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)

− λ_oε_oμ_o ^∂Φ/_∂t − 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”.

Remark

Let the potentials A_o, Φ_o be defined by

(3a) A_o := A + ÑΨ , Φ_o := Φ − Ψ_t ,

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

(3b) − λ_oε_oμ_o Ψ_tt + ΔΨ = S .

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

(1') −ÑΦ_o − ^∂A_o/_∂t = E ,

(2') Ñ×A_o = B ,

(3') − λ_oε_oμ_o ^∂Φ_o/_∂t − Ñ·A_o = 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 A_o, Φ_o as well, which fulfil the homogenous Lorentz condition (3').

Thus, van Vlaenderen's "seventh field component S" cannot generate electromagnetic phenomena that could not be generated without S. The "seventh field component S" can be omitted without loss of generality. The introduction of S by the equation (3) is not wrong but completely unnecessary.

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 λ_o=1 and introduce ¹/_c² := ε_oμ_o. The check of van Vlaenderen's equations (4),(5) yields (see the Appendix below)

(4') ¹/_c² Φ_tt − ΔΦ + S_t = ρ/ε_o

and

(5') ¹/_c² A_tt − ΔA − ÑS = μ_oJ ,

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

div E = − ΔΦ − (div A)_t = − ΔΦ + (S + ¹/_c²Φ_t)_t = ¹/_c²Φ_t − ΔΦ + S_t ,

therefore from (4')

(6') div E = ρ/ε_o .

Similarly we obtain by applying Ñ× to (2)

(9') Ñ×B − ¹/_c² E_t = μ_oJ ,

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 −S_t 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/_c² Φ_tt − ΔΦ = ρ/ε_o (4)

¹/_c² A_tt − ΔA = μ_oJ , (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 − S_t = ρ/ε_o generalised Gauss Law (6)

Ñ×B − ¹/_c² E_t + ÑS = μ_oJ 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
http://home.wanadoo.nl/raccoon/

[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”
http://www2.mathematik.tu-darmstadt.de/~bruhn/Besprechung-Oschman.htm

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 = – B_t ,

(A2) curl B = εμ E_t + 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 + A_t) = 0 ,

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

(A7) E = – grad Φ – A_t .

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

μ J = curl B – εμ E_t

(A8) = curl curl A – εμ (– A_tt – grad Φ_t)

= grad (div A + εμ Φ_t) – (ΔA – ¹/_c² A_tt) .

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

(A9) S := – (div A + εμ Φ_t) ,

then we obtain the vector potential representation of the current density

(A10) μ J = ¹/_c² A_tt – ΔA – grad S .

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

(A11) ρ/ε = div E = ¹/_c² Φ_tt – ΔΦ + S_t .

(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 = ρ_t + div J.

das ist die Kontinuitätsgleichung der el. Stromdichte als Kompatibilitätsbedingung:

(K) div J + ρ_t = 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 A → A' 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 − U_t die Felder B und E, nach (A5) und (A7) berechnet, unverändert lässt. Wie hängt S von diesem Übergang ab?

Die Größe S = – div A – ¹/_c² Φ_t ändert sich durch die Ersetzungen

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

S' = – div(A + grad U) – ¹/_c² (Φ − U_t)_t = S + ¹/_c² U_tt – 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/_c² U_tt – 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/_{4π ∫} 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/_c² Φ_t = 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.

Added on August 9, 2005:

Appendix C: Coulomb Gauge

Let λ_o = 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 A_o, Φ_o be defined by

(3a) A_o := A + ÑΨ , Φ_o := Φ − Ψ_t ,

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·A_o = 0.

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

(1') −ÑΦ_o − ^∂A_o/_∂t = E ,

(2') Ñ×A_o = B ,

(3') Ñ·A_o = 0 .

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

− ΔΦ_o = ρ/ε ,

a solution of which is given by the integral

Φ_o(x,t) = ¹/_{4πε ∫} ^ρ(y,t)/_|x–y| dy .

And the Ampère Law yields

μJ = Ñ×B − ¹/_c² E_t = Ñ×(Ñ×A_o) − ¹/_c² E_t = − ΔA_o + ¹/_c² A_{o tt} + ¹/_c² Φ_{o t} ,

that is the wave equation

¹/_c² A_{o tt} − ΔA_o = μJ − ¹/_c² ÑΦ_{o 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

A_o(x,t) = ^μ/_{4π ∫} [J(y,t – ^|x–y|/_c)/_|x–y| − ¹/_μc² ÑΦ_{o t}] dy

is source-free indeed.