Primer: Koen J. van Vlaenderen and the "scalar field" of electrodynamics

Step 4 study document. Date 2026-10-04. Built from arbiter.md (authoritative verdicts, including its section 8 "QE revision"), the three reviews (physicist.md, mathematician.md, contrarian.md), the deep reads in deep/ (including the QE follow-up reads: 2015 pp.16-21, the 2015 far-field dipole check, and the covariance of the 2003 power theorems), and the Investigator's sites.md and research/ notes. This primer gives no new verdicts.

Revised 2026-10-04 after qe.md (MINOR): the static-charge power verdict changed (section 4.2), a verdict on the "far-field inconsistency" claim was added (section 4.3), the author's further cited evidence was restored to the evidence table, and example.py gained Part F.

Source tags: [read: ...] from the author's papers (via the deep reads); [web] with URL; [memory] recalled from training, with confidence; [check] a calculation redone for this primer (see example.py). The papers spell the name "van Vlaenderen"; the folder's "Vlaenederen" is a typo.

What was read: three papers, all with a text layer, so there is no source gap. They are "A charged space as the origin of sources, fields and potentials" (arXiv, 1999), "A generalisation of classical electrodynamics for the prediction of scalar field effects" (arXiv physics/0305098, 2003), and "Classical Electrodynamics in agreement with Newton's third law of motion" (January 2015). The author reports no experiment of his own. Every experiment he cites was done by someone else, and most of those primary papers were not read by any reviewer. Where this primer relies on a cited experiment, it is relying on the author's description, the Investigator's saved copies, or abstracts. Each case says which.

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1. The idea in one paragraph

Ordinary electromagnetism can be written with two "potentials": a voltage-like scalar potential and a vector potential. The physical fields come from derivatives of these, and the potentials themselves have some built-in slack. You can change them in a certain way without changing any field (this is "gauge freedom"). Textbooks use that slack to set one particular combination of potentials to zero, because nothing measurable seems to depend on it. Van Vlaenderen's idea is that this combination, which he calls S (later the "scalar magnetic field" B_l), is a real physical field that textbook theory has thrown away by habit. If you keep it, he argues, the equations allow new things: electric waves that oscillate along their direction of travel with no magnetic part; a force that pushes a current along its own wire; a balance of forces between pieces of a circuit that obeys Newton's third law exactly; and, more speculatively, ways to pull energy from a natural background of such waves, as he believes Tesla and T. Henry Moray did. His 2015 paper adds a third wave type that would travel much faster than light. He says himself that quantitative experimental tests are still needed [read: 2003 p.1, p.11].

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2. What you need to know first

Maxwell's equations in differential form. These are four equations linking the electric field E and magnetic field B to charge density rho and current density J. Gauss's law: div E = rho/eps0. No monopoles: div B = 0. Faraday's law: curl E = -dB/dt. Ampere-Maxwell law: curl B = mu0 J + mu0 eps0 dE/dt. Here eps0 = 8.854e-12 F/m (permittivity of free space), mu0 = 4 pi e-7 H/m (permeability of free space), and c = 1/sqrt(eps0 mu0) = 2.998e8 m/s. Read Griffiths, Introduction to Electrodynamics (4th ed.), chapter 7.

Potentials and gauge freedom. Because div B = 0, you can write B = curl A for a vector potential A. Faraday's law then lets you write E = -grad Phi - dA/dt, with Phi the scalar potential (volts). If you replace Phi by Phi - d(lambda)/dt and A by A + grad(lambda) for any function lambda(x,y,z,t), E and B do not change. That is gauge freedom. Two common choices: the Coulomb gauge (div A = 0) and the Lorenz gauge (div A + (1/c^2) dPhi/dt = 0). In the Lorenz gauge, Phi and A each obey a simple wave equation with sources. Griffiths chapter 10.

Conservation laws for fields. The fields carry energy, with density u = (eps0/2)E^2 + B^2/(2 mu0), and an energy flow, the Poynting vector (1/mu0) E x B. They also carry momentum, with density g = eps0 E x B. Poynting's theorem says the field's energy goes down at exactly the rate it does work on charges. The momentum theorem says the same for force. Griffiths chapter 8, which includes a short section on how magnetic forces between moving charges seem to break Newton's third law and how field momentum restores momentum conservation.

Current elements and the Biot-Savart law. A short piece of wire carrying current I along a length dl (a "current element", strength I dl in A·m) makes a magnetic field B = (mu0/4pi) I dl x r̂ / r^2 at distance r. A second element feels a force I' dl' x B. Griffiths chapter 5.

Helmholtz decomposition. Any reasonable vector field that falls off at infinity splits uniquely into a curl-free ("longitudinal") part and a divergence-free ("transverse") part. Van Vlaenderen's 2015 paper uses this split everywhere. One point the paper does not state: the two parts are non-local. If a current flows only inside a wire, its two parts each spread through all of space and cancel outside the wire [memory, high]. Griffiths Appendix B.

Electromagnetic mass and the 4/3 problem. If an electron is modelled as a small charged ball, its field energy U gives a mass U/c^2. The momentum carried by its field when it moves gives a different mass, 4/3 of that. This old puzzle is resolved in mainstream physics by adding non-electrical binding stresses (Poincaré stresses) or by defining field energy-momentum covariantly (Fermi, Rohrlich) [memory, high]. Feynman, Lectures on Physics Vol. II, chapter 28 ("Electromagnetic Mass").

Rest mass and four-forces (only for section 4.2). In special relativity a particle's energy and momentum form one package, and its rest energy mc^2 is part of that package. A force can change the particle's speed (kinetic energy and momentum), and in principle it could also change its rest mass. The ordinary electric and magnetic forces never change rest mass: the magnetic force does no work, and the electric force's work all goes into kinetic energy [memory, high]. A force that pushes energy into a charge sitting still, without moving it, has nowhere to put that energy except the charge's rest (internal) energy. That is all you need for section 4.2. Taylor and Wheeler, Spacetime Physics, or Griffiths ch. 12, covers the background.

No tensors or quantum field theory are needed for the core of this primer. One reviewer remark (section 4.2) mentions a "Lagrangian", which you can read as the single function from which a theory's field equations and its energy follow. You do not need it to follow the argument.

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3. The textbook view

Write the two Maxwell equations that contain sources in terms of the potentials. With S defined as

S = -(1/c^2) dPhi/dt - div A (units: tesla, the same as B) [read: 2003 eq. 7]

substitution gives [read: 2003 eqs 8-9; SymPy-checked by the deep reader]:

(1/c^2) d^2Phi/dt^2 - Laplacian Phi + dS/dt = rho/eps0 (1/c^2) d^2A/dt^2 - Laplacian A - grad S = mu0 J

Here "Laplacian" is div grad. These hold in any gauge. In the Lorenz gauge you choose S = 0 and are left with clean wave equations for Phi and A that travel at c.

Two textbook facts matter for everything that follows.

1. S is not measurable in Maxwell's theory. Under a gauge change, E and B stay the same but S shifts by the wave operator acting on lambda [read: 2003 eq. 14]. You can always pick lambda to make S zero. Nothing in E or B, and so nothing in any force on a charge, depends on S. The gauge invariance of E and B follows from their definitions alone, before any field equation is used (mathematician, [check]).

2. If charge is conserved, S cannot be produced. Take the divergence of the A equation and use the Phi equation. The result is (1/c^2) d^2S/dt^2 - Laplacian S = -mu0 (div J + d rho/dt) [read: 2003 eq. 24]. Charge conservation makes the right side zero. So S at most obeys a source-free wave equation, and no ordinary circuit, antenna or charge can generate it. Van Vlaenderen's own 2001 preprint says the same: "A single point charge cannot be a source for the scalar field" [read: research/files/wayback-raccoon, via Investigator].

Newton's third law in magnetostatics (the textbook account). The Biot-Savart-plus-Lorentz force between two current elements is often called the Grassmann force. It is not, in general, equal and opposite between two isolated elements. For whole closed circuits the totals are equal and opposite [memory, high; checked numerically in example.py Part E]. For an open element (current flowing into a growing pile of charge) the mechanical forces do not balance. The textbook resolution is that the field itself carries momentum, g = eps0 E x B. The piled-up charge makes an E field that grows with time, so the field momentum changes, and mechanical momentum plus field momentum is conserved [memory, high; Griffiths ch. 8]. Section 5 works this out with numbers.

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4. Where the inventor departs

4.1 The 2003 paper: keep S and give it source terms

His starting points are real. Adding a term to a field equation by analogy is how Maxwell got the displacement current, and the author says he is following "the spirit of J.C. Maxwell" [read: 2003 p.3]. The algebra is correct. Every identity in the 2003 paper was checked symbolically [read: deep/A-generalisation..., SymPy]. Equations 18-19 are exactly the potential wave equations with no gauge condition imposed. He also has a fair point about textbook habit: a gauge condition is a choice about how to describe nature, and the 2003 paper argues that the reasons for treating S as unphysical are circular [read: 2003 p.4].

The point where the reviewers say the argument stops holding is the next step. Because E and B are gauge-invariant from their definitions alone, the circularity argument does not show that S is physical. Making S physical is a new postulate, which the author concedes in his reply to Bruhn: "S is introduced not for mathematical reasons ... for describing new effects" [read: reply to Bruhn, research/files]. And because his own eq. 24 says charge conservation makes S source-free, the theory only predicts something new where charge is not conserved locally. The 2003 paper proposes that quantum tunnelling, for example at Josephson junctions, is such a place [read: 2003 p.5].

4.2 The power and energy bookkeeping (the "static charge power source")

Divide eq. 33 by mu0. The field loses energy at the rate J·E + c^2 rho S. This field-side statement is an identity: it holds for any potentials once rho and J are defined by eqs 18-19 [read: 2003 pp.6-7; check, example.py Part F1]. The question is who receives the c^2 rho S.

The paper's words. It calls (rho/eps0)S "applied power by means of static charge rho and a dynamic scalar field S", says a scalar field "can turn an object with charge density rho into an electrical power source", calls this "a remarkable prediction", and notes that "rumours exist" of it having been achieved, though the author is "not aware of any published scientific experiments" [read: 2003 pp.7, 10]. It never says where the energy comes from or goes to on the matter side. It mentions no rest mass, internal energy or equation of motion for matter [read: deep read, full-text search]. It gives no magnitude for S [read: 2003 pp.8-10].

The first analysis (arbiter section 1.1, kept in the record). The paper's force on matter, rho E + J x B + J S, does work at the rate J·E + rho v^2 S for charge moving at speed v (with J = rho v). The difference, (c^2 - v^2) rho S, goes to no named place. For a charge at rest it is c^2 rho S. Example: rho = 1e-6 C/m^3 and S = 1e-6 T gives 8.99e4 W/m^3 that leaves the field and is received by nothing [arbiter 1.1, check]. This analysis assumed, without saying so, that the charged matter's rest mass is fixed.

The QE revision (arbiter section 8.2). The paper's power and force, taken together, are exactly a relativistic four-force with an extra part S times (c rho, J). That extra part points along the charge's own "four-velocity", and a force of that kind changes rest mass and nothing else. For a point charge q the result is dm/dtau = qS (tau is the time on the charge's own clock) [read: deep read of 2003, follow-up section; check, example.py Part F2]. This gives two readings, and both stay in the record: - Reading 1 (rest mass may change). Field plus matter energy and momentum are conserved exactly. The c^2 rho S is an exchange between the S field and the internal (rest) energy of the charged matter, with sign following S. It is then not a free source: it draws on the carrier's finite internal energy. - Reading 2 (rest mass fixed, as in ordinary physics). Nothing can absorb the c^2 rho S. A charge at rest feels no spatial force J S (J = 0), so the energy goes nowhere. This is the imbalance computed above.

The paper adopts neither reading. Size, with a hypothetical S since the paper gives none: q = 1 microcoulomb in S = 1e-6 T gives c^2 q S = 9.0e4 W, while dm/dt = qS = 1e-12 kg/s, far too small to weigh [check, example.py Part F3]. So even a tiny S would make a charged body heat or cool measurably, and the absence of reported anomalous heating of charged bodies bounds S in ordinary laboratories; nobody has computed that bound [arbiter 8.2]. There is also a tension the paper does not resolve: its own eq. 24 says S is sourced only where charge is not conserved, yet its design advice names ordinary capacitors and Tesla-type high-voltage circuits, where charge is conserved, as sources [read: 2003 pp.1, 4-5, 10; deep read E].

Arbiter's verdict (changed in the QE revision): Not testable as stated, because no S source, magnitude or matter-side law is given. Both conditionals stay: with fixed rest mass, Internally inconsistent; with rest mass changing at qS, consistent, and it makes a definite prediction (a charged body in an S field gains or loses internal energy at c^2 rho S). A calorimetry measurement on a static charged body in a controlled S environment is the test that discriminates [arbiter 8.2, 8.5].

The energy of S itself. The physicist adds a further point that the arbiter reports but did not re-derive: in the standard field theory with this kind of scalar, the S-wave's energy has the opposite sign to S^2/(2mu0) [physicist, check; memory, high that such a scalar mode has indefinite sign]. The arbiter's label: Not testable as stated, on paper, because the sign depends on which conserved quantity is called energy; it becomes Internally inconsistent if the canonical-energy argument is re-derived and accepted [arbiter 8.3]. The follow-up deep read also found that the theory's energy-momentum bookkeeping is not symmetric (the S part of the momentum density is +eps0 E S, while the S part of the energy flow divided by c^2 is -eps0 E S). This bears on angular momentum, which the paper does not discuss [read: deep read G; arbiter 8.2].

4.3 The 2015 paper: Newton's third law and three waves

Again, the starting point is real. The Grassmann force between isolated current elements really is non-reciprocal [Established]. This has been discussed since Ampère, and Griffiths mentions it, as the author notes [read: 2015 p.4]. His reciprocal force kernel has the same form as one proposed by Whittaker in 1910 [memory, medium-high]. It is a legitimate mathematical object.

The departures the reviewers identify: - The premise that open-circuit magnetostatics has no field momentum is stated, not derived. The paper's own momentum density (eq. 4.34) contains eps0 E x B_t, which changes in time when charge piles up [read: deep read 2015, 4.7]. - Eq. 4.22, presented as what selects a >> c, is the continuity equation rewritten. It holds for any a. The values b = c and a >> c come from requiring the force law to take the form J B_l [deep read 2015, 4.5-4.6]. - B_l is zero for every closed, steady circuit (it is proportional to the integral of d rho/dt / r) [deep read 2015, 2.2, check]. So the new force acts only where charge accumulates. The contrarian notes this raises a question the paper does not resolve: does the theory predict any Aharonov-Bohm force from a closed solenoid at all? - The 4/3 agreement needs a self-energy twice the standard (eps0/2)E^2 value that the paper's own eq. 4.33 gives (section 5.6 below).

Why the author thinks open circuits matter. The paper argues that the classic force experiments of Ampère, Gauss and Weber "are all examples of open circuit currents that are not divergence free", because "batteries or capacitor banks were used as electric current sources and current sinks" with time-varying charge at the terminals. It says "the only example of a truly closed circuit stationary current" is a superconductor current induced by an approaching magnet [read: 2015 p.16]. No sources, dates or numbers are given, and no role has checked this against the primary histories. The arbiter lists it as Never properly tested as an argument [arbiter 8.4]. It is the author's reason why the J B_l term would matter in real experiments, if the term were real.

The second headline conclusion: "Lorenz-gauge Maxwell is far-field inconsistent". The 2015 paper's conclusions name two results: the Maxwell-Lorentz force law disagrees with Newton's third law, and "Maxwell's electrodynamics theory in the Lorenz gauge is also far field inconsistent" [read: 2015 p.21]. The argument [read: 2015 pp.9-10]: - The Jefimenko formula for E (eq. 3.18, the Lorenz-gauge solution of Maxwell's equations) is an integral with four terms. Two of them, the charge-rate term (rho-dot) and the curl-free-current term (J_l-dot), fall off as 1/r and point along r. The author calls them "longitudinal far fields". - "Since far fields are defined as two mutually inducing fields", these terms have no partner field, so the theory is "far field inconsistent". This definition is the author's own; no source is cited. - His "simple Maxwell" field (eq. 3.16: an instantaneous Coulomb field plus a retarded transverse part) has no such terms and is "far field consistent". This is the Coulomb-gauge field.

What a calculation shows [read: deep read 2015 section 7, SymPy/NumPy; arbiter 8.1]. Take an oscillating point dipole, a source that conserves charge. Each of the two longitudinal 1/r terms is really there as a piece of the integrand: in the deep reader's units each has radial size 2.261763. But they have opposite signs, so they cancel exactly. The two formulas give the same E and the same B at every point and time. What remains at large r is the ordinary transverse dipole radiation. The general reason is the Helmholtz split from section 2: a field that vanishes at infinity splits only one way, and Gauss's law fixes its curl-free part to be the instantaneous Coulomb field. This is the subject of J.D. Jackson, "From Lorenz to Coulomb and other explicit gauge transformations", Am. J. Phys. 70, 917 (2002), whose stated purpose is to show that the fields are the same whatever the gauge [web: https://www.semanticscholar.org/paper/From-Lorenz-to-Coulomb-and-other-explicit-gauge-Jackson/cf3f1a498297aa7df026b6799b11e3cb923475e2, abstract only].

The paper's side, stated fairly: the cancellation needs charge conservation and integrals over all space. In the author's own Section 4 theory, J_l travels at its own speed b, and then the terms no longer cancel. That is a different theory with new constants, not an inconsistency inside Maxwell's [arbiter 8.1]. Note also that the paper uses "longitudinal" in two senses: radial (pointing along r) and curl-free (the Helmholtz sense). They are not the same thing.

Arbiter's verdict (added in the QE revision): Refuted by calculation, for the claim as made about Maxwell's theory. The lesson for you: the Coulomb and Lorenz gauges describe one physical field, and individual terms of an integral are not separate physical fields. The author could revive the claim only by naming a measurable field that differs between 3.16 and 3.18 [arbiter 8.1, 8.5].

Related work named in the 2015 paper (leads, not endorsements) [read: 2015 refs; arbiter 8.4]: Nikolaev (Tomsk, 2004), who gave B_l the name "scalar magnetic field"; the Marinov motor and Phipps; Moyssides, Peoglos and Hillas on longitudinal forces; and Caroline H. Thompson's Phi-wave aether.

4.4 The 1999 paper

This paper treats the quantum wave function psi as a real charge density and "derives" Maxwell's equations from charge conservation. The reviewers found that the derivation defines the sources as the wave operator applied to the potentials, which is the Lorenz-gauge route run backwards. The self-potential equation has a tachyon-like sign that the paper does not discuss. The "fractal dimension 3/2 of charge" is the exponent of metres when charge is expressed in mechanical units (in Gaussian units charge has dimensions sqrt(g·cm^3)/s), so it is a units fact rather than a discovery. The arbiter's label is Refuted by calculation as a physical finding [arbiter claim 6, 8.3]. No numbers or experiments appear in this paper.

The paper builds on four outside ideas [read: 1999 pp.1, 6-7]: - W.A. Hofer's Material Wave Theory: psi is a physical wave, and a particle carries "intrinsic potential energy". The paper takes psi real and its charge-density reading from Hofer. - Haisch, Rueda and Puthoff (Phys. Rev. A 49, 678, 1994): stochastic electrodynamics (SED), in which inertia arises from a charge's interaction with the zero-point field (ZPF), the random electromagnetic background that quantum theory says remains even at absolute zero. - V. Petkov: the mass of a charge equals its electrostatic energy divided by c^2. - R.L. Collins: momentum as charge times vector potential, mu = qA.

The abstract promises to unify "two mass models". They are joined like this [read: 1999 p.6, page image; deep read 4.10]. Model 1 (electrostatic mass, Petkov and Collins) is mc^2 = qPhi, or rho Phi = (mass density) c^2. Model 2 (ZPF mass, SED) pictures a massless charged "parton" moving at speed c on a fractal path confined to a small volume. Setting the parton speed to c in the paper's energy equation gives rest energy = rho Phi, split half kinetic and half field energy. The reviewers note that no ZPF quantity enters any equation (no spectrum, no cut-off, no force calculation), so the "unification" is an interpretation of one identity, not a calculation of ZPF energy. The arbiter's label for the mass-model unification is Not testable as stated. The SED/ZPF inertia idea recurs in other projects in this review, so it is worth knowing by name.

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5. Worked example: two current elements, Newton's third law, and where the missing momentum goes

This example takes the author's strongest starting point, the non-reciprocal Grassmann force, and computes it three ways: textbook mechanical forces, the author's corrected forces, and textbook forces plus field momentum. Everything is reproduced in example.py.

5.1 Set-up

• Element 1 sits at the origin, carrying current I1 along +y over a short length l1. Its strength is p1 = I1 l1 ŷ.
• Element 2 sits at (d, 0, 0), carrying current I2 along +x over length l2. Its strength is p2 = I2 l2 x̂.
• Both are open: charge piles up at each element's + end at rate I and drains from its - end at rate I. Picture a short wire joining two small capacitor plates that are being charged.
• Numbers: I = 10 A, l = 0.1 m (so each element has strength 1 A·m), d = 0.1 m. Write k = mu0 I1 l1 I2 l2 / (4 pi d^2) = (1e-7)(1)(1)/(0.01) = 1.0e-5 N.

5.2 Textbook (Grassmann) forces

Biot-Savart field of element 1 at element 2: the vector from 1 to 2 is d x̂, so B1 = (mu0/4pi) p1 x x̂ / d^2 = (mu0/4pi)(I1 l1/d^2)(ŷ x x̂) = -(mu0 I1 l1 / 4pi d^2) ẑ. Force on 2: F2 = p2 x B1 = I2 l2 x̂ x (-(...) ẑ) = +k ŷ, because x̂ x ẑ = -ŷ.

Field of element 2 at element 1: the vector from 2 to 1 is -d x̂, which is parallel to p2, so B2 ∝ x̂ x (-x̂) = 0. Force on 1: F1 = 0.

Sum of mechanical forces = +k ŷ = 1.0e-5 N. Newton's third law fails for the pair. This is the fact the author starts from, and it is correct [check, example.py Part A].

5.3 The author's correction: the force J B_l

His scalar magnetic field from an element is B_l = (mu0/4pi) p·r / r^3, with r the vector from the element to the field point [read: 2015 eq. 2.5]. - At element 1 (r = -d x̂): B_l = (mu0/4pi) I2 l2 (x̂·(-d x̂))/d^3 = -mu0 I2 l2/(4 pi d^2). - The extra force on element 1 is p1 B_l = -k ŷ, along element 1's own wire. That is why it is called "longitudinal". - At element 2: p1·r = ŷ·(d x̂) = 0, so no extra force.

New totals: F1 = -k ŷ and F2 = +k ŷ. Sum = 0. The kernel is reciprocal, as the paper says [check, Part B].

5.4 The textbook account: field momentum

The charges at the ends of each element are growing. That growing charge makes an electric field, and together with the steady magnetic fields it gives a changing field momentum. A standard result is that, for slowly varying systems, the field momentum shared between a charge q and other sources is q A(position of q), with A in the Coulomb gauge [memory, high; Griffiths ch. 8 problems; Calkin 1966]. For a current element, the Coulomb-gauge vector potential is the Darwin form [memory, high]:

A(r) = (mu0/8pi) [ p/s + (p·s) s / s^3 ], s = r - (position of element)

The currents are steady, so A does not change in time. Only the charges change, so

dP_field/dt = I1 [A2(+end of 1) - A2(-end of 1)] + I2 [A1(+end of 2) - A1(-end of 2)] ≈ (p1·grad) A2 at element 1 + (p2·grad) A1 at element 2.

(Each element's own self-term cancels between its two ends by symmetry.) Doing the two derivatives (example.py Part C, SymPy):

dP_field/dt = -k ŷ = -1.0e-5 N.

So F1 + F2 + dP_field/dt = k ŷ + 0 - k ŷ = 0. Total momentum is conserved with no new force.

5.5 What the comparison teaches

Both bookkeepings balance. They put the missing -k ŷ in different places: - Author: a real mechanical push of 1.0e-5 N on element 1, along its own wire. - Textbook: no push on element 1. The field absorbs the momentum.

That difference is measurable in principle: put a force gauge on element 1. This is the arbiter's suggested decisive test for claim 3 (an open-circuit geometry designed so that the two accounts differ).

Part D of example.py shows why such a test is hard. After only 1 microsecond at 10 A, each end holds q = 1e-5 C. The ordinary Coulomb force from element 1's end charges on element 2 is then about 231 N, roughly 2 x 10^7 times the 1e-5 N effect being looked for. Real "open" currents are short pulses into capacitors, so the magnetic effect has to be separated from very large and changing electric forces. The arbiter's verdict on the longitudinal force is "Never properly tested", and this is one reason.

Part E closes the loop. For two closed square loops (side 0.1 m, 1 A each, centres 0.25 m apart), the Grassmann and Whittaker totals agree to discretisation error: 9.485e-9 N of repulsion. Each also obeys Newton's third law. The author's extra force makes no difference for closed circuits, so any test has to involve charge piling up.

Run: .venv/bin/python "review/van Vlaenederen Koen/example.py" (it prints all parts and ends "All checks passed.").

5.6 Two smaller calculations from the reviews

• LES wave (2003). S = S0 cos(kx - wt) with E = -c S0 cos(kx - wt) x̂ and B = 0 solves the extended equations. For S0 = 1 microtesla, E has amplitude 300 V/m and the peak energy flux c S0^2/mu0 = 239 W/m^2. That is the arbiter's corrected figure; the mathematician's 0.24 W/m^2 dropped a factor of 1000 [arbiter 1.1].
• Rutherford 4/3 (2015). Electron as a charged shell of radius 2.818e-15 m. Field energy / c^2 = 4.554e-31 kg. Momentum mass from eps0 E x B = 6.073e-31 kg (the standard 4/3 of the energy mass). Extra eps0 B_l E momentum = 3.036e-31 kg. Total = 9.109e-31 kg, which is twice the energy mass. So the "agreement" in eqs 4.35-4.36 needs the self-energy doubled, which comes from Rutherford's unrefereed "energy density correction" [read: 2015 p.15; arbiter 1.2]. The paper's own eq. 4.33 gives the standard energy.

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6. Evidence table

Verdicts are the arbiter's (arbiter.md sections 3-4, as revised in section 8). "Replicated?" means an independent repeat found on record.

Context that stays in the record (from the Investigator and arbiter section 5): - Bruhn's critiques. G.W. Bruhn (TU Darmstadt), a mathematician and critic with a stake in the dispute, made two points. S is superfluous in standard theory, and the modified sources conserve charge only if S solves the homogeneous wave equation. Both are correct algebra, and the author replied. Bruhn also wrote "Maybe that some reasons can be found for the validity of condition (II)" [web: https://www2.mathematik.tu-darmstadt.de/~bruhn/vV191005.html]. - Priority dispute. The author's 2001 paper with Waser was received 13 Nov 2001, and Monstein-Wesley published 15 Aug 2002. The 2003 paper says their energy formulas came from his theorem without citation. The reviewers find those formulas also follow from the plain wave equation. Monstein and Wesley's own position is known here only through the 2003 paper. - Influence. The 2001 paper has about 99 citations [web: Google Scholar, via Investigator]. The same S field is developed in mainstream journals by Minotti and Modanese (EPJ C 83:1086, 2023), who cite van Vlaenderen and Waser, and by Arbab, Hively, Simulik, and Tomilin-Panchelyuga. None of them reports a detection. - Where the record goes dark. - Ethergy B.V. (KvK 59508701, active) [web: https://opencorporates.com/companies/nl/59508701] is the author's affiliation from 2016 to 2018. It has no patents, products or papers found. A 2023 Prague lecture (not transcribed) discusses converting the scalar field into electricity [web: https://www.youtube.com/watch?v=Egv98PRKrwc]. Competing explanations are a one-person research company, trade secrecy, a device that failed quietly, or absorption under another name. The records that would decide are the KvK filings, Espacenet/WIPO searches, and the lecture's Q&A. - Publication moved from arXiv and the Hadronic Journal to viXra (2015) and HRPUB (2016). No refusal or referee record exists either way. - A Google legal-removal notice appeared on a Moray/Fletcher affidavit query. It is a lead to follow (LumenDatabase), not evidence.

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7. What a learner should take away

Solid. - The algebra of the 2003 and 2015 papers is correct as algebra. Every identity checked out. - The Grassmann force between isolated open current elements really is non-reciprocal (section 5.2). - Maxwell's displacement current is fixed by charge conservation only in its curl-free part. The author's remark that the transverse part needed Hertz's experiment is mathematically right [read: 2015 p.8]. - The Monstein-Wesley pattern was rebuilt by sceptics, who found it and could not explain it. That is a genuine open anomaly, although it is not a test of S.

Wrong, and why the mistake is instructive. - Gauge freedom is not a circular argument. E and B are unchanged by a gauge change because of how they are defined, so S cannot be shown physical by logic alone. It has to be postulated and then detected. - The Newton's-third-law problem is a bookkeeping question: who holds the momentum? Section 5 shows the textbook answer (the field) and the author's answer (a new mechanical force) both balance the books. The mistake is the premise that the field holds none, which the paper's own eq. 4.34 contradicts. The useful lesson is that "conservation of momentum" and "Newton's third law for mechanical forces" are different statements once fields carry momentum. - Equations that are always true (continuity, eq. 4.22) cannot select a physical constant such as a >> c. - The static-charge power term shows why every new coupling needs a full energy budget, including the matter side. Field energy lost must show up somewhere. The first analysis found no receiver; the QE revision found one (the charge's rest energy) that works only if rest mass is allowed to change. The lesson is to state that assumption out loud: the same equations are consistent or inconsistent depending on it (section 4.2). - Reading individual terms of one integral as separate physical fields leads to the "far-field inconsistency" claim. The Lorenz- and Coulomb-gauge formulas look different term by term but give the identical field (section 4.3). - The 4/3 match shows how a borrowed correction (Rutherford's doubled energy) can quietly carry the result.

Genuinely open. - Is there any consistent coupling of S to matter that conserves energy, and is S ever sourced in nature (tunnelling, non-equilibrium plasmas)? Reading 1 of section 4.2 (rest mass changing at dm/dtau = qS) balances on paper, but neither the author nor anyone else has written that matter-side law down or tested it [arbiter 8.2]. The Minotti-Modanese programme is the mainstream-journal version of this question. - Why is the Monstein-Wesley pattern there? A repeat beyond lambda/(2pi) with ground-current controls would help. - Would a force measurement on one open current element, isolated from the electrostatic forces, show the push J B_l or not? - What did the 2023 lecture show, and was any device built? - The author's own named test, induction law 3.2: drive sinusoidal divergent (curl-free) currents in a primary and look for a secondary curl-free electric field E_l and currents 90 degrees or more out of phase. He writes: "no experiments have been done yet" and "The first experimenter who verifies this law also verifies the existence of the scalar magnetic field Bl and the classical physicality of the vector potential A" [read: 2015 p.17]. No amplitude, frequency or geometry is given, so a bench test would first need the expected size computed. This is separate from the Aharonov-Bohm question, where B_l is zero for a closed solenoid current. - Strebkov's single-wire line: the author's own discriminator (with or without a ground return) could be applied by inspecting or rebuilding the system and measuring the field orientation [arbiter 8.5]. - Does a static charged body in a controlled environment heat or cool in a way that bounds S? The static-charge power term predicts it would if S existed (section 4.2).

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8. Check yourself

1. Under the gauge change Phi -> Phi - d(lambda)/dt, A -> A + grad(lambda), why do E and B stay the same? What happens to S?
2. Using eq. 24, explain why a theory with physical S and locally conserved charge predicts no S from an ordinary antenna.
3. In section 5, element 1 lies along y and element 2 along x. Swap the elements' directions (element 1 along x at the origin, element 2 along y at (d,0,0)). Which element now feels the Grassmann force, and in which direction?
4. A static charge density rho = 2e-6 C/m^3 sits in S = 5e-7 T. Per the paper's eq. 33, at what rate per cubic metre does the field give up energy? What receives it?
5. Why does a closed-loop experiment such as two coils carrying steady currents not discriminate between the Grassmann and the author's force law?

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9. Further reading

Mainstream. - D.J. Griffiths, Introduction to Electrodynamics, 4th ed. Chapter 5 (Biot-Savart, Lorentz force), chapter 7 (Maxwell's equations), chapter 8 (conservation laws, field momentum, Newton's third law in electrodynamics), chapter 10 (potentials, gauge transformations), Appendix B (Helmholtz theorem). - R.P. Feynman, Lectures on Physics Vol. II, chapter 27 (field energy and momentum) and chapter 28 (electromagnetic mass and the 4/3 problem). - J.D. Jackson, Classical Electrodynamics, 3rd ed., chapter 6 (gauges, Poynting theorem). Also J.D. Jackson, "From Lorenz to Coulomb and other explicit gauge transformations", Am. J. Phys. 70, 917-928 (2002) [web: https://www.semanticscholar.org/paper/From-Lorenz-to-Coulomb-and-other-explicit-gauge-Jackson/cf3f1a498297aa7df026b6799b11e3cb923475e2]. - E.F. Taylor and J.A. Wheeler, Spacetime Physics (rest mass, energy-momentum), for section 4.2. - Caprez, Barwick, Batelaan, PRL 99, 210401 (2007) [web: https://arxiv.org/pdf/0708.2428].

The author and related work. - K.J. van Vlaenderen, "A charged space as the origin of sources, fields and potentials", arXiv physics/9910022 (1999). - K.J. van Vlaenderen and A. Waser, "Generalisation of classical electrodynamics to admit a scalar field and longitudinal waves", Hadronic Journal 24 (2001) 609-628 [web: http://www.tuks.nl/pdf/Reference_Material/Waser/GeneralisationOfClassicalElectrodynamics.pdf]. - K.J. van Vlaenderen, "A generalisation of classical electrodynamics for the prediction of scalar field effects", arXiv physics/0305098 (2003). - K.J. van Vlaenderen, "Classical Electrodynamics in agreement with Newton's third law of motion" (2015); fuller version viXra 1512.0297 [web: https://vixra.org/abs/1512.0297]. - K.J. van Vlaenderen, "General Classical Electrodynamics", Universal Journal of Physics and Application 10(4) (2016) 128-140 [web: https://www.hrpub.org/journals/article_info.php?aid=5192]. - SEM2023 lecture, "Scalar field theory and technologies" [web: https://www.youtube.com/watch?v=Egv98PRKrwc]. - Minotti and Modanese, "Aharonov-Bohm electrodynamics" line, EPJ C 83:1086 (2023) [web: https://arxiv.org/abs/2310.13968] and the detection-circuit proposal [web: https://arxiv.org/abs/2403.10580]. - Butterworth et al., J. Sci. Exploration 27(1) 13-23 (2013) [web: https://journalofscientificexploration.org/index.php/jse/article/view/477]; Monstein and Wesley, EPL 59 (2002) 514. - Related work named in the 1999 paper [read: 1999 refs]: W.A. Hofer, Beyond Uncertainty: the internal structure of electrons and photons, quant-ph/9611009; B. Haisch, A. Rueda and H.E. Puthoff, Phys. Rev. A 49, 678 (1994) (inertia from the zero-point field); V. Petkov, physics/9909019 (mass of a charge as its electrostatic energy). - G.W. Bruhn, critique pages [web: https://www2.mathematik.tu-darmstadt.de/~bruhn/vV191005.html].

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Answers

1. E = -grad Phi - dA/dt gains +grad(d lambda/dt) from the Phi change and -d(grad lambda)/dt from the A change. These cancel because derivatives commute. B = curl A gains curl grad lambda = 0. S gains (1/c^2) d^2(lambda)/dt^2 - Laplacian lambda, the wave operator on lambda, with a sign that depends on convention. So S can be set to anything that solves the matching wave equation, including 0.
2. Eq. 24's source is -mu0 (div J + d rho/dt). An antenna's currents and charges obey continuity, so the source is zero. S then satisfies a source-free wave equation and nothing in the antenna drives it.
3. Now p1 = I1 l1 x̂ at the origin and p2 = I2 l2 ŷ at (d,0,0). At element 2, B1 ∝ x̂ x x̂ = 0, so element 2 feels nothing. At element 1, the vector from 2 to 1 is -d x̂, so B2 ∝ ŷ x (-x̂) = +ẑ, and F1 = p1 x B2 ∝ x̂ x ẑ = -ŷ. Its size is again k = 1e-5 N with the section 5 numbers. The roles are swapped, and the sum is again nonzero.
4. c^2 rho S = (8.988e16)(2e-6)(5e-7) = 8.99e4 W/m^3. The paper's force does no work on a stationary charge, so if the charge's rest mass is fixed nothing receives it, and the books do not balance (Reading 2). If rest mass may change at dm/dtau = qS, the energy goes into the charge's rest (internal) energy, here (8.99e4 W/m^3)/c^2 = 1.0e-12 kg per cubic metre per second (Reading 1). The paper does not say which. That is why the arbiter's revised verdict is "Not testable as stated", with "Internally inconsistent" holding only under fixed rest mass.
5. For a closed steady circuit B_l = 0 everywhere, because it is proportional to the integral of d rho/dt / r and d rho/dt = 0. So the extra force J B_l vanishes, and the two laws give identical total forces (example.py Part E: 9.485e-9 N both ways). A discriminating test needs charge accumulation, that is, an open circuit.