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I am reading Weinberg's Gravitation and Cosmology. On page 99, the author defines the covariant Levi-Civita tensor density as $$\epsilon_{\rho\sigma\eta\xi}=g_{\rho\mu}g_{\sigma\nu}g_{\eta\lambda}g_{\xi\kappa}\epsilon^{\mu\nu\lambda\kappa}\tag{4.4.10}$$ and gives the relation $$\epsilon_{\rho\sigma\eta\xi}=-g\epsilon^{\rho\sigma\eta\xi}\tag{4.4.11}.$$ I understand where the $g$ in$(4.4.11)$ comes from. But why is there a minus sign, since it seems, as an example, $$\epsilon_{0123}=g_{0\mu}g_{1\nu}g_{2\lambda}g_{3\kappa}\epsilon^{\mu\nu\lambda\kappa}=g=\epsilon^{0123}.$$ Is that a typo?

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The key is the sign in the definition at the start of the section: $$g \equiv -\mathrm{Det} g_{\mu\nu} \tag{4.4.1}.$$

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