updated by a Remark on Evans' flaw of thinking on July 03, 2006

some typos removed on July 04, 2006

Survey on Evans/Eckardt's Coulomb Resonance:

Resonance Nowhere

Gerhard W. Bruhn, Darmstadt University of Technology


Quotations from Evans/Eckardt's web-note [1] and [6] are displayed in black with equation labels [1;(nn)]/[6;(nn)] at the right margin.


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.

I. Evans' spherical version

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

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

Application of the Euler transform r = et yields

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

Eigenvalues of the left hand side differential operator:

                λ² + (1−A) λ − A = 0         yields         λ1 = −1         and         λ2 = A        REALs

Eigenfunctions:                                                 Φ1 = e−t         and         Φ2 = eAt         if A≠−1
                                                                                                              Φ2 = te−t         if A=−1

Eigenfunctions of [1;(18)]:                               Φ1 = 1/r          and         Φ2 = rA          if A≠−1
                                                                                                              Φ2 = 1/r ln r     if A=−1

Result         NO RESONANCE with an oscillating driving inhomogenity.

II. Evans' modified version

Ñ²Φ − A/z ∂Φ/∂z + A/z² Φ = ρo/εo cos(κz) ,                                                                                 [1;(77)]

where Ñ²Φ due to Evans has the meaning ∂²Φ/∂z² , hence

                                        (z ∂/∂z)² Φ − (1+A) z ∂Φ/∂z + A Φ = z² ρo/εo cos(κz) .

Application of the Euler transform r = et yields

                                                Φ· · − (1+A) Φ· + A Φ = e2t ρo/εo cos(κet) .

Eigenvalues of the left hand side differential operator:

                λ² − (1+A) λ + A = 0         yields         λ1 = 1         and         λ2 = A        REALs

Eigenfunctions:                                                 Φ1 = et         and         Φ2 = eAt         if A≠1
                                                                                                             Φ2 = te−t         if A=1

Eigenfunctions of [1;(77)]:                               Φ1 = z          and         Φ2 = zA          if A≠1
                                                                                                            Φ2 = z ln z       if A=1

Result         NO RESONANCE with an oscillating driving inhomogenity.

Remark on Evans' flaw of thinking

In some of his web-notes instead of the (real) Euler transform z = exp(x) Evans used the complex transform z = exp(iκx) not recognizing that his transform due to the factor i would exchange real and imaginary axes of the complex x-plane. Of course, then the eigenvalues calculated above would transform to imaginary eigenvalues. However, afterwards he had to use an imaginary variable x, x=it, which would let his "oscillations" disappear. So no resonance as well, if Evans would use his transform properly:

On June 26, 2006 Evans circulated a third version with the following note:

SUMMARY, 23 June 2006
THE RESONANT COULOMB LAW

In one dimension (z)

d²Φ/dz² + 1/z dΦ/dz − Φ/z² = − ρo/εo cos(κz),                                                                 [6;(1)]

Using the Euler method this reduces to

d²Φ/dx² + κ² Φ = ρ/εo Re(e2iκx cos(eiκx))                                                                 [6;(2)]

where ...

These equations have been checked for analytical correctness and eq.(2) has been solved numerically by Dr. Horst Eckardt of the Siemens Company in Munich.

Awful if true! Analytical correctness! cf. [2].

Dr. Eckardt requested on June 27, 2006 by email for his opinion refused any discussion and broke off further contacts electronically.

From later circulars by Evans I learned what is meant by "Euler method": He applies the transform

                                                                z = eiκx

without recognizing the peculiarities of that complex transform:

Since z is a real variable the new variable x must be chosen such that the image z = eiκx is real, and that is not the case if Evans assumes x to be real too.

(I) For real x the image z = eiκx is a point on the complex unit circle |z|=1,
hence no real value in general.

(II) To obtain a real value of z = eiκx the original x must be chosen at the
imaginary axis, x=it, where t is real.

Instead of the correct real Euler transform z = eκt, t=1/κ log z, which maps zÎR → tÎR, Evans and Eckardt use the complex transform z = eiκx, x=1/iκ log z, which maps zÎR → xÎiR . The variable x is not a real variable and therefore inadmissible.

To obtain a real differential equation instead of the complex de.[6;(2)] a further transform

x = it , t = −i x

must be applied to [6;(2)] to obtain

d²Φ/dt² − κ² Φ = − ρo/εo Re(e2κt cos(eκt)) = − ρo/εo e2κt cos(eκt)                                                

where obviously no oscillating eigensolutions exist. The eigensolutions are

Φ1 = eκt ,         Φ2 = e−κt ,

corresponding to the eigensolutions z and 1/z of the original de.[6;(1)]. Hence there are no oscillating solutions as well. The oscillations in eq.[6;(2)] were an artifact merely due to incorrect application of transformation rules by Evans and Eckardt.


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://www2.mathematik.tu-darmstadt.de/~bruhn/NoO3-symmetry.pdf

[4] G.W. Bruhn; Refutation of Myron W. Evans’ B(3) field hypothesis
        http://www2.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

[5] A. Lakhtakia; Negative experimental evidence, Sect.5 in
        http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[6] M.W. Evans; SUMMARY, 23 June 2006: THE RESONANT COULOMB LAW,
        http://www2.mathematik.tu-darmstadt.de/~bruhn/aresonantcoulomblaw26thJune06.pdf



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