## Remarks on Evans' paper #100 - Section 2.

### G.W. Bruhn, Darmstadt University of Technology

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.

### 1. Dualizing the 1st Bianchi identity?

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

Dμ Tκμν = Rκμμν                                                                                                 (13)

and

Dμ T~ κμν = R~ κμμν                                                                                            (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.

### 2. Concerning the 2nd Bianchi identity

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

Dρ Ù Rκσμν + Dμ Ù Rκσνρ + Dν Ù Rκσρμ = 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

Rab = d Ù ωab + ωac Ù ωcb                                                           [2,(3.138)]

to apply d Ù which yields

d Ù Rab = d Ù d Ù ωab + d Ùac Ù ωcb) = 0 + (d Ù ωac) Ù ωcb − ωac Ù (d Ù ωcb)

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

d Ù Rab = d Ù d Ù ωab + d Ùac Ù ωcb) = 0 + (d Ù ωac) Ù ωcb − ωac Ù (d Ù ωcb)
= (Rac − ωas Ù ωsc) Ù ωcb − ωac Ù (Rcb − ωcs Ù ωsb) = Rac Ù ωcb − ωac Ù Rcb

or

d Ù Rab + ωac Ù Rcb − ωcb Ù Rac = 0                                                 [2,(3.140)]

However, from

Dμ Xab = ∂μ Xab + ωμac Xcb − ωμcb Xac                                           [2,(3.128)]

we obtain by left multiplication by dxμÙ due to dxμDμ = D, dxμωμac = ωac, dxμωμcb = ωcb, the result

D Ù Xac = d Ù Xac + ωac Ù Xcb − ωcb Ù Xac

valid for arbitrary tensor valued forms Xab = Xab μν dxμ Ù dxν

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

### References

[1.1] M.W. Evans, A Review of Einstein-Cartan-Evans (ECE) Field Theory (Introduction of Paper #100),
http://www.atomicprecision.com/blog/2007/12/27/introduction-to-paper-100/wp-filez/a100thpaperintroduction.pdf .

[1.2] M.W. Evans, Geometrical Principles (Section 2 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory
,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection2.pdf .

[1.3] M.W. Evans, The Field (Section 3 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory
,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection3.pdf .

[1.4] M.W. Evans, Aharonov Bohm and Phase Effects in ECE Theory (Section 4 of Paper #100:
A Review of Einstein-Cartan-Evans (ECE) Field Theory
,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection4.pdf .

[1.5] M.W. Evans, Tensor and Vector Laws of Classical Dynamics and Electrodynamics (Section 5 of Paper #100) ,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection5.pdf .

[1.6] M.W. Evans, Spin Connection Resonance (Section 6 of Paper #100) ,
http://www.atomicprecision.com/blog/wp-filez/a100thpapersection6.pdf .

[1a] M.W. Evans, Development of the Einstein Hilbert Field Equation . . .,
http://www.aias.us/documents/uft/a103rdpaper.pdf .

[1b] M.W. Evans, Proof of the Hodge Dual Relation,
http://www.atomicprecision.com/blog/wp-filez/a100thpapernotes16.pdf .

[1c] M.W. Evans, Some Proofs of the Lemma,
http://www.atomicprecision.com/blog/wp-filez/acheckpriortocoding5.pdf .

[1d] M.W. Evans, Geodesics and the Aharonov Bohm Effects in ECE Theory,
http://www.aias.us/documents/uft/a56thpaper.pdf .

[2] S.M. Carroll, Lecture Notes on General Relativity,
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf, 1997.

[3] S.M. Carroll, Spacetime and Geometry,
http://xxx.lanl.gov/PS_cache/gr-qc/pdf/9712/9712019v1.pdf, 1997.

[4] F.W. Hehl and Y.N. Obukhov, Foundations of Classical Electrodynamics −
Charge, Flux and Metric
,
Birkhäuser 2003, ISBN 0-8176-4222-6, ISBN 3-7643-4222-6

[5] G.W. Bruhn, Consequences of Evans' Torsion Hypothesis,

[6] G.W. Bruhn, Remarks on Evans' paper #100 - Section 2,
onMwesPaper100-2.html .

[7] M.R. Spiegel, Vector Analysis,
in Schaum's Outline Series, McGraw-Hill.

[8] G.W. Bruhn, Evans' "3-index Î-tensor" ,
Evans3indEtensor.html .

[9] G.W. Bruhn, Comments on Evans' Duality,
EvansDuality.html .

[10] G.W. Bruhn, F.W. Hehl, A. Jadczyk , Comments on ``Spin Connection Resonance
in Gravitational General Relativity''
, ACTA PHYSICA POLONICA B Vol. 39/1 (2008)
pdf . html

[11] G.W. Bruhn, Remarks on Evans/Eckardt’sWeb-Note on Coulomb Resonance,,
RemarkEvans61.html .

(08.01.2008) An Editorial Note by G. 't Hooft in Found. Phys.

(25.01.2008) Remarks on Evans' Web Note #100-Section 6: SCR

(27.12.2007) Remarks on Evans' Web Note #103

(19.12.2007) Myron now completely confused

(14.12.2007) Evans' Central Claim in his Paper #100

(10.12.2007) How Dr. Evans refutes the whole EH Theory