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Show that this field is orthogonal to each vector field.

  • Thread starter gotmilk04
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  • #1
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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.
 

Answers and Replies

  • #2
fzero
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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]?
 
  • #3
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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]?
 
  • #4
fzero
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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.
 

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