## Remarks on Evans' paper #100 - Section 4.

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

08.01.2008

(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 should continue reviewing paper #100 with its section 4. However, this section is a subset of Evans' more detailed paper 56 [1g]. Therefore we shall comment here on this paper finding a further collection of older elementary flaws of that "theory".

### 1. Ryder's "Error"

We read on p.35 of Evans' paper 56 [1g]:

. . . This error in the standard model can be illustrated for example by the error in Eq.(3.105) of a standard model textbook such as ref. [16] (L.H. Ryder, Quantum Field Theory, Cambridge UP). The task is to evaluate:

Δδ = Ñ X · dr = S Ñ×ÑX := 0                                                                 (2.72)

As we have seen, Eq.(2.72) must be the correct result by the Stokes Theorem. The function X in the example we are considering here (Eq.3.105 of ref. [16]) is

X = ½ BR² θ                                                                                                 (2.73)

where B is magnetic flux density, R is a radius and θ is a cylindrical polar coordinate. The cylindrical polar coordinates are related [17] to the Cartesian coordinates by:

θ = tan−1 y/x,     r = (x² + y²)½ ,     x = r cos θ, y = r sin θ                             (2.74)

as is well known. The left hand side of Eq.(2.72) is evaluated between θ = 0 and θ = 2π. Therefore:

x = r cos 2π = r, y = r sin 2π = 0,     x = r cos 0 = r, y = r sin 0 = 0                 (2.75)

and the integral is:

Δδ = rr /∂x (tan−1 y/x) dx + rr /∂y (tan−1 y/x) dy = 0   (Errors #1 and #2)     (2.76)

This is the correct result as given by the Stokes Theorem (2.72). However, in ref. [16] it is incorrectly asserted that:

Ñ X · dr =? [X]θ=0 ≠ 0                                                                                 (2.77)

The correct way to evaluate Eq. (2.77) is:

θ=0 Ñ X · dr = xx Ñ X · dr = 0                 (Error #3)                                 (2.78)

because

x = r cos θ,     r = (x² + y²)½                                                                               (2.79)

and:

if θ = 2π, r = x,     if θ = 0, r = x                                                                         (2.80)

The whole of the argument on the AB effect in ref. [16] is therefore incorrect following the occurrence of this error, both mathematically and physically. The magnetic AB effect is not due to multiply connected spaces in special relativity. It is due to general relativity as we have argued. Thus ECE theory is preferred to the standard model, philosophically, mathematically and experimentally.

That was an awful example of Evans' New Math, again far away from all Math that is taught to the freshmen at the universities. We give a short reasoning for the errors committed by the author:

(Errors #1 and #2): We comment here on Error #1, Error #2 is analogue: In traditional Math we have no rule of the type

ab   /∂x f   dx = f(b) − f(a).

The correct rule Evans had in mind is the fundamental theorem of calculus:

### ∫ab   d/dx f   dx = f(b) − f(a)

with an essential distinction: The differential operator must be the "total" operator d/dx, not the operator of partial differentiation /∂x.

(Error #3): The author should take advice from n-dimensional Calculus of the "Line integral", e.g. in [7, Chap.5]. We have the rule:

### ∫abÑf · dr = f(b) − f(a) .

valid without further comments only for one-valued functions.

Evans, however, applies this rule incorrectly to a function that is not one-valued: The function f = ½ BR² θ changes its value when the origin O is surrounded once in anti-clockwise sense by the value ½ BR² 2π = BR² π ≠ 0 (if B≠0).

The above mentioned errors show that Dr. Evans has some shortcomings in his knowledge of Calculus that could be removed by studying a textbook like Vector Analysis in Schaum's Outlines Series [7, p.82 ff.]. Then he possibly would not attempt in future to attack prestigious scientists by using wrong arguments.

### 2. The 3-index Î-tensor in four dimensions

On p.31 of [1g] Evans introduces the "three index, totally antisymmetric unit tensor Îabc" in four dimensions by means of Eq. (2.51). However, in tensor analysis it is well-known that such a tensor cannot be defined. The definition (2.51) does not behave covariantly under coordinate transforms. Therefore the conclusions from Eq.(2.52) to Eq.(2.66) are irrelevant as based on a wrong assumption. For more information the reader is kindly requested to have a look at [8] and [9].

It is possible to trace back Evans' "three index, totally antisymmetric unit tensor Îabc" to an erroneous interpretation of his standard textbook, S.M. Carroll's Lecture Notes [2]. On [2, p.20] Carroll is going to rewrite the Maxwell equations.

ÎijkjBk − ∂0Ei = 4π Ji
. . .
ÎijkjEk − ∂0Bi = 0
. . .                                                                 (1.74)

. . . Meanwhile, the three-dimensional Levi-Civita tensor Îijk is defined just as the four-dimensional one, although with one fewer index.

Fij = Îijk Bk .                                                 (1.75)

In his book [3, p.29] S.M. Carroll adds the hint:

(normalized so that Î123 = Î123 = 1)

i.e. no index 0 occurs. And, in addition, the attentive reader (not Dr. Evans) of Carroll's note will have remarked that Latin indices are used in the eqs. (1.74) and (1.75) while elsewhere Greek indices appear.

What is the meaning of this tiny distinction?

There is one: On [2, p.3-4] S.M. Carroll defines:

. . . Greek ... indices running from 0 to 3, with 0 generally denoting the time . . .

. . . Latin ... indices to stand for the space components alone . . .

This tiny distinction was overlooked by Dr. Evans and lead him to the erroneous introduction of a "three index, totally antisymmetric unit tensor Îabc". As a consequence Evans has introduced "tensors" in his "covariant" theory which don't transform covariantly under (local) Lorentz transforms: The "tensors" Bi, Ei are no four-dimensional tensors. They are parts of the four-dimensional tensors Fμν and Fμν respectively and transform as such under (local) Lorentz transforms.

### 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, Birkhäuser 2003

[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