# Calculation involving angular momentum

1. Aug 28, 2009

### serali

First of all, I'm sorry but I couldn't find a better title. And I'm sorry if I'm using the wrong forum.

I'm going through W. G. Dixon "Extended Bodies in general relativity" (Appearing in "Isolated Gravitating Systems in General Relativity", J. Ehlers)

To summarize the problem, we have an (antisymmetric) angular momentum tensor $$S^{\kappa \lambda}$$ and a timelike unit vector $$n^\kappa$$ such that $$n_\kappa S^{\kappa \lambda} =0$$

I'm trying to get
Code (Text):
$$S^{\kappa [ \lambda} S^{\rho ] \sigma} = \frac{1}{2}S^{\rho \lambda} S^{\kappa \sigma}$$
where square brackets denote antisymmetrization.

Dixon says "by $$n_\kappa S^{\kappa \lambda} =0$$, $$S^{\kappa \lambda}$$ has matrix rank at most 2 since it cannot have 3 in virtue of antisymmetry. Consequently $$S^{[\kappa \lambda} S^{\rho ] \sigma}$$ must be identically zero, which implies that ...." (above equation)

I can see that $$S^{\kappa \lambda}$$ has 3 linearly independent components, and (only by explicit calculation) showed that it has at most rank 2, but I was not able to show the rest.
Is it possible to show the first claim without calculation? And how should I proceed for the rest of the problem?