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 Exercise [19.05] 
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Joined: 26 Mar 2010, 04:39
Posts: 109
Post Exercise [19.05]
Here is exercise 19.5


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24 Jun 2010, 01:31

Joined: 12 Jul 2010, 07:44
Posts: 154
Post Re: Exercise [19.05]
Note that the sign change depends on the Lorentzian signature of the metric because

\epsilon^{abcd}\varepsilon_{abcd} \equiv 4!

but

\varepsilon^{abcd}\varepsilon_{abcd} = g^{ae}g^{bf}g^{cg}g^{dh}\varepsilon_{abcd}\varepsilon_{efgh} = 4! g^{00}g^{11}g^{22}g^{33} = 4! |\boldsymbol g|

so

\varepsilon^{abcd} \equiv |\boldsymbol g|\epsilon^{abcd}

i.e. the sign change is caused by the determinant of the metric.

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Note that the comment in the caption to figure 19.1 that
Quote:
"this sign change is absorbed by an effective index reversal."

is a potential source of confusion here. Index reversal has nothing to do with it, and in fact this is wrong, since the reversal of 4 indices is an even permutation that does not cause a sign change anyway.

See my post in the "Errors, typos, and corrections" forum here.


04 Mar 2012, 06:04
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