Evans' Stubbornness: The 2nd Bianchi again

Response to a recent blog note

Gerhard W. Bruhn, Darmstadt University of Technology

Sept 15, 2008
Quotations from Evans' writings in

Subject: Further Misrepresentation by Bruhn Date: Sun, 14 Sep 2008 11:41:24 EDT

Bruhn has attempted further misrepresentation of Cartan geometry by attempting once more to assert that D ^ R = 0 is valid for all connections, when it is valid only for the symmetric conenction.

Indeed!!! Evans is confirming here once more his total misunderstanding of Cartan's differential geometry: As can be seen from the text books, e.g. the books [1, p.93] and [2, p.489] by S.M. Carroll for general torsion we have

                D Ù Ta := d Ù Ta + ωab Ù Tb = Rab Ù qb ,                                 (1st Bianchi identity)

                D Ù Rab := d Ù Rab + ωac Ù Rcb − ωcb Ù Rac = 0 .                 (2nd Bianchi identity)

So as usual he attempts to misrepresent my rebuttal, which is a quotation of Carroll, page 81 of his online notes following Carroll’s eq. (3.88). In respect of the second Bianchi identity Carroll writes “Notice that for a general connection there would be additional terms involving the torsion tensor”. These torsion terms are given by the correct derivative identity of ECE theory:

D ^ (D ^ T) := D ^ (R ^ q)

As I wrote: This is a trivial (and therefore useless) implication of the 1st Bianchi identity, and has nothing to do with the 2nd Bianchi identity.

In one of his latest misrepresentations, Bruhn asserts that the quotation at the foot of Carroll, page 91 refers to the Bianchi identity. It does NOT, it refers to the tetrad postulate. Earlier Bruhn had attempted to misrepresent the tetrad postulate itself. Now he accepts it.

Carroll's remark in his Lecture Notes Chap.3, Eq.(3.141) is preliminary to his calculations on the subsequent pages too where the 2nd Bianchi identity Eq.(3.141) is derived for general torsion. And Carroll writes there in addition:
''The first of these is the generalization of Rρ[σμν] = 0 while the second is the Bianchi identity Ñ[λ|Rσ|μν] = 0. (Sometimes both equations are called Bianchi identities.)''

Further confirmation can be found by independent calculation or in the text books by S.M. Carroll [2, p.488 ff.] or by F.W. Hehl and Y.N. Obukhov [3, p.208, eq.(C.1.69)].

Bruhn states that he is trying to force me into debate, so I will not debate, merely correct. The more this goes on, the more the evidence piles up against him, and the more he is ignored.

That's your problem, Myron, not mine.

Your corrections are welcome. However, since your ''corrections'' are always going astray:

Before writing once - think TWICE!

Some proposals: What about the Poincaré Lemma d Ù d Ù ω = 0 ?

Or your 3-D Îabc tensor in 4D tensor calculus?

Or what about your dualization of the 1st Bianchi?

Or ... or ...

Now he has been reduced to outright dishonesty and misquotation, thus bringing TU Darmstadt into disrepute. He is out of touch with reality if he seriously expects me to debate him now. It takes me a few minutes only to correct him, in order to further build evidence against him, in public.

Civil List Scientist

Wishful thinking!


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

[2] S.M. Carroll, Spacetime and Geometry,
      Addison Wesley 2004, ISBN 0-8053-8732-3

[3] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics - Charge, Flux, and Metric,
      Birkhäuser 2003, ISBN 0-8176-4222-6