M.W. Evans’ reactions on my note [2]

(quotations from MWE’s emails in black italics, comments by G.W. Bruhn in blue)

In a short note [2] I recently showed by simple insertion that a basic statement of Myron W. Evans (MWE) in [1], that

«if there exists a symmetric metric, then there must exist an antisymmetric metric»,

leads to a contradiction already in the simplest case, the R3 case, which MWE had used to exemplify his assertion:

By simply inserting MWE’s matrices (qkr(S)) and (qij(A)) defined in [1; (2.4) and (2.6)] into MWE's equations (2.8) and (2.9) respectively one obtains

            ω1 = qkr(S) dxk dxr = dx12 + dx22 + dx32                                     (2.8')
and
            ω2 = − ˝ qij(A) dxi^dxj = dx1^dx2 + dx2^dx3 + dx3^dx1 .             (2.9')

MWE’s assertion at the top of p.23 in [1] was that ω1 = ω2 , i.e. detailed
(*)            dx12 + dx22 + dx32 = dx1^dx2 + dx2^dx3 + dx3^dx1 ,

an evidently wrong equation, which he has recently confirmed by email (see below).

MWE has blocked direct email communication with me immediately after my first criticisms. Probably he is afraid of a precise discussion of his statements - as is use everywhere else in the scientific world. However, he gave feedbacks via his AIAS email list. These feedbacks, which will be discussed below, consist of

1) Polemics (collected in the Appendix),
            which don’t help to find the truth and therefore will be ignored without reply.

2) Unproven assertions and statements ,
            which do not meet my objection that the simple consequence (*) from his theory [1] is evidently erroneous:

Email We 11-03-04 11:24:
« In general the symmetric metric used by Einstein is well known to be (e.g. Carroll):
            gμν = qaμ qbν ηab
This is the inner or dot product of two tetrads, a scalar and thus a zero form (ω1). Here ηab is the Minkowski metric of the tangent bundle spacetime. >From the same two tetrads we can construct straightforwardly a wedge product of two tetrads:
            gcμν = qaμ ^ qbν
This is an antisymmetric tensor (i.e. a differential one-form No! a 2-form). This is denoted qij(A) in Bruhn's eqn. (2.9). Its dual is well known to be an axial vector:
            qk = εijk qij(A)
where εijk is the Levi-Civita symbol in 3-D. (Bruhn's eqns, (2.8) and (2.9) are in 3-D).
The wedge product dxi^dxj in Bruhn's eqn. (2.9) is similarly an antisymmetric tensor whose dual is another axial vector, and finally the dot product of two axial vectors in Brun's eqn. (2.9) is a scalar, ω2 and thus another zero form. »

That long explanation cannot invalidate the fact that MWE's theory [1] leads to the erroneous consequence (*).
Besides: «Bruhn’s equations» are MWE’s original equations taken from MWE’s article [1].

Analysis of the last break:
« The wedge product dxi^dxj in Bruhn's eqn. (2.9) is similarly an antisymmetric tensor »            OK
« whose dual is another axial vector, »            OK
« and finally the dot product of two axial vectors is a scalar»             OK - BUT
there is no dot product of two axial vectors in MWE’s eqn. (2.9)
hence the conclusion « ω2 is another zero form» is not justified:
ω2 is a differential 2-form in the sense of S.M. Carroll [3; p.21] (due to ω2 = ˝ εik dxi^dxk), while ω1 doesn’t have this property.
Thus, ω1 and ω2 cannot be equated.

« Bruhn has chosen to deliberately misinterpret eqn. (2.9) in a trivial way in order to give a false impression that there exists a trivial error in my work.»
We have just seen, that the given impression is quite correct. Where is the « misinterpretation of eqn. (2.9)»?

In addition MWE declares in his

Email Thu 11-4-04 14:02
«ONE LINER FOR GERHARD BRUHN.
To add to my collection of one liners, the following is for Gerhard Bruhn:
            ω1 = − qij(A) dxi^dxj . . . » (In MWE's sense a factor ˝ is missing here.)
i.e. MWE confirms here again explicitly
            dx12 + dx22 + dx32 = dx1^dx2 + dx2^dx3 + dx3^dx1 ,
which is evidently erroneous: On the right-hand side we have a 2-form according to Carroll [3; p.21](due to ω2 = ˝ εik dxi^dxk) while the left-hand side – since ω1 contains no wedge products and is no antisymmetric tensor – is no 2-form. That is the type mismatch in (*) that I have criticized in my short note [2]. This and nothing else!
Email We 11-3-04 11:33:
« PS for www.aias.us
PS Rigorously, an antisymmetric tensor in 3-D is a differential two-form dual to an axial vector, a differential one-form. In 4-D an antisymmetric tensor is a differential two-form dual to another two-form. These are all VERY well known points, but worth repeating in view of the attacks being made on me personally by Bruhn. Background to my comments is available in the graduate course by Carroll on www.aias.us. Significantly, Bruhn does not attack Carroll, showing clearly that his motives are personal, and unscientific. (Carroll and I say the same thing exactly, Bruhn attacks me, but not Carroll.)
MWE »
I do not criticize Carroll since in Carroll’s book [3] there is no justification for MWE’s assertion ω1 = ω2. Carroll introduces differential p-forms at p.21 as «completely antisymmetric (0,p) tensors» and the wedge product with (1.79) while the (Minkowski-) metric can be found on p.4 and p.25, equ. (1.95), and more generally on p.47 without any reference to differential forms or the wedge product. Therefore MWE’s assertion « Carroll and I say the same thing exactly» is untrue.
Email Fr 11-5-04 11:40:
« TEMPLATE APPENDIX. … The reasons why Bruhn has destroyed himself is as follows:
1) In A. Einstein, "The Meaning of Relativity" (Princeton, 1921), we find:
            gμν gμν = 4                                     (1)
… It is seen that the double contravariant covariant summation yields a scalar, the number 4. Bruhn asserts that this is vector. Reductio ad absurdum.»
One of MWE's unproven assertions: Any reference to my note [2] is missing and would not invalidate the consequence (*) of MWE's theory [1]. Besides: I cannot remember any statement of mine against this equ. (1).
« 2) In an earlier note, which appeared entirely without my knowledge on the net, Bruhn asserts (apparently) that the well known complex circular basis is incorrect. The circular basis is well knwon (e.e. B. Silver, "Irreducible Tensorial Sets") and is
            e(1)×e(2) = i e(3)*                         (2)
                        et cyclicum
(see numerous works listed on www.aias.us). I have already given this refutation.»
This is very imprecise. True is that I had criticized some of MWE’s statements concerning the method of complex representation in [4], not the method itself, which makes a great difference. For a refutation of MWE's «refutation» see [4].

References

[1]            M.W. Evans: http://www.aias.us/book01/chap2-cr3.pdf
[2]            G. W. Bruhn: http://www2.mathematik.tu-darmstadt.de/~bruhn/contra-E-1.jpg
[3]            S.M. Carroll: http://arxiv.org/pdf/gr-qc/9712019
[4]            G. W. Bruhn: http://www2.mathematik.tu-darmstadt.de/~bruhn/Refutation_of_EVANS_REFUTATION.htm


Appendix: A Collection of MWE’s Polemics

which don’t help to find the scientific truth and therefore will be ignored without reply:

Email 7-13-2004
« … Bruhn seems to be a typical example of an arrogant and defunct establishment.
Myron W. Evans AIAS Director»
Email We 11-03-04 11:24:
« Bruhn's dishonesty and harassment is part of a campaign being carried on by one or two unprofessional individuals. I have been of the receiving end of this for some years. I have learned that the only way to deal with these cranks is to block them. Obviously I do not regard them as scientists. . . .
Bruhn has chosen to deliberately misinterpret eqn. (2.9) in a trivial way in order to give a false impression that there exists a trivial error in my work. In legal terms this is defamation (false information spread to many parties) and first degree harassment (he does it repeatedly). I warn colleagues to be alert to further attempts of this kind, and to calmly report Bruhn to his Chair at the Technical University of Darmstadt with a request for appropriate disciplinary action against him (bringing the University into disrepute). Therefore I demand Dr Bruhn's apology and immediate resignation. His conduct is unprofessional. »

The reader himself may decide whether my internet articles with concern to MWE’s work

            http://www2.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm
            http://www2.mathematik.tu-darmstadt.de/~bruhn/Refutation_of_EVANS_REFUTATION.htm
            http://www2.mathematik.tu-darmstadt.de/~bruhn/PHYSICAL_OPTICS1.htm
            http://www2.mathematik.tu-darmstadt.de/~bruhn/contra-E-1.jpg

contain any comparable personal attacks against MWE.