Show that this field is orthogonal to each vector field.

[gotmilk04]

Homework Statement

If a, b, and c are any three vector fields in locally Minkowskain 4-manifold, show that the field ε_{ijkl}a^{i}b^{k}c^{l} is orthogonal to \vec{a}, \vec{b}, and \vec{c}.

Homework Equations

The Attempt at a Solution

I know I have to show that multiplying the field by each individual vector field equals 0, but I don't know how to go about doing this.

 

[fzero]

The tensor \epsilon_{ijkl} is totally antisymmetric. In particular, \epsilon_{ijkl}=-\epsilon_{jikl}. What does that imply about \epsilon_{ijkl}a^i a^j?

 

[gotmilk04]

So then ε_{ijkl}a^{i}a^{j}= -ε_{jikl}a^{i}a^{j}?

 

[fzero]

So then ε_{ijkl}a^{i}a^{j}= -ε_{jikl}a^{i}a^{j}?

Yes, but also note that we can swap the indices that we're summing over:

\epsilon_{jikl} a^i a^j = \epsilon_{ijkl} a^j a^i .

You might want to do this in steps if it's not completely obvious (first change i to m, j to n, then n to i, m to j).

After you figure it out, put it all back together in the expression that you started with.