## 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)                 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:

### "... this works ok logically, but is it valid as physics??"

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:

### 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.

### 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,