until recently) last true relativist," in that he expressed the action between two charges purely in terms of their scalar separation r and its time derivatives, with no reference to any external frame. The velocity in the later Lorentz Force law is by contrast relative to an observer — what O'Rahilly called a "schesic" velocity, requiring a third body. Weber's force law is
- FW = (ee′/r2)[1 − (1/2c2)(dr/dt)2 + (r/c2)(d2r/dt2)]
in e.s.u., directed along the instantaneous intercharge line, and derived from a velocity-dependent Weber potential VW = (ee′/r)[1 − (1/2c2)(dr/dt)2]. Weber showed that applied to a two-fluid conductor model his law yields Ampère's ponderomotive law between current elements. Phipps notes that the two-fluid model is not physically correct, but reports that Paul Wesley has shown that the correct model — mobile negative electrons against a fixed positive lattice — also gives the Ampère law when the Weber force is used. The Weber potential and force law can therefore be regarded as observationally confirmed at least to order c−2.
The Helmholtz objection and the modernized potential
Helmholtz objected in 1872 that the negative sign in Weber's law permits nonphysical negative-mass behaviour at relative speeds exceeding c; the objection went unanswered in Weber's lifetime. Phipps' remedy is to replace the bracketed expansion with a square root,
- V = (ee′/r)√(1 − β2), β = (1/c)(dr/dt)
whose r-derivative gives a force law containing a 1/(1 − β2) factor and not subject to negative-mass effects, with the relative charge velocity dr/dt explicitly restricted to less than c. Both agree with Weber's originals to order c−2 and depart at higher order precisely so as to remove Helmholtz's difficulty. A limiting relative velocity between any two bodies composed of charges is thus made explicit in both the force law and the potential.
Two "derivations", and a choice between them
Section 3 offers what Phipps carefully puts in quotation marks as a derivation, from an energy postulate. Taking τ as the source charge's proper time and t as laboratory time, he postulates that the product of the potential energy with the corresponding proper-time differential is invariant, Vdτ = V′dτ′, justified "both from charge symmetry and from dimensional considerations" and by analogy with the invariance of an energy-time product (if Vτ = h, then the product is invariant). Since the primed charge is at rest in the laboratory, V′ is the Coulomb energy; combining with the timelike invariant gives exactly the modernized potential, and the Coulomb law is recovered when dr/dt = 0.
Section 4 pursues the older, Newtonian view that force rather than energy is fundamental, defining an invariant force Finv = m0d2r/dτ2 and relating it to the laboratory force by Flab = γ−1Finv. This yields Flab = (ee′/r2)√(1 − β2) — which differs from the energy-based result by the omission of the d2r/dt2 term. Phipps' "present guess" is that the energy-based law is correct and the force-based one wrong, since the latter is not obviously derivable from a potential and the situation is certainly conservative; he adds that this casts doubt on a chapter of his own book Heretical Verities, and insists the issue "should be settled by experiment, not by guesswork."
Testability and plasma physics
Because conduction-electron drift speeds are of order millimetres per second, the c−2 order is the only one probeable with solid conductors; it is Weber's original law that presents itself for testing, and Phipps thinks the combination of (dr/dt)2 and d2r/dt2 terms should be verifiable with modern sensitive detection, though separating their individual effects is harder. Charges in vacuum could probe the higher-order distinction between the two laws.
The paper's sharpest practical claim concerns plasma. Where charges move in closed loops the Ampère and Lorentz laws are predictively equivalent; otherwise they are "not even approximately equivalent." Plasma is precisely the domain where charges interact without necessarily moving in closed loops, so any plasma calculation using the Lorentz force must implicitly violate Newton's third law at the level of charge-on-charge action. On that criterion, Phipps writes, "despite billions poured into tokamaks," plasma physics "has not yet begun," and he asks that someone with supercomputer resources recompute a known plasma configuration with the Weber law and compare.
He also records the disagreement among his allies. Graneau holds that the Ampère law applies only to metallic conductors, the Lorentz law being preferable for charges in vacuum; Phipps, with Paul Wesley and Andre K T Assis, holds that a Weber-type law applies "either universally or not at all," as does Newton's third law. A third school (Jean-Pierre Vigier and Rambaut) accepts noncovariant forces but denies that they overthrow Einstein's physics. Phipps closes with a Beerbohm paraphrase — "You cannot make a physicist by standing a sheep on its hind legs. But by standing a flock of sheep in that position you can make a crowd of physicists" — and with the clarification that nothing here strikes at causality, only at the idea that all effects must be preceded by speed-c retarded causes.
Assessment
The paper's real strength is that it identifies a genuine and rarely stated structural fact: Maxwell's equations do not determine the force between charges, the supplementary force law is a separate empirical commitment, and Ampère's and Lorentz's forms of it are experimentally distinguishable in exactly the situations where the current path is not closed. That is a legitimate open question, not a rhetorical one, and Phipps states its test conditions precisely. His treatment of the Helmholtz objection is also a real technical contribution rather than a debating point — the square-root modification removes the negative-mass pathology while preserving agreement with Weber at the only order then measurable, and it makes the limiting relative velocity explicit rather than imposing it. He is unusually candid about the weak points, flagging his own Section 4 as probably wrong, conceding that a schesic (frame-dependent) treatment is unavoidable once more than two charges or any real apparatus is involved, and admitting that the radiation objection to dispensing with fields is "well taken" and "not decisive" only if the reader chooses to make it so.
The difficulties are correspondingly clear. The central derivation is asserted, and Phipps says so by putting "derivation" in quotation marks: the postulate Vdτ = V′dτ′ has no independent motivation beyond symmetry and dimensional plausibility, and since it is precisely what generates the desired square-root potential, the argument is closer to a reverse-engineering than to a derivation from first principles. Worse, the two routes — energy-based and force-based — start from equally plausible premises and give different laws, and the paper resolves the conflict by the author's "present guess." That is honest but it leaves the main result resting on a preference.
The empirical case is also narrower than the rhetoric suggests. The Ampère-versus-Lorentz question turns on wire-fragmentation and railgun experiments whose interpretation was contested at the time and remains so; Phipps himself cites Christodoulides, Jolly and Ternan as dissenters, and the standard counter-analysis is that for closed circuits the two laws give identical net forces, so that observed longitudinal effects must be attributed to internal stresses rather than to a failure of the Lorentz law. Nothing in the paper resolves that, and no calculation with an open-circuit configuration is actually carried out.
The largest gap is the treatment of relativity. Phipps keeps the time-dilation invariant while rejecting the spacelike one, and the rejection is argued mostly by ridicule — the sign-change analogy with tachyons, the "flock of sheep" — rather than by confronting the experiments that bear on the spacelike side. The Ehrenfest paradox is invoked to show that length contraction lacks universality, but the paper does not address the Michelson–Morley null result, the Kennedy–Thorndike experiment, or the Ives–Stilwell measurements, all of which conventionally constrain the combination of length and time behaviour rather than time behaviour alone. Nor does it engage the electron-optical and accelerator practice in which the retarded (Liénard–Wiechert) fields are used quantitatively and successfully. A Euclidean-space kinematics with an invariant "now" is a coherent programme, and Phipps had developed it at length in Heretical Verities (1987), but this paper takes it as given rather than defending it, so a reader who does not already accept that book will find the electrodynamics resting on unexamined ground. Within its own terms — as a proposal for a testable modernization of the Weber force, with a specific experimental programme attached — the paper is careful and well posed, and the plasma-physics suggestion at the end is a concrete challenge that could in principle be settled by computation.