The B(3) Field Controversy

Scientific Paper
Title	The B(3) Field Controversy
Read in full	Link to paper
Author(s)	Geoffrey Hunter
Keywords	Stokes Theorem, Myron Evans, B(3) Magnetic Field
Published	2000
Journal	Apeiron
Volume	7
Number	1-2
No. of pages	12
Pages	17-28

Read the full paper here

Abstract

The controversy generated by the theory of the B(3) magnetic field is considered in the light of a paradox raised by E. Comay and its repudiation by M. W. Evans and S. Jeffers. Their arguments are examined and assessed.

Overview

This paper is an unusual item in the dissident literature: a referee's report grown into an article. Hunter, a chemist at York University in Toronto, was asked by Myron W Evans in December 1996 to review the proofs of Evans and Jeffers' published reply to E. Comay's criticism of the B(3) field. He found the reply mathematically defective, communicated that to Evans, and later published the analysis. His stated purpose is not to adjudicate the physics of B(3) directly but "to calibrate the credibility of the opponents in the controversy" — to establish, in public and in detail, which side of a long-running dispute was actually doing the mathematics correctly.

The B(3) theory, originating with Evans in 1992 and developed across the four volumes of The Enigmatic Photon, holds that a circularly polarised electromagnetic wave carries a longitudinal magnetic field B(3), parallel to the direction of propagation, defined as a conjugate cross-product of the two transverse magnetic components. Its amplitude is proportional to that of the transverse field but, unlike the transverse field, independent of angular frequency ω; it vanishes for linear polarisation; and it is said to be unaccompanied by any real electric field. Hunter notes at the outset that the direct experimental tests — Rikken (1995) and Akhtar Raja and colleagues (1997), the latter conducted by Evans's own colleagues at the University of North Carolina — produced only negative results, and that these negative results were nevertheless rebutted rather than accepted. That pattern is what prompts the paper.

The argument

Comay's counter-example

Comay's 1996 Chemical Physics Letters comment argued that "Evans's modified electrodynamics yields physically unacceptable results". His construction is a closed path PQRSP in the radiation zone of a rotating electric dipole. PQ runs radially outward along the rotation axis (the z-axis) for one wavelength; RS runs radially inward along the x-axis, in the plane of rotation, also one wavelength; QR and SP are quarter-circle arcs of concentric circles one wavelength apart, centred on the dipole. The radius is large compared with the dipole length, so the whole path lies in the far field, where propagation is radially outward everywhere.

Radiation from a rotating dipole is circularly polarised along the axis, linearly polarised in the plane of rotation, and elliptically polarised in between. Comay then evaluates the line integral of the total magnetic field around PQRSP:

• Along PQ, B(3) is parallel to the path and the polarisation is circular, so B(3) contributes.
• Along QR and SP, B(3) is perpendicular to the path and contributes nothing.
• Along RS, the radiation is linearly polarised, so B(3) = 0 by the theory's own assertion.
• The transverse components contribute nothing along the radial segments, and their contributions along the two arcs cancel exactly — the arcs are a wavelength apart and hence in phase, and although the inner arc is shorter, the far-field amplitude is larger there by the same ratio.

So the closed line integral is non-zero, entirely on account of B(3) along PQ. Applying Stokes's theorem and the vacuum Maxwell equation ∇×B = ∂E/∂t converts this into ∂/∂t ∫E·ds = constant ≠ 0, whence ∫E·ds = at + b. The electric flux through the surface therefore grows linearly without limit instead of oscillating — "contrary to the physics of a radiation field whose intensity is stationary." It also shows B(3) is associated with an electric field, contradicting assertion (C) of the theory. Comay's own summary of the method is that "one counter-example justifies a refutation of a theory."

Hunter interposes one caveat in Comay's favour and one against. Against: B(3) theory elsewhere proposes adding a photon-mass term −ξ²A to two of Maxwell's equations, with a mass "of the order of 10⁻⁵¹ kg", so Comay's use of the unmodified equation might be challenged. In favour: the same source states that "the three field components B(1), B(2) and B(3) obey the Maxwell equations in free space", which licenses exactly what Comay did — and in any case the Evans–Jeffers reply never raised this objection.

Misattributions in the reply

Hunter first documents three misdescriptions. The reply's opening sentence credits Comay with a "definition" of B(3) that is in fact Evans's own, merely quoted; the same misattribution recurs later as "Comay's own [sic] definition". The reply says Comay "asserted the curl of B(3) to be non-zero", whereas Comay deduced the non-vanishing of the curl of the total field B from a computed line integral. And the reply's abstract states that Comay erred "precisely at the point where he uses the Cartesian form of Stokes's theorem" — but, Hunter observes, the Cartesian form of the theorem "simply isn't mentioned anywhere in his Comment."

Stokes's theorem, done carefully

The technical core is a deliberately elementary derivation, which Hunter admits "may appear to be needlessly elementary to some readers." He writes the Cartesian form of Stokes's theorem as quoted by Evans and Jeffers, relates it to the general form ∮F·dl = ∫∫ curl F·n ds via the determinant expression for the curl and the identity nds = i dydz + j dxdz + k dxdy, and then specialises to the case both disputants use — a planar surface in the x–z plane. There dy = 0 on the path and n is parallel to j, leaving only

∮(F_xdx + F_zdz) = ∫∫(∂F_x/∂z − ∂F_z/∂x)dxdz

Against this he checks the reply's equations (4) and (5) term by term, and lists which terms are present that should be absent and which are missing. For their equation (5), covering the transverse components, he finds the B_ydy line term should be absent, the B_zdz line term should be present, the surface term ∂B_y/∂z dydz should be absent and ∂B_z/∂x dxdz should be present — and since the segment under evaluation is PQ along the z-axis, even the surviving B_xdx term should be absent.

The open-path objection

Hunter's strongest point is conceptual rather than algebraic. Both reply equations attempt to equate a line integral taken along the open segment PQ with a surface integral. But along PQ, dx = dy = 0 and the enclosed area is zero, so the "surface" integral vanishes identically "regardless of which derivatives of B(3)_z are being integrated." Stokes's theorem relates a line integral to a surface integral only when the path is closed and bounds a non-zero area; computing the contribution of a single open segment is legitimate, but it cannot be equated with a surface integral. Hunter adds that the reply also switched to a rectangular path ABCDA that Comay never used, one corner of which sits at the origin — that is, at the centre of the rotating dipole, and hence not in the radiation zone at all — while attributing to Comay assertions about that path.

Conclusion and diagnosis

Because the reply's own abstract identifies the Cartesian Stokes theorem as the kingpin of the rebuttal, and that is precisely where it fails, Hunter concludes the reply "has no credibility as an effective rebuttal". He remarks on the incongruity between the "prodigious scholarly achievement" of the B(3) literature and "the elementary nature of the errors". His own diagnosis, developed elsewhere, is that B(3) is "a case of mistaken identity": it is proportional to one of the four Stokes parameters of the field, and a Stokes parameter characterises the polarisation state, not an independent field component. He cites Comay's further observation that the cross-product definition makes B(3) proportional to charge squared in dipole radiation, whereas a physical field must be linear in charge — which is also why B(3) does not obey superposition. A genuine longitudinal field, he allows, remains possible, but would require a different definition.

Assessment

The paper is a model of a particular kind of criticism and deserves credit for it. Hunter does not argue from authority or from the negative experiments alone; he reproduces both disputants' equations, states the theorem being applied, specialises it to the geometry both sides chose, and compares term by term, explicitly so that "any reader of this article to verify the correctness of the inferences". He is also scrupulous in giving the other side its best case, raising the photon-mass modification of Maxwell's equations as a possible defence of B(3) that the reply itself failed to use. And the decisive objection is unanswerable on its own terms: an open line segment bounds no area, so no application of Stokes's theorem to it can be meaningful. Two supporting pieces of arithmetic also check out. His reduction of the Cartesian Stokes theorem for a surface in the x–z plane is correct, since the y-component of the curl is indeed ∂F_x/∂z − ∂F_z/∂x. And Comay's arc-cancellation, which Hunter endorses, is exact for the reason given: arc length scales as r while far-field amplitude scales as 1/r, so the product is radius-independent.

One of his own term-by-term charges is weaker than he presents it. Against the reply's equation (4) — which retains ∂B(3)/∂y dydz and −∂B(3)/∂x dzdx — Hunter objects both to the presence of the first term and to the absence of a term ∂B(3)/∂z dxdz. But for a field with only a z-component, those are exactly the two terms of the general Cartesian Stokes expression that survive, and the term Hunter calls missing occupies the ∂F_x/∂z slot of his own equation (10) — a slot that is identically zero when F_x = 0. The reply's first term is redundant on a surface in the x–z plane rather than wrong, and Hunter's "missing" term is a mis-substitution. His parallel list of four errors in the reply's equation (5) does hold up, and the open-path objection covers both equations regardless, so the conclusion survives; but the charge of "algebraic errors" is overstated at that one point. It is worth noting that Hunter himself hedges the adjacent complaint with "appear to be" and "in this author's opinion".

The paper's larger limitation is one it acknowledges by design. It establishes that a particular published rebuttal was defective; it does not by itself establish that B(3) is wrong, and Hunter is careful to say that a longitudinal field in radiation remains "of course possible". The substantive physical case — that B(3) is a Stokes polarisation parameter rather than a field component, that its definition makes it quadratic in charge, and that it violates superposition — is stated compactly and referred to his own separate 1999 paper and to Comay's later work rather than argued here. Those are the points that actually decide the matter, and a reader who wants them must go elsewhere. The strongest empirical fact in the paper is likewise mentioned only in passing: the null results of Rikken and of Akhtar Raja et al., searching for the I^1/2 and I^3/2 intensity-dependent terms predicted by the 1992 formula, found nothing. On the paper's own terms, though, the case is made and made cleanly: the mathematics offered in defence of B(3) in that exchange does not work, and it fails at a level that any undergraduate text on vector calculus would settle.

See also

• Geoffrey Hunter
• Myron W Evans
• Stanley Jeffers
• Jean-Pierre Vigier
• Maxwell's Equations
• Electrodynamics
• Photon
• Light
• Apeiron