Review of the FoPL paper [1]

The Evans Lemma of Differential Geometry

Gerhard W. Bruhn, Darmstadt University of Technology


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 Dμ. 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.

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,


[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 [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)]                 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,


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


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


[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 [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 = 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 ...)


[1] M.W. Evans, The Evans Lemma of Differential Geometry,
      FoPL 17 433 ff. (2004)

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

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


(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