Notes 2: The ECE Lemma

Evans' notes deal again with the "proof" of the ECE Lemma (formerly denoted as Evans Lemma, now even attributed to Einstein and Cartan) following the same line of argumentation as in his recent Bruhn rebuttal. Following a request of Gareth Evans for pointing to "the exact point of fraud" (in Evans' rebuttals) I skip some well-known lines of Evans' notes and point to the essential eq. (8):

         q^a_λ := R q^a_λ                 (8)

Evans considers eq. (8) with a=0,1,2,3 and λ=0,1,2,3 as definition of the scalar R. It 's hard to grasp for Evans (I can understand this well, since this objection touches the basis of his Lemma, a question of To Be Or Not To Be) that eq. (8) contains 16 different conditions for the one scalar R, which will not agree in general and thus, a value R fulfilling all 16 conditions will not exist.

The reader is therefore requested for patience when we completely list the sixteen equations now:

         q^o_o := R q^o_o ,          q^o₁ := R q^o₁ ,          q^o₂ := R q^o₂ ,          q^o₃ := R q^o₃ ,

         q¹_o := R q¹_o ,          q¹₁ := R q¹₁ ,          q¹₂ := R q¹₂ ,          q¹₃ := R q¹₃ ,

         q²_o := R q²_o ,          q²₁ := R q²₁ ,          q²₂ := R q²₂ ,          q²₃ := R q²₃ ,

         q³_o := R q³_o ,          q³₁ := R q³₁ ,          q³₂ := R q³₂ ,          q³₃ := R q³₃ ,

Evans did never show, that each of these 16 equations determines the same value of R, and so the scalar R is not defined by that way, unless Evans (or someone else) proves the compatibility of these 16 equations.

This is the GAP in Evans' proof of his Lemma.

Subsequently Evans attempts to justify his argumentation by comparison. Though this is irrelevant for closing Evans' gap, we shall have a look at and comment on it: His (16-fold overdetermined) definition of the scalar R is:

        R q^a_λ = ∂^μ(Γ^ν_μλq^a_ν−ω^a_μbq^b_λ)                 (9)

i.e.

        R := ¼q^λ_a ∂^μ(Γ^ν_μλq^a_ν−ω^a_μbq^b_λ)                 (10)

because:

        q^a_λq^λ_a = 4 .                 (11)

That step from eq. (9) to eq. (10) is only possible if one knows in advance that there exists a common solution R of the sixteen scalar equations contained in eq. (9).

The definition (11) is in all ways similar to
the definition used by Einstein:

        R = g^μν R_μν                 (12)

In eq. (10), as is eq. (12), summation over
repeated indices is used.

The Exact Point of Fraud

this occurs when Bruhn states that there are
many definitions of R in eq. (10). If this were
true, Einstein's own definition of R in eq.(12) would
be many definitions. This is of course pure nonsense.
In eq. (12) R_μν is the Ricci tensor, g^μν the
metric and R the scalar curvature.

Eq. (12) is the analogue to eq. (10).

But, where, Myron, is the previous analogue to eq. (9)? Missing!!!