 ## Review of the FoPL paper  The Evans Lemma of Differential Geometry

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

#### Abstract

The Evans Lemma is basic for Evans GCUFT or ECE Theory . Evans has given two proofs of his Lemma. The first proof in  is shown to be invalid due to dubious use of the covariant derivative Dμ. A second proof in [2, Sec.J.3] is wrong due to a logical error.

### Note

The page numbers of the web copy mentioned in  start with 1 instead of 433 (= 432+1). Equations from Evans' publication [p] appear with equation labels [p,(nn)] in the left margin. Quotations from Evans' contributions appear in black.

### 1. Evans' first proof of his Lemma

The Evans Lemma is the assertion of proportionality of the matrices (∇qμa) and (qμa) with a proportionality factor R:

(oqμa) = R (qμa) .

Quotation from [1, p.432+8]

The Evans lemma is a direct consequence of the tetrad postulate. The proof of the lemma starts from covariant differentiation of the postulate:

[1,(36)]                 Dμ (∂μqaλ + ωaμbqbλ − Γνμλqaν) = 0.

Using the Leibnitz rule, we have

[1,(37)]                 (Dμμ)qaλ + ∂μ(Dμqaλ ) + (Dμωaμb)qbλ
+ ωaμb(Dμqbλ ) − (Dμ Γνμλ)qaν − Γνμλ (Dμqaν ) = 0,

and so

[1,(38)]                 (Dμμ)qaλ + (Dμωaμb)qbλ − (Dμ Γνμλ)qaν = 0,

because

[1,(39)]                 Dμqaλ = Dμqbλ = Dμqaν = 0.

Eq. [1,(36)] is formally correct, however, the decomposition in Eq.[1,(37)] yields undefined expressions: What e.g. is the meaning of the terms Dμωaμb and Dμ Γνμλ ? Note that the terms ωaμb and Γνμλ both are no tensors and so the covariant derivative Dμ is not applicable. Therefore we skip over the rest of .

### 2. Evans' second proof of his Lemma

Evans himself felt it necessary to give another proof in [2, p.514], now avoiding the problem of undefined terms.

#### J.3 The Evans Lemma

The Evans Lemma is the direct result of the tetrad postulate of differential geometry:

[2,(J.27)]                 Dμqaλ = ∂μqaλ + ωaμbqbλ − Γνμλqaν = 0.

using the notation of the text. It follows from eqn. (J.27) that:

[2,(J.28)]                 Dμ(Dμqaλ) = ∂μ(Dμqaλ) = 0,

i.e.

[2,(J.29)]                 ∂μ (∂μqaλ + ωaμbqbλ − Γνμλqaν) = 0,

or

[2,(J.30)]                 o qaλ = ∂μνμλqaν) − ∂μaμbqbλ) .

Define:

[2,(J.31)]                 R qaλ := ∂μνμλqaν) − ∂μaμbqbλ)

to obtain the Evans Lemma:

[2,(J.32)]                 oqaλ = R qaλ

As simple as wrong: Note that Eq.[2,(J.31)] represents a set of 16 equations each of which for one fixed pair of indices (a,μ) (a,μ = 0,1,2,3). Each equation is a condition to be fulfilled by the quantity R. These 16 conditions for R will not agree in general. Thus, the author Evans, when giving the "definition" Eq.[2,(J.31)], ignored the possible incompatibility of the sixteen definitions of R contained in his "definition" of R by Eq.[2,(J.31)]. Therefore this proof of the Evans Lemma in [2, Sec. J.3] is invalid.

### There is no proof of the Evans Lemma, neither in the article  nor in [2, Sec. J.3].

In his note [3, p.2] Evans gives a variation of this "proof". There he defines R directly and applies his "Cartan Convention":

[3, (9)]                 R = qλaμνμλqaν − ωaμbqbλ)

and use (the "Cartan Convention")

[3, (10)]                 qλaqaλ = 1

to find

[3, (11)]                 o qaλ = R qaλ .

i.e. from the correct Eq. [2,(J.30)] he erroneously concludes

qaλ R = (qaλ qλa) ∂μνμλqaν − ωaμbqbλ) = 1 · o qaλ .

We learn from this that one can "prove" every nonsense, if one has the suitable error at hand: To ignore the rules of tensor calculus on hidden indices. (see also Evans' New Math in Full Action ...)

### References

 M.W. Evans, The Evans Lemma of Differential Geometry,
FoPL 17 433 ff. (2004)
http://www.aias.us/documents/uft/a7thpaper.pdf

 M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics; Arima 2006

 M.W. Evans, Some Proofs of the Lemma,
http://www.atomicprecision.com/blog/wp-filez/acheckpriortocoding5.pdf