 ## Evans' Central Claim in his Paper #100

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

In his web-paper  M.W. Evans proudly presents the

Theorem:

If             R Ù q = D Ù T                                 [1,(1)]
then         R~ Ù q = D Ù T~ .                            [1,(2)]

where the indices are suppressed by Evans. The correct version of the quoted claim is

(1)             If     Rab Ù qb = DÙTa     then     R~ ab Ù qb = DÙT~ a .

Evans' claim is followed by a fuzzy "proof" the shortcomings of which will be revealed below. And by a lot of far reaching "consequences" . . .

### However, that all is nothing but hot air, caused by incorrect calculations.

At first Evans refers to a representation of the lhs of eq. [1,(1)], of Rab Ù qb, by a cyclic sum using qb = qbρ dxρ and Rab qbρ = Raρ:

Rλρμν + Rλνρμ + Rλμνρ = cyclic sum of definitions (3)                                 [1,(7)]

taken from

(2)                 Rab Ù qb = Ra[ρμν] dxρÙdxμÙdxν = ½ (Raρμν + Raνρμ + Raμνρ) dxρÙdxμÙdxν

for a = λ.

Then Evans introduces the coefficients of the Hodge dual of the curvature 2-form Rab:

The Hodge dual of eq. [1,(3)] is:

R~ λραβ = ½ ||g||½ Îαβμν Rλρμν                                 [1,(5)]

This Evans' original equation can be replaced with the equivalent (but more convenient) equation

(3)                 R~ ab αβ = Îαβμν Rab μν

which would yield

(4)                 R~ ab αβ dxαÙdxβ = Rab μν Îαβμν dxαÙdxβ = Rab μν (dxμÙdxν)~ = (Rab μν dxμÙdxν)~.

This means that the 2-form R~ ab αβ dxαÙdxβ is the Hodge dual of the 2-form Rab μν dxμÙdxν for each pair of fixed indices a,b.

However, this equation has to be applied to eq. [1,(7)], which seems to be simple due to the short hand notation of that equation. But its correct detailed notation in eq. (2) reveals a problem: In contrast to Evans' belief the transition to

R~ λρμν + R~ λνρμ + R~ λμνρ = cyclic sum of definitions (14)                                 [1,(15)]

is not possible by a mere multiplication of eq. [1,(7)] by an Î-factor due to the wrong positioned index ρ at the two right terms in eq. [1,(7)]. In other words: The multiplication of ½ (Rλρμν + Rλνρμ + Rλμνρ) by

(6)                 (dxμÙdxν)~ Ù dxρ = Îμναβ dxαÙdxβ Ù dxρ

yields

½ (Raρμν + Raνρμ + Raμνρ) Îμναβ dxαÙdxβ Ù dxρ
(7)                                = ½ Rab μν Îμναβ dxαÙdxβ Ùqb + ½ (Raνρμ + Raμνρ) Îμναβ dxαÙdxβ Ù dxρ
= ½ R~ab μν Ùqb + ½ (Raνρμ + Raμνρ) Îμναβ dxαÙdxβ Ù dxρ

The first summand yields half of the aspired 3-form R~ab Ù qb whilst the red marked summands cannot be rewritten as duals of Rab due to the occurrence of the index ρ at an improper second or third position.

### Therewith Evans' proof in  has broken down without any chance of repair.

This breaking off does not mean that there would not occur other irreparable problems while checking other parts of Evans' proof, e.g. for the transformation of D Ù Ta. But further refutations are left to the reader.

Final Remark
There is an error between Evans' eqs. [1,(19)] and [1,(20)]: The latter is NOT a consequence of the preceeding equation as asserted by Evans:

DρT~μλν + DνT~ρλμ + DμT~νλρ = − (R~μλν + R~ρλμ + R~νλρ)                 [1,(19)]

even if the symbol ~ is removed. That is due to the repeated upper and lower indices μ in eq.[1,(20)]. This repetition does not occur in [1,(19)]:

DμTλμν = − Rλμμν                                                                                 [1,(20)]

Both sides of this equation are two-tensors (index μ contracted) while eq.(19) displays four-tensors. Since 2 ≠ 4 both equations cannot be equivalent.

This remark is important since in a further note [2, eqs.(25-26) and (39-40)] falsly considers eq. [1,(20)] as the equivalent of the 1st Bianchi identity (the first equ. of (1)).

### References

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

 M.W. Evans, The Fundamental Origin of the Bianchi Identity of Cartan Geometry and ECE Theory,
http://www.atomicprecision.com/blog/wp-filez/a102ndpaper.pdf

 W.A. Rodrigues, Differential Forms on Riemannian (Minkowskian) and Riemann-Cartan Structures
and some Applications to Physics
, arXiv