The Faraday Induction Law In Covariant Ether Theories
| Scientific Paper | |
|---|---|
| Title | The Faraday Induction Law In Covariant Ether Theories |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | Faraday induction law, covariant ether theories, Lorentz ether theory, Thomas-Wigner rotation |
| Published | 2004 |
| Journal | Apeiron |
| Volume | 11 |
| Number | 2 |
| No. of pages | 21 |
Read the full paper here
Abstract
The non-invariance of the Faraday induction law, revealed in [1] through calculation of an e.m.f. along a mathematical line, was further analyzed for integration over a conducting closed circuit within special relativity theory [2]. Now this problem is considered within the framework of covariant ether theories [3]. A physical meaning of the non-invariance of the Faraday induction law is revealed, and a possible experimental scheme for measuring an absolute velocity of Earth has been proposed.
Overview
This is the third instalment of a sequence Alexander Kholmetskii published in Apeiron and Physica Scripta between 1997 and 2004, and it is the one that draws the sharpest conclusion. In the earlier papers he argued that the Faraday induction law ε = −d/dt ∫B·dS does not follow from the Maxwell–Faraday equation and is not Lorentz invariant; that the non-invariance shows up physically, not merely for "mathematical lines", when the induced e.m.f. receives no contribution from the internal fields of the conduction electrons; and that a concrete arrangement exists in which the two cases can be separated — a charged parallel-plate condenser in relative motion with respect to a conducting loop one side of which lies inside it.
Here Kholmetskii re-analyses that arrangement in the framework he calls covariant ether theories (CETs), of which Lorentz ether theory (LET) is the simplest member. His claim is strong and specific: worked in special relativity, the problem yields an e.m.f. in one inertial frame and no e.m.f. in another, which he regards as a violation of causality; worked in LET, the e.m.f. is present in both. The difference traces entirely to how the two theories read the Thomas–Wigner rotation. Since the discrepancy is a first-order-in-v effect on a measurable voltage, he ends by proposing a bench-top experimentum crucis — which, he notes, "rejects a wide-spread opinion that these two theories are indistinguishable at the experimental level."
The argument
Covariant ether theories
CETs keep three of special relativity's foundations — homogeneity of space-time, isotropy of space, causality — but replace the Einstein relativity principle with the general relativity principle, which permits a preferred frame K0. In K0 the geometry is pseudo-Euclidean with a Galilean metric. Because motion cannot alter the geometry of empty space, it stays pseudo-Euclidean for every moving observer, but the metric tensor in a moving frame is no longer Galilean: the space-time is given an oblique-angled metric.
The methodological pivot of the paper follows: in an oblique-angled metric the true (physical) values differ in general from the values measured in experiment. Kholmetskii therefore carries two sets of four-vectors, (xi)ph and (xi)ex, related by a matrix B that depends only on absolute velocity, and shows that whatever admissible transformation A the physical four-vectors obey, the measured four-vectors always obey the Lorentz transformation L, with B = A(v)L−1(v). Different choices of A give different CETs; the general principles do not fix it.
Choosing A = G, the Galilean matrix, reproduces Lorentz ether theory exactly. One obtains an absolute time dilation by √(1−v2/c2), an absolute contraction of moving scales along the velocity (the FitzGerald–Lorentz hypothesis), and the Galilean addition law for the physical light velocity — the full set of Lorentz's postulates in modern form. Crucially, transformations between two moving frames K and K" never use a direct relative velocity: Nature composes v1 ⊕ v2 through the absolute frame, and a rotation-free Lorentz transformation between them is therefore impossible — the Thomas–Wigner angle Ω appears in the measured coordinates while being absent from the physical ones.
The condenser-and-loop problem
A rectangular conducting loop lies with its long side AB inside a charged flat condenser, its leads entering through tiny holes so the internal field is undisturbed. K1 rides with the loop, K2 with the condenser; in K0 the loop moves at v along x and the condenser at {v,u}. Since the field transformations are the same in both theories when K0 is taken as absolute, the fields in K0 to order c−2 are
E0x = −uvE/2c2, E0y = E(1 + v2/2c2), B0z = vE/c2.
A magnetic field therefore exists in K0, and the area ABDC threaded by it shrinks with time, so dΦ/dt ≈ −uvEL/c2 and the induction law demands an e.m.f. Computing it directly from ε = ∫(E + v×B)·dl requires the geometrical term DB − AC = Lφ = uvL/2c2, arising because scale contraction tilts the condenser plates through φ = uv/2c2 relative to x0. The result is ε = uvEL/c2, in full agreement with the flux rule.
Transforming to the loop frame K1, the magnetic field vanishes (B1z ≈ 0) and only E1x ≈ −uvE/2c2 survives. In SRT the observer in K1 sees the Thomas–Wigner rotation Ω ≈ uv/2c2 as a genuine spatial rotation of the condenser — and a rotating charged condenser induces no e.m.f. in a loop passing through it. Formally, DB − AC reverses sign, DB − AC = −uvL/2c2, and ε = 0. Kholmetskii calls this "quite contradictory": the e.m.f. exists in K0 and disappears in the laboratory frame K1.
Thomas–Wigner rotation as an illusion
The paper's most original section argues that in LET the rotation is not real. In the physical space-time of LET the transformations are rotation-free Galilean ones, so no spatial rotation occurs. What the K1 observer measures is the joint artefact of absolute scale contraction and the anisotropy of the physical light velocity in a moving frame (c+ = c − v, c− = c + v).
Kholmetskii spells out the measurement. Two clocks Cl1 and Cl2 separated by L along x1 record when the moving axis x2 touches them; with the true angle φ ≈ −uv/2c2 the true interval is Δt = −Lv/2c2, so Cl1 fires first. Each clock emits a light pulse toward a time analyser midway between them. Because of the light-speed anisotropy the analyser records
Δτ = −Lv/2c2 + L/2(c−v) − L/2(c+v) ≈ +Lv/2c2,
of the opposite sign. The observer, who does not know his frame's light velocity is anisotropic (the measured velocity is c in every CET), infers a positive inclination and concludes that both x2 and y2 are turned through the same positive Ω — that is, that the frame has simply rotated. Kholmetskii regards it as "an important advantage of LET" that the rotation is thereby explained without invoking any torque, whereas in SRT it is unexplained; he adds that Thomas precession, usually cited as its experimental confirmation, "finds alternative explanations in LET."
The consequence for the induction problem is immediate. In LET calculations one must use the true position of the condenser plate, for which DB − AC = +uvL/2c2 as in K0, giving ε = uvEL/c2 in K1 as well. The e.m.f. exists for every inertial observer and causality is restored.
The proposed experiment
A charged condenser is oscillated harmonically, y = y0sinωt, with the side AB of a multi-turn loop, at rest in the laboratory, passing through it. In SRT the relative velocity u is collinear with E, no magnetic field arises, and no e.m.f. is induced — ever. In LET the laboratory itself moves at absolute velocity v, the motion diagram of Fig. 2 applies, and Eq. (25) gives εm = ωx0vUL/l0c2 = 2ωLvU/c2. With ω = 6×102 Hz, U = 4 kV, L = 0.2 m, plate gap l0 = 2 mm, amplitude x0 = 1 mm and v ≈ 10−3c, this is about 0.8 µV per turn; 100 turns and a gain of 103 bring it to roughly 80 mV, "which can be easily measured by oscilloscope." If CETs are right, the amplitude should show sidereal and annual variation as the Earth's absolute velocity swings.
Assessment
The paper is admirably concrete. Rather than arguing about interpretation, Kholmetskii identifies one geometrical quantity — the sign of DB − AC, the tilt of the condenser plate relative to the loop — on which the two theories are made to disagree, and then converts that sign into a laboratory voltage with a full numerical estimate. That is exactly the right shape for a foundational dispute, and his signal estimate is honest about the amplification needed. His physical account of the Thomas–Wigner rotation is also genuinely illuminating on its own terms: in SRT the rotation is a group-theoretic fact with no mechanical story attached, and Kholmetskii's derivation of it from scale contraction plus one-way light-speed anisotropy at least supplies a mechanism, which is more than the orthodox treatment offers. The framework's core structural result — that whatever the physical transformation A, the measured four-vectors obey Lorentz — is a clean statement of why the Michelson–Morley class of null results does not by itself dispose of an ether.
The difficulties, however, are serious and largely internal. The paper's own construction guarantees that measured four-vectors transform by L in every CET; the proposed experiment measures a voltage, which is a measured quantity. It is not shown anywhere in the paper that the observable e.m.f. can differ between LET and SRT once that theorem is applied consistently — the difference is introduced by choosing to use the "true" plate position in the LET calculation and the "illusionary" one in the SRT calculation, and the licence for that step is asserted rather than derived. This is the paper's weakest joint, and everything downstream, including the experiment, hangs on it.
The charge of a causality violation in SRT is also overstated. That an e.m.f. exists in one frame and vanishes in another is the ordinary frame-dependence of a field split, not a causal paradox; causality concerns the ordering of events with timelike separation, and no such ordering is inverted here. What SRT does require is that the physically detectable quantity — the voltmeter reading — be the same in every frame, and if Kholmetskii's own SRT calculation yields ε = 0 in K1 and ε ≠ 0 in K0 for the same voltmeter, the natural inference is an error in one of the two calculations rather than a defect in relativity. The paper does not pursue that possibility, nor does it check its K0 result against a direct computation of the force on the conduction electrons in K1.
Against the proposed experiment stand the modern one-way anisotropy bounds — the cryogenic optical resonator and rotating Michelson–Morley experiments of the 2000s, which constrain fractional anisotropy of the speed of light to the 10−16–10−17 level, and the Ives–Stilwell-type limits on preferred-frame effects. Kholmetskii would answer that these test measured light velocity, which his framework holds to be isotropic by construction, and that answer is consistent — but it also means the anisotropy doing the work in Section 3 is, in every other context, unobservable in principle. A theory whose distinguishing ingredient is undetectable except through one first-order-in-v geometric term in one particular circuit is doing a great deal of work through a very narrow channel. Finally, the paper reports no attempt to perform the experiment, and as far as the record shows it has not been performed. That leaves an interesting proposal in the same condition as the objection it answers: undecided.