Review of the FoPL paper [1]

The Evans Lemma of Differential Geometry

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.

1. Evans' first proof of his Lemma

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 (∂_μq^a_λ + ω^a_μbq^b_λ − Γ^ν_μλq^a_ν) = 0.

Using the Leibnitz rule, we have

[1,(37)]                 (D^μ∂_μ)q^a_λ + ∂_μ(D^μq^a_λ ) + (D^μω^a_μb)q^b_λ
                                                + ω^a_μb(D^μq^b_λ ) − (D^μ Γ^ν_μλ)q^a_ν − Γ^ν_μλ (D^μq^a_ν ) = 0,

and so

[1,(38)]                 (D^μ∂_μ)q^a_λ + (D^μω^a_μb)q^b_λ − (D^μ Γ^ν_μλ)q^a_ν = 0,

because

[1,(39)]                 D^μq^a_λ = D^μq^b_λ = D^μq^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^μω^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].

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)]                 D_μq^a_λ = ∂_μq^a_λ + ω^a_μbq^b_λ − Γ^ν_μλq^a_ν = 0.

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

[2,(J.28)]                 D^μ(D_μq^a_λ) = ∂^μ(D_μq^a_λ) = 0,

i.e.

[2,(J.29)]                 ∂^μ (∂_μq^a_λ + ω^a_μbq^b_λ − Γ^ν_μλq^a_ν) = 0,

or

[2,(J.30)]                 o q^a_λ = ∂^μ(Γ^ν_μλq^a_ν) − ∂^μ(ω^a_μbq^b_λ) .

Define:

[2,(J.31)]                 R q^a_λ := ∂^μ(Γ^ν_μλq^a_ν) − ∂^μ(ω^a_μbq^b_λ)

to obtain the Evans Lemma:

[2,(J.32)]                 oq^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.

There is no proof of the Evans Lemma, neither in the article [1] nor in [2, Sec. J.3].

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_μλq^a_ν − ω^a_μbq^b_λ)

and use (the "Cartan Convention")

[3, (10)]                 q^λ_aq^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^λ_a) ∂^μ(Γ^ν_μλq^a_ν − ω^a_μbq^b_λ) = 1 · o q^a_λ .

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