In his web-paper [1] M.W. Evans proudly presents the

Theorem:

If
R Ù q
=
D Ù T
[1,(1)]

then
^{ }
R^{~ } Ù q
=
D Ù T^{~} .
^{ }
[1,(2)]

where the indices are suppressed by Evans. The correct version of the quoted claim is

(1)
If
R^{a}_{b }Ù q^{b}
=
DÙT^{a}
then
R^{~ a}_{b }Ù q^{b}
=
DÙT^{~ a} .

Evans' claim is followed by a fuzzy "proof" the shortcomings of which will be revealed below. And by a lot of far reaching "consequences" . . .

At first Evans refers to a representation of the lhs of eq. [1,(1)], of
R^{a}_{b }Ù q^{b},
by a cyclic sum using
q^{b}
=
q^{b}_{ρ} dx^{ρ}
and
R^{a}_{b }q^{b}_{ρ} =
R^{a}_{ρ}:

R^{λ}_{ρμν} +
R^{λ}_{νρμ} +
R^{λ}_{μνρ}
= cyclic sum of definitions (3)
[1,(7)]

taken from

(2)
R^{a}_{b }Ù q^{b}
=
R^{a}_{[ρμν]}
dx^{ρ}Ùdx^{μ}Ùdx^{ν}
= ½
(R^{a}_{ρμν} +
R^{a}_{νρμ} +
R^{a}_{μνρ})
dx^{ρ}Ùdx^{μ}Ùdx^{ν}

for a = λ.

Then Evans introduces the coefficients of the Hodge dual of the curvature 2-form
R^{a}_{b}:

The Hodge dual of eq. [1,(3)] is:

R^{~ λ}_{ρ}^{αβ}
= ½ ||g||^{½}
Î^{αβμν}
R^{λ}_{ρμν}
[1,(5)]

This Evans' original equation can be replaced with the equivalent (but more convenient) equation

(3)
R^{~ a}_{b αβ}
=
Î_{αβ}^{μν}
R^{a}_{b μν}

which would yield

(4)
R^{~ a}_{b αβ}
dx^{α}Ùdx^{β}
=
R^{a}_{b μν}
Î_{αβ}^{μν}
dx^{α}Ùdx^{β}
=
R^{a}_{b μν}
(dx^{μ}Ùdx^{ν})^{~}
=
(R^{a}_{b μν}
dx^{μ}Ùdx^{ν})^{~}.

This means that the 2-form
R^{~ a}_{b αβ}
dx^{α}Ùdx^{β}
is the Hodge dual of the 2-form
R^{a}_{b μν}
dx^{μ}Ùdx^{ν}
for each pair of *fixed* indices a,b.

However, this equation has to be applied to eq. [1,(7)], which seems to be simple due to the short hand notation of that equation. But its correct detailed notation in eq. (2) reveals a problem: In contrast to Evans' belief the transition to

R^{~ λ}_{ρμν} +
R^{~ λ}_{νρμ} +
R^{~ λ}_{μνρ}
= cyclic sum of definitions (14)
[1,(15)]

is not possible by a mere multiplication of eq. [1,(7)] by an
Î-factor due to the *wrong positioned* index
ρ
at the two right terms in eq. [1,(7)]. In other words: The multiplication of
½
(R^{λ}_{ρμν} +
R^{λ}_{νρμ} +
R^{λ}_{μνρ})
by

(6)
(dx^{μ}Ùdx^{ν})^{~}
Ù dx^{ρ}
=
Î^{μν}_{αβ}
dx^{α}Ùdx^{β}
Ù dx^{ρ}

yields

½
(R^{a}_{ρμν} +
R^{a}_{νρμ} +
R^{a}_{μνρ})
Î^{μν}_{αβ}
dx^{α}Ùdx^{β}
Ù dx^{ρ}

(7)
= ½
R^{a}_{b μν}
Î^{μν}_{αβ}
dx^{α}Ùdx^{β}
Ùq^{b}
+ ½
(R^{a}_{νρμ} +
R^{a}_{μνρ})
Î^{μν}_{αβ}
dx^{α}Ùdx^{β}
Ù dx^{ρ}

= ½
R^{~}^{a}_{b μν}
Ùq^{b}
+ ½
(R^{a}_{νρμ} +
R^{a}_{μνρ})
Î^{μν}_{αβ}
dx^{α}Ùdx^{β}
Ù dx^{ρ}

The first summand yields half of the aspired 3-form
R^{~}^{a}_{b}
Ù
q^{b}
whilst the
red marked summands cannot be rewritten as duals of R^{a}_{b} due to the occurrence
of the index ρ at an improper second or third position.

This breaking off does not mean that there would not occur other irreparable problems
while checking other parts of Evans' proof, e.g. for the transformation of
D Ù T^{a}. But further refutations are left to the
reader.

**Final Remark**

There is an error between Evans' eqs. [1,(19)] and [1,(20)]:
The latter is NOT a consequence of the preceeding equation as asserted by Evans:

D_{ρ}T^{~}_{μ}^{λ}_{ν} +
D_{ν}T^{~}_{ρ}^{λ}_{μ} +
D_{μ}T^{~}_{ν}^{λ}_{ρ} =
−
(R^{~}_{μ}^{λ}_{ν} +
R^{~}_{ρ}^{λ}_{μ} +
R^{~}_{ν}^{λ}_{ρ})
[1,(19)]

even if the symbol ~ is removed.
That is due to the repeated upper and lower indices **μ**
in eq.[1,(20)]. This repetition does not occur in
[1,(19)]:

D_{μ}T^{λμν} =
− R^{λ}_{μ}^{μν}
[1,(20)]

Both sides of this equation are *two*-tensors (index μ contracted) while eq.(19)
displays *four*-tensors. Since 2 ≠ 4 both equations cannot be equivalent.

This remark is important since in a further note [2, eqs.(25-26) and (39-40)] *falsly*
considers eq. [1,(20)]
as the equivalent of the 1st Bianchi identity (the first equ. of (1)).

[1] M.W. Evans, *Proof of the Hodge Dual Relation*,

http://www.atomicprecision.com/blog/wp-filez/a100thpapernotes16.pdf

[2] M.W. Evans, *The Fundamental Origin of the Bianchi Identity
of Cartan Geometry and ECE Theory*,

http://www.atomicprecision.com/blog/wp-filez/a102ndpaper.pdf

[3] W.A. Rodrigues, *Differential Forms on Riemannian (Minkowskian)
and Riemann-Cartan Structures
and some Applications to Physics*,
arXiv

(27.12.2007)
**Remarks on Evans' Web Note #103
**

(19.12.2007)
**Myron's New Questionable Developments of Cartan Geometry
**

(10.12.2007)
**How Dr. Evans refutes the whole EH Theory
**

(20.11.2007)
**Remarks on Evans' papernotes #100
**