Remarks on Evans' paper #100 - Section 2.

G.W. Bruhn, Darmstadt University of Technology


(Quotations from Evans' papers are displayed in

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)


                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


                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

D Ù Rab = 0         valid without any assumptions on torsion.


[1.1] M.W. Evans, A Review of Einstein-Cartan-Evans (ECE) Field Theory (Introduction of Paper #100), .

[1.2] M.W. Evans, Geometrical Principles (Section 2 of Paper #100:
      A Review of Einstein-Cartan-Evans (ECE) Field Theory
, .

[1.3] M.W. Evans, The Field (Section 3 of Paper #100:
      A Review of Einstein-Cartan-Evans (ECE) Field Theory
, .

[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
, .

[1.5] M.W. Evans, Tensor and Vector Laws of Classical Dynamics and Electrodynamics (Section 5 of Paper #100) , .

[1.6] M.W. Evans, Spin Connection Resonance (Section 6 of Paper #100) , .

[1a] M.W. Evans, Development of the Einstein Hilbert Field Equation . . ., .

[1b] M.W. Evans, Proof of the Hodge Dual Relation, .

[1c] M.W. Evans, Some Proofs of the Lemma, .

[1d] M.W. Evans, Geodesics and the Aharonov Bohm Effects in ECE Theory, .

[2] S.M. Carroll, Lecture Notes on General Relativity,, 1997.

[3] S.M. Carroll, Spacetime and Geometry,, 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,
      ECEcontradictions.html .

[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.

(29.01.2008) Remarks on Evans' Web Note #100-Section 7: The Sagnac Effect

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

(16.01.2008) Remarks on Evans' Web Note #100-Section 5: EM field

(08.01.2008) Remarks on Evans' Web Note #100-Section 4: The Aharonov Bohm effect

(05.01.2008) Remarks on Evans' Web Note #100-Section 3: Field and Wave equation

(01.01.2008) Remarks on Evans' Web Note #100-Section 2: Torsion and Bianchi identity

(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