Volume element as n-form

1. LAHLH

411
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

$$\epsilon\equiv\epsilon_{\mu_1\mu_2...\mu_n}dx^{\mu_1}\otimes dx^{\mu_2}\otimes...\otimes dx^{\mu_n} =\frac{1}{n!}\epsilon_{\mu_1\mu_2...\mu_n}}dx^{\mu_1}\wedge dx^{\mu_2}\wedge...\wedge dx^{\mu_n}$$

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

Formula (1.81) in Carroll's "Lecture Notes on General Relativity" (1997) tells you:

$$A\wedge B=A\otimes B-B\otimes A$$

So, what is it that causes you the problem?

3. LAHLH

411
Oh I was unaware of this formula.

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

Which I guess can be written $$(A\wedge B)_{\mu\nu} =(A\otimes B)_{\mu\nu}-(B\otimes A)_{\mu\nu}$$, leading to the equation you stated: $$(A\wedge B) =(A\otimes B)-(B\otimes A)$$