Carroll Refutes Bruhn

Sept 10, 2008

_{Quotations from Evans' blog in }**black**

What is going on on Evans blog site is a kind of discussion on his ECE theory as offered by me seversal times in the past. If Evans would stop his polemics then this could be the beginning of a (somewhat one-sided) fruitful exchange of scientific ideas. Let's see.

Subject: Carroll Refutes Bruhn Date: Wed, 10 Sep 2008 07:14:43 EDT

Sean Carroll’s online notes also refute Bruhn’s latest attempt to misrepresent Cartan geometry. Carroll taught to graduates at Harvard, UCSB and Chicago, and is a frequent commentator on science in the US. In his online lecture notes, p. 82, following his eq. (3.88) he states correctly that the second Bianchi identity of the standard faction (all remaining two of them) is true if and only if the torsion vanishes.

On p.81 of his Lecture Notes on GRT for sake of simplicity Carroll considers the *torsion free* case.
In *that* case he derives the (second) Bianchi identity in the form

D_{[λ}R_{ρσ]μν} = 0 ,
(3.88)

(with Ñ instead of D) adding the remark that for a general connection there would be additional terms involving the torsion tensor.

However, Evans' laughter is *premature*:

I never considered Carroll's form (3.88) of the Bianchi identity.
However, a few pages later, in [1, p.93],
Carroll derives the Bianchi identity for general torsion:

dÙR^{a}_{b} +
ω^{a}_{c}ÙR^{c}_{b} −
R^{a}_{c}Ùω^{c}_{b} = 0 .
(3.141)

This is the version of the Bianchi identity that was asserted by me too. And if Evans should try to doubt this I recommend him to read the last two lines on p.91. There Carroll writes:

In his latest postings Bruhn had given a false proof of the second Bianchi identity, a deliberately contrived misrepresentation I unravelled this morning in my blog posted refutation (n’th refutation, n goes to infinity). I am amused at the way in which Bruhn tries to avoid criticising Carroll, who uses the precise same Celtic geometry as Cartan and myself.

As I wrote yesterday: M.W. Evans is not E. Cartan, and as proven by his web note, not S.M. Carroll. Therefore it is Evans' work that I am criticizing, which is far away from the prestigious work of E. Cartan or from S.M. Carroll's Lecture Notes.

Anyway, Bruhn mysteriously arrives at D ^ R = 0 which is the second Bianchi identity WITHOUT torsion. The only trouble is that friend Bruhn “forgot” to mention this fact, and his proof is contrived garbage. We are told that it holds for all connections, while the truth is that it holds only for the symmetric connection. It does not hold in general, and in looking closely at it there are mysterious jumps and omissions.

This is nice: Now Evans is criticizing Cartan's work, since the above general version of the Bianchi identy is standard in non-coordinate differential geometry that was introduced already by E. Cartan.

The true second Bianchi is derived in paper 88 and is D ^ (D ^ T) := D ^ (R ^ q) the ECE derivative identity. The neglect of torsion is curtains for the Einstein field equation (paper 93, which the busy bee accepts to be correct, or is he going to change his mind again).

British Civil List Scientist

This, the ''ECE derivative identity'' or ''true second Bianchi'', is a trivial implication of the 1st Bianchi identity
DÙT^{a} =
R^{a}_{b}Ùq^{b}
and has nothing to do with the 2nd Bianchi identity
DÙR^{a}_{b} = 0 .

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

http://xxx.lanl.gov/pdf/gr-qc/9712019,