An AIAS dissident and attentive reader of [2] has sent me an inquiry about Evans' definition of a "3-index, totally antisymmetric unit tensor in 4D" given in [1, Eq.(51)]. He proposed a representation of Evans' "3-index Î-tensor in 4D" as a contracted 4D Levi-Civita Î-tensor as follows:
(1) Îijk := Cs Îsijk
where the composition vector C is given by its components
(2) (Cs | s=0,1,2,3) := (+1,−1,+1,−1) .
The proof of the agreement with Evans' listing in [1, Eq.(51)] is left to the reader.
Hint: Consider the cases
0Ï{i,j,k},
1Ï{i,j,k},
2Ï{i,j,k} and
3Ï{i,j,k}
separately to obtain from Eqs.(1-2)
Îijk = Î0ijk if 0Ï{i,j,k},
Îijk =
Îi1jk
if 1Ï{i,j,k},
(3)
Îijk =
Îij2k
if 2Ï{i,j,k},
Îijk =
Îijk3
if 3Ï{i,j,k},
Compare this result with Evans' appropriately reordered listing.
My correspondent wrote:
Indeed, that's the question, better, it's a question of tensor calculus.
Tensor calculus means that Equ. (1) must transform covariantly under say Lorentz transforms.
(1') Î'i'j'k' = C's' Îs'i'j'k' .
However, as can be seen from (2), the components of the composition vector C will be transformed into some other vector components C's' which disagree with (2) in general. Thus, under Lorentz transforms Evans' "3-index totally antisymmetric unit Î-tensor" in 4D will lose its form given by [1, Eq.(51)].
In other words:
In that sense the defining Eqs. (1-2) of Evans' "3-index Î-tensor in 4D" don't belong to tensor calculus.
[1] M.W. Evans, Geodesics and the Aharonov-Bohm effect in ECE theory,
http://www.aias.us/documents/uft/a56thpaper.pdf
[2] G.W. Bruhn, Comments on Evans' Duality,
http://www2.mathematik.tu-darmstadt.de/~bruhn/EvansDuality.html