## M.W. Evans "proves" the Evans Lemma again

### Gerhard W. Bruhn, Darmstadt University of Technology

On Mon, 12 Mar 2007 02:48:40 EDT M.W. Evans once more felt it be necessary to give (repeat) a "proof" of his "Evans Lemma".

We quote the essential part of that "proof" starting with

o qaλ = ∂μνμλ qaν − ωaμb qbλ)                                 (8)

Now define

R = qλaμνμλ qaν − ωaμb qbλ)                                 (9)

(Note that all indices here are dummy (or umbral) indices.)

and use

qaλ qλa = 1                                                                 (10)

This is one of Evans' favorite errors: The left hand expression is the trace of the 4×4 unit matrix and thus has the value 4.

However, regardless if 1 or 4, what follows now is another wrong step that Evans applies in hopeless situations: He wants to resolve Eq.(9) for the expressions

Qaλ := ∂μνμλ qaν − ωaμb qbλ)

which appear on the right hand side of Eq.(8). So he multiplies Eq.(9) by qaλ hoping that so the factor qλa in Eq.(9) would be compensated and removed due to Eq.(10).

#### However, this operation would require free indices a and λ which both are dummies.

Therefore Evans' conclusion is a fallacy.

### The proof of his Evans Lemma is wrong.

And there is no hope to correct that error: Eq.(9) is one linear equation for 16 unknowns which - as is well-known - cannot uniquely be resolved for the 16 unknowns Qaλ.

### Further remarks

(i) The Eqs. (46-47) are invalid for using a non-existing 3-index Î-tensor in 4D, see

(ii) Equ.(74) falsely attributed to Bruhn can originally be found in Evans GCUFT book vol.1 as Eq.(1.59): In the first part of this book (Chap.2 - 7) Evans uses the version (1.59).