MW Evans

Summary of Bruhn rebuttals

with comments by Gerhard W. Bruhn

Preface

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://www.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

gμν gμν=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 (gμν) and (gμν) are mutually inverses.

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

qμa qaμ = 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 (qμa) and (qaμ) are mutually inverses:

qμa qaμ = 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:

Xa = qμa Xμ                                                                 (5)

where Xa are components in the tangent bundle and Xμ 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 TP at some point P of the 4-D spacetime manifold. Thus X is a vector of dimension 4.

2) Xa and Xμ 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 (qμa) 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 (qμa) as e.g. used by Carroll later in his book (and some formulations there) is at least misleading.


The covariant derivative of Xν is defined by:

Dμ Xν = ∂μ Xν + Γμνλ Xλ                                                                 (6)

The covariant derivative of Xa is defined by:

Dμ Xa = ∂μXa + ωμab Xb                                                                 (7)

The basis elements are defined by:

ea = qaσσ                                                                 (8)

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

Dμ qaν = ∂μqaν + Γμνλ qaλ                                                                 (9)

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

Dμ qν = ∂μqν + Γμνλ qλ                                                                 (2.182)

i.e. written with reinserted tetrad index a:

Dμ qaν = ∂μqaν + Γμνλ qaλ                                                                 (2.182')

which is identical with eq. (9) of this "rebuttal". In
    http://www.mathematik.tu-darmstadt.de/~bruhn/MWEsFurtherErrors.html
I wrote:

... Evans' reference [1] is ambiguous: The covariant derivation operator Dρ 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'))

Dρqaμ = ∂ρqaμ + Γρμσqaσ                                                                 (1.59')

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

(I)                                                 ∂νqaμ − ωνba qbμ + Γνμλqaλ = 0

we obtain from

Dνqμ = ∂νqμ + Γμνλ qλ = 0                                                         (2.182)

the result

ωμab = 0     for all index combinations.

Thus, Dρqμ = 0 is valid, if and only if the manifold M possesses a tetrad field such that ωμab = 0 everywhere on M. Such special manifolds are called teleparallel.

So Evans criticises his own equation [1, (2.182)].


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:

Dμ qνa = 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://www.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://www.mathematik.tu-darmstadt.de/~bruhn/anotherdetailedbruhnrebuttal.pdf

[4] G.W. Bruhn, Comments on M.W. Evans' Rebuttal of arXiv 0607190 V1,
    http://www.mathematik.tu-darmstadt.de/~bruhn/EvansRebuttal.html



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