Fitzgerald Contraction, Larmor Dilation, Lorentz Force, Particle Mass and Energy as Invariants of Galilean Electrodynamics

Scientific Paper
Title	Fitzgerald Contraction, Larmor Dilation, Lorentz Force, Particle Mass and Energy as Invariants of Galilean Electrodynamics
Read in full	Link to paper
Author(s)	Horst E Wilhelm
Keywords	Maxwell equations, Fitzgerald contraction, inertial frames
Published	1994
Journal	Apeiron
Volume	1
Number	18
No. of pages	12
Pages	9-19

Read the full paper here

Abstract

By means of the generalized, Galilei covariant Maxwell equations for inertial frames Σ(r,t,w) with substratum velocity w, Fitzgerald contraction = o 1- - co 2 2 1 2 bv wg of rods, Larmor dilation t = t o 1- - co 2 2 1 2 bv wg of clock periods, and velocity dependence of particle mass m = mo 1- - co 2 2 1 2 bv wg are shown to be Galilei-invariant vacuum substratum effects, where v - w = v°= inv is the respective object velocity relative to the substratum frame Σ (r°,t°,0). The Lorentz force transferred through the substratum is Galilei-invariant, F = e[E°+ w × B + (v – w) × B ] = e(E° + v° × B°) = inv. The kinetic energy K(v°) of high-velocity particles is given by the Galilei-invariant mass-energy relation K(v°) + Eo = mo(v°) co 2 , where Eo = mo co 2 (mass-energy equivalence). The Galilean measurement process in inertial frames Σ(r,t,w) is explained considering physical length contraction of measuring rods and rate retardation of measuring clocks, as well as synchronization of clocks in absolute time. Crucial experiments underlying Galilean electrodynamics are discussed briefly.

Overview

Horst E. Wilhelm, then at the Department of Materials Science and Engineering at the University of Utah, wrote this 1994 Apeiron paper as one instalment of a long programme on what he calls G-covariant (Galilei-covariant) electrodynamics. The programme rests on two assumptions he regards as experimentally supported: that Maxwell's equations hold in their ordinary form in one particular frame, the frame of the electromagnetic wave carrier Σ°, and that the Galilean transformations relate the space and time coordinates of all inertial frames. From these he had earlier derived a generalised, Galilei-covariant form of Maxwell's equations for a frame Σ(r,t,w) moving with substratum velocity w. The present paper applies that apparatus to five specific quantities: rod length, clock rate, the Lorentz Force, particle mass, and kinetic energy.

The recurring result is the same in every case. Each quantity depends not on the object's velocity v relative to the observer, as special relativity has it, but on the combination v − w = v°, the object's velocity relative to the substratum — and because v − w is unchanged by a Galilean transformation, each quantity is an invariant with a single objective value that all observers agree on. So a rod moving relative to the ether really is shorter, by a definite amount; a clock moving relative to the ether really does run slow, by a definite amount; and a fast particle really is more massive. Wilhelm's central complaint against special relativity is not that these effects are wrong but that relativity makes them observer-relative, which he regards as physically incoherent: "the observer does not interact with the rod". His most memorable formulation of the objection turns Einstein's own polemic against the Copenhagen interpretation back on him — when a "mouse" travelling at half the speed of light looks at the universe, where would it get the energy to contract the radii of the galaxies?

For evidence that a preferred frame exists at all, Wilhelm points to the 2.7 K microwave background of Penzias and Wilson and to the dipole anisotropy found by Conklin (1969) and Henry (1971): a frame in which the background is isotropic is an absolute frame, and terrestrial frames, which see it anisotropic, are not. He gives the terrestrial substratum velocity as w ≈ 3 × 10⁵ m/s. He also cites Sagnac's 1913 effect, the Aharonov-Bohm effect, the dielectric Cerenkov effect and anomalous induction in homopolar generators without relative motion between conductor and magnet as further support.

The argument

Length contraction from a moving charge's potential

The derivation begins from the potential of a charge in uniform motion in a frame with ether velocity w. Its equipotential surfaces are ellipsoids of revolution about the direction of v − w = v°, shortened along that axis by the factor [1 − (v − w)²/c_o²]^1/2. Since matter is built of nuclei and electrons, the contraction of their equipotential surfaces produces, in equilibrium, a Fitzgerald contraction of the body itself:

ℓ = ℓ_o[1 − (v − w)²/c_o²]^1/2 = inv

A rod has its greatest length when at rest in the ether. Wilhelm then works out the general case of a rod at angle θ to v°, obtaining tan θ = tan θ_o[1 − v°²/c_o²]^−1/2 and a length formula reducing to ℓ_o[1 − v°²/c_o²]^1/2 at θ = 0 and to ℓ_o at θ = π/2. The contrast he draws with relativity is the reciprocity paradox: two identical rods in frames Σ and Σ′ each measure the other as shorter, and special relativity "cannot decide which of the two identical rods is longer or shorter". In the substratum theory the two rods have different velocities relative to the ether and therefore different, definite lengths.

Clock retardation and synchronisation

Following Jánossy, Wilhelm builds a light clock from two mirrors on a rod. In the ether frame light propagates isotropically at c_o; in a frame with ether velocity w it does not, and

c(φ) = (c_o² − w²sin²φ)^1/2 + w cos φ

so that c(0) = c_o + w, c(π) = c_o − w and c(π/2) = (c_o² − w²)^1/2. Combining the anisotropic up-rod and down-rod light speeds with the angle-dependent contracted rod length, the two-way period comes out independent of the clock's orientation:

ν = ν_o[1 − (v − w)²/c_o²]^1/2 = inv

Only a clock moving relative to the wave carrier is retarded; a clock with v = w runs at the maximum rate ν_o. Wilhelm treats this as the necessary condition for absolute time, t = t′ = t° = inv, and it disposes of the twin paradox in a definite way: "the twin who moves faster relative to the ether ages slower". Synchronisation of distant clocks is straightforward in the ether frame by the usual light-signal exchange; in a frame with ether flow it requires w to be known in magnitude and direction, except for the special case of a baseline perpendicular to w, where the two-way travel time suffices.

Lorentz force and potentials

The force on a charge is written first as F = e(E + v × B), which appears observer-dependent, and then rearranged as

F = e[E + w × B + (v − w) × B]

in which E + w × B = E°, B = B° and v − w = v° are each Galilei-invariants. The Lorentz force is therefore "an ether excitation", the same for all inertial observers. Recasting in terms of the potentials, Wilhelm shows Φ − w·A = Φ° and A = A° are likewise invariant, and that the "generalized" electric field E + w × B has a more fundamental status than the ordinary E. He adds the polemical remark that field forces in an empty vacuum without a carrier reveal "a lack of understanding of elementary physics".

Electromagnetic mass

A charged particle moving through the substratum excites a field whose energy grows as (1 − v°²/c_o²)^−1/2. Starting from a rest-frame charge density ρ_o, contracted along the motion, Wilhelm reduces the wave equation for the potential to a Poisson equation, obtains a self-similar solution with similarity variable ζ° = γ(z° − v°t°), and integrates the field momentum c_o⁻²∫E° × H° d³r°. Assuming spherical symmetry at rest, this gives p° = m°(v°)v° with

m°(v°) = m_o[1 − v°²/c_o²]^−1/2, m_o = 4U_o/3c_o²

He notes candidly that the notorious factor 4/3 becomes 1 once the stresses holding the charge together are included, citing Fermi (1923), and that the spin field has been neglected. Both p and m are Galilei-invariants because v − w is.

Kinetic energy and mass-energy

Differentiating m²(c_o² − v°²) = m_o²c_o² and integrating the work integral gives

K = mc_o² − m_oc_o², K + E_o = mc_o², E_o = m_oc_o²

with mass-energy equivalence obtained without relativistic space-time, a derivation he attributes in principle to Lewis (1908). The important and unusual step is the definition K = K(v°): kinetic energy is measured relative to absolute space. Wilhelm draws the consequence explicitly — K reduces to ½m_ov°², not to the classical ½m_ov² of the observation frame, and the two agree only when w ≪ v ≪ c_o. He justifies this by arguing that giving a body kinetic energy in the laboratory means accelerating it against the gravitational field of all the masses of the universe, an appeal to Mach's Principle in all but name.

The Michelson-Morley null result

The paper's most interesting single calculation closes the loop. In a frame with ether flow, a measuring rod is contracted and a measuring clock is retarded, so the measured distance Δz_m and measured two-way time Δt_m are not the true values. Correcting both and inserting into the two-way travel time Δt = 2Δz c_o/(c_o² − w²) gives, exactly,

c_o = 2Δz_m/Δt_m

The two-way light speed comes straight out of the raw readings "without any corrections whatsoever", because the contraction and retardation cancel identically. The algebra is correct as stated. Wilhelm's gloss is that Michelson and Morley hit on this formula "instinctively", and that relativists misread the null result: from the non-observation of an effect in one experiment one cannot conclude the effect does not exist.

The conclusions restate four "non-relativity" theses — of space, time, mass and velocity — and note that in quasi-ether frames with negligible w the formulae reduce to the relativistic ones, which is why "the STR gives approximately correct results on the Earth". The paper ends with the G-invariant equation of motion d[m(v − w)]/dt = e[E + w × B + (v − w) × B].

Assessment

The technical work here is competent and, on the points that can be checked, correct. The angular formulae for rod contraction and anisotropic light propagation are right; the light-clock period does come out orientation-independent; the electromagnetic-mass derivation is a clean rederivation of the classical Lorentz-Abraham result, and Wilhelm is honest about both the 4/3 factor and the neglected spin field; the Michelson-Morley cancellation reproduces exactly. The paper also has real virtues of exposition — it lays out an internally coherent Lorentzian ether theory with its measurement conventions stated openly, which is more than most ether papers manage, and it correctly identifies clock synchronisation as the place where the one-way light speed enters.

Its central weakness is a confusion between two quite different senses of "preferred frame". A frame in which the microwave background is isotropic certainly exists; so does a frame in which the interstellar medium is at rest, or the Earth. None of this bears on special relativity, whose postulate concerns the form of the laws, not the distribution of matter. A universe filled with a fluid picks out a rest frame for the fluid without making Maxwell's equations frame-dependent. Wilhelm's claim in his introduction that the Penzias-Wilson and Conklin-Henry results "alone refute" special relativity does not follow, and nothing later in the paper repairs the inference.

The second difficulty is that the paper's own results undercut its main thesis. Equation (68) shows that rod contraction and clock retardation cancel exactly in a two-way light-speed measurement, so that no such measurement can reveal w. That is precisely the empirical equivalence between Lorentzian ether theory and special relativity that Builder and Jánossy had concluded, and which Wilhelm dismisses as "strange" in his introduction on the grounds that v − w ≠ v for all frames. But that is a statement about the theory's internal variables, not about anything measurable. The remaining first-order effect — anisotropy of the one-way light speed — is exactly the quantity that cannot be measured without first synchronising distant clocks, and Wilhelm concedes that synchronisation in a frame with ether flow requires w to be known in advance. The circularity is his own, and he does not address it.

Third, the redefinition of kinetic energy as K(v°) has a consequence the paper does not pursue. Since |v − w|² = v² − 2v·w + w², the difference between K and the classical ½m_ov² is a term linear in the momentum plus a constant. For non-relativistic laboratory processes this is harmless — momentum conservation carries it along automatically, which is why nothing has ever been noticed — so at low velocities the redefinition is empirically empty rather than wrong. At high velocities it is not harmless at all. Because velocities compose Galileanly, v° = v − w can exceed c_o, and m°(v°) then becomes imaginary. The theory therefore places a hard singularity at a laboratory speed of c_o + w in one direction and c_o − w in the other, a difference of about 600 km/s on Wilhelm's own value of w. Counter-circulating beams in the same storage ring would then reach different limiting energies, and would do so with an annual modulation as the Earth's velocity through the substratum changes. This is measurable and has been measured: the LEP electron and positron beam energies, determined by resonant depolarisation to about one part in 10⁵ of 45 GeV, were the same for both directions, and the storage-ring measurements of the muon anomalous magnetic moment agree with the relativistic time dilation factor γ ≈ 29.3 for muons circulating in both senses. No directional asymmetry of the required size appears.

Fourth, the recurring rhetorical argument — that the observer "does not interact" with the rod, so cannot affect it — misstates what relativity claims. Relativity does not say the observer changes the rod; it says length is a relation between an object and a frame, in the way that the x-component of a vector is a relation between the vector and a choice of axes. Rotating one's axes does not shorten a stick, and no energy is required to do it. The mouse polemic is vivid but rests on this misreading, and repeating it does not make the reciprocity of length contraction a contradiction: the two observers are measuring different things, namely the separation of events that each judges simultaneous.

Fifth, the supporting experimental list is thinner than presented. The Sagnac effect is a rotation effect and is predicted by special relativity for a rotating (non-inertial) apparatus with no ether required; the Aharonov-Bohm effect is a quantum phase effect involving the potentials and has no bearing on a substratum; unipolar induction is a long-standing puzzle of interpretation rather than of prediction, since the standard treatment reproduces the observed EMF. Wilhelm asserts, but does not here demonstrate, that G-covariant electrodynamics will explain the Fizeau and Hoek results; that is promised for a later paper.

What survives is worth stating plainly. As a Lorentzian ether theory — the tradition of Lorentz, FitzGerald, Larmor, and in this wiki's own literature Herbert E Ives, Paul Wesley and Ludwik Kostro — Wilhelm's construction is consistent and reproduces the standard formulae exactly in the substratum frame and to excellent approximation elsewhere. Anyone who finds the operational reading of relativity philosophically unsatisfying can adopt it and lose nothing at ordinary velocities. What it does not do, on the evidence assembled in this paper, is produce a single quantity that differs measurably from the relativistic prediction in a way that has been confirmed — and where it does differ measurably, in the high-velocity regime where Galilean composition breaks the γ factor, the measurements go against it.