updated by an Appendix 1 on June 24, 2006

updated by a Preliminary Remark on June 27, 2006

updated by a Some remarks added on June 29, 2006


Remarks on Evans/Eckardt's Web-Note on Coulomb Resonance

Gerhard W. Bruhn, Darmstadt University of Technology


Quotations from Evans/Eckardt's web-note [1] are displayed in black with equation labels [1;(nn)] at the right margin.

Abstract
The following note shows the mathematical inconsistencies of the web-note under review: The authors assert that the addition of a hypothetical spin connection term in case of spherical symmetry leads to resonance effects for the modified Coulomb potential equation. However, the direct evaluation of the authors' original equation [1;(18)] gives merely real eigenvalues of the corresponding linear de. (5) and hence no resonance with an oscillatory driving term. The authors ignore this result. They modify their equation [1;(18)] by falsely replacing the radial unit vector er with a constant unit vector k, which then (after unnecessary further manipulations) yields complex eigenvalues, an complete "artifact" that has nothing to do with the original equation [1;(18)].


Preliminary Remark

The note of M.W. Evans and H. Eckardt to be discussed below is part and consequence of Evans' ECE Theory (former GCUFT) which is essentially based on Evans' O(3) hypothesis. However that hypothesis lacks from being not Lorentz invariant [3] and not fulfilling the linear superposition principle of em waves [4], and therefore is no valid theory of physics. Negative experimental evidence of the O(3) hypothesis can be found in [5; Sect.5] contributed by A. Lakhtakia. So the note [1] is invalid from a general point of view.

However, M.W. Evans doesn't care for objections against his considerations. Therefore we review the note [1] on "Coulomb Resonance" here independently showing that even that small part of Evans' general theory contains several calculation and other errors that make the whole note wrong in itself.



Without taking any liability for Evans' previous assumptions and calculations we start our check of the web-note [1] with the de.

Ñ²Φ − Ñ·ω) = − ρ/εo ,                                                                         [1;(10)]

where the authors assume radial variance only and

ω = ωr er ,                                                                                         [1;(13)]

ω·Ñ Φ = ωr ∂Φ/∂r ,                                                                                 [1;(14)]

Φ Ñ·ω = Φ/ /∂r (r²ωr) ,                                                                                 [1;(15)]

Ñ²Φ = 1/ /∂r (r²∂Φ/∂r) = ∂²Φ/∂r² + 2/r ∂Φ/∂r ,                                                                 [1;(16)]

ωr = A/r ,                                                                                         [1;(17)]

to obtain the result (we agree to)

∂²Φ/∂r² + (2−A) 1/r ∂Φ/∂rA/ Φ = − ρ/εo ,                                                                 [1;(18)]

or multiplied by r²

(1)                                                 r² ∂²Φ/∂r² + (2−A) r ∂Φ/∂r − A Φ = − r² ρ/εo ,

and using the identiy

(2)                                 r² ∂²Φ/∂r² = r /∂r (r ∂Φ/∂r) − (r ∂r/∂r) ∂Φ/∂r = r /∂r (r ∂Φ/∂r) − r ∂Φ/∂r

to obtain from [1;(18)]

(3)                                         (r /∂r)² Φ + (1−A) r ∂Φ/∂r − A Φ = − r² ρ/εo .

As is well-known [3] this Euler de. can be reduced to a linear de. by means of the transformations

(4)                                 t = ln r , r = et         r /∂r = r d/dr = d/dt .

Therefore eq.(3) yields the linear de.

(5)                                 Φ· · + (1−A) Φ· − A Φ = − e2t ρ/εo .

Instead of going this obvious way Evans starts a rather complicated and dubious consideration at the end of which he arrives at

Ñ²Φ − A/z ∂Φ/∂z + A/ Φ = ρo/εo cos(κz) .                                                                 [1;(77)]

where the former r was substituted with z and now apparently the term Ñ²Φ has the meaning Φzz, i.e.

∂²Φ/∂z² A/z ∂Φ/∂z + A/ Φ = ρo/εo cos(κz) .                                                                 [1;(77')]

Comparing this with Evans' correct eq. [1;(18)] the reader should note the important differing coefficients.

By applying the Euler transform (4) we obtain Evans' version of the linear de. (5)

(5')                                 Φ· · − (1+A) Φ· + A Φ = ρo /εo cos(κet),

Comparison with

x· · + 2βx· + ωo² x = α cos ωt                                                                 [1;(78)]

shows that the right hand sides of eqns. [1;(77)] and (5') are erroneous and should correctly read as ρo /εocos κt . That is only a minor error. Much more important is the difference that appears by calculation of the characteristic numbers of the correct linear de.(5). We obtain from

(7)                                                 λ² − (1+A) λ + A = 0

the real eigenvalues λ1 = A and λ2 = 1. The corresponding eigensolutions are therefore non-oscillatory.

To express that in Evans' terms: The device described by eq. (5') is no "damped oscillator". A damped oscillator would require conjugate complex eigenvalues of the corresponding characteristic equation (7) of (5'), however, the real eigenvalues λ1 = A and λ2 = 1 are real whatever the (real) value of A should be. There is no difference to the special case A=0 (the pure Coulomb operator) where even Evans admits it to be resonance-free.

The oscillating cos-ansatz "α cos ωt" at the right hand side of eq.[1;(78)] cannot be in resonance with its non-oscillatory eigensolutions Φ1=eAt and Φ2=et . Thus, there is NO RESONANCE at all.

Remark The exception case A=1 yields a real double eigenvalue λ=1 with the eigenfunctions Φ1 = et and Φ2 = t et the latter of which is increasing due to the "eigen-resonance" λ12.


Which are the reasons of the authors' completely deviating results?

The resonance equation

Ñ²Φ − ω·Ñ Φ −(Ñ·ω) Φ = − ρ/εo ,                                                                 [1;(74)]

is identical with eq.[1;(10)] except an application of the Leibniz product rule. Then the authors erroneously assume

ω = A/z k ,                                                                                 [1;(75)]

while the combination of their eqns.[1;(13)] and [1;(17)] yields

ω = A/r er .                                                                         [1;(13+17)]

This shows that the authors, without giving any explanation, have ignored the radial direction of er and replaced er by the constant vector k , while they did not ignore the variance of the factor 1/r by replacing it with 1/z. That yields

(8)                                                 Ñ· er/r = Ñ· r/ = 3/2/ = + 1/

while

(9)                                                         Ñ· k/z = 1/ .

This difference explains the deviating results of Evans and Eckardt, but, no doubt, their justification for using eq.(9) is missing.


Appendix 1: The Eigensolutions of Evans' De. [1;(18)]

Eigensolutions of the de. [1;(18)] are solutions of the corresponding homogeneous de.

(A1-1)                                                 ∂²Φ/∂r² + (2−A) 1/r ∂Φ/∂rA/ Φ = 0

As the reader can check by elementary calculation the de.(A1) has the eigensolutions

(A1-2)                                                 Φ1 = 1/r         and         Φ2 = rA.

The eigenspace consists just of all linear combinations of the solutions (A2) none of which is oscillating. Thus:

There is NO RESONANCE with oscillating driving inhomogenities

in contradiction to the authors' assertions later on in [1] and supplied by complicated and erroneous calculations. The authors' results have nothing to do with their own equation [1;(18)].



References

[1] M.W. Evans and H. Eckardt; Space-time Resonances in the Coulomb Law,
        http://www.aias.us/documents/uft/a61stpaper.pdf

[2] Mathematik-Online-Lexikon: Euler-Differentialgleichung
        http://mo.mathematik.uni-stuttgart.de/inhalt/aussage/aussage773/

[3] G.W. Bruhn; On the Lorentz Variance of the Claimed O(3)-Symmetry Law,
        http://www.mathematik.tu-darmstadt.de/~bruhn/NoO3-symmetry.pdf

[4] G.W. Bruhn; Refutation of Myron W. Evansí B(3) field hypothesis
        http://www.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

[5] A. Lakhtakia; Negative experimental evidence, Sect.5 in
        http://www.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html




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