25.01.2008


(Quotations from Evans' papers are displayed in black.)

With the introduction to his paper #100 [1.1]. Evans announces a review of major themes of his ECE theory. We continue reviewing paper #100 with its section 6 [1.6] which is a somewhat revised version of Evans' papers [1e] and [1f] of that ECE "theory". However, the fact of giving wrong assertions of alleged "spin connection resonance" (SCR) remained unchanged. The flaws were already reported in our papers [10] and [11]: There is no SCR - nothing but Evans' New Math.

>From his ECE theory - dubious due to the extensive use of New Math (see e.g. the Comments on the previous Sections Sect.2, Sect.3, Sect.4, Sect.5) Evans has derived a potential equation for the case of spherical symmetry. Regardles of its dubious origin we accept this equation in the following to discuss Evans' further developments:

In paper 90 of www.aias.us this method was made more general by considering the equation

                ^∂²Φ/_∂r² + (²/_r+ω_r) ^∂Φ/_∂r + ^Φ/_r² (2rω_r + r² ^∂ω_r/_∂r) = − ^ρ/_Îo                           (142)

When the spin connection is defined as:

                ω_r = ω_o² r − 4β log_er − ⁴/_r                                                                       (143)

Eq.(142) becomes a simple resonance equation in r itself:

                ^∂²Φ/_∂r² + 2β ^∂Φ/_∂r + ω_o² Φ = − ^ρ/_Îo                                                         (144)

There is freedom of choice of the spin connection. . . .

Freedom of choice??? Eq.(143) is just a condition for ω_r (to be discussed in Sect.3).
With concern of the terms ω_o² and κ_o² there is a discrepancy between the Eqs.(143 - 144) on the one hand and Eqs.(145 - 147) on the other hand: We'll replace Eq.(143) with:

                ω_r = κ_o² − 4β log_e² r − ⁴/_r .                                                                     (147)

and compare the Eqs.(142) and (145) directly (ignoring the dubious Eq.(144)), this in accordance with the corresponding Eqs. (2-6) in Evans' SCR paper [1e],

                ω_r = 2 (β − ¹/_r),         κ_o² = ⁴/_r (β − ¹/_r) + ^∂ω_r/_∂r                                 (146) / [2,(3-4)]

In paper 92 of the ECE series (www.aias.us) Eq.(142) was further considered and shown to reduce to an Euler Bernoulli resonance equation

                ^d²x/_dr² + 2β ^dx/_dr + κ_o² x = A cos(κr)                                                 (145)

in which β plays the role of a friction coefficient, κ_o is a Hooke's law wave number and in which the right hand side is a cosinal driving term. . . .

The latter remark on the role of β and κ_o could be understood as a hint to the case of constant coefficients 2β and κ_o², which should be discussed first, referring to the Eqs.(146):

                ω_r = 2 (β − ¹/_r),         κ_o² = ⁴/_r (β − ¹/_r) + ^∂ω_r/_∂r                                 (146)

If β=const then Eq.(146-1) yields ^∂ω_r/_∂r = ²/_r², hence from Eq.(146-2) κ_o² = ^4β/_r − ²/_r², which shows that κ_o² is not constant.

Therefore the resonance theory of linear differential equations with constant coefficients does not apply to Eq.(144).

In Section 3 of the web-paper [1e] with the title "Development of spin connection resonance in the Coulomb Law" the authors attempt to determine a "direct solution of the SCR equation for the Coulomb Law". they consider the differential equation

(15)                 ^d/_drΦ /_r − ^Φ/_r² + ^d²/_dr²Φ = − ^ρ/_Îo

(which belongs to the case a=−1) and append a lengthy numerical evaluation. However, in the web-paper [1.6] under review here the author Evans has come to better insight. He now remarks (of course, without losing any word on the flaw in [1e]):

Reduction to the standard model Coulomb law occurs when:

                β = ¹/_r                                                                                                 (148)

when

                ω_r = 0 ,                 κ_o² = 0 .                                                                 (149)

In general there is no reason to assume that condition (148) always holds. The reason why the standard model Coulomb law is so accurate in the laboratory is that it is tested off resonance. ...

In both this paper [1.6] and the preceeding web-paper 92 [1e] the author(s) claim (cf. [1e, p.3]) immediately after having displayed the Eqs.(146)/(4) we read:

Solving these equations defines the condition under which the spin connection gives the simple resonance equation:

                ω_r = κ_o² − 4β log_e² r − ⁴/_r .                                                                 (147)/[1e,(5)]

Reduction to the standard model Coulomb law occurs when:

                β = ¹/_r                                                                                                 (148)

when

                ω_r = 0,         κ_o² = 0                                                                             (149)

Under the conditions (148) and (149) the eqs.(146) are fulfilled while from Eq.(147) we obtain

0 = 0 − ⁴/_r log_e r − ⁴/_r                 or                 0 = log r + 1

which is only fulfilled if the independent variable is fixed to the value r=¹/_e.

The other special case considered by Evans, the case of an undamped oscillator, gives a contradiction in Eq.(147) as well. Therefore Eq.(147) is not compatible with the Eqs.(146). That may go back to a simple calculation error. This is no question of importance.