## A Remark on van Vlaenderen's Seventh Field Component

### Gerhard W. Bruhn, Darmstadt University of Technology

#### Summary

Recently Koen van Vlaenderen has introduced a "seventh field component" by generalizing the so-called Lorentz gauge condition with the aim of reviving Tesla's longitudinal electromagnetic waves. However, as we shall show below, that "seventh field component" is not appropriate for generating new solutions E, B of the Maxwell equations. It is completely superfluous. More, Van Vlaenderen's basic equations (1), (2), (3) contradict his equations (4), (5), (6) and (9) in general. Hence Van Vlaenderen's derived "Tesla"-results are not correct.

Quotation from van Vlaenderen's article [1]:

### The scalar field: a 7th field component

Oliver Heaviside reduced Maxwell's equations to a few vector/scalar equations of 6 field components. Heaviside did not like the concept of electromagnetic potentials, because these potentials are unphysical and abstract in comparison with the “measurable” fields. Yet the potentials are very useful in order to simplify many calculations and useful for the prediction of new physical effects, such as the Aharonov-Bohm effect. Here it is shown that extra scalar field terms, that are defined in terms of the electromagnetic potentials, can be added to the Maxwell/Heaviside field equations of Classical Electrodynamics (CED). The scalar field is introduced in the spirit of Oliver Heaviside, who stressed the importance of fields, and of James Clerc Maxwell, who predicted vacuum waves by adding a new displacement current term to the Ampère law.

First we define the fields in terms of the electromagnetic potentials Φ and A.

ÑΦ − A/∂t = E the electric field                                                     (1)

Ñ×A = B the magnetic field                                                   (2)

− λoεoμo ∂Φ/∂tÑ·A = S the scalar field, a seventh field component              (3)

The official theory of electrodynamics presumes that always S=0, and this is called “gauge condition”.

### Remark

Let the potentials Ao, Φo be defined by

(3a)                                Ao := A + ÑΨ ,                 Φo := Φ − Ψt ,

where Ψ is some function at our disposal. Let Ψ be a solution of the inhomogenous wave equation

(3b)                                 − λoεoμo Ψtt + ΔΨ = S .

Then inserting (3a-b) into (1), (2), (3) yields

(1')                                             −ÑΦoAo/∂t = E ,

(2')                                                           Ñ×Ao = B ,

(3')                                 − λoεoμo ∂Φo/∂tÑ·Ao = 0 .

### Conclusion

Each electrodynamic field (E,B) that is generated by potentials A, Φ , which fulfil the inhomogenous equation (3) can be generated by potentials Ao, Φo as well, which fulfil the homogenous Lorentz condition (3').

### Further discussion

The attained result is not all new in electrodynamics. It means that electrodynamics, i.e. the set of possible solutions of the Maxwell equations, is gauge independent. (cf. Anhang B)

Therefore van Vlaenderen's "generalised Gauss Law" (6) and his "generalised Ampère Law" (9) (see below) cannot be correct in the sense of being compatible with the Maxwell equations. There is a contradiction between the Maxwell compatible equations (1), (2), (3) on one hand and van Vlaenderen's "generalized" Laws (4), (5), (6), (9) on the other hand.

To check that we restrict our consideration to the case λo=1 and introduce 1/ := εoμo. The check of van Vlaenderen's equations (4),(5) yields (see the Appendix below)

(4')                                                 1/ Φtt − ΔΦ + St = ρ/εo

and

(5')                                                 1/ Att − ΔA ÑS = μoJ ,

the red terms are missing in (4) and (5) (see below). From (1) we obtain

div E = − ΔΦ − (div A)t = − ΔΦ + (S + 1/Φt)t = 1/Φt − ΔΦ + St ,

therefore from (4')

(6')                                                 div E = ρ/εo .

Similarly we obtain by applying Ñ× to (2)

(9')                                                 Ñ×B1/ Et = μoJ ,

i.e. the red terms in (4') and (5') have cancelled out: We have arrived at two of the original Maxwell equations, the Gauss Law and the Ampère Law, again.

Thus, correct calculation yields the unchanged Maxwell equations (6') and (9') instead of van Vlaenderen's equations (6) and (9) containing the wrong additional terms −St and +ÑS respectively.

Instead of starting with the correct consequences of his own ansatz (1), (2), (3), which would lead to (4'), (5') for arbitrary functions S, van Vlaenderen puts the equations

1/ Φtt − ΔΦ = ρ/εo                                                                 (4)

1/ Att − ΔA = μoJ ,                                                                 (5)

at the top of his considerations. Both equations are (for S ≠ const) not derivable from the Maxwell equations as can be seen by comparing his "generalised" laws (6) and (9) with the correct Maxwell laws (6') and (9'):

div E − St = ρ/εo generalised Gauss Law                                  (6)

Ñ×B1/ Et + ÑS = μoJ   generalised Ampère Law                               (9)

### No miracle that this procedure yields equations and effects that cannot be attained from the basis of the Maxwell equations.

The erroneous equations (6) and (9) have consequences, of course, for the rest of the article. We can state here that van Vlaenderen's longitudinal Tesla effects are nothing but the consequences of his erroneous equations (4) and (5).

### References

[1]            Koen van Vlaenderen: Generalised Classical Electrodynamics
for the prediction of scalar field effects

[2]            K.J. van Vlaenderen and A. Waser: Electrodynamics with the scalar field
http://www.info.global-scaling-verein.de/Documents/ElectrodynamicsWithTheScalarField03.pdf

[3]            Gerhard W. Bruhn: Besprechung des Abschnitts "Scalar Waves"
im Buch "Energy Medicine - The Scientific Basis”

### Appendix A: Potential representations of the fields of Electrodynamics

Under the assumption of constant material coefficients ε, μ (we omit the subscript o in the following) the electrical charge density ρ(x,t), the current density J(x,t), the electric field vector E(x,t) and the magnetic field vector H(x,t) have to fulfil the "inhomogeneous" Maxwell-equations

(A1)                                                        curl E = – Bt ,

(A2)                                                        curl B = εμ Et + μJ ,

(A3)                                                       ε div E = ρ ,

(A4)                                                        div B = 0 .

(A4) yields the (local) existence of a vector potential A such that:

(A5)                                                       B = curl A .

Inserting this into (A1) gives

(A6)                                                       curl (E + At) = 0 ,

which proves the existence of a (local) potential Φ such that

(A7)                                                           E = – grad Φ – At .

>From (A2) we obtain by using (A5) and (A7):

μ J = curl B – εμ Et

(A8)                                                = curl curl A – εμ (– Att – grad Φt)

= grad (div A + εμ Φt) – (ΔA1/ Att) .

We now introduce the "generalised" Lorentz-function S by

(A9)                                                        S := – (div A + εμ Φt) ,

then we obtain the vector potential representation of the current density

(A10)                                                      μ J = 1/ Att – ΔA – grad S .

Similarly (A3) and (A9) yield the potential representation of the charge distribution ρ:

(A11)                                                     ρ/ε = div E = 1/ Φtt – ΔΦ + St .

(A10) and (A11) are the correct substitutes for van Vlaenderen's wrong equations (4) and (5).

### Anhang B

Einige Bemerkungen des Autors vV lassen vermuten, dass er etwas an der Maxwell-Theorie missverstanden hat, die zugegebenermaßen hinsichtlich der Lorentz-Bedingung (s.u.) auch ein wenig raffiniert ist.

### Skizze der Maxwell-Lösungstheorie

(1) Beliebige Potentiale Φ,A liefert nach (A7) und (A5) eine Lösung E,B der beiden
homogenen Maxwell-Gleichungen (A1) und (A4), unabhängig davon, welches S bei vV's
Gleichung (A9) herauskommt. Die Gleichungen bestimmen (A2) und (A3) bestimmen dann die
zugehörigen Dichte ρ und Stromdichte J. Die Gleichung (A9) definiert das zugehörige S, das
keineswegs 0 sein muss. D.h. die Lorentz-Bedingung S=0 ist i.a. verletzt.

Die Annahme (1) gegebener Potentiale Φ,A ist aber nicht realistisch, weil in der Realität nicht Φ,A, sondern Dichte ρ und Stromdichte J vorgegeben sind. Allerdings sind beide Größen nicht ganz unabhängig vorschreibbar: Die Bildung von div (A2) gibt wegen div curl = 0 und mit (A3)

0 = 1/ε (div E)t + div J = ρt + div J.

das ist die Kontinuitätsgleichung der el. Stromdichte  als Kompatibilitätsbedingung:

(K)                   div J + ρt = 0 ,

(2) Die Maxwell-Gleichungen (A1) und (A4) sind für jede Potentialwahl erfüllt.
Die Maxwell-Gleichungen (A2) und (A3) ergeben dagegen Bestimmungsgleichungen
für die Potentiale Φ,A, das sind die Gleichungen (A10), (A11), das sind bei gegebenen
Dichten ρ,J "Wellengleichungen" (A10), (A11) für die Potentiale Φ,A.
Diese sind aber über S nach (A9) miteinander gekoppelt, was eine Auflösung effektiv
verhindert.

Aber in der Wahl des Potentials A hat man noch Freiheit: Statt A ist auch A' = A + grad U wieder ein mögliches Vektorpotential. Allerdings würde der Übergang AA' allein sich auch in (A7) als Abänderung von E bemerkbar machen, was nicht sein darf. Man mache sich aber klar, dass die zusätzliche Änderung Φ → Φ' = Φ − Ut die Felder B und E, nach (A5) und (A7) berechnet, unverändert lässt. Wie hängt S von diesem Übergang ab?

Die Größe S = – div A1/ Φt ändert sich durch die Ersetzungen

(G)                   AA' = A + grad U ,     Φ → Φ' = Φ − Ut

zu

S' = – div(A + grad U) – 1/ (Φ − Ut)t = S + 1/ Utt – ΔU.

Man nennt die Ersetzung (G) eine Eichtransformation (gauge transformation) oder auch Umeichung.

#### Die Umeichung S → S' hat auf die physikalisch beobachtbaren Größen E, B, J und ρ keinen Einfluss.

Wählt man also das noch freie U als Lösung der inhomogenen Wellengleichung

1/ Utt – ΔU = – S,

so erhält man S' = 0.

Folgerung: Unter den möglichen äquivalenten Potentialen A', Φ' gibt es ein ausgezeichnetes Paar mit S' = 0. Anders ausgedrückt: Im Fall S≠0 ist stets eine Umeichung auf S'=0 möglich.

Diese besondere Umeichung S → S' = 0, also das Erfüllen der Lorentz-Bedingung S' = 0, dient der Entkopplung der Gleichungen (A10), (A11), ohne beobachtbare Größen zu verändern.

Für die (entkoppelten) Wellengleichungen (A10), (A11) (jetzt mit S = 0) kann man Lösungen in Integralform angeben:

Φ(x,t) = 1/4πε ρ(y,t – |x–y|/c)/|x–y| dy,

A(x,t) = μ/ J(y,t – |x–y|/c)/|x–y| dy .

Natürlich muss für die Brauchbarkeit dieser Lösungen auch das Bestehen der Lorentz-Bedingung

div A + 1/ Φt = 0

überprüft werden. Es zeigt sich nach einiger Rechnung, dass hierfür das Bestehen der aus physikalischen Gründen geforderten Kontinuitätsbedingung (K) der el. Stromdichte hinreichend ist.

### Zusammenfassung des Lösungsverfahrens

Zu gegebener Dichte ρ und Stromdichte J mit erfüllter Kontinuitätsgleichung (K) werden die Integrale für die Potentiale Φ und A berechnet. Mit deren Hilfe liefern dann (A5) und (A7) die Felder E und B.

### Literatur

E. Schmutzer, Grundlagen der Theoretischen Physik, Teil 1 S.588 ff.

### Appendix C: Coulomb Gauge

Let λo = 0. Then vV's ansatz is

ÑΦ − A/∂t = E the electric field                                                     (1)

Ñ×A = B the magnetic field                                                   (2)

Ñ·A = S the scalar field, a seventh field component              (3)

The official theory of electrodynamics presumes that always S=0, and this is called “gauge condition”.

We shall see that the condition S = 0 can be assumed without loss of generality by performing a suitable gauge transformation:

Let ρ and A be given densities and let the potentials Ao, Φo be defined by

(3a)                                Ao := A + ÑΨ ,                 Φo := Φ − Ψt ,

where Ψ is some function at our disposal. Let Ψ be a solution of the Poisson equation

(3b)                                                 ΔΨ = S = − Ñ·A .

Our gauge transform transforms S → S'= − Ñ·Ao = 0.

Then inserting (3a-b) into (1), (2), (3) yields

(1')                                             −ÑΦoAo/∂t = E ,

(2')                                                           Ñ×Ao = B ,

(3')                                                            Ñ·Ao = 0 .

Inserting this into the Maxwell (Gauss) equation Ñ·E = ρ/ε yields the Poisson equation

− ΔΦo = ρ/ε ,

a solution of which is given by the integral

Φo(x,t) = 1/4πε ρ(y,t)/|x–y| dy .

And the Ampère Law yields

μJ = Ñ×B1/ Et = Ñ×(Ñ×Ao) − 1/ Et = − ΔAo + 1/ Ao tt + 1/ Φo t ,

that is the wave equation

1/ Ao tt − ΔAo = μJ1/ ÑΦo t ,

a source-free solution of which is to be found due to the gauge condition (3').

E. Schmutzer showed on p.607 that the integral solution

Ao(x,t) = μ/ [J(y,t – |x–y|/c)/|x–y|1/μc² ÑΦo t] dy

is source-free indeed.