Volume element as n-form

  1. Hi,

    I'm reading Sean Carroll's text, ch2, and believe I understood most of the discussion on Integration in section 2.10. However in equation 2.96, he states that

    [tex] \epsilon\equiv\epsilon_{\mu_1\mu_2...\mu_n}dx^{\mu_1}\otimes dx^{\mu_2}\otimes...\otimes dx^{\mu_n} =\frac{1}{n!}\epsilon_{\mu_1\mu_2...\mu_n}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge...\wedge dx^{\mu_n}[/tex]

    I don't quite understand this equality. For example just taking n=2, the LHS is [tex] dx^0\otimes dx^1-dx^1\otimes dx^0[/tex] (which isn't zero because tensor products don't commute). Where on the RHS one would have [tex] \frac{1}{2}\left(dx^0\wedge dx^1-dx^1\wedge dx^0\right)=dx^0\wedge dx^1[/tex], by the antisymmetry of the wedge product.

    So I'm at a loss to understand this part of 2.96 despite understand the following lines.

    Thanks alot for any replies.
  2. jcsd
  3. Formula (1.81) in Carroll's "Lecture Notes on General Relativity" (1997) tells you:

    [tex]A\wedge B=A\otimes B-B\otimes A[/tex]

    So, what is it that causes you the problem?
  4. Oh I was unaware of this formula.

    Do you mean (1.80) in this edition of Carroll? namely [tex] (A\wedge B)_{\mu\nu}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}[/tex]

    Which I guess can be written [tex] (A\wedge B)_{\mu\nu} =(A\otimes B)_{\mu\nu}-(B\otimes A)_{\mu\nu}[/tex], leading to the equation you stated: [tex] (A\wedge B) =(A\otimes B)-(B\otimes A)[/tex]
  5. Yes - that is another way of writing it.
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