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[itex]_{ijkl}[/itex]a[itex]^{i}[/itex]b[itex]^{k}[/itex]c[itex]^{l}[/itex] is orthogonal to [itex]\vec{a}[/itex], [itex]\vec{b}[/itex], and [itex]\vec{c}[/itex].

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

 

[gotmilk04]

So then ε[itex]_{ijkl}[/itex]a[itex]^{i}[/itex]a[itex]^{j}[/itex]= -ε[itex]_{jikl}[/itex]a[itex]^{i}[/itex]a[itex]^{j}[/itex]?

 

[fzero]

So then ε[itex]_{ijkl}[/itex]a[itex]^{i}[/itex]a[itex]^{j}[/itex]= -ε[itex]_{jikl}[/itex]a[itex]^{i}[/itex]a[itex]^{j}[/itex]?

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

[itex]\epsilon_{jikl} a^i a^j = \epsilon_{ijkl} a^j a^i .[/itex]

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.