On Koen van Vlaenderen's Seventh Field Component Again

Gerhard W. Bruhn, Darmstadt University of Technology


Summary In my first review [2] of van Vlaenderen's article [1] I stated that his theory is not compatible with the Maxwell theory. Van Vlaenderen replied in [3] that he intended a modification of the Maxwell equations consciously. Therefore in the following this modification is checked. It turns out that van Vlaenderen modifies the inhomogenities of two Maxwell equations, the charge density ρ and the current density J. We obtain the following result: While ρ and J - as is well-known - fulfil the conservation law of charges, the same is not true for van Vlaenderen's modifications, which makes the physical relevance of van Vlaenderen's theory questionable.

As can be seen from van Vlaenderen's response [3] to my web article [2] his theory consists in a modification of the well-known Maxwell equations of electrodynamics. His basic equations are

Φ − At = E     the electric field                                                 (1)

A = B     the magnetic field                                               (2)

which are equivalent to the "homogeneous" Maxwell equations

(1')                                                curl E + Bt= 0 ,

(2')                                                       div B = 0 ,

and the inhomogenious differential equations

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

1/c Att − ΔA = μoJ                                                                   (5)

where 1/c = εoμo. Due to (1) and (4) we obtain

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

where S denotes van Vlaenderen's "seventh field component" given by

λoεoμo ∂Φ/∂tA = S     the scalar field, a seventh field component              (3)

in the case λo=1 under consideration. Similarly (2) and (5) yield

(5')                               curl B1/c Et = 1/c Att − ΔA + St = μoJ − grad S

The red marked terms in (4'-5') show the deviations from the corresponding inhomogeneous Maxwell equations if a non-constant function S is used.

The deviations mean that the local charge ρ and the current J of the charge are modified: Instead of the true charge and current van Vlaenderen uses modified charge and current fields:

(6')                               ρ' := ρ + εoSt ,           J' := J1/μograd S

As is well-known charge density ρ and current density J of the Maxwell equations fulfil the conservation law for charges

(7')                                                             ρt + div J = 0 .

The check for the conservation law of van Vlaenderen's modified charge and current fields yields

(8')                               ρ't + div J' = εoStt1/μoΔS = 1/μo (1/cStt − ΔS) .

Thus we have the results:

(I) Van Vlaenderen's modified charge and current fields (6') do not fulfil the conservation law (7') of charges in general.

(II) Van Vlaenderen's modified charge and current fields (6') fulfil the conservation law (7') of charges if and only if his seventh field component S is a solution of the wave equation

1/cStt − ΔS = 0 .

The result (I) is a strong argument against the physical validity of van Vlaenderen's modification of the Maxwell equations. Maybe that some reasons can be found for the validity of condition (II).

Remark Van Vlaenderen has repeatedly remarked that he thinks the usual gauge theory to be of circular logic. I cannot follow his ideas and refer to the literature, e.g. to my article [4]. If necessary this topic could be discussed in another article.


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

[2]            Gerhard W. Bruhn: A Remark on van Vlaenderen's Seventh Field Component

[3]            Koen van Vlaenderen: Gerhard Bruhn's Remark on my scalar field theory is wrong

[4]            Gerhard W. Bruhn: Gauge Theory of the Maxwell Equations

[5]            Ernst Schmutzer, Grundlagen der Theoretischen Physik, Teil 1 p.588 ff.