The Mystery of the Minus Sign in Geometry, Topology, and Physics

[nenyan]

I am reading GEOMETRY, TOPOLOGY AND PHYSICS written 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.

 

[vanhees71]

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 $\mathrm{diag}(-1,1,1,1)$).

 

[nenyan]

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.