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 Sci Advisor HW Helper PF Gold P: 2,606 Symmetry of the metric ($g^{\mu \nu}=g^{\nu \mu}$) and total antisymmetry of $\epsilon_{\alpha \mu \beta \nu}$ imply that $\epsilon_{\alpha \mu \beta \nu}g^{\mu \nu}=0$ (together with all possible permutations of indices). So all terms of that form vanish.