Special Theory of Relativity and the Lorentz Force

Scientific Paper
Title	Special Theory of Relativity and the Lorentz Force
Read in full	Link to paper
Author(s)	Vesselin C Noninski
Keywords	Special Theory of Relativity, Lorentz force, Lorentz transformations
Published	2003
Journal	PhilSci-Archive
No. of pages	10

Read the full paper here

Abstract

Efforts to demonstrate that the application of Lorentz transformations leads to the form Y' = [Y - (v/c)N] (at least to the leading orders of approximation), which has been elevated to the criterion for the validity of STR , has been the main point in the founding paper on Special Theory of Relativity (STR). The text that follows shows that the said criterion for the validity of STR set forth in the mentioned paper is not fulfilled, despite the implication in it that it is.

Overview

This is a short critical paper aimed at a single passage of a single document: §6 of Einstein's 1905 paper "On the Electrodynamics of Moving Bodies", in the Dover translation of The Principle of Relativity (pages 37–65), where the transformed Lorentz force on a moving unit charge is derived and where Einstein compares what he calls the "old manner of expression" with the "new manner of expression" of that force. Noninski's claim is that the two expressions do not in fact agree, not even "to the first order of small quantities" as Einstein asserts, and that because Einstein himself elevated that agreement to the test of the theory, the failure is fatal rather than cosmetic.

The paper's strategy is deliberately narrow and deliberately old-fashioned. Noninski does not appeal to experiment, to a rival aether theory, or to any later formulation of relativity. He argues that "paper [1] still contains everything one needs to know about the fundamentals of the Special Theory of Relativity", that its derivations are used "in exactly the same form as it has been at its inception", and therefore that an internal defect in the 1905 text cannot be waved away as obsolete. He also refuses in advance the reply that experiment settles the matter: "no experimental finding can undo the inherent lack of physical meaning of the theory", and any agreement with measurement would be "a fortuitous coincidence at best". The argument is presented as work done, as Einstein's was, with "a pencil and a paper".

The argument

The force in the laboratory frame

For a unit test charge q = 1 moving along the x-axis with velocity v relative to the source of the electric field, the force is F = E + (1/c)v×B, which Noninski writes in components using (X,Y,Z) for the electric and (L,M,N) for the magnetic field vector:

F_x = X , F_y = Y − (v_x/c)N , F_z = Z + (v_x/c)M

Direct transformation of the force

Rather than transforming Maxwell's equations, as Einstein does, Noninski proposes to transform the force expression itself. He simply re-expresses the same laboratory field components X, Y, Z, L, M, N as functions of the co-moving coordinates (ξ, η, ζ, τ), noting that the charge q = 1 "remains unchanged despite the motion (an experimental fact)" and that v_x ≠ 0 still, "because we are still in the laboratory frame". Setting this alongside the form demanded by the principle of relativity in the primed frame, he concludes:

X' = X , Y = Y − (v/c)N , Z = Z + (v/c)M  (his eq. 6)

He stresses that no Lorentz transformation was used to obtain this, and gives a physical reason why it must hold: "the same charge residing in the same electro-magnetic circumstances should experience only one force despite the viewpoints from which this force is observed."

The result Einstein actually obtains

Einstein's route — transform Maxwell's equations with the Lorentz transformation and compare with the postulated primed form — yields instead

Y = β[Y − (v/c)N] , Z = β[Z + (v/c)M] , with β = (1 − v²/c²)^−1/2

The whole dispute is the factor β. Noninski expands it by the binomial series, keeping two terms, β ≈ 1 + (1/2)v²/c², and multiplies out:

Y' = Y − (v/c)N + (1/2)(v²/c²)Y − (1/2)(v³/c³)N

Dropping the third-order term leaves his eq. (13): Y' = Y − (v/c)N + (1/2)(v²/c²)Y.

The comparability objection

Noninski's decisive move is to observe that the second and third terms of eq. (13) can be of the same size. If (v/c)N ≈ (1/2)(v²/c²)Y, then "neglecting one of these terms inevitably requires neglecting of the other term as well", and eq. (13) collapses to Y' = Y. That, he says, is manifestly not of the form Y = Y − (v/c)N. Conversely, if (1/2)(v²/c²)Y dominates, one gets Y = Y + (1/2)(v²/c²)Y, again the wrong form. Only in the special case where the Y-term is negligible against (v/c)N do the two expressions agree, and Noninski holds that a special case is not enough — least of all "at speeds where the effects of STR should become detectable".

He illustrates the alleged fallacy with the algebraic analogy y = A − Bx + (1/2)Ax² where Bx = (1/2)Ax², so that dropping only the last term to recover y = A − Bx is illegitimate; and, more polemically, with the claim that Einstein's licence would permit writing 1 − 1 = 1.

Wider claims

Two further threads run through the paper without being developed in it. First, Noninski asserts that Einstein's "Definition of Simultaneity" (p. 38 of the same translation) "lacks physical meaning", that the light-ray device used there is simply "not appropriate for such exploration", and that this is "the ultimate generator of the errors" because the Lorentz transformation is derived from it — with the consequence that "the claim for the relativity of time is untenable". Second, he reports a private communication from G. Pellegrini, who agreed the algebra is wrong but held the point to be minor since a transformed force need not retain its form; Noninski rejects this on the strength of his own eq. (6). He promises a companion piece in which a train is destroyed by an explosion triggered by two events simultaneous in the train, while a ground observer applying Einstein simultaneity concludes the train is intact. He closes by noting that problems in special relativity necessarily propagate into the general theory, so that tests of the latter should be preceded by scrutiny of the former.

Assessment

What is attractive here is the discipline of the method. Noninski restricts himself to one text, quotes its page numbers, reproduces its equations, and asks only whether the paper does what it says it does. That is a legitimate and under-used form of criticism, and his target is well chosen: Einstein's §6 comparison of the "old" and "new" manners of expression really is offered as a consistency check, and the factor β really is there in the transformed force. It is also fair to insist, as he does, that the historical text is not superseded in its fundamentals.

The load-bearing arithmetic, however, does not support the conclusion, and it can be checked directly from the paper's own equations. The difference between eq. (6) and the Lorentz-transformed eq. (7) is (β − 1)[Y − (v/c)N] = (1/2)(v²/c²)Y − (1/2)(v³/c³)N + …, which is smaller than the leading term Y by a factor of order v²/c² unconditionally. Einstein's claim of agreement to first order in v/c is therefore correct as stated. Noninski's special case does not touch that difference at all: if (v/c)N ≈ (1/2)(v²/c²)Y, then the magnetic term is itself second order, and a first-order truncation of eq. (6) gives Y' = Y just as a first-order truncation of eq. (7) does. The two expressions agree in that regime rather than diverging. His eq. (14) is derived by expanding eq. (7) to first order and then comparing it against an unexpanded eq. (2) — the two sides of the comparison are truncated at different orders, and that inconsistency, not the factor β, produces the apparent contradiction. The "1 − 1 = 1" analogy does not model the algebra, because nothing in the expansion involves cancelling terms of equal magnitude and opposite sign.

The deeper difficulty is that eq. (6) is asserted rather than derived. Two independent assumptions are folded into it: that the field components carry over unchanged when their arguments are relabelled in co-moving coordinates, and that the force on the charge is frame-independent because "the same charge… should experience only one force despite the viewpoints from which this force is observed." The second is precisely the Newtonian premise that relativity abandons; in the relativistic account the transverse component of force is not invariant, and the factor β that Noninski treats as an intruder is exactly the expression of that non-invariance. Assuming force invariance and then finding that the Lorentz transformation fails to preserve force is not an internal inconsistency in Einstein's paper but a restatement of the disagreement between the two frameworks. It should be added that Einstein's §6 quantity is a force per unit charge on a charge whose velocity is being transformed as well, so a careful comparison must transform the velocity in the force law too — a step the paper does not perform.

Finally, the paper's methodological stance limits what it can establish. Ruling experiment out of court in advance — "no experimental finding can undo the inherent lack of physical meaning" — means the argument must stand or fall on internal consistency alone, and that is a high bar the algebra above does not clear. The claims about simultaneity and about the relativity of time are announced rather than argued, with the supporting text promised elsewhere, so they cannot be weighed here. Against this, the empirical record the paper declines to engage is specific and quantitative: the v²/c² effects at issue are the ones measured directly in muon lifetime dilation in storage rings and in Ives–Stilwell transverse Doppler experiments, and the velocity-dependent force law itself is what magnet design in high-energy accelerators relies on. A criticism of the 1905 derivation would need to say what those measurements are measuring instead.

See also

• Vesselin C Noninski
• Lorentz Force
• Lorentz Transformation
• Simultaneity
• Maxwell's Equations
• Special Relativity
• Albert Einstein
• Speed of Light