APPENDIX 10: REBUTTAL OF G. BRUHN'S COMMENTS ON THE
LORENTZ COVARIANCE OF THE B CYCLIC THEOREM.

July 19, 2008
_{(Quotations from Evans' original text in} black, comments in blue)

It is well known that special relativity demands that any equation of physics be covariant under the Lorentz transformation. It has been shown elsewhere {8} that the B Cyclic Theorem is correctly Lorentz invariant. This is self-evident from the fact that Theorem is, within Lorentz invariant phase factors, the frame of reference itself, and it is well known that a frame of reference is Lorentz covariant. Bruhn {14} incorrectly asserts that a frame of reference must be Lorentz invariant. . . .

This is a distortion of facts. And Evans does not specify the spot of that alleged flaw. − However, let's see below how Evans attempts to prove his Theorem which asserts that if the cyclic relations (10.1) are valid in a inertial frame S then the primed version of (10.1) is valid also in another inertial frame S' moving at constant velocity v (|v|<c) relative to the un-primed frame. Therewith the transformation rules of the electromagnetic field for that Lorentz boost must be taken into account. Then to prove or disprove the Theorem is merely a matter of patience and skill [2].

However, an impatient reader needs not to check all the following tedious calculations in detail. Dr. Evans has provided a very surprising conclusion just before eq.(10.22) which make all further efforts superfluous.

. . .

The B Cyclic Theorem is {8}:

                B⁽¹⁾×B⁽²⁾ = i B^(o) B⁽³⁾                                 (10.1)
                                et cyclicum

where the plane wave magnetic flux densities are:

                B⁽¹⁾ = B^(2)* = ^B(o)/_2^½ (i−ij) e^iΦ
                B⁽³⁾ = B^(3)* = B^(o) k.                                 (10.2)

Here, the electromagnetic phase factor is

                exp(iΦ) = exp(i(ωt−κz))                                 (10.3)

where ω is the angular frequency at instant t, κ the wave number at position z. it is well known that the phase factor is Lorentz invariant. The frame of reference being used is the complex circular basis {8}

                e⁽¹⁾×e⁽²⁾ = i e^(3)*                                 (10.4)

whose unit vectors are related to the Cartesian unit vectors by

                e⁽¹⁾ = e^(2)* = ¹/_2^½ (i−ij) e^iΦ
                e⁽³⁾ = e^(3)* = k.                                 (10.5)

where * denotes ''complex conjugate''.

The general rule {13} (ref. {13} is S.M. Carroll's book Spacetime and Geometry, eqs.(2.17-19). The same can be found in Carroll's Lecture Notes [5], eqs.(2.11-13)) for the covariant transformation of a basis vector is well known, and is:

                ê_(μ') = (^∂xμ/_∂x^μ') ê_(μ)                                 (10.6)

                ∂_μ' = ^∂xμ/_∂x^μ' ∂_μ                                 (Carroll [5] (2.11))

where

                x^μ = (ct, x, y, z)                                     (10.7)

and where the prime denotes a different frame of reference moving arbitrarily with respect to the original (unprimed) frame of reference. The rule is found in innumerable textbooks and shows that basis elements such as unit vectors are changed under the transformation. In other words they are COVARIANT. The Lorentz transform is a special case:

                ê_(a') = Λ_a'^a(x) ê_(a)                                     (10.8)

The complete vector field {13}:

                V = V^μ ê_(μ)                                                 (10.9)

is invariant under the transformation:

                V = V^μ ê_(μ) = V^μ' ê_(μ')                                 (10.10)

                V^μ ∂_μ = V^μ' ∂_μ                                     (Carroll [5] (2.12))

Therefore the rule for general coordinate transformation {13} is:

                V^(μ') = (^∂xμ'/_∂x^μ) V^(μ)                                 (10.11)

                V^μ' = ^∂xμ'/_∂x^μ V^μ                                 (Carroll [5] (2.13))

The Lorentz boost considers a frame of reference moving at velocity in an axis with respect to another frame. If this is the Z axis then:

                V = V_z k = V_z' k'                                 (10.12)

from eq. (10.10) In general the Lorentz transformation produces the result

                i × j = k
                        ↓                                                         (10.13)
                i' × j' = k'

in the Cartesian basis . . .

That's a mere claim, no proof! What is required here for the application of the eqs. (10.6) and (10.11) are the transformation rules of coordinates, i.e. x^μ = x^μ(x^μ') for the eq.(10.6) and x^μ' = x^μ'(x^μ) for the eq.(10.11). These well known eqs. using x instead of z can be found at Wikipedia. Hence we have the coordinate transformation to be used here:

                t' = γ (ct−βz)/c,         z' = γ (z−vt)         (to be supplied by x' = x and y' = y)

where β = v/c and γ = (1−β²)^−½. Hence for

                x⁰ = ct, x¹ = x, x² = y, x³ = z         and         x^0' = ct', x^1' = x', x^2' = y', x^3' = z',

we obtain the transformation x^μ' = x^μ'(x^μ)

                x^0' = γ (x⁰− β x³),         x^1' = x¹,         x^2' = x²,         x^3' = γ (x³− β x⁰),         (L)

and the inverse transformation x^μ = x^μ(x^μ') is

                x⁰ = γ (x^0'+ β x^3'),         x¹ = x^1',         x² = x^2',         x³ = γ (x^3'+ β x^0'),         (L')

Now we can check by calculation Evans' eq. (10.12) where he asserts k | | k'. This is evidently wrong as can be seen by the well known Lorentz transformation graph e.g. at p.7 of Carroll's Lecture notes [5]:

Evans has not realized that the vectors k and k' are 4-dimensional.

The same holds for Evans' next assertion

. . . or:

                e⁽¹⁾ × e⁽²⁾ = i e^(3)*
                                ↓                                                         (10.14)
                e^(1)' × e^(2)' = i e^(3)*'

in the complex circular basis. These results follow directly from Eq.(10.8)

This is wishful thinking only: The vectors e⁽¹⁾ and e⁽²⁾ remain unchanged as being linear combinations of the invariant vectors i and j, but not so the vector e⁽³⁾ = k which is NOT equal to k' = e^(3)', not even e^(3)' | | e⁽³⁾.

In the B Cyclic Theorem (10.1) the following definitions are used:

                B⁽¹⁾ = B^(2)* = B^(o) e^iΦ e⁽¹⁾
                                                                                (10.15)
                B⁽³⁾ = B^(3)* = B^(o) k.

It is well known that the electromagnetic phase is Lorentz invariant, because is a scalar, so:

                e^iΦ = e^iΦ'.                                                 (10.16)

Therefore we need only consider the Lorentz covariance of B^(o) and the frame itself (the unit vectors). If the boost takes place in z the complete vector field to be considered is:

                B⁽³⁾ → B⁽³⁾                                                 (10.17)

Probably a prime is missing at the right hand side.

This means that:

                B^(o) k = B^(o)' k'
                                                                                                (10.18)
                B^(o) i = B^(o) i , B^(o) j = B^(o) j .

The first line (10.18)₁ does not hold since as we have seen the unit vectors k and k' are not parallel. The second line is trivial, perhaps there are missing primes. Whatsoever, the eqs. (10.18) do not describe the transformation conditions valid for the magnetic flux that were derived in the eqs.(5-9) in [2]. Especially the following equ.(10.19) is wrong: Due to [2], eq.(7) the quantity B^(o) must be invariant: B^(o)' = B^(o). And just this discrepancy will lead Evans to a very curious conclusion below.

It has been shown that in this case:

                B^(o)' = (^1−v/c/_1+v/c)^½ B^(o)                                                 (10.19)

(Ref.{8}, eq.(3.111)), where v is the velocity or the primed frame with respect to the unprimed frame. The Cartesian frame after Lorentz transformation is:

                i×j = k'                                                                 (10.20)

This is not true: we have

                i×j = k ≠ k' = i'×'j'                                                 (10.20')

where × and ×' are different cross products defined in the different subspaces of spacetime spanned by {i,j,k} and {i',j',k'} respectively.

and the complex circular frame after Lorentz transformation is:

                e⁽¹⁾ × e⁽²⁾ = i e^(3)*'                                                 (10.21)

Wrong again! We have

                e⁽¹⁾ × e⁽²⁾ = i e^(3)* ≠ i e^(3)*' = e^(1)' ×' e^(2)'                 (10.21')

The frames are distorted, indicating the length contraction, but the overall form of the cyclic relations (10.20) and (10.21) is the same as the original cyclic relations (10.1) and (10.4). This is what is meant by COVARIANCE.

Furthermore, as indicated in ref.(8), the velocity v is zero, because the fields are already propagating at c, and cannot propagate any faster. Thus:

                B^(o)' = B^(o)                                                 (10.22)

The above is elementary and well known. Why should such an elementary error by Bruhn be published.

The remark before eq.(10.22) tops all! After having discussed in full the effect of a Lorentz boost on Evans' Cyclical Theorem he tells us that all was a joke only: The velocity v of the primed frame of reference relative to the unprimed frame must be ZERO to yield the validity of his Covariance Theorem for his Cyclical basis:

Evans' ''Covariance Theorem'' is valid only for Lorentz boosts of velocity v=0.