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

−ÑΦ − A_t = E the electric field (1)

Ñ×A = B the magnetic field (2)

which are equivalent to the "homogeneous" Maxwell equations

(1') curl E + B_t= 0 ,

(2') div B = 0 ,

and the inhomogenious differential equations

¹/_c² Φ_tt − ΔΦ = ^ρ/_{ε_o} (4)

¹/_c² A_tt − ΔA = μ_oJ (5)

where ¹/_c² = ε_oμ_o. Due to (1) and (4) we obtain

(4') div E = ¹/_c² Φ_tt − ΔΦ + S_t = ^ρ/_{ε_o} + S_t

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

λ_oε_oμ_o ^∂Φ/_∂t − Ñ·A = S the scalar field, a seventh field component (3)

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

(5') curl B − ¹/_c² E_t = ¹/_c² A_tt − ΔA + S_t = μ_oJ − 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 + ε_oS_t , J' := J − ¹/_{μ_o}grad S

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

(7') ρ_t + div J = 0 .

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

(8') ρ'_t + div J' = ε_oS_tt − ¹/_{μ_o}ΔS = ¹/_{μ_o} (¹/_c²S_tt − 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/_c²S_tt − 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.

References

[1] Koen van Vlaenderen: Generalised Classical Electrodynamics
for the prediction of scalar field effects
http://home.wanadoo.nl/raccoon/

[2] Gerhard W. Bruhn: A Remark on van Vlaenderen's Seventh Field Component
http://www2.mathematik.tu-darmstadt.de/~bruhn/Refutation-raccoon.html

[3] Koen van Vlaenderen: Gerhard Bruhn's “Remark” on my scalar field theory is wrong
http://home.wanadoo.nl/raccoon/refutation-Bruhn-Remark.html

[4] Gerhard W. Bruhn: Gauge Theory of the Maxwell Equations
http://www2.mathematik.tu-darmstadt.de/~bruhn/Maxwell-Theory.html