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Need clarification on the product of the metric and Levi-Civita tensor

  1. May 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Hi all, I'm having trouble evaluating the product [itex]g_{αβ}ϵ^{αβγδ}[/itex]. Where the first term is the metric tensor and the second is the Levi-Civita pseudotensor. I know that it evaluates to 0, but I'm not sure how to arrive at that.

    3. The attempt at a solution
    My first thought process was that every permutation will include at least 2 indices which are equal, (because every permutation will have at least 2 zeroes) which makes every term zero. That seems too trivial however. Sorry if this sounds totally nonsensical, but I haven't been able to find a resource that really clarifies this.
  2. jcsd
  3. May 4, 2014 #2
    I think you are close to the right answer, but I can't tell because your language is a little too vague. Permutations in which indices? The summation occurs only over the α and β, so these are the only indices you need to worry about. Recall as well that g_{αβ} is symmetric, and if ε^{αβγδ} ≠ 0, then ε^{αβγδ} = -ε^{βαγδ}. Also, ε^{αβγδ} = 0 if any of the α,β,γ,δ are equal.
  4. May 5, 2014 #3
    That's very simple. The metric is symmetric and the Levi-Civita is anti-symmetric.
    gαβ = gβα, and
    εαβγδ = -εβαγδ
    gαβεαβγδ = gβα(-εβαγδ) = - gαβεαβγδ
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