MW Evans

Summary of Bruhn rebuttals

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_μ^a) as e.g. used by Carroll later in his book (and some formulations there) is at least misleading.

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 = ∂_μX^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 = ∂_μq_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 = ∂_μq^ν + Γ_μ^ν_λ q^λ                                                                 (2.182)

i.e. written with reinserted tetrad index a:

D_μ q_a^ν = ∂_μq_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]:

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.