##
Evans' "3-index, totally antisymmetric unit tensor"

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

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)
Î_{ijk} :=
C^{s} Î_{sijk}

where the composition vector C is given by its components

(2)
(C^{s} | 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)

Î_{ijk} =
Î_{0ijk}
if 0Ï{i,j,k},

Î_{ijk} =
Î_{i1jk}
if 1Ï{i,j,k},

(3)

Î_{ijk} =
Î_{ij2k}
if 2Ï{i,j,k},

Î_{ijk} =
Î_{ijk3}
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'} Î_{s'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:

### Evans' "3-index, totally antisymmetric unit tensor" in 4D does not transform *covariantly*,
it is NO TENSOR.

In that sense the defining Eqs. (1-2) of Evans' "3-index
Î-tensor in 4D" don't belong to tensor calculus.

### The *missing tensor property* of Evans'
"3-index Î-tensor in 4D"
is fatal for Evans' ECE theory as was listed in [2].

### References

[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://www.mathematik.tu-darmstadt.de/~bruhn/EvansDuality.html