On the Relativistic Transformation of Force
| Scientific Paper | |
|---|---|
| Title | On the Relativistic Transformation of Force |
| Read in full | Link to paper |
| Author(s) | Alexander L Kholmetskii |
| Keywords | force transformation law, electromotive force, causality principle |
| Published | 2005 |
| Journal | Apeiron |
| Volume | 12 |
| Number | 2 |
| No. of pages | 23 |
Read the full paper here
Abstract
The paper analyzes the relativistic law of transformation of force and some accompanied physical difficulties. We focus our attention on the complex systems, consisting of a number of sub-systems i with the velocities ui in a laboratory frame. We establish an analogy between the total force in such system and e.m.f. in a closed deforming circuit with respect to the force transformation law. It has been concluded that for these systems the relativistic law of transformation of force contradicts the causality principle.
Overview
Alexander Kholmetskii's paper is a narrowly targeted internal-consistency argument against Special Relativity. He opens by rejecting the common position among critics that relativity is logically and mathematically perfect and can therefore only be defeated experimentally; his own view, developed across several Apeiron papers, is that "an intrinsic problem of compatibility of SRT with the causality principle might exist". The vehicle here is the relativistic law by which force transforms between inertial frames, applied not to a single particle but to a complex system — one composed of sub-systems moving at different velocities ui in the laboratory.
The point of attack is that such a transformation is unusual in relativistic physics. Ordinary rotation-free transformations depend on a single vector parameter, the relative velocity v of the two frames. The total force on a complex system depends on v and on the whole set of sub-system velocities ui — a multi-parametric transformation. Kholmetskii constructs cases in which the total force vanishes in the laboratory frame but does not vanish for a moving observer, verifies them independently with the Lorentz force law, and then transfers the same structure to the electromotive force in a deforming closed circuit, where he argues the contradiction with causality becomes unavoidable.
The argument
What "causality principle" means here
Kholmetskii states the causality principle (CP) in two clauses: the cause–consequence order of events is absolute, and events that can cause essential inferences (such as a collision of particles) are absolute. He prefers this formulation because "just the absolute events lie on the basis of all measurements in space-time, giving a physical interpretation to the Lorentz transformations". He then adds what he calls indirect consequences, which are the ones the paper actually uses: an electromotive force in a circuit must have the same sign for all inertial observers and must vanish simultaneously for all of them, and the same must hold for a torque on a mechanical system. These indirect requirements are not tied to relativity's measuring procedures but, he argues, must be satisfied by any correct theory of space-time — and all of them reduce to the transformation of force.
Existing difficulties with force
Before his own case he surveys the known trouble spots. The general transformation law, obtained from the space-time and energy–momentum four-vectors and applicable to any interaction, is quoted from Møller. Since force is an awkward notion in microphysics it is analysed in macrophysics, where electromagnetic forces are given by F = q(E + u × B) and where "mechanical forces" must be added because electromagnetic forces between spinless particles cannot stabilise an isolated system. He notes pointedly that the nature of these mechanical forces "cannot be clearly determined; there is only the general requirement" that they transform correctly.
He recalls the Trouton–Noble experiment, which sought a couple on a suspended charged capacitor arising from the Earth's motion and found no significant rotation, and Cornille's more recent version in which the condenser was not shielded from external electric field and a rotation was observed — attributed by others to the Earth's magnetic field. Kholmetskii's verdict is that "the problem at the whole seems non-resolved in full up to date."
He then reproduces Endean's sliding-friction paradox. A particle slides on a surface with normal force N, friction Ff = kN, over length l. Both frames agree on the friction magnitude and on the dissipated power Ffu, but disagree on the duration — l/u in the surface frame, reduced by γ in the particle frame — so they compute different total heat. Kholmetskii adds a new twist not previously considered: for an observer K′ moving along the normal at velocity v, N is unchanged, the friction force is reduced by 1/γ, and an additional normal-direction "friction" force F′y = kNuv/c2 appears, whose physical origin "is unclear". In the ultra-relativistic case with k > 1 this force can formally exceed N, which would imply the vanishing of both the friction force and N, and hence no heat dissipated at all for that observer. His conclusion is not that relativity fails here but that for a moving observer friction "cannot be simply expressed as kN" — the mechanical stress distribution at the sliding contact and the energy–momentum of the short-range normal-force field both transform, making the problem very complicated.
He lists five outstanding problems (force depending on acceleration and needing three initial conditions; transformation of mechanical versus electromagnetic momentum; field and charge-density transformations, including Jefimenko's alternative account of length contraction and time dilation; self-force and "hidden" momentum in non-radiative systems) and sets four of them aside to concentrate on the fifth: the force on a complex non-radiative system of charges moving at different velocities.
Two sub-systems: the table of four cases
The model system has sub-system 1 at rest in the laboratory frame K and sub-system 2 moving at velocity u. An external force F acts on the first and −F on the second, so the total force vanishes in K. Restricting v, u and F to two mutually orthogonal directions gives four combinations. In two of them — case 2 (v ∥ u ⊥ F) and case 4 (v ⊥ u, F ⊥ v) — the total force in K′ is not zero. Kholmetskii is careful about a subtlety: because of the relativity of simultaneity one cannot in general add forces applied to a system for different observers, so he stipulates that either the forces are static or they are applied at the same spatial point.
Verification by the Lorentz force law
He then builds explicit electromagnetic realisations so the abstract result can be checked. Two point charges +q and −q occupy the same coordinates inside a neutral insulating tube in which they can move freely along one axis; a charge +Q at rest in K supplies the external field; −q has initial velocity u. In frame K the resting Q produces only an electric field, the forces on +q and −q are qE and −qE, and the net force on the tube is zero.
In K′ the source Q moves, so it produces both E′y = E/(1 − v2/c2)1/2 and Bz = vE/[c2(1 − v2/c2)1/2]. The velocity of the positive charge is v; that of the negative charge follows the relativistic composition (u + v)/(1 + uv/c2). Computing F′ = q(E′ + u′ × B′) for each and summing gives
F′t = qEuv/[c2(1 + uv/c2)] × (1 − v2/c2)1/2
in "full agreement" with the table's case 2. A second geometry, with the field along x and the frame moving along y, reproduces case 4 and yields F′yt = −qEvu/c2. In both, the tube feels no net force in K but a non-vanishing force in K′.
Why the mechanical cases are not conclusive
Kholmetskii does not claim victory here, and this is one of the paper's better moments. The derivation tacitly assumes equality of action and reaction for the normal forces between charges and tube. That equality certainly holds in K, but in K′ "is a matter of separate analysis", because for electromagnetic interaction the relationship between active and reactive forces is frame-dependent, owing to the transformation of the field's energy and momentum. Since the normal forces arise from a short-range 1/rn (n > 2) interaction that is itself electromagnetic, its field carries energy and momentum that must be transformed too. "Since this problem is very complicated, we cannot conclude yet that the results of this section indicate a violation" of the indirect consequences of CP.
The deforming circuit
The decisive case, on his account, is electromagnetic throughout. An e.m.f. in a closed circuit is ε = ∮ f · dl, and the flux rule ε = −d/dt ∫ B · dS holds for both fixed and deforming circuits. A deforming circuit is exactly a complex system: an infinite set of segments dl(r) with velocities u(r, t), so its e.m.f. transforms multi-parametrically.
Kholmetskii inserts an important methodological point about measurement. Einstein's interpretation has each observer using his own instruments, but for Faraday's law the voltmeter is normally an inherent part of the circuit and every observer reads the same instrument. Only in the voltmeter's rest frame is the loop integral taken at a single instant; to relate an integration at fixed t′ to the actual reading one must use t′ = γ(t − v·r/c2) at constant t, so t′ differs for different points of the circuit. This is Cullwick's rule for retarded (advanced) e.m.f., and Kholmetskii notes explicitly that he had not explained it properly in his own earlier papers.
The worked example is a rectangular loop A-B-C-D whose side AB slides along BC and AD at constant velocity toward the fixed side CD, with a constant force per unit charge f along y at every point. In K the e.m.f. vanishes. Treating the loop as a resting fragment plus a moving segment puts it in case 2 of the table, and the transformed e.m.f. is
ε′ = fLu/[γ c(1 + uv/c2)] · v/c
with L the length of AB — the same result reached in his earlier paper from the Lorentz force law. Since no mechanical forces or short-range fields are involved, he takes this as a genuine contradiction with CP.
He reinforces it with a general argument from Marx's proof that the flux rule is Lorentz invariant. In the product B · dS dt, which dominates for slowly varying or constant B, only the second factor has fixed sign for all observers; the magnetic field, being a component of the field tensor, can change sign between frames. So the product, and hence the e.m.f., is an alternating quantity — whereas voltage, a quantity of fixed sign, should transform as U′ = U(1 − v2/c2)1/2-type law. The discrepancy between the two transformation laws is what he means by the non-invariance of Faraday's law; he stresses it is a discrepancy between the two equations, not a violation of the flux rule itself.
Practical caveat
Kholmetskii ends the section with an unusually frank retraction. The constant f assumed in the example cannot be realised in a real conducting circuit: a constant magnetic field will not give f = const when different parts move at different velocities, and a constant electric field will not either, because conduction electrons redistribute to cancel the internal field. "Unfortunately, such a re-distribution of conduction electrons was not correctly analyzed in the papers [24, 25], and the experimental schemes, proposed in those papers, should be essentially complicated."
Assessment
The paper is careful in ways that similar critiques often are not, and that carefulness is its main virtue. Kholmetskii distinguishes what he has shown from what he has not: he explicitly refuses to draw a conclusion from the mechanical tube examples because the action–reaction equality for the short-range normal forces has not been established in the moving frame, and he identifies exactly why (the energy–momentum of the binding field transforms too). He then corrects his own earlier publications twice — once on the retarded-e.m.f. rule, which he says was not properly explained before, and once on the redistribution of conduction electrons, which invalidates the experimental schemes he had previously proposed. He also verifies his abstract table against an independent route, the Lorentz force law applied field-by-field, rather than trusting a single formalism. The choice of target is well judged: multi-parametric transformations of the total force on an extended system genuinely are less well explored than the single-parameter case, and the "indirect" causality requirement — that an e.m.f. should not change sign between observers — is a reasonable thing to demand of a space-time theory.
The difficulties are real too. The strongest is that the objection may be a restatement of a known and accepted feature of relativistic dynamics rather than a contradiction. That the sum of three-forces on a spatially extended system is not frame-invariant when the parts move at different velocities is precisely why the relativity of simultaneity forbids naive summation — a point Kholmetskii raises himself in a footnote and then sets aside by stipulating that forces be static or co-located. In the circuit case that stipulation cannot hold: the segments are at different places and in relative motion, which is exactly the configuration where "the total force at an instant" is frame-dependent by construction. The paper does not demonstrate that the resulting non-zero ε′ corresponds to any observable that a single voltmeter would actually register, and Kholmetskii's own retraction concedes that the configuration cannot be built. A contradiction with causality that appears only in a physically unrealisable circuit, and only under a summation convention the theory itself warns against, is not yet a contradiction.
Second, the "indirect consequences of CP" are assumptions the author introduces, not consequences derived from his own two-clause statement of causality. Requiring an e.m.f. to have the same sign for all observers is a substantive extra postulate; a relativist would simply deny it, on the grounds that e.m.f. is not the kind of quantity whose sign the causal ordering of events protects — unlike, say, the sign of the interval between a cause and its effect. The argument therefore risks assuming what it sets out to prove.
Third, the paper is a survey of difficulties as much as a proof, and the surveyed items are cited rather than resolved. The Trouton–Noble situation is left as "non-resolved in full", with Cornille's positive result and its magnetic-field explanation both reported and neither adjudicated. The friction paradox is raised, extended with a new anomalous normal force, and then explicitly handed off as "very complicated". These are honest admissions, but they mean the reader is left with a list of places where the force transformation law is hard to apply, and one worked case whose conclusiveness the author has partly undercut himself. The constructive suggestions at the close — reconsidering Weber's force, or re-analysing the transformation law on the basis of covariant ether theories — are deferred to later papers, so no alternative is offered here.