Levi-Civita Symbol and indefinite metrics

  • #1
112
20
Let ##(M,g)## be an ##n##-dimensional pseudo-Riemannian manifold of signature ##(n_+, n_-)## and define the Levi-Civita symbol via
$$\varepsilon_{i_1 \dots i_n} \, \theta^{i_1 \dots i_n} = n! \, \theta^{[1 \dots n]} =
\theta^1 \wedge \dots \wedge \theta^n$$
where ##\theta^1, \dots, \theta^n## is a basis of covectors and the brackets denote the antisymmetrization operation.
With this definition I was able to prove the following formula for the determinant of a tangent space endomorphism (i.e. a "matrix") ##A##
$$\det A = n! \, A^{1}{}_{[1} \cdots A^{n}{}_{n]}.$$
My first question is: How do I compute
$$\varepsilon_{i_1 \dots i_n} \varepsilon^{i_1 \dots i_n} \equiv
\varepsilon_{i_1 \dots i_n} \, g^{i_1j_1} \cdots g^{i_n j_n}\, \varepsilon_{j_1 \dots j_n}$$
in a formal manner?
How do I compute the other common identities for the Levi-Civita Symbol like
$$\varepsilon_{i_1 \dots i_k j_{1} \dots j_{n-k}} \varepsilon^{j_{1} \dots j_{n-k} l_1 \dots l_k} = \, ? $$

I have browsed loads of differential geometry books, but none do this seemingly basic thing explicitly.
 

Answers and Replies

  • #2
112
20
Noone has any idea? Actually, I was trying to prove this to understand the Hodge operator.

Does anyone have a good reference?

The Levi-Civita symbol as defined here is just the standard one:

$$\varepsilon_{i_1 \dots i_n} = n! \, \delta^{[1}_{i_1} \cdots \delta^{n]}_{i_n}$$
 
  • #3
robphy
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Have you seen the determinant expressions in http://en.wikipedia.org/wiki/Levi-Civita_symbol ?
Are you specifically looking for something that uses metrics with general signatures (so there may be factors of (-1) for each timelike direction)?

Look at the google book page from this google search: de felice alternating tensors
 
Last edited:

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