How do I construct the Maxwell tensor $\bf{^*F}$ from Fadaray one $\bf{F}$ in a non-flat spacetime?

Question

In the book Gravitation (Misner, Throne and Wheeler), it's said that to consider the line element of the flat space on the derivation of Maxwell tensor $\bf{^*F}$ from the Fadaray tensor $\bf{F}$ (text before equation 4.15).

How do I construct the Maxwell tensor $\bf{^*F}$ from Fadaray one $\bf{F}$ in a non-flat spacetime, once the unique equation presented in Gravitation is using the "Levi-Civita tensor" $\epsilon_{\mu\nu\alpha\beta}$ by

$^*F_{\alpha\beta}=\frac{1}{2}F^{\mu\nu}\epsilon_{\mu\nu\alpha\beta}$

and not the metric tensor $g_{\mu\nu}$?

Answer 1

$F_{\alpha \beta}=\frac 1 2 \epsilon_{\alpha \beta \mu\nu}F^{\mu\nu}$ is general tensor equation so it is true in all frame both flat and curved spacetime. We can write it also as $F_{\alpha \beta}=\frac 1 2 \sqrt{|g|}\varepsilon_{\alpha \beta \mu\nu}F^{\mu\nu}$ ,where $\varepsilon_{\alpha \beta \mu\nu}$ is a Levi-Civita symbol. In the case of flat spacetime you just change $g\longrightarrow \eta$

Answer 2

Once $^*$ is the Hodge star operator, its definition in non-flat spacetime

$(\star \eta)_{i_1,i_2,\ldots,i_{n-k}} = \frac{1}{(k)!} \eta^{j_1,\ldots,j_k}\,\sqrt {|\det g|} \,\epsilon_{j_1,\ldots,j_k,i_1,\ldots,i_{n-k}}$

say that we only need the determinant of the metric tensor $\bf{g}$ to change from $\bf{F}$ to $\bf{^*F}$.

Thanks for @ACuriousMind and @0celo7 for the help.