## Rejection of Evans' "Refutation of Comment by Jadczyk et Alii"

### Gerhard W. Bruhn, Darmstadt University of Technology

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

### Preface

In the Introduction to his note [1] Evans complains somewhat deceitful about not being informed that the authors of the critical note [3] would post their note on arXiv. That is deceitful since Dr Evans has blocked himself against critical email messages electronically in order not to be disturbed by the truth. More, on his blog he has repeatedly urged his AIAS supporters to do the same. So it is very difficult to inform Evans about critical scientific views.

### 1. Evans' Resonance effects

Evans writes on p.7 of [2]:

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

d²Φ/dr² + 1/r /dr1/ Φ = − ρ/εo                                                                 (24)

where

ρ = ρo cos(κrr)                                                                                                 (25)

### This is a distortion of facts:

It must be stated that equ.(24) differs essentially from the original equation [1,(65)] which is

d²Φ/dr²1/r /dr + 1/ 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:

### Evans' ODE in his APPB paper [1] gives no "Resonance".

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/dr²1/r /dr + 1/ Φ = − ρ(0)/εo cos(κr)                                                 [1,(65)]

These changes yield altered eigenfunctions which are now

Φ1 = r                 and                 Φ2 = 1/r

(ii) Which is the meaning of the subscript r in κr? 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 κr 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:

### Resonance of eq.(24) with an oscillatory driving term (25) is impossible

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

κr r = exp(iκrR)                                                                                                 (26)

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

d²Φ/dR² + κr² Φ = ρ/εo                                                                                 (27)

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

### A miracle?

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(iκR) 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.

#### Therefore the transform (26) is inadmissible.

Removing the factor i from (24), i.e. considering the modified transform

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)

### Evans' flaw of thinking:

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(+iκR). Evans' eigenfunctions sin(κR), cos(κR) to eq. (27) are linear combinations of exp(+iκR). 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.

### 2.1 Cartan Geometry

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.

### 2.2 The new Bianchi identity

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).

### 2.3 Does d commute with the Hodge dual?

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

d Ù Rab = Rac Ù ωcb − ωac Ù Rcb                                                                 (4)

it can be deduced that

d Ù R~ ab = R~ ac Ù ωcb − ωac Ù R~ cb .                                                 (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:

### 2.4 Consequences of Evans' Torsion Hypothesis

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 Fa is not Lorentz invariant.

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

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

- Evans' em-fields Fa cannot exist in torsion-free spacetime manifolds (the spacetime manifolds of GRT).

### 3. Schwarzschild or Minkowski metric ? (by A. Jadczyk)

We wrote in our paper:

Raising the index o of ωo in the term ∂io)ab, as Evans does, is illegitimate, because the metric component goo of the Schwarzschild metric, which Evans considers, is not a constant function of the variables xi.

While raising tetrad-indices a,b is done with the Minkowski metric, raising the coordinate-index o in ωo 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.

### References

[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",

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

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

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