## Remarks on Evans' paper #100 - Section 6

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

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.

### 1. Evans' "Euler Bernoulli resonance equation of general type"

>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/rr) ∂Φ/∂r + Φ/ (2rωr + r² ∂ωr/∂r) = − ρ/Îo                           (142)

When the spin connection is defined as:

ωr = ωo² r − 4β loger − 4/r                                                                       (143)

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

∂²Φ/∂r² + 2β ∂Φ/∂r + ωo² Φ = − ρ/Îo                                                         (144)

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

Freedom of choice??? Eq.(143) is just a condition for ωr (to be discussed in Sect.3).
With concern of the terms ωo² and κo² 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:

ωr = κo² − 4β loge² 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],

ωr = 2 (β − 1/r),         κo² = 4/r (β − 1/r) + ∂ωr/∂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 + κo² x = A cos(κr)                                                 (145)

in which β plays the role of a friction coefficient, κo 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 κo could be understood as a hint to the case of constant coefficients 2β and κo², which should be discussed first, referring to the Eqs.(146):

ωr = 2 (β − 1/r),         κo² = 4/r (β − 1/r) + ∂ωr/∂r                                 (146)

If β=const then Eq.(146-1) yields ∂ωr/∂r = 2/, hence from Eq.(146-2) κo² = /r2/, which shows that κo² is not constant.

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

The term "Euler Bernoulli", however, has another meaning: It means that Eq.(145) after division by r² should have the form of an Euler differential equation:

d²x/dr² + b/r dx/dr + a/ x = A/ cos(κr)                                                 (145')

The comparison with Eq.(145) yields conditions to be fulfilled by constant coefficients a,b:

b = 2β r ,                 a = r² κo² .

where β and κo² have to satisfy the Eqs.(146). We ask for suitable constants a,b: At first the quantity β can be eliminated from the Eqs.(146) to obtain

a = r² κo² = r² (4/r ½ωr + ∂ωr/∂r) = 2r ωr + r² ∂ωr/∂r = /∂r (r² ωr) .

Since a is constant we can integrate this differential equation to obtain

a r + c = r² ωr

where c is some constant. This yields

a + c/r = r ωr =(146-1) 2 (βr − 1) = b − 2.

and by coefficients matching

c = 0         and         b = a + 2 .

Therefore Eq.(145) can be written equivalently as

d²x/dr² + (a+2)/r dx/dr + a/ x = A/ cos(κr) .                                                 (145'')

Resonance will occur if the inhomogenity A/ cos(κr) of Eq.(145'') belongs to the eigenspace of its associated homogeneous equation

d²z/dr² + (a+2)/z dx/dr + a/ z = 0 .

It is easy to determine eigensolutions by exponential ansatz z = rλ: The characteristic equation is

λ(λ−1) + (a+2) λ + a = 0 ,                 i.e.                 λ² +(a+1) λ + a = 0 ,

with the solutions λ1 = −1 and λ2 = −a . The corresponding eigensolutions are

z1 = 1/r                 and                 z2 = 1/ ra .                     if a≠1 ,

and

z1 = 1/r                 and                 z2 = 1/ r log r                 if a=1 .

>From this result we may conclude that the driving term, the inhomogenity A/ cos(κr) of Eq.(145'') cannot cause resonance since it does not belong to the eigenspace that is spanned by the eigensolutions z1 and z2 .

So we come to the conclusion:

### 2. A remark to the SCR web-paper 92

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 /rΦ/ + /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

ωr = 0 ,                 κo² = 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. ...

### 3. A remark on a "simple resonance equation"

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:

ωr = κo² − 4β loge² r − 4/r .                                                                 (147)/[1e,(5)]

Reduction to the standard model Coulomb law occurs when:

β = 1/r                                                                                                 (148)

when

ωr = 0,         κo² = 0                                                                             (149)

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

0 = 0 − 4/r loge 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.

### 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