The Evans Lemma of Differential Geometry

The Evans Lemma is basic for Evans GCUFT or ECE Theory [2]. Evans has given
two proofs of his Lemma. The first proof in [1] is shown to be invalid due to
dubious use of the covariant derivative D^{μ}. A second proof in
[2, Sec.J.3] is wrong due to a logical error.

The page numbers of the web copy mentioned in [1] 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**.

The Evans Lemma is the assertion of proportionality of the matrices
(∇q_{μ}^{a}) and (q_{μ}^{a})
with a proportionality factor R:

(oq_{μ}^{a}) = R (q_{μ}^{a}) .

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)]
D^{μ} (∂_{μ}q^{a}_{λ} +
ω^{a}_{μb}q^{b}_{λ} −
Γ^{ν}_{μλ}q^{a}_{ν}) = 0.

Using the Leibnitz rule, we have

[1,(37)]
(D^{μ}∂_{μ})q^{a}_{λ} +
∂_{μ}(D^{μ}q^{a}_{λ} ) +
(D^{μ}ω^{a}_{μb})q^{b}_{λ}

+
ω^{a}_{μb}(D^{μ}q^{b}_{λ} )
− (D^{μ} Γ^{ν}_{μλ})q^{a}_{ν} −
Γ^{ν}_{μλ} (D^{μ}q^{a}_{ν} ) = 0,

and so

[1,(38)]
(D^{μ}∂_{μ})q^{a}_{λ} +
(D^{μ}ω^{a}_{μb})q^{b}_{λ} −
(D^{μ} Γ^{ν}_{μλ})q^{a}_{ν} = 0,

because

[1,(39)]
D^{μ}q^{a}_{λ} =
D^{μ}q^{b}_{λ} =
D^{μ}q^{a}_{ν} = 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
D^{μ}ω^{a}_{μb} and
D^{μ} Γ^{ν}_{μλ} ?
Note that the terms
ω^{a}_{μb} and
Γ^{ν}_{μλ}
both are no tensors and so the covariant derivative D^{μ}
is not applicable. Therefore we skip over the rest of [1].

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

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

[2,(J.27)]
D_{μ}q^{a}_{λ} =
∂_{μ}q^{a}_{λ} +
ω^{a}_{μb}q^{b}_{λ} −
Γ^{ν}_{μλ}q^{a}_{ν} = 0.

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

[2,(J.28)]
D^{μ}(D_{μ}q^{a}_{λ}) =
∂^{μ}(D_{μ}q^{a}_{λ}) = 0,

i.e.

[2,(J.29)]
∂^{μ} (∂_{μ}q^{a}_{λ} +
ω^{a}_{μb}q^{b}_{λ} −
Γ^{ν}_{μλ}q^{a}_{ν}) = 0,

or

[2,(J.30)]
**o** q^{a}_{λ} =
∂^{μ}(Γ^{ν}_{μλ}q^{a}_{ν})
−
∂^{μ}(ω^{a}_{μb}q^{b}_{λ}) .

Define:

[2,(J.31)]
R q^{a}_{λ} :=
∂^{μ}(Γ^{ν}_{μλ}q^{a}_{ν})
−
∂^{μ}(ω^{a}_{μb}q^{b}_{λ})

to obtain the Evans Lemma:

[2,(J.32)]
**o**q^{a}_{λ} =
R q^{a}_{λ}

**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.

**Additional remark**

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 =
q^{λ}_{a} ∂^{μ}(Γ^{ν}_{μλ}q^{a}_{ν}
− ω^{a}_{μb}q^{b}_{λ})

and use (the "Cartan Convention")

[3, (10)]
q^{λ}_{a}q^{a}_{λ} = 1

to find

[3, (11)]
**o** q^{a}_{λ} = R q^{a}_{λ} .

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

q^{a}_{λ} R =
(q** ^{a}_{λ}**
q

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

[1]
M.W. Evans, The Evans Lemma of Differential Geometry,

FoPL 17 433 ff. (2004)

http://www.aias.us/documents/uft/a7thpaper.pdf

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

[3]
M.W. Evans, Some Proofs of the Lemma,

http://www.atomicprecision.com/blog/wp-filez/acheckpriortocoding5.pdf

(25.06.2007)
**The consequences of the invalidity of the Evans Lemma
**

(19.06.2007)
**A Lecture on New Math given by Dr Horst Eckardt and Dr Myron W. Evans
**

(27.05.2007)
**Commentary on Evans' recent remark on the ECE Lemma**

(09.04.2007)
**Review of the Evans Lemma**

(12.03.2007)
**Evans "proves" the Evans Lemma again**