Comments on Chap.18 of Evans' GCUFT Book

Gerhard W. Bruhn, Darmstadt University of Technology

(Quotations from Evans in black)

[1, Chapter 18] deals with the purpose of deriving "O(3) electrodynamics (i.e. in Minkowskian gravitation-free spacetime) in detail from the generally covariant unified field theory".

If Evans would succeed in proving O(3) symmetry from the general field theory, then he could argue that his O(3) hypothesis would be invariant (covariant) under local coordinate transforms. However, it is well known that the O(3) symmetry is not invariant under local Lorentz transforms [5],[6].

The chapter contains a very strange statement about the general solution of the homogeneous Maxwell equations in GCUFT:

Each solution (B,E) is decomposable into the sum of three pairwise orthogonal plane waves (B(a),E(a)) (a = 1,2,3),

(1)                                 B = B(1) + B(2) + B(3) ,                 E = E(1) + E(2) + E(3) ,

the third of which is constant with vanishing electric field. (cf (18.69) below)

However, everybody who once dealt with Maxwell solutions knows that things are not so easy as M.W. Evans believes. Already the case of the third wave (B(3),E(3)) travelling as plane wave in x- or y-direction cannot be excluded and would not contradict the general field equations while the above decomposition theorem were violated.

Let's follow Evans' considerations first: Evans uses complex representations denoting the conjugate complex by the suffix *, so especially

(2)                                 B(2) = B(1)* ,                 E(2) = E(1)* ,

Referring to "geometric duality" (?) Evans assumes the coupling

                                B(1) = i E(1)/ c ,         E(1) = − ic B(1)                                 (18.53)

to fulfil the homogeneous Maxwell equations

                                Ñ · B(1) = 0 ,         Ñ · E(1) = 0 ,
                                B(1)/∂t + Ñ × E(1) = 0                                                 (18.54)
                                Ñ × B(1)1/ E(1)/∂t = 0

The electric and magnetic fields for index (1) can be expressed as the well known transverse plane waves

                                E(1) = E(o) (ii j) eiΦ/sqr(2)
                                B(1) = B(o) (i i + j) eiΦ/sqr(2)

where Φ denotes the phase factor ωt−κz, ω/κ = c. The vectors i, j, k are the usual orthonormal spatial basis of an inertial frame of the Minkowskian spacetime. Instead of which Evans believes the "complex cyclical basis"

                                e1 := (ii j)/sqr(2) ,                 e2 := e1* = (i + i j)/sqr(2) ,                 e3 := k

to be more advantageous.

By considering the conjugate complex of the eqns. (18.53-55) we obtain the analogue for the plane wave (B(2),E(2)) .

                                B(2) = − i E(2)/ c ,         E(2) = ic B(2)                                 (18.60)

and the homogeneous Mexwell equations

                                Ñ · B(2) = 0 ,                 Ñ · E(2) = 0 ,
                                B(2)/∂t + Ñ × E(2) = 0                                                 (18.61)
                                Ñ × B(2)1/ E(2)/∂t = 0

satisfied by

                                E(2) = E(o) (i + i j) eiΦ/sqr(2)
                                B(2) = B(o) (− i i + j) eiΦ/sqr(2)

Finally: The field equations of index (3) are fond from the geometrical duality

                                B(3) = i E(3)/ c ,         E(3) = − ic B(3)                                 (18.67)

and are

                                Ñ · B(3) = 0 ,                 Ñ · E(3) = 0 ,
                                B(3)/∂t + Ñ × E(3) = 0                                                 (18.68)
                                Ñ × B(3)1/ E(3)/∂t = 0

with the trivial solutions (B(o) = const Î R)

                                B(3) = B(o) k ,
                                E(3) = − ic B(3) ,                                                                 (18.69)
                                Re (E(3)) = 0 .

Evans writes:

The fields of index (3) are missing entirely from Maxwell Heaviside field theory (spacetime with no torsion) and are the fundamental spin fields of general relativity (spacetime with torsion).


These duality relations, field equations and fields of O(3) electrodynamics all follow from the fundamental definition (18.32)

                                d Ù Fa = μo ja .                                                                 (18.32)


                                a,b,c = (1),(2),(3) .                                                                 (18.19)

That's a distortion of facts and only half the truth: Of course, the Maxwell equations admit an additional "longitudinal" solution (18.69), i.e. a wave travelling in z-direction which means B(3) = B(3)(z,t) || k and E(3) = E(3)(z,t) || k. Under this assumption only constant solutions of the Maxwell eqns. (18.68) are possible. The special form of the solution (18.69), i.e. the size of both B(3) and E(3) is Evans' additional assumption of cyclical symmetry, which is NOT implied by the field equations.

It is true and well known that in the homogeneous case ja = 0 the Maxwell equations (18.54), (18.61) and (18.68) can be derived from the general Maxwell equations

                                d Ù F = 0                 and                 d Ù *F = 0,

                                (where * denotes the Hodge duality operator)

[2, (1.91) and (1.93)], [3, (2.85) and (2.87)], [4, (10.10*)] by replacing F with Fa, but nothing else. Especially neither the "duality" eqns. (18.53), (18.60) and (18.67) nor the specific form (18.55), (18.62) and (18.69) of the solutions are prescribed by the field equations. That confirms the NON-Lorentz invariance of the O(3) symmetry [5],[6]. Evans' statements above are only hopeful wishing which is nowhere confirmed by Evans' calculations. Thus,

we cannot confirm that O(3) electrodynamics is implied by the field equation (18.32).



[2] S.M. Carroll, Lecture Notes on General Relativity, arXiv 1997

[3] S.M. Carroll, Spacetime and Geometry, Addison Wesley 2004

[4] E. Zeidler in Teubner Taschenbuch der Mathematik Teil II, 8.Aufl. 2003

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

[6] G. W. Bruhn, On the Non-Lorentz-Invariance of M.W. Evans’ O(3)-Symmetry Law,