 ## Introduction

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  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  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(1) and B(2) that was pointed out in my papers  and . However, one equation before (equ. (3.110)) Evans states the transformation rule

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)

### I wonder how Evans will explain this contradiction - I cannot.

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.

## No Lorentz Invariance of M.W. Evans' 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 . 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  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) (iij) eiΦ ,                                                 (1.43-1)
B(2) = 1/sqr(2) B(0) (i+ij) eiΦ ,                                               (1.43-2)
B(3) = B(0) k ,                                                                       (1.43-3)

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

B(1) × B(2) = iB(0)B(3)* ,                                                         (1.44-1)
B(2) × B(3) = iB(0)B(1)* ,                                                         (1.44-2)
B(3) × B(1) = iB(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) (iij) eiΦ   ,   A(2) = 1/κ B(2) = 1/sqr(2) B(0) (i+ij) eiΦ

while the vector potential of the longitudinal field B(3) is

(2)                                                 A(3) = ½ B(3)× (xi + yj) = ½ B(0) (xj − yi) .

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 : 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

     M.W. Evans, Generally Covariant Unified Field Theory,
the geometrization of physics
; Web-Preprint,
http://www.atomicprecision.com/new/Evans-Book-Final.pdf

    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

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

    G.W. Bruhn, No Lorentz Invariance of M.W. Evans' O(3)-Symmetry Law,
http://arxiv.org/pdf/physics/0607186

    G.W. Bruhn, No Lorentz property of M W Evans’ O(3)-symmetry law,
Phys. Scr. 74 (2006) 537–538

## Appendix: The detailed calculations

### A.1 The vector potentials yielding the B components

(A.1.1)         Φ = ω t − κ z     Þ     Ñ eiΦ = ieiΦ ÑΦ = − k iκ eiΦ

The transversal components

(A.1.2)        A(1) = 1/κ 1/sqr(2) B(0) (iij) eiΦ

Þ     B(1) = Ñ×A(1) = − 1/κ 1/sqr(2) B(0) (iijÑ eiΦ = 1/κ 1/sqr(2) B(0) (iijiκ k eiΦ = 1/sqr(2) B(0) (iij) eiΦ = 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+ij) eiΦ

The longitudinal component (B(0) = const assumed, Ñx = i, Ñy = j)

(A.1.4)        A(3) = ½ B(0) (xj − yi)     Þ     B(3) = Ñ×A(3) = ½ B(0)(i×jj×i) = B(0) k .

(A.1.5)        A(3) invariant     Þ     B(0) invariant     Þ     B(3) invariant .

### A.2 The Lorentz transform of the wave number κ

(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.

### A.3 The Lorentz transform of Evans' first cyclic equation

The right hand side iB(0)B(3)* was shown above to be invariant. The left hand side transforms as was shown in equ. (4), i.e. is not invariant. Thus, Evans' first cyclic equation is not invariant.