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 ^{∂ωr}/_{∂r}) =
− ^{ρ}/_{Îo}
(142)

When the spin connection is defined as:

ω_{r} = ω_{o}² r − 4β log_{e}r −
^{4}/_{r}
(143)

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

^{∂²Φ}/_{∂r²} +
2β ^{∂Φ}/_{∂r} +
ω_{o}² 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_{o}² and κ_{o}²
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_{o}² − 4β
log_{e}² 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_{o}² = ^{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} +
κ_{o}² 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_{o}², which should be discussed first, referring to
the Eqs.(146):

ω_{r} = 2 (β − ^{1}/_{r}),
κ_{o}² = ^{4}/_{r} (β −
^{1}/_{r}) + ^{∂ωr}/_{∂r}
(146)

If β=const then Eq.(146-1) yields ^{∂ωr}/_{∂r}
= ^{2}/_{r²}, hence from Eq.(146-2)
κ_{o}² = ^{4β}/_{r} − ^{2}/_{r²},
which shows that κ_{o}² is not constant.

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}/_{r²} x = ^{A}/_{r²} cos(κr)
(145')

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

b = 2β r ,
a = r² κ_{o}² .

where β and κ_{o}² 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 κ_{o}² = r² (^{4}/_{r} ½ω_{r} +
^{∂ωr}/_{∂r}) =
2r ω_{r} + r² ^{∂ωr}/_{∂r} =
^{∂}/_{∂r} (r² ω_{r}) .

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

a r + c = r² ω_{r}

where c is some constant. This yields

a + ^{c}/_{r} = r ω_{r}
=_{(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}/_{r²} x = ^{A}/_{r²} cos(κr) .
(145'')

*Resonance* will occur if the inhomogenity ^{A}/_{r²} cos(κr)
of Eq.(145'') belongs to the eigenspace of its associated homogeneous equation

^{d²z}/_{dr²} + ^{(a+2)}/_{z} ^{dx}/_{dr} +
^{a}/_{r²} z = 0 .

It is easy to determine eigensolutions by exponential ansatz z = r^{λ}:
The characteristic equation is

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

with the solutions
λ_{1} = −1 and λ_{2} = −a . The corresponding
eigensolutions are

z_{1} = ^{1}/_{r}
and
z_{2} = ^{1}/_{ ra} .
if a≠1 ,

and

z_{1} = ^{1}/_{r}
and
z_{2} = ^{1}/_{ r} log r
if a=1 .

>From this result we may conclude that the driving term, the inhomogenity
^{A}/_{r²} cos(κr)
of Eq.(145'')
cannot cause resonance since it does not belong to the eigenspace
that is spanned by the eigensolutions z_{1} and z_{2} .

So we come to the conclusion:

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}/_{dr²}Φ
=
− ^{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_{o}² = 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_{o}² − 4β
log_{e}² r
− ^{4}/_{r} .
(147)/[1e,(5)]

Reduction to the standard model Coulomb law occurs when:

β = ^{1}/_{r}
(148)

when

ω_{r} = 0,
κ_{o}² = 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.

[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*,

ECEcontradictions.html .

[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.
**

(29.01.2008)
**Remarks on Evans' Web Note #100-Section 7: The Sagnac Effect
**

(25.01.2008)
**Remarks on Evans' Web Note #100-Section 6: SCR
**

(16.01.2008)
**Remarks on Evans' Web Note #100-Section 5: EM field
**

(08.01.2008)
**Remarks on Evans' Web Note #100-Section 4: The Aharonov Bohm effect
**

(05.01.2008)
**Remarks on Evans' Web Note #100-Section 3: Field and Wave equation
**

(01.01.2008)
**Remarks on Evans' Web Note #100-Section 2: Torsion and Bianchi identity
**

(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
**