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.