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".

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 = ∫_r^{r ∂}/_∂x (tan^{−1 y}/_x) dx + ∫_r^{r ∂}/_∂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=0^2π ≠ 0                                                                                 (2.77)

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

                    ∫_θ=0^2π Ñ X · dr = ∫_x^x 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

                ∫_a^{b   ∂}/_∂x f   dx = f(b) − f(a).                

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

∫_a^{b   d}/_dx f   dx = f(b) − f(a)

It is possible to trace back Evans' "three index, totally antisymmetric unit tensor Î_abc" 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.

                Î^ijk ∂_jB_k − ∂₀Eⁱ = 4π Jⁱ
                . . .
                Î^ijk ∂_jE_k − ∂₀Bⁱ = 0
                . . .                                                                 (1.74)

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

                F^ij = Î^ijk B_k .                                                 (1.75)

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

(normalized so that ι²³ = 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 Î_abc". As a consequence Evans has introduced "tensors" in his "covariant" theory which don't transform covariantly under (local) Lorentz transforms: The "tensors" Bⁱ, E_i are no four-dimensional tensors. They are parts of the four-dimensional tensors F^μν and F_μν respectively and transform as such under (local) Lorentz transforms.