Quotes from Evans' note [2] are displayed in **black**.

Evans writes on p.7 of [2]:

Finally we are told that there exist no resonance solutions to the equation:

^{d²Φ}/_{dr²} +
^{1}/_{r} ^{dΦ}/_{dr} −
^{1}/_{r²} Φ = −
^{ρ}/_{εo}
(24)

where

ρ = ρ_{o} cos(κ_{r}r)
(25)

^{d²Φ}/_{dr²} −
^{1}/_{r} ^{dΦ}/_{dr} +
^{1}/_{r²} Φ = −
^{ρ(0)}/_{εo} cos(κr)
[1,(65)]

Therefore, whatever Evans wants to say here, **it has nothing to do
with his paper [1] under review.** So our former result stands *uncontradicted*:

We are going to discuss Evans' *new expositions* now:

There are the following changes compared with the original equation [1, eq.(65)]:

(i) Two signs have changed. The original equation is

^{d²Φ}/_{dr²} −
^{1}/_{r} ^{dΦ}/_{dr} +
^{1}/_{r²} Φ = −
^{ρ(0)}/_{εo} cos(κr)
[1,(65)]

These changes yield altered eigenfunctions which are now

Φ_{1} = r
and
Φ_{2} = ^{1}/_{r}

(ii) Which is the meaning of the subscript r
in κ_{r}? This index is missing
in the original equation [1,(65)] and nowhere explained in Evans' rebuttal.
However, from equ. (27) it can be seen, that the subscript r does not indicate
a dependancy of κ on the variable r, since otherwise the differentiation
would have caused additional terms. Therefore it is justified
to write κ instead of κ_{r} in the following.

Obviously the driving term designed by Evans shall be oscillatory. However, no linear combination of the eigenfunctions is oscillatory too. Thus, the driving term cannot belong to the eigenspace:

Some remarks to Evans' subsequent change of the variable {1}:

κ_{r} r = exp(*i*κ_{r}R)
(26)

is only possible for κ = κ_{r}, since an r-dependent κ_{r}
would have caused additional terms in the subsequent differential equation (27). Evans:

^{d²Φ}/_{dR²} +
κ_{r}² Φ =
^{ρ}/_{εo}
(27)

which now has oscillatory eigenfunctions cos(κR) and sin(κR).

Of course, not: The transform (26) is complex: **Reals are not mapped to reals.**

Therefore, Evans' assumption
of R being real and travelling along the real axis yields the point
κr = exp(*i*κR)
to travel along the unit circle of the complex plane, while κr
should move along the real axis.

That *wrong* path of κr along the complex unit circle
causes periodicity that disappears when instead κr moves along the real
axis.

κ r = exp(κx) (26')

would leave the positive real axis invariant and leads to the differential equation

^{d²Φ}/_{dx²} −
κ² Φ =
^{ρ}/_{εo}
(27')

with the eigenfunctions exp(__+__κx). Both eigenfunctions are
again non-oscillatory and cannot generate an oscillatory driving
term by linear combination. Hence, the differential equation
does not show any resonance effects, like the original de. (24)

From the eqs. (26')/(27') one can get to Evans' eqs. (26)/(27) by the simple additional transform

x = *i* R
(26'')

which is a rotation of the complex plane by an angle of 90°. By that transform
the eigenfunctions exp(__+__κx) to eq.(27') transform to
exp(__+__*i*κR). Evans' eigenfunctions sin(κR), cos(κR)
to eq. (27) are linear combinations of exp(__+__*i*κR). From eq.(26) Evans' flaw
of thinking becomes obvious again: The *real* variable x is exchanged by the
*imaginary* variable R = −*i* x , and that exchange is inadmissible,
since Evans considers R to be *real*. Especially, for Evans' numerics the variable R
was used varying on the *real* axis.

Evans' flaw of thinking was already pointed out in July 2006 in two web papers [4,5].

Evans' remarks on the eqs. (28)/(30) contain further math errors, without interest in the given context.

Evans writes on p.3 of [2]:

... The perpetrators assert that Cartan geometry is "undefined". This alone is enough to arose suspicion, because Cartan geometry is standard text book material {4}.

The reference {4} points to Carroll's book "Spacetime and Geometry". A quick look at the Index of that book shows that the term "Cartan geometry" does not occur. So what, Dr Evans? It's an Evans-typical wishful thinking beyond the borders of reality.

Evans continues:

There is a basic error in eq.(6) ...

Indeed, there is a typo in eq.((6)). "= 0" is missing. However, Evans hasn't recognized
that he is just criticizing his *own* equation [1,(6)] written as ((6)) with double
parenthesis to distinct Evans' equations in [3] from our own equations.

Evans then asserts that the traditional second Bianchi identity {4}:

D Ù R = 0 (3)

is a special case of eq.(2). ...

D Ù (D Ù T) := D Ù (R Ù q) (2)

It is true that Evans' eq.(2) is a (trivial) consequence of the first Bianchi identity

D Ù T := R Ù q . (1)

However, the eqs. (1) and (2) are NOT equivalent: Whenever eq.(1) is satisfied then eq.(2) as well. But the reverse is not true: If eq.(2) is fulfilled then we may conclude that the terms D Ù T and R Ù q differ by a form Ψ that satisfies the condition D Ù Ψ = 0,

D Ù T := R Ù q + Ψ. where D Ù Ψ = 0 (1')

This means that **eq.(1) cannot be deduced from eq.(2)**.

We arrive at Evans' "proof" of the claim that from

d Ù R^{a}_{b} =
R^{a}_{c} Ù ω^{c}_{b}
−
ω^{a}_{c} Ù R^{c}_{b}
(4)

it can be deduced that

d Ù R^{~ a}_{b} =
R^{~ a}_{c} Ù ω^{c}_{b}
−
ω^{a}_{c} Ù R^{~ c}_{b} .
(6)

It is easily be seen that Evans' proof of eq.(6) is wrong since it is 'folklore' that the operators d and ~ do not commute. However, it seems that Evans has never before heard of that song, so we must sing it here once more:

Evans' proof is based on the assertion

d Ù (Î R) = Î (d Ù R) (13)

in other words, Evans assumes that d and the Hodge dual ^{~} commute,
(dÙR)^{~} =
(dÙR^{~}).
However, that is nonsense!

We remind the reader of the fact that in
n-dimensional case the Hodge dual F^{~} of a p-form F is an (n-p)-form.

deg(F^{~}) = n − deg(F) = n − p .

We have n=4, deg(R^{~})=deg(R)=2 and deg(d)=1.

Hence, considering the degrees of the forms in Evans' "rule"
(dÙR)^{~} =
(dÙR^{~})
we obtain on the l.h.s.

deg((dÙR)^{~}) =
4−deg(dÙR) = 4 − [deg(d)+deg(R)] = 4 − [1+2] = 1 ,

while the r.h.s. yields

deg(d Ù R^{~}) = deg(d)+deg(R^{~})= 1 + 2 = 3

Thus, Evans' commutation "rule" is wrong:

Some serious consequences of Evans' torsion hypothesis can be found in [6]:

- Evans' representation of the experimentally measurable field form F by his
4-vector valued field form F^{a} is not Lorentz invariant.

- Evans' concept of free space is not Lorentz invariant.

- Evans' representation of the field forms F^{a} (a=1,2,3) are wrong since based on
non-existing 3-index Î tensors in 4D.

- Evans' em-fields F^{a} cannot exist in torsion-free spacetime manifolds
(the spacetime manifolds of GRT).

We wrote in our paper:

Raising the index o of ω_{o} in the term
∂_{i}(ω_{o})^{a}_{b}, as Evans does,
is illegitimate, because the metric component g^{oo} of the Schwarzschild metric,
which Evans considers, is *not* a constant function of the
variables x^{i}.

While raising tetrad-indices a,b is done with the Minkowski metric, raising the
coordinate-index o in ω_{o} is done with the Schwarzschild metric .
Therefore Evans shows again his complete ignorance, unless he wants to confuse his readers and
believers intentionally, which cannot be excluded when taking into account
his pathologically aggressive behavior towards scientific criticism.

Evans is blocking e-mails systematically from those scientists who find errors in his papers, and then he assures of getting no objections against his strange views and developments.

[1] M.W. Evans, *Spin connection resonance in gravitational general relativity*,

Acta Physica Polonica **B38** (2007) 2211-2220

[2] M.W. Evans, *Refutation of Comment by Jadczyk ei Alii*,

a90thpaper.pdf

[3] G.W. Bruhn, F.W. Hehl, A. Jadczyk, *Comments on "Spin Connection Resonance in Gravitational General
Relativity"*,

http://arxiv.org/pdf/0707.4433
see also
Comments on Evans' SCR Paper

[4] G.W. Bruhn, *Remarks on Evans/Eckardt's Web-Note on Coulomb Resonance
*

http://www.mathematik.tu-darmstadt.de/~bruhn/RemarkEvans61.html

[5] G.W. Bruhn, *No Coulomb Resonance (Survey of Evans/Eckardt's Web-Note)*

http://www.mathematik.tu-darmstadt.de/~bruhn/Coulomb-R-Survey.html

[6] G.W. Bruhn, *Consequences of Evans' Torsion Hypothesis*

http://www.mathematik.tu-darmstadt.de/~bruhn/ECEcontradictions.html