Why I missed a minus sign?

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  • Thread starter nenyan
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I am reading GEOMETRY, TOPOLOGY AND PHYSICS writen by MIKIO NAKAHARA (second edition). I have a problem on page 400.
I wonder why the sign is minus in Eq. (10.85).

And the same problem appears on page 405 in Eq.(10.108). I think it should be minus one half.
 

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  • #2
vanhees71
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Have you considered that with the standard definition (at least in the HEP community) ##\epsilon^{\mu \nu \rho \sigma}=\text{sign}(\mu,\nu,\rho,\sigma)## the covariant components of the Levi-Civita (pseudo-)tensor reads
$$\epsilon_{\alpha \beta \gamma \delta}=\eta_{\alpha \mu} \eta_{\beta \nu} \eta_{\gamma \rho} \eta_{\delta \sigma} \epsilon^{\mu \nu \rho \sigma}=\det \eta \epsilon^{\alpha \beta \gamma \delta}=-\epsilon^{\alpha \beta \gamma \delta},$$
because ##\eta=\mathrm{diag}(1,-1,-1,-1)## (or in the east-coast convention ##\mathrm{diag}(-1,1,1,1)##).
 
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Have you considered that with the standard definition (at least in the HEP community) ##\epsilon^{\mu \nu \rho \sigma}=\text{sign}(\mu,\nu,\rho,\sigma)## the covariant components of the Levi-Civita (pseudo-)tensor reads
$$\epsilon_{\alpha \beta \gamma \delta}=\eta_{\alpha \mu} \eta_{\beta \nu} \eta_{\gamma \rho} \eta_{\delta \sigma} \epsilon^{\mu \nu \rho \sigma}=\det \eta \epsilon^{\alpha \beta \gamma \delta}=-\epsilon^{\alpha \beta \gamma \delta},$$
because ##\eta=\mathrm{diag}(1,-1,-1,-1)## (or in the east-coast convention ##\mathrm{diag}(-1,1,1,1)##).
Thank you! I see.
 

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