Recently Evans discovered two papers of mine that concern the basis of ECE theory: His O(3)-Symmetry Law. He replied with a series of 'rebuttals' which are a mixture of polemics and confused verbal attempts of refutations *). Though in 2000 he had written a paper [2] starting with an elegant idea of deciding the question of Lorentz invariance of the O(3)-Symmetry Law by means of the invariance of the ED potentials. Evans result then was the assertion that everything is fine: The O(3)-Symmetry is Lorentz invariant.

However, that paper contains a spot where Evans *verbally* (without calculation)
concludes the desired invariance from the invariance of the ED potentials, ignoring the fact
that the calculation of the **B**-field components by curl-differentiation of the corresponding
vector potentials produces the wave number κ as factor in the transversal components
**B**^{(1)} and **B**^{(2)} while the
longitudinal component **B**^{(3)} remains independent of κ.
And the wave number κ is no invariant (Doppler effect).
Hence it becomes questionable whether the first of Evans' cyclic equations could remain invariant
where only the left hand side contains κ while the right hand side remains invariant.
So it is very easy to repair Evans' shortcoming. That was done in an arXiv-paper [4] the web version
of which is attached here containing the calculations appended in full detail.

*) Among his polemics Evans gives a hint to his book "The Enigmatic Photon, Vol. 4" at

http://www.atomicprecision.com/blog/2006/12/26/jackson-on-the-lorentz-transformation/

where equ. (3.111) is really equivalent to the result of the Lorentz transform applied to the transversal components

**B**^{(3)}**'** =
**B**^{(3)} , . . .
(3.110-3)

thus also
|**B**^{(3)}**'**| = |**B**^{(3)}|
which immediately leads to a *contradiction in Evans' book itself* as on p.5 we find the equation

|**B**^{(1)}| =
|**B**^{(2)}| =
|**B**^{(3)}| = *B*^{(0)}
(1.2a)

to obtain

*B*^{(0)}**'** =
|**B**^{(3)}**'**| =
|**B**^{(3)}| = *B*^{(0)}

in contrast to Evans' equ.

*B*^{(0)}**'** =
[^{(1−v/c)}/_{(1+v/c)}]^{½} *B*^{(0)} .
(3.111)

My conclusion is that the proof of the Lorentz invariance in "The Enigmatic Photon" is enigmatic itself.

The other hint to eq. (11.16) in J.D. Jackson's "Classical Electrodynamics" leads to the well-known Lorentz transform of coordinates. It's Evans' secret why.

O(3)-Symmetry Law

Abstract

In 1992 M.W. Evans proposed a so-called O(3) symmetry of electromagnetic fields by adding a
constant longitudinal ghost field to the well-known transversal plane em waves. Evans considered this symmetry
as a new law of electromagnetics. In 2000 he tried to show the Lorentz invariance of
O(3) symmetry of electromagnetic fields in this Journal [1]. However, this proof lacks from
a simple calculation error. As shall be shown below a correct calculation yields no Lorentz invariance.
This result is of importance as later on, since 2002,
this O(3) symmetry became the center of Evans' CGUFT which he recently renamed as ECE Theory.
A law of Physics must be invariant under admissible coordinate transforms, namely under Lorentz transforms.
Therefore, to check the validity of Evans' O(3)-symmetry law, we apply a longitudinal Lorentz transform
to Evans' plane em wave (the ghost field included). As is well-known from SRT and recalled here the transversal amplitude
decreases while the additional longitudinal field remains unchanged. Thus, Evans' O(3) symmetry
cannot be invariant under (longitudinal) Lorentz transforms: **Evans' O(3) symmetry
is no valid law of Physics.**

In the following text quotations from Evans' book [1] appear in
**black**
with equation labels (1.nn)
at the right margin.

The assertion of O(3) symmetry is a central concern of M.W. Evans' considerations since 1992: He claims
that each plane circularly polarized electromagnetic wave **B** is accompanied by a constant longitudinal
field **B**^{(3)}, a so-called "ghost field".

Evans considers a circularly polarized plane electromagnetic wave propagating in z-direction, cf. [1; Chap.1.2] . Using the electromagnetic phase

Φ = ω t − κ z , (1.38)

where κ = ω/c, Evans describes the wave relative to his complex circular basis [1; (1.41)]. The magnetic field is given by

**B**^{(1)}
= ^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ} ,
(1.43-1)

**B**^{(2)}
= ^{1}/_{sqr(2)} *B*^{(0)}
(**i**+*i***j**) e^{−iΦ} ,
(1.43-2)

**B**^{(3)}
=
*B*^{(0)} **k** ,
(1.43-3)

satisfying Evans' "cyclic O(3)-symmetry relations"

**B**^{(1)} × **B**^{(2)}
= *i**B*^{(0)}**B**^{(3)}* ,
(1.44-1)**B**^{(2)} × **B**^{(3)}
= *i**B*^{(0)}**B**^{(1)}* ,
(1.44-2)

**B**^{(3)} × **B**^{(1)}
= *i**B*^{(0)}**B**^{(2)}* .
(1.44-3)

Especially equ.(1.43-3) defines Evans' ghost field **B**^{(3)}
which is coupled by the relations (1.44) with the transversal **B** components given by the
two eqns.(1.43-1,2).
Evans' "**B Cyclic Theorem**" is the statement that each plane circularly polarized wave (1.43-1,2)
is accompanied by a longitudinal field (1.43-3), and the associated fields fulfil the Cyclic equations (1.44-1,2,3).
Evans considers this **O(3) hypothesis** as a **Law of Physics**.

In the article [2: p.14] M.W. Evans therefore tries to prove the Lorentz invariance of the O(3) hypothesis
(1.44) by referring to the invariance of the vector potential **A** under Lorentz transforms.
That is a good method obtaining the transform of ED fields if one calculates *correctly*.

So the reader should first check that the vector potentials of the transversal components
**B**^{(1)} and **B**^{(2)}
of the plane wave under consideration are given by

(1)
**A**^{(1)} =
^{1}/_{κ} **B**^{(1)} =
^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ}
,
**A**^{(2)} =
^{1}/_{κ} **B**^{(2)} =
^{1}/_{sqr(2)} *B*^{(0)}
(**i**+*i***j**) e^{−iΦ}

while the vector potential of the *longitudinal* field **B**^{(3)} is

(2)
**A**^{(3)} =
½ **B**^{(3)}× (x**i** + y**j**)
= ½ *B*^{(0)} (x**j** − y**i**)
.

The invariance of the vector potential **A**^{(3)}
yields the invariance of
**B**^{(3)}
and the factor *B*^{(0)}
in the eqs. (1.43) and (1.44)
for *longitudinal* Lorentz transforms, i.e. between
inertial frames K, K' where K' moves relative to K with velocity **v** | | **k**
and β = |**v**|/c.

What Evans has obviously ignored in [2]: Frequency ω and wave number κ are *no invariants*.
Under *longitudinal* Lorentz transforms we have the well-known Doppler effect:

(3)
ω' = sqr(^{1−β}/_{1+β}) ω ,
κ' = sqr(^{1−β}/_{1+β}) κ .

Therefore the invariance of the vector potentials doesn't transfer to the transversal field components
**B**^{(1)} and **B**^{(2)} .
We obtain from (1) taking into account the invariance of **A**

(4)
**B**'^{(1)}×**B**'^{(2)} =
κ'² **A**'^{(1)}×**A**'^{(2)}
=
^{κ'²}/_{κ²} κ²
**A**^{(1)}×**A**^{(2)}
=
^{1−β}/_{1+β}
**B**^{(1)}×**B**^{(2)} ,

i.e. the expression
**B**^{(1)}×**B**^{(2)}
at the left hand side
of (1.44-1) does not remain invariant while **B**^{(3)}
is invariant due to (2). Hence equ.(1.44-1),
if valid in the inertial system K, *cannot be valid* also in the inertial system K':
Evans' cyclical symmetry (1.44) is *not Lorentz invariant*. Thus,

References

[1]
M.W. Evans, *Generally Covariant Unified Field Theory,
the geometrization of physics*;
Web-Preprint,

http://www.atomicprecision.com/new/Evans-Book-Final.pdf

[2]
M.W. Evans, *On the Application of the Lorentz Transformation in O(3) Electrodynamics*

APEIRON Vol. 7 Nr.1-2, 2000, 14-16

http://redshift.vif.com/JournalFiles/Pre2001/V07NO1PDF/V07N1EV1.pdf

[3]
G.W. Bruhn and A. Lakhtakia, *Commentary on Myron W. Evans' paper
"The Electromagnetic Sector ..." *,

http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansChap13.html

[4]
G.W. Bruhn, *No Lorentz Invariance of M.W. Evans' O(3)-Symmetry Law*,

http://arxiv.org/pdf/physics/0607186

[5]
G.W. Bruhn, *No Lorentz property of M W Evans’ O(3)-symmetry law*,

**Phys. Scr. 74** (2006) 537–538

(A.1.1)
Φ = ω t − κ z
Þ
Ñ e^{iΦ} =
*i*e^{iΦ} ÑΦ =
− **k** *i*κ e^{iΦ}

The transversal components

(A.1.2)
**A**^{(1)} =
^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ}

Þ
**B**^{(1)} =
Ñ×**A**^{(1)} =
− ^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**)×Ñ e^{iΦ}
= ^{1}/_{κ}
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**)×*i*κ **k** e^{iΦ}
=
^{1}/_{sqr(2)} *B*^{(0)}
(**i**−*i***j**) e^{iΦ} =
κ **A**^{(1)}

(A.1.3)
**A**^{(2)} =
**A**^{(1)}*
Þ
**B**^{(2)} =
Ñ×**A**^{(2)} =
Ñ×**A**^{(1)}* =
**B**^{(1)}* =
κ **A**^{(1)}* =
^{1}/_{sqr(2)} *B*^{(0)}
(**i**+*i***j**) e^{−iΦ}

The longitudinal component (*B*^{(0)} = const assumed,
Ñx = **i**, Ñy = **j**)

(A.1.4)
**A**^{(3)} =
½ *B*^{(0)} (x**j** − y**i**)
Þ
**B**^{(3)} =
Ñ×**A**^{(3)} =
½ *B*^{(0)}(**i**×**j** − **j**×**i**)
= *B*^{(0)} **k** .

(A.1.5)
**A**^{(3)} invariant
Þ
*B*^{(0)} invariant
Þ
**B**^{(3)} invariant .

(A.2.1) Φ invariant Þ ωt − κz = Φ = Φ' = ω't' − κ'z'

where due to the velocity c of the wave fronts

(A.2.2) ω = κ c and ω' = κ' c

where t' = γ (t − v/c˛ z), z' = γ (z − vt),
β = v/c, γ = (1−β˛)^{−½} .

Coefficients matching in (A.2.1) yields the transformation rule

(A.2.3)
κ = κ' [(1+β)/(1−β)]^{½}.