# Antisymmetrization leads to an identically vanishing tensor

1. Jul 19, 2009

### 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?

2. Jul 19, 2009

### tiny-tim

Hi jason12345!

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

3. Jul 19, 2009

### jason12345

Re: antisymmetrisation

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

4. Jul 19, 2009

### 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.

5. Jul 20, 2009

### jason12345

Re: antisymmetrisation

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.