Antisymmetrization leads to an identically vanishing tensor

[jason12345]

This comes from Andersons's Principles of Relativity Physics:

"Of course, for fifth- or higher-rank tensors antisymmetrization leads to an
identically vanishing tensor"

But I don't understand why, even if it's "of course". So can someone show me why?

 

[tiny-tim]

Hi jason12345!

Because any tensor A_abcde has elements which look like A_xyztx (or worse), and if A is antisymmetric, then this must be zero.

 

[jason12345]

Hi jason12345!

Because any tensor A_abcde has elements which look like A_xyztx (or worse), and if A is antisymmetric, then this must be zero.

Antisymmetrising A_abcde means writing it as 1/5!(A_abcde + ... - A_bacde -.. ) so that the indices abcde are merely permutated, whereas you have A_xyztx which has indices x repeated. Why did you change the labelling from abcde to xyz and then t?

Thanks for your interest.

 

[tiny-tim]

Because the only candidates for the indices for the elements of the matrix are x y z and t (or 1 2 3 and 4, or whatever the four basis elements are) …

each element of the matrix has to have each of a b c d and e equal to x y z or t.

 

[jason12345]

Because the only candidates for the indices for the elements of the matrix are x y z and t (or 1 2 3 and 4, or whatever the four basis elements are) …

each element of the matrix has to have each of a b c d and e equal to x y z or t.

Thanks, I understand what you're saying now :), although I still say it isn't obvious since Anderson was defining Tensors as general geometrical objects upto this point, it seems, rather than applying them to the space-time manifold.