Usually Evans replies to scientific criticisms of his work by means of crude polemics. Therefore we have some progress since now Evans feels prompted to reply to [4] with a kind of scientific argumentation. The occurring shares of polemics should be ignored as the price of this discussion. Evans' handwritten "rebuttal" [5] is contained in a circular he had sent to his supporters, and some of them have forwarded it to me. The first part of the "rebuttal" is a repetition of [1, Chap.14.2] which now gives opportunity to detect some math errors in that chapter. The second part of Evans' rebuttal gives the impression that Evans has some problems with understanding of counter arguments.

This is a repeat of previous rebuttals of G. Bruhn a retired individual in Darmstadt.

1. The "dubious alternative" method is in fact the same method as taught in every reputable University. The definitions (1.2), (1.7) and (1.8) are the same definitions as used in ECE theory. So what is Bruhn trying to say?

Bruhn says that such an erroneous and confuse introduction to a theory related to Einstein's Relativity Theory as in the Evans' book [1] can hardly be taught even in universities of minor rank.

2. The assertion by Bruhn as going from his eq. (2.1) to (2.2) is not made in ECE theory. This is an example of disinformation. The correct method is already given in M.W. Evans, "Generally Unified Field Theory" (Abramis Academic, 2005 and 2006) Vol. 1 pp.261 ff.

Start with the Einstein equation

R_{μν} − ½ R g_{μν} = k T_{μν}
(1)

Introduce

R_{μν} = R_{μ}^{a} q_{ν}^{b} η_{ab}
(2)

T_{μν} = T_{μ}^{a} q_{ν}^{b} η_{ab}
(3)

g_{μν} = q_{μ}^{a} q_{ν}^{b} η_{ab}
(4)

Nothing against Evans eqs. (1-4). However, should it be that Evans does not know the equivalence of his eq. (1) with Bruhn's eq.

(B1)
R^{μν} − ½ R g^{μν} = k T^{μν}

which can easily be shown by raising or lowering the indices as usual in tensor calculus?

**Besides, eq. (B1) stems from Evans' GCUFT book Vol.1, Chap.3, eq.(3.3).**

eq. (4) is the standard decomposition of the metric into the product of two tetrads (e.g. Carroll).

The expression "product of two tetrads" shows that the author has not understood what is meant by
"tetrad" in differential
geometry. A tetrad is an alternative (orthonormal) vector basis defined in the tangential spaces of a metric
manifold instead of the coordinate dependent usual tangent vectors which are not orthonormal in general.
Carroll correctly introduces a tetrad as "vierbein" = four legs where each of the "legs" represents one of
the orthonormal basis vectors. That concept goes back to
Darboux (around 1880) who introduced a so-called "moving frame" (= repère mobile) to differential
geometry. Therefore the notation "tetrad" for the coefficient matrix
(q_{μ}^{a})
as e.g. used by Carroll later in his book is at least misleading.

This method is used in eqs. (2) and (3) to define R_{μ}^{a} and T_{μ}^{a},
which are < the coefficients of > vector valued one
forms < R^{a} = R_{μ}^{a} dx^{μ} and T^{a} = T_{μ}^{a} dx^{μ} >.
One of the Evans field equations is

G_{μ}^{a} = − ¼ R q_{μ}^{a}
(5)

T_{μ}^{a} = ¼ T q_{μ}^{a}
(6)

Eqs. (5) and (6) are derived from the definition of R and T originally used by Einstein:

R = g^{μν} R_{μν} ,
T = g^{μν} T_{μν} .
(7)

Use the Einstein convention

g^{μν} g_{μν} = 4 ,
(8)

This is no "convention" but a consequence of the fact that the 4×4 matrices
(g^{μν}) and
(g_{μν}) are mutually inverses.

and the Cartan convention

q_{μ}^{a} q_{a}^{μ} = **1** ,
(9)

Our poor É. Cartan did never deliver such a wrong "convention" which can be found in Evans' book
[1,(14.20)] and should be called "Evans convention". The correct equation follows from
the fact that the 4×4 matrices (q_{μ}^{a}) and (q_{a}^{μ})
are mutually inverses:

(B2)
q_{μ}^{a} q_{a}^{μ} = 4 ,

This "Evans convention" (... = **1**) is no typo
by Evans; it appears at several spots of his work, e.g. below in the "proof" of
eq. (11) by means of "eq."(12).
And to attribute the *trivialities* (8) and (9) to Einstein and Cartan respectively
seems to be somewhat akward showing that the author's understanding
of the matter has some gaps.

to obtain

R = g^{μν} R_{μν} =
q_{μ}^{a} q_{b}^{ν} η^{ab}
R_{μ}^{a} q_{ν}^{b} η_{ab} ,
(10)

This equation from [1, (14.21)] is formally wrong: In a correct equation we have the following formal rules to be satisfied:

1. At each summand of a tensor equation
each index can appear *at most once* in the upper index position and
*at most once* in the lower position.

2. Indices that appear twice at a summand are to be summed over (Einstein
summation convention) and therefore called *summation indices*.

3. Indices that appear at *only one* (upper or lower) position must be
the same at each of the summands on both sides of the equation.

The correct equation instead of eq. (10) is

(B3)
R = g^{μν} R_{μν} =
q_{a}^{μ} q_{b}^{ν} η^{ab}
R_{μ}^{c} q_{ν}^{d} η_{cd} ,

where we have used eq. (2).

Multiply both sides of eq. (10) by q_{μ}^{a} to obtain:

R_{μ}^{a} = ¼ R q_{μ}^{a}
(11)

This advice is wishful thinking: All indices in eq. (B3) (= eq. (10) corrected) are summation indices,
and so the advice is *not feasible* since the multiplication would yield
a formally incorrect equation.

But some simplifications of eq. (10) after correction, i.e. of eq. (B3),
are possible: Using q_{b}^{ν}q_{ν}^{d}
= δ_{b}^{d} we obtain

q_{b}^{ν}η^{ab}q_{ν}^{d}η_{cd}
= δ_{b}^{d}η^{ab}η_{cd}
= η^{ab}η_{cb} = δ_{c}^{a}

and thus

(B4)
R = q_{c}^{μ} R_{μ}^{c} .

As one can see from eq. (11) Evans aim is to resolve this equation for
R_{μ}^{c}.
Since Evans' solution (11) shows proportionality of the matrices
(R_{μ}^{a}) and (q_{μ}^{a}),
so eq. (11) (= eq.(5)) is *wrong in general* (and eq. (6) as well).

This is because the R.H.S. of eq. (10) is:

(η^{ab}η_{ab})
(q_{b}^{ν}q_{ν}^{b})
(q_{μ}^{a}R_{μ}^{a})
= 4 q_{μ}^{a} R_{μ}^{a} .
(12)

Formally incorrect equation from [1, (14.21)]:
The expression (q_{μ}^{a}R_{μ}^{a})
is not well formed. Perhaps it should be read as
(q_{a}^{μ}R_{μ}^{a}).
Then even if one evaluates the first two brackets separately
the left hand side yields the **contradiction**:

4 · 4 · (q_{a}^{μ}R_{μ}^{a})
= 16 (q_{a}^{μ}R_{μ}^{a})
≠ 4 (q_{a}^{μ}R_{μ}^{a}).

Evans by using his (wrong) "Evans convention" obtains an equality.

Multiply both sides of eq. (12) by q_{μ}^{a}

Inadmissible though in [1, p.262 bottom line]! The indices μ and a in eq. (12) are both summation indices (dummies).

to obtain:

R_{μ}^{a} = ¼ R q_{μ}^{a}
(13)

So:

G_{μ}^{a} =
R_{μ}^{a} − ½ R q_{μ}^{a}
= − ¼ R q_{μ}^{a}
(14)

This is eq. (5), **Q.E.D.**

As was shown above this "proof" lacks from several irremovable inconsistencies,
hence it is **no proof**.

Using the *correct* eqs. (1-3) that are accepted by Evans too
it is easy to show that the
Evans eqs. (5), (6) and (13) are *invalid in general*:
We plug Evans eq. (13) into eq. (2) to obtain

R_{μν} =
¼ R q_{μ}^{a}
q_{ν}^{b} η_{ab} = ¼ R g_{μν} ,

and likewise by plugging (6) into (3)

T_{μν} =
¼ T q_{μ}^{a}
q_{ν}^{b} η_{ab} = ¼ T g_{μν} ,

which means that Evans restricts his consideration to *very special* cases where the matrices
(R_{μν}), (T_{μν}) and (g_{μν}) are
*proportional*, which is not fulfilled in general.

The contracted form of eq. (1) is

R = − k T (15)

(Einstein, "The Meaning of Relativity"). Multiply both sides of eq. (15) by q_{μ}^{a}:

R q_{μ}^{a} = − k T q_{μ}^{a}
(16)

Substituting eqs. (5) and (6) into eq. (16) gives:

G_{μ}^{a} = k T_{μ}^{a}
(17)

Write eq. (17) in the form:

¼ R q_{μ}^{a} − ½ R q_{μ}^{a}
= ¼ k T q_{μ}^{a} .
(18)

However, as shown above, the eqs. (5), (6) are invalid in general.

Multiply both sides of eq. (18) by q_{ν}^{b} η_{ab}

¼ R q_{μ}^{a}q_{ν}^{b} η_{ab}
−
½ R q_{μ}^{a}q_{ν}^{b} η_{ab}
=
¼ k T q_{μ}^{a}q_{ν}^{b} η_{ab} .
(19)

using eqs.(2) - (4), and (5) and (6), this is eq. (1), **Q.E.D.**

Fulfilled in the *very special* case of proportionality of the matrices
(R_{μν}), (T_{μν}) and (g_{μν}).

Starting from equ. (16) we may construct

R q_{μ}^{a} q_{ν}^{b} η_{ab}
= − k T q_{μ}^{a} q_{ν}^{b} η_{ab}
(20)

i.e.

R g_{μν}
= − k T g_{μν}
(21)

We may also construct the **wedge product**

R q_{μ}^{a} Ù q_{ν}^{b}
= − k T q_{μ}^{a} Ù q_{ν}^{b}
(22)

The remark on page 3 of Bruhn has also been corrected many times before. We may define the quantity:

R_{μν}^{c A}
:= R q_{μ}^{a} Ù q_{ν}^{b}
(23)

analogously with:

q_{μν}^{c(A)}
:= q_{μ}^{a} Ù q_{ν}^{b}
(24)

**I assume that this is what Bruhn is trying to say.**

The definition of the wedge product that I use is the same as that used by everyone else, (e.g. Carroll) ...

No! Bruhn says that eq. (24) is *incorrect* since the right hand side shows the upper
indices a, b which don't appear on the left hand side, while there appears an index c only
on the left hand side.
A definition of c is missing. Such a notation does not occur anywhere
and especially nowhere in Carroll's book [2]. A correct equation would be

q_{μν}^{ab(A)}
:= q_{μ}^{a} Ù q_{ν}^{b} .

...

**There is no type mismatch in equ. (23) and (24).**

The term q_{μν}^{c(A)}
shows *one* upper index (c) while the other side shows *two* other upper indices (a,b).
**That is type mismatch.** It would imply that both sides of eq. (24) transform *differently*
under changes of the tetrad field used, e.g. under local Lorentz transforms.

b) **There is no "illegal removal" of indices.**

This is misinformation by Bruhn who tries to create confusion.

An example of illegal suppression of indices is the eq.

G^{μν} = G^{(0)} (R^{μν(A)} − ½ R q^{μν(A)})
(3.29)

in [1], which *seems* to be well-formed on first view.
However, the correct version of that equation is

G^{μν} = G^{(0)}
(R_{ab}^{μν(A)}
− ½ R q_{ab}^{μν(A)}) .

This notation with written tetrad indices a,b makes the **type mismatch evident**.

**e**_{1} × **e**_{2} = **e**_{3}*
(26)

et cyclicum

Bruhn ommits the star (denoting complex conjugate).

He cannot even copy out my work correctly.

"Bruhn's equations (3.10)/(8)" as indicated by the equation label are taken from [1]/[2], i.e.
**identical with Evans' eqs. [1,(3.10)] and [2, (8)]** respectively.
Bruhn is *not responsible* for the numerous inconsistencies in Evans' work.
Evans is criticising here *his own* book [1]: As was precisely remarked in [4] **Evans'** own eq. (3.10)
in [1] is

**e**_{1} × **e**_{2} = **e**_{3}
**e**_{2} × **e**_{3} = **e**_{1}
**e**_{3} × **e**_{1} = **e**_{2}
(3.10)

The same line can also be found in Evans' FoPL article [2; (3)] that appeared two years before his book. So Evans had sufficient time to correct it if anything should have seemed wrong to him in this article.

c)The line element is

ds² = g_{μν} dx^{μ}dx^{ν}
(27)

The line element ds² is an invariant under coordinate transforms which can be written as

ds² = g_{μν} dx^{μ}Ädx^{ν}
= ½ g_{μν}
(dx^{μ}Ädx^{ν} +
dx^{ν}Ädx^{μ})

because of the symmetry of the matrix (g_{μν}) using the tensor
product Ä.

Bruhn incorrectly asserts that the Hodge dual of a two-form cannot be defined. This is, I assume, what he is trying to do on his page 5. It is well known that the Hodge dual of a two-form in 4-D is another two-form (e.g. Carroll).

That's a distortion of facts: Carroll considers **no symmetric** two-forms,
only *alternating* two-forms. For a two-form A the Hodge dual *A is defined as an
*alternating* two-fom. One important rule for p-forms A in n-D is
**A = (−1)^{1+p(n−p)} A,
which immediately implies that A is alternating like **A. The form ds² is **not alternating**.

e) There is a **deliberate misinformation** by Bruhn concerning tensors q^{μν(S)} and
q^{μν(A)}. He misrepresents my definitions, and makes non-consequential deductions.
My definitions are as follows:

q_{ij}^{(S)} = [ h_{1} ] [ h_{1} h_{2} h_{3} ]
(28)

[ h_{2} ]

[ h_{3} ]

= [ h_{1}² h_{1}h_{2} h_{1}h_{3} ]

[ h_{2}h_{1} h_{2}² h_{2}h_{3} ]

[ h_{3}h_{1} h_{3}h_{2} h_{3}² ]

Q.E.D.??? What has Evans demonstrated here?

The objection here is the well-known fact that matrices defined by outer products of vectors like (28)
have **vanishing determinant** since they have parallel line vectors: Each line is parallel to the vector
[ h_{1} h_{2} h_{3} ]. However, the metric must have a nonsingular matrix, since
its inverse is needed in tensor calculus too.

Bruhn is again spreading **disinformation. The remarks at the ?or? of his page 5 make no sense in
Cartan geometry.** ECE is based on Cartan geometry.

Maybe it wasn't Evans' intention, however, his eq. (3.7)/(5) in [1]/[2]

ds = |d**r**| = |^{∂r}/_{∂u1}du_{1}
+ ^{∂r}/_{∂u2}du_{2}
+ ^{∂r}/_{∂u3}du_{3}|
(3.7)/(5)

yields a *positive definite* line element ds² (≥0); i.e. this method is unsuitable for defining
a indefinite (locally Minkowskian) metric.

Finally, the obscure disinformation on page 6 of Bruhn makes no sense in light of the self consistent mathematics of this rebuttal.

With this concluding remark Evans tries to avoid a discussion about his assumption of proportionality
between R_{a}^{μ} and q_{a}^{μ} as was needed for the proof
of his Evans field equation (11) above.

References

[1] M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics; Arima 2005

[2] M.W. Evans, A Generally Covariant Field Equation for Gravitation and Electromagnetism,

Foundations of Physics Letters Vol.16 No.4, 369-377

[3] S.M. Carroll, Lecture Notes on General Relativity,

http://arxiv.org/pdf/gr-qc/9712019

[4] G.W. Bruhn, The Central Error of MYRON W. EVANS’ ECE Theory - a Type
Mismatch,

http://arxiv.org/pdf/physics/0607190

[5] M.W. Evans, Another detailed Bruhn rebuttal (hand written),

http://www.mathematik.tu-darmstadt.de/~bruhn/anotherdetailedbruhnrebuttal.pdf