Updated on June 13, 2006: Appendix

On the Lorentz Behavior of M.W. Evans'
O(3)-Symmetry Law

Gerhard W. Bruhn, Darmstadt University of Technology


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. Later on, since 2002, this O(3)-symmetry became the center of his 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 manuscript [1] appear in black with equation labels (1.nn) at the right margin.

The claim of O(3)-symmetry is a central concern of Evans' considerations since 1992. The reader will find a historical overview in [2; Sect. 5] written by A. Lakhtakia. Among a lot of papers Evans has written five books on "The Enigmatic Photon" that deal with the claimed O(3)-symmetry of electromagnetic fields.

In [1; Chap.1.2] Evans considers a circularly polarized plane electromagnetic wave propagating in z-direction. 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)], see also [3; Appendix 1] and [4]. The magnetic field is given by

B(1) = B(0)q'(1) = 1/sqr(2) B(0) (iij) eiΦ ,                                           
B(2) = B(0) q'(2) = 1/sqr(2) B(0) (i+ij) eiΦ ,                               (1.43)
B(3) = B(0) q'(3) = B(0) k ,                                                                

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

B(1) × B(2) = iB(0)B(3)* ,                                                                  
B(2) × B(3) = iB(0)B(1)* ,                                                         (1.44)
B(3) × B(1) = iB(0)B(2)* .                                                                  

Due to M.W. Evans the corresponding electric field is given by

E(1) = − E(0)/sqr(2) (ii+j) eiΦ ,                                                          

E(2) = E(0)/sqr(2) (iij) eiΦ ,                                                   (1.85)

E(3) = −iE(0) k .                                                                             

The relation between E(0) and B(0) is

E(0) = c B(0) ,                                                                 (1.87)

i.e. E(0) = B(0), if we assume c=1 below.

Due to B(2) = B(1)* and E(2) = E(1)* the complex fields (1.43) and (1.85) belong to the real fields

                                B = B(1) + B(2) + B(3) = Bx i + By j + Bz k
                                E = E(1) + E(2) + E(3)= Ex i + Ey j + Ez k .

Inserting of (1.43) and (1.85) and coefficient matching yields

(1)                 Bx = B(0) sqr(2) cos Φ ,         By = B(0) sqr(2) sin Φ ,             Bz = B(0) ,
(2)                 Ex = E(0) sqr(2) sin Φ ,         Ey = − E(0) sqr(2) cos Φ ,         Ez = E(0) .

Summing the equations in (1) with combination factors 1,+i and comparing with (1.43) yields

(3)                 (Bx + i By) (iij)= 2 B(1),         (Bxi By) (i+ij) = 2 B(2) .

and therefore for further use in rewriting of the first equation of (1.44)

(4)                 B(1) × B(2) = ½ (Bx + i By) (Bxi By) i k = ½ (Bx˛ + By˛) i k .

while the last equations of (1.43) and (1) yield

B(0)B(3)* = ik B(0)˛ = ik Bz˛ .

Thus, one of "Evans' cyclic symmetry relations", the first rule of (1.44), is equivalent to

(5)                                                                 ½ (Bx˛ + By˛) = Bz˛ .

The first two equations of (1.43) and (1.85) describe a circularly polarized plane wave propagating in z-direction. The third equations, however, contain Evans' O(3)-Law from 1992, saying that the well-known plane wave is always accompanied by a constant longitudinal magnetic "ghost" field B(3), the size of which - this is important - is given by the third equation of (1.43), or by the first equation of (1.44), which is in real formulation our equation (5).

If Evans' O(3)-Law were a Law of Physics then it must be invariant under the admissible coordinate transforms, i.e. under Lorentz transforms.

Therefore we consider the wave as observed from other coordinate systems S' in constant motion v = v k relative to our original Cartesian coordinate system S. The transformation rules for the electromagnetic field are well-known (β=v/c, γ=1/sqr(1−β˛) and c=1 assumed)):

(6)                                 Ex' = γ (Ex − β By),         Ey' = γ (Ey + β Bx),         Ez' = Ez ,
(7)                                 Bx' = γ (Bx + β Ey),         By' = γ (By − β Ex),         Bz' = Bz .

We shall check the first rule of Evans' O(3) symmetry Law (1.44) in our equivalent real formulation (5). Therefore we are now going to transform the wave (1-2) into the coordinate frame S' by means of the transformation rules (6-7) to obtain

(8)       Bx' = (1−β)γ B(0) sqr(2) cos Φ = (1−β)γ Bx ,       By' = (1−β)γ B(0) sqr(2) sin Φ = (1−β)γ By .       Bz' = Bz ,

which shows that due to

(9)                ½ (Bx'˛ + By'˛) = 1−β/1+β ½ (Bx˛ + By˛) = 1−β/1+β Bz˛ = 1−β/1+β Bz'˛ < Bz'˛         (0 < β < 1)

Evans' first O(3)-symmetry relation (5) in S', the equation ½ (Bx'˛ + By'˛) = Bz'˛, is not fulfilled:

Evans' cyclical O(3)-symmetry is not Lorentz invariant and hence no law of Physics.


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

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

[3]    G.W. Bruhn, Comments on Myron W. Evans’ "REFUTATION"

[4]    V.V. Dvoeglazov, Comment on the 'Comment on the Longitudinal Magnetic Field ...’
        by E. Comay ...

[5]    M.W. Evans, On the Application of the Lorentz Transformation in O(3) Electrodynamics
                APEIRON Vol. 7 Nr.1-2, 2000, 14-16


In his article [5: p.14] M.W. Evans tries to prove the Lorentz invariance of the O(3) hypothesis 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 first the reader should check that the vector potentials of the transversal components B(1) and B(2) of the plane wave under consideration are given by

(A1)                 A(1) = 1/κ B(1) = 1/κ 1/sqr(2) B(0) (iij) eiΦ   ,   A(2) = 1/κ B(2) = 1/κ 1/sqr(2) B(0) (i+ij) eiΦ

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

(A2)                                                 A(3) = ½ B(3)× (xi + yj) .

The invariance of A(3) yields the invariance of B(3) as well.

What Evans has ignored: Frequency ω and wavenumber κ are no invariants. Under longitudinal Lorentz transforms we have the well-known Doppler effect:

(A3)                                 ω' = sqr(1−β/1+β) ω   ,   κ' = sqr(1−β/1+β) κ .

Therefore we obtain from (A1) taking into account the invariance of A

(A4)                 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 does not remain invariant while B(3), as was shown above, remains invariant. Hence the first equation of (1.44), if valid in the inertial system S, cannot be valid also in the inertial system S'. Thus, Evans' cyclical symmetry (1.44) is not Lorentz invariant. Thus,

M.W. Evans article [5] is wrong as well.