 ## APPENDIX 2: PROOF OF THE LORENTZ INVARIANCE OF THE B CYCLIC THEOREM.

### Gerhard W. Bruhn, Darmstadt University of Technology

July 20, 2008
(Quotations from Evans' original text in

One of the most serious errors by Hehl is his uncritical citation of a claim by Bruhn concerning the B Cyclic Theorem {1-12}, available for fifteen years. This alone brings into serious doubts Hehl's impartiality. . . .

B(1) = B(o)/2½ (iij) eiΦ                                 (2.1)

B(2) = B(o)/2½ (i+ij) eiΦ                                 (2.2)

B(3) = Bo k                                                     (2.3)

then

B(1)×B(2) = i B(o) B(3)                                   (2.4)
et cyclicum

is Lorentz invariant. . . .

The proof of the Lorentz invariance is simple. . . .

Under a Lorentz transformation the unit vectors i, j and k do not change, and so the B Cyclic Theorem is Lorentz invariant, Q.E.D. Bruhn falsely claims that this is not so.

This is an ambiguous statement: Of course, vectors are geometrical objects in 4-dimensional spacetime and do not change. What changes under a Lorentz transform is the frame of reference given by its basis vectors i, j, k and a unit vector h in the time axis. Under a Lorentz transform these basic vectors are replaced with other vectors h', i', j', k' the relation of which to the former basis vectors is of interest:

For a Lorentz boost in the z axis we have i' = i, j' = j, however k' = γ (k + βh) and h' = γ (h + βk) where γ = (1−βý)−½ and β = v/c.

Therefore the reference vectors h and k change to other vectors h' and k'. In this sense Evans' statement is wrong. However, Evans continues after that concluding Q.E.D. What for???

Since he feels that the behavior of the Cyclical relations must be derived from the transformation of the magnetic flux.

The Lorentz transformation of a magnetic field is found in any textbook to be:

B' = γ (B1/ v×E)                                                 (2.7)

where v is a velocity and where:

γ = (1−vý/cý)−½.                                                 (2.8)

The Lorentz transform of an electric field is:

E' = γ (E + v×B)                                                 (2.9)

It is seen that i, j and k do not change, they are the same for B' and B and for E' and E. The B(o) factor changes, but is cancelled out on both sides of Eq.(2.4) to give Eq.(2.5). This was first shown nearly a decade ago{1-12} but was ignored by both Bruhn and Hehl. Bruhn's unrefereed websites consist entirely of trivially erroneous comments such as this, and Hehl cites these trivial errors without citing the correct rebuttals, long available on www.aias.us. This seriously misrepresents ECE theory.

This argumentation is dubious. The behavior of the electromagnetic field vectors is somewhat more complicated: The eqs. (2.7) and (2.9) are incomplete as giving only the transformation of the transversal parts (^v). The longitudinal parts ( | | v) are missing. And just the missing components are most important:

It is well known from the textbooks that the size of the longitudinal components is not affected by the Lorentz boost: The size of the longitudinal component is given by the cofactor B(o) of k in eq.(2.3), and by the cofactor B(o)' of k' in the primed system. Therefore we have:

B(o)' = B(o)

However, since ''nearly a decade'' Evans repeats the following equation (also as eq.(10.19) in Appendix 10 of this ''Annalen'' submittal ):

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

In Appendix 10 of ref. he concludes from this evident discrepancy between both equations that v must be zero !!! And gives an explanation for this that tops all:

Evans' original explanation: ''. . . the velocity v is zero, because the fields are already propagating at c, and cannot propagate any faster.''

Did he never take notice of the relativistic addition theorem of velocities in case that one of the summands is c?

w(u,v) = u+v/1+uv/cý         Þ         w(c,v) = c+v/1+cv/cý = c .

### References

 M.W. Evans, APPENDIX 10: REBUTTAL OF G. BRUHN'S COMMENTS
ON THE LORENTZ COVARIANCE OF THE B CYCLIC THEOREM
,
Part of web-paper #89,
http://www.aias.us/documents/uft/a89thpaper.pdf

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

 G.W. Bruhn, No Lorentz Property of MW Evans' O(3)-Symmetry Law,
Physica Scripta 74 2006, pp.1-2

 G.W. Bruhn, On the Non-Lorentz-Invariance of M.W. Evans O(3)-Symmetry Law,
Foundations of Physics, 38, 1, pp.3-6