 ## A remark on Evans' recent web article on the ECE Lemma

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

There is some little progress but a fatal error left in

and its attachment

For the first time the former "Cartan Convention" is displayed correctly:

qλa qaλ = 4                                 (12)

But any remark about the wrong former version qλa qaλ = 1 (e.g. in the GCUFT book vol.1) is missing, and no word about the designation "Cartan Convention", it's a "normalisation" now.

However, already "Definition" (11)

R qλa = ∂μνμλqνa − ωaμbqλb)                                 (11)

is inadmissible : Equ. (11) is to be satisfied for all pairs of indices a,λ = 0,1,2,3. Therefore Equ. (11) represents 4×4 = 16 (sixteen) definitions of R, and nobody tells us

### why these different definitions of R should agree.

Thus, the conclusion

. . . and multiply both sides of eq. (11) by qaλ to obtain

R = ¼ qaλμνμλqνa − ωaμbqλb)                                 (13)

is inadmissible. This value of R cannot fulfil the 16 equations (11) in general. Myron should check that by plugging the expression (13) into the definitions (11) (by hand *) or by computer algebra). He should tell us the result.

### There is no ECE Lemma.

Part 2 of Evans' paper is based on the existence of the (non-existing) value of R and is obsolete therefore.

*) Applying his New Math (handling of Eqs. (22 - 23)) Evans would probably execute the following "self check" by hand:

Plugging R (eq.(13)) into eq.(11) yields

R qλa = [¼ qaλμνμλqνa − ωaμbqλb)] qλa

= [¼ qaλqλa]μνμλqνa − ωaμbqλb)

= [¼ 4] ∂μνμλqνa − ωaμbqλb)

= μνμλqνa − ωaμbqλb)

and so he feels fully justified.

Really! So he did!!! In the handwritten document http://www.atomicprecision.com/blog/wp-filez/acheckpriortocoding5.pdf we read

. . .

i.e.

o qαa = ∂μνμλqνa − ωaμbqλb)                                                       (8)

Now define

R = qaλμνμλqνa − ωaμbqλb)                                                       (9)

and use the famous EC (= Evans Convention)

qλaqaλ = 1 (!!!)                                                                                 (10)

to find by using the eqs. (8-10)