01.01.2008

(Quotations from Evans' papers are displayed in black.)

With the introduction to his paper #100 . Evans announces a review of major themes of his ECE theory. We shall start reviewing paper #100 with its section 2 finding a collection of older and recent flaws of that "theory". All these errors are beyond repair.

We read on p.9 of Evans' paper 100 [1.2]:

. . . The second advance in basic geometry is the inference {1-12} of the Hodge dual of the Bianchi identity. In short hand notation is:

                D Ù T^~ = R^~ Ù q                                                                                                 (9)

The proof of this assertion as given by Evans in his papernote 100(16) [1b] is wrong as was shown in [5].

In addition Evans writes on p.9/10 of paper #100 - Section 2 [1.2]:

. . . that the Bianchi identity and its Hodge dual are the tensor equations

                D_μ T^κμν = R^κ_μ^μν                                                                                                 (13)

and

                D_μ T^{~ κμν} = R^{~ κ}_μ^μν                                                                                            (12)

Both equations have (due to summation over the index μ) 2-tensors on both sides, while the tensor equivalents of the 1st Bianchi identity and its alleged dual Eq.(9) must have 4-tensors. Thus, the eqs. (13) and (12) cannot be the tensor duals of the 1st Bianchi identity and of Eq.(9) respectively.

. . . Therefore the neglect of torsion makes EH theory internally inconsistent, so standard model general relativity and cosmology are also internally inconsistent at a basic level.

As was pointed out above the author Evans failed to prove the existence of internal inconsistencies in EH theory.

We read on p.6 of Evans' paper [1a]:

. . . Similarly the "second Bianchi identity" of standard model general relativity is:

                D Ù R = 0                                                                                                 (8)

and in tensor notation this becomes

                D_ρ Ù R^κ_σμν + D_μ Ù R^κ_σνρ + D_ν Ù R^κ_σρμ = 0 .                                           (9)

Again this neglects torsion arbitrarily, and . . .

And in Evans' paper #100/2 [1.2]:

In the course of development of ECE theory a similar problem was found with what is referred to in the standard model literature as "the second Bianchi identity". In shorthand notation this is given {13} as:

                D Ù R = 0                                                                                                 (15)

but again this neglects torsion. . . .

The reader should have a look in Evans' standard textbooks [2, p.93, Eqs.(3.139-141] and [3, p.488 f., Eqs.(3.28)-(3.32)] where Equ.(8)/(15) is proven in case of non-vanishing torsion. We follow here [2] applying slight modifications of notation. The reader is also recommended to read the more systematic and deeper book by F.W. Hehl [4, p.208].

S.M. Carroll writes

                R^a_b = d Ù ω^a_b + ω^a_c Ù ω^c_b                                                           [2,(3.138)]

to apply d Ù which yields

                d Ù R^a_b = d Ù d Ù ω^a_b + d Ù (ω^a_c Ù ω^c_b) = 0 + (d Ù ω^a_c) Ù ω^c_b − ω^a_c Ù (d Ù ω^c_b)

where d Ù ω^._. can be eliminated by means of Equ. [2,(3.138)] to obtain

                d Ù R^a_b = d Ù d Ù ω^a_b + d Ù (ω^a_c Ù ω^c_b) = 0 + (d Ù ω^a_c) Ù ω^c_b − ω^a_c Ù (d Ù ω^c_b)
                             = (R^a_c − ω^a_s Ù ω^s_c) Ù ω^c_b − ω^a_c Ù (R^c_b − ω^c_s Ù ω^s_b) = R^a_c Ù ω^c_b − ω^a_c Ù R^c_b

or

                d Ù R^a_b + ω^a_c Ù R^c_b − ω^c_b Ù R^a_c = 0                                                 [2,(3.140)]

However, from

                D_μ X^a_b = ∂_μ X^a_b + ω_μ^a_c X^c_b − ω_μ^c_b X^a_c                                           [2,(3.128)]

we obtain by left multiplication by dx^μÙ due to dx^μD_μ = D, dx^μω_μ^a_c = ω^a_c, dx^μω_μ^c_b = ω^c_b, the result

                D Ù X^a_c = d Ù X^a_c + ω^a_c Ù X^c_b − ω^c_b Ù X^a_c

valid for arbitrary tensor valued forms X^a_b = X^a_{b μν} dx^μ Ù dx^ν

By applying this to [2,(3.140)] we obtain the second Bianchi identity

D Ù R^a_b = 0         valid without any assumptions on torsion.