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 indirectly with a kind of scientific argumentation (see also [4]). The occurring shares of polemics should be ignored as the price of this discussion. Evans' handwritten "rebuttal" is contained in a circular he had sent to his supporters, G. Giacchetta was so kind as to rewrite it to pdf, and some of the supporters being interested in discussions have forwarded it to me.

G. Bruhn has posted trivially erroneous comments which are rebutted as follows:

1. Bruhn attempt to refute the well known complex circular basis:

**e**^{(1)} × **e**^{(2)} = i **e**^{(3)}
(1)

by asserting that **e**^{(3)} is "undefined". However, it is well known that the basis (1) is
equivalent to:

**i** × **j** = **k**
(2)

where:

**e**^{(3)} = **k**
(3)

Therefore, if **e**^{(3)} is undefined, so is **k**.

REDUCTIO AD ABSURDUM

Evans does not reveal the source of his claim. Where has he found that? My first posting on that
topic

http://www2.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

gives a clear description of Evans' circular basis.

2. Bruhn asserts that the well known Einstein summation convention applied to double indices does not produce a scalar. However, it is well known that

g^{μν} g_{μν}=4
(4)

(Einstein, "Meaning of Relativity" (Princeton, 1921))

REDUCTIO AD ABSURDUM

Again: Evans does not reveal the source of his claim. Where is the origin of that story about his "Einstein convention"?

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

Evans knows still another "convention", the "Cartan convention"

q_{μ}^{a} q_{a}^{μ} = **1** .
(4a)

Our poor E. 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:

q_{μ}^{a} q_{a}^{μ} = 4 .

This "Evans convention" (... = **1**) is no typo
by Evans; it appears at several spots of his work.
And attributing the *trivialities* (4) and (4a) to Einstein and Cartan respectively
seems to be somewhat akward showing that the author's understanding
of the matter has some gaps.

3. Bruhn asserts that the well known tetrad postulate is incorrect. However, the tetrad postulate is defined by the tetrad. The latter is defined by:

X^{a} = q_{μ}^{a} X^{μ}
(5)

where X^{a} are components in the tangent bundle and X^{μ} are components in the
base manifold. Here, X may be a vector of any dimension, or a basis element of any
kind.

That's a lot of misinformation:

1) X is no vector of *any* dimension but a vector in the tangent space T_{P}
at some point P of the 4-D spacetime manifold. Thus X is a vector of dimension 4.

2) X^{a} and X^{μ} are the **represention coefficients** of the vector X relative
to the tetrad frame and the coordinate frame respectively at P.

3) Eq. (5) is the linear relation between both kinds of representation coefficients.

4) The coefficients matrix (q_{μ}^{a}) is not the tetrad.

5) A tetrad is an alternative (orthonormal) vector basis defined in the tangential spaces of a metric
manifold * M* 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

The covariant derivative of X^{ν} is defined by:

D_{μ} X^{ν} = ∂_{μ} X^{ν} +
Γ_{μ}^{ν}_{λ} X^{λ}
(6)

The covariant derivative of X^{a} is defined by:

D_{μ} X^{a} =
∂_{μ}X^{a} + ω_{μ}^{a}_{b} X^{b}
(7)

The basis elements are defined by:

e_{a} = q_{a}^{σ} ∂_{σ}
(8)

Bruhn uses equation (8) in his "refutation" of the tetrad postulate but asserts erroneously that:

D_{μ} q_{a}^{ν} = ∂_{μ}q_{a}^{ν}
+ Γ_{μ}^{ν}_{λ} q_{a}^{λ}
(9)

That's a distortion of facts: In Evans GCUFT book [1] Vol. 1 we find at p.46 the equation

D_{μ} q^{ν} = ∂_{μ}q^{ν}
+ Γ_{μ}^{ν}_{λ} q^{λ}
(2.182)

i.e. written with reinserted tetrad index a:

D_{μ} q_{a}^{ν} = ∂_{μ}q_{a}^{ν}
+ Γ_{μ}^{ν}_{λ} q_{a}^{λ}
(2.182')

which is identical with eq. (9) of this "rebuttal". In

http://www2.mathematik.tu-darmstadt.de/~bruhn/MWEsFurtherErrors.html

I wrote:

... Evans' reference [1] is ambiguous:
The covariant derivation operator D_{ρ} is available in Evans' text
in *two* versions [1; p.56-58] and [1; p.91]:

In the first part of his book manuscript [1] (Chap.2 - 7) he uses the version (1.59) (after tacit correction of an index exchange error at Γ here named (1.59'))

D_{ρ}q_{a}^{μ}
=
∂_{ρ}q_{a}^{μ}
+
Γ_{ρ}^{μ}_{σ}q_{a}^{σ}
(1.59')

which, as we shall see, restricts the spacetime
manifold to an uninteresting special case: Using the **well-known tetrad identity**

(I)
∂_{ν}q_{a}^{μ}
− ω_{ν}^{b}_{a}
q_{b}^{μ}
+ Γ_{ν}^{μ}_{λ}q_{a}^{λ}
= 0

we obtain from

D_{ν}q^{μ}
= ∂_{ν}q^{μ}
+ Γ^{μ}_{νλ} q^{λ}
= 0
(2.182)

the result

ω_{μ}^{a}_{b}
= 0 for all index combinations.

Thus, D_{ρ}q^{μ} = 0 is valid, if and only if
the manifold ** M** possesses a tetrad field such that
ω

Equation (9) is erroneous because q is not an arbitrary covariant vector for each a.

Indeed, as being identical with Evans' equation [1, (2.182)].

The tetrad is defined by - and constrained by equations (5) to (8), from which follows the tetrad postulate:

D_{μ} q_{ν}^{a} = 0
(10)

Bruhn arrives erroneously at a different result from equation (10).

REDUCTIO AD ABSURDUM

Where? Source missing again. − For more information the reader should have a look at

http://www2.mathematik.tu-darmstadt.de/~bruhn/ECE-Sketch.html

An accumulation of trivial errors of this kind means that Bruhn is either incompetent or being deliberately deceptive.

I think the above discussion showed the contrary.

References

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

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

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

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

http://www2.mathematik.tu-darmstadt.de/~bruhn/anotherdetailedbruhnrebuttal.pdf

[4] G.W. Bruhn, Comments on M.W. Evans' Rebuttal of arXiv 0607190 V1,

http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansRebuttal.html