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

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 **e**_{r} 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**·**(Φ**ω**)
=
− ^{ρ}/_{εo} ,
[1;(10)]

where the authors assume *radial* variance only and

**ω** = ω_{r} **e**_{r} ,
[1;(13)]

**ω·**Ñ Φ =
ω_{r}
^{∂Φ}/_{∂r} ,
[1;(14)]

Φ Ñ**·ω** =
^{Φ}/_{r²}
^{∂}/_{∂r} (r²ω_{r}) ,
[1;(15)]

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

ω_{r} = ^{A}/_{r} ,
[1;(17)]

to obtain the result (we agree to)

^{∂²Φ}/_{∂r²} +
(2−A) ^{1}/_{r} ^{∂Φ}/_{∂r}
− ^{A}/_{r²} 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 = e^{t}
r ^{∂}/_{∂r} =
r ^{d}/_{dr} = ^{d}/_{dt} .

Therefore eq.(3) yields the linear de.

(5)
Φ^{· ·} +
(1−A) Φ^{·}
− A Φ =
− e^{2t} ^{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}/_{z²} Φ =
^{ρo}/_{εo} cos(κz) .
[1;(77)]

where the former r was substituted with z and now apparently the term
Ñ²Φ
has the meaning Φ_{zz}, i.e.

^{∂²Φ}/_{∂z²}
− ^{A}/_{z} ^{∂Φ}/_{∂z}
**+**
^{A}/_{z²} 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(κe^{t}),

Comparison with

x^{· ·} + 2βx^{·} + ω_{o}² 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 }/_{εo}cos 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 λ_{1} = A and
λ_{2} = 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
λ_{1} = A and λ_{2} = 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.

**Remark** The exception case A=1 yields a real double eigenvalue λ=1 with the eigenfunctions
Φ_{1} = e^{t} and Φ_{2} = t e^{t}
the latter of which is increasing due to the "eigen-resonance" λ_{1}=λ_{2}.

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} **e**_{r} .
[1;(13+17)]

This shows that the authors, without giving any explanation, have ignored
the *radial* direction of
**e**_{r} and replaced **e**_{r}
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}/_{r²}
=
^{3}/_{r²} − ^{2}/_{r²}
= + ^{1}/_{r²}

while

(9)
Ñ**·** ^{k}/_{z}
= **−** ^{1}/_{z²} .

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

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

(A1-1)
^{∂²Φ}/_{∂r²} +
(2−A) ^{1}/_{r} ^{∂Φ}/_{∂r}
− ^{A}/_{r²} Φ = 0

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

(A1-2)
Φ_{1} = ^{1}/_{r}
and
Φ_{2} = r^{A}.

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

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. Evans’ 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