A Fatal Error in M.W. Evans' EVANS Lemma

Gerhard W. Bruhn, Darmstadt University of Technology

Quotations from Evans' articles or book appear in black

In his GCUFT book Vol.1 [5;Chap.J.3] M.W.Evans considers the differential equation

o q_λ^a = ∂^μ(Γ_μ^ν_λ q_ν^a) − ∂^μ(ω_μ^a_b q_λ^b) .                                                 (J.30)

As abbreviation for the right hand side of (J.30) we introduce the quantities

(1)                                                 R_λ^a := ∂^μ(Γ_μ^ν_λ q_ν^a) − ∂^μ(ω_μ^a_b q_λ^b) ,

which allows us to write the differential equation (J.30) in the much shorter form

(2)                                                 o q_λ^a = R_λ^a .

The statement of the Evans Lemma is the assertion that a scalar factor R exists such that

R q_λ^a = ∂^μ(Γ_μ^ν_λ q_ν^a) − ∂^μ(ω_μ^a_b q_λ^b) .                                                 (J.31)

or in our abbreviated notation

(3)                                                 R q_λ^a = R_λ^a .

However, Evans believes the existence of R to be obvious, which, as we shall see below, is a fatal error.

R shall be the proportionality factor between the two 4×4 matrices, Q = (q_λ^a) and P = (R_λ^a).

Q is the invertible tetrad matrix Q = (q_λ^a), the inverse of which is called here Q⁻¹ = (q_a^λ), hence q_a^λ q_μ^a = δ_μ^λ and for later use

(4)                                                 q_a^λ q_λ^a = δ_λ^λ = 1 + 1 + 1 + 1 = 4

under Einstein summation convention.

Evans' Lemma means that a factor R exists such that R Q = P, i.e. both matrices are elementwise proportional with the same propertionality factor R for all matrix elements.

Is the EXISTENCE of the proportionality factor R obvious?

Of course, such a factor R cannot exist in general, since that one real value R would be overdetermined by the matrix equation R Q = P, which is equivalent to 4×4 = 16 scalar equations. Thus the degree of overdetermination is 16:1.

Nevertheless, M.W. Evans knows such a factor. From [6; Equ.(2)] we learn

R = q_a^λ R_λ^a .                                                                 [6;(2)]

The solution of that enigma:

We have to distinguish between the special case a where the matrix P is proportional to the Matrix Q and the general case b where we have NO propertionality of P to Q.

Case a
If both matrices are proportional, P = R Q, then the factor R can be determined: We "multiply" the indirect definition of R, the equation (J.31), by q_a^λ to obtain

R q_λ^a q_a^λ = q_a^λ R_λ^a

and due to (4) we obtain R·4 = q_a^λ R_λ^a or

(5)                                                                 R = ¼ q_a^λ R_λ^a .

Comparison with [6;(2)] shows that the factor ¼ is missing there. A minor error only, that occasionally happens when handling tensor equations. Of course, to be corrected.

Case b
What, if the matrix P is NOT proportional to Q, which is the general case: Then there doesn't exist any proportionality factor R. The equations R q_λ^a = R_λ^a determine different values R depending on the indices λ and a.

But the corrected equ. (5) determines some value R nevertheless. What, if we use that value?

"No proportionality" means that however the value R is chosen at least one index position (a,λ) can be found such that R q_λ^a ≠ R_λ^a. Then for that special index position we have

o q_λ^a = R_λ^a ≠ R q_λ^a

This means that:

The Evans Lemma is not fulfilled in general.

If P is NOT proportional to Q, then (5) yields an irrelevant value.

Evans' "nice" ECE Lemma does not hold with the consequence that the EVANS Field Equation is wrong too. One essential column of the EVANS-Cartan-Einstein-Theory (ECE) has broken down.

Due to a simple flaw of thinking! Think first, write later.

References

[1]   M.W. Evans, SUMMARY OF BRUHN REBUTTALS,
        http://www2.mathematik.tu-darmstadt.de/~bruhn/asummaryofbruhnrebuttals.pdf

[2]   G.W. Bruhn, Refutation of Myron W. Evans’ B⁽³⁾ field hypothesis,
        http://www2.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

[3]   G.W. Bruhn, Myron W. Evans' Most Spectacular Errors,
        http://www2.mathematik.tu-darmstadt.de/~bruhn/MWEsErrors.html

[4]   M.W. Evans, Generally Covariant Unified Field Theory, Vol.1,
        Arima Publishing 2005

[5]   M.W. Evans, Some Subsidary Relations from the ECE Lemma (pdf),
        http://www2.mathematik.tu-darmstadt.de/~bruhn/anecelemmacrosscheck.pdf

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