How do we compute Hodge duals?

The motivation for this question is to try to come up with a general expression for $(\star F)_{\mu\nu}$, the $\mu,\nu$ component of the Hodge dual of the Field strength tensor, which is of great importance in Electrodynamics. In the course I am taking, the instructor has defined it contravariantly, as $$(\star F)^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$$ Where $F_{\alpha\beta}$ are the components of the normal field strength tensor, $$F_{\alpha\beta}=\nabla_\alpha A_\beta-\nabla_\beta A_\alpha$$ And we have defined the raised Levi-Civita symbol using the generalized Kronecker delta: $$\varepsilon^{ijkl}=\delta^{i~j~k~l}_{1~2~3~4}$$ However, there are some problems with this equation. As far as I know, it only holds when the metric is simple as possible, i.e $\boldsymbol{\eta}=\operatorname{diag}(-1,1,1,1)$ the standard Minkowski metric. I don't think it holds in other coordinate systems e.g polar coordinates. Also, I thought the Hodge star was a map that took $k$ forms to $d-k$ forms, where $d$ is the dimension of the space. But my instructor's equation seems to have the Hodge dual being a fully contravariant tensor... are the components he is listing actually the components of $(\star\mathbf{F})^{\sharp}$, its index-raised counterpart?

With this motivation, I searched the internet for answers and stumbled across this post on physics SE. Given a $d$ dimensional manifold, the author of the post defines the Hodge dual of a totally skew-symmetric $k$ form $\omega$ as having components

$$(\star \omega)_{\mu_1\dots\mu_{d-k}}=\frac{1}{k!}\epsilon^{\nu_1\dots\nu_k}{}_{\mu_1\dots\mu_{d-k}}\omega_{\nu_1\dots\nu_k}$$

However, the author does not make it very clear what the object $\epsilon^{\nu_1\dots\nu_k}{}_{\mu_1\dots\mu_{d-k}}$ is. He does write $$\epsilon_{\mu_1\dots\mu_d}=\sqrt{-g}~\varepsilon_{\mu_1\dots\mu_d}$$ But not only does this look very different to$\epsilon^{\nu_1\dots\nu_k}{}_{\mu_1\dots\mu_{d-k}}$ in terms of the number and structure of the indices, it also raises other concerns, like what if $g=\det \mathbf{g}$ is a positive number? Will we start getting imaginary components of the Hodge dual, even when we are working on a real manifold?

My request: Could someone please provide some other formula for the Hodge dual of a totally skew symmetric $k$ form on a $d$ dimensional manifold, in terms of well known objects, such as the metric, Christoffel symbols, Kronecker delta, Levi-Civita symbol, etc? I searched the Wikipedia page for answers but I found it very dense and difficult to understand.

Solution 1:

Since the Hodge dual maps $k$ forms to $n-k$ forms, and covariant skew-symmetric tensors are forms, the contravariant expression for $(\star F)$ must be interpreted as $(\star F)^\sharp$.

The expression for Hodge dual in OP is correct. What appears on the RHS is the Levi-Civita tensor, with a mixture of up/ down indices. The Levi-Civita symbol, which has the same components in all co-ordinates, is related to it by

$$ \epsilon^{\text{tensor}}_{ij\dots}=\sqrt{|g|}\varepsilon^{\text{symbol}}_{ij\dots} $$

Since this is a tensor, indices may be raised and lowered with the metric as necessary

$$ \epsilon^{\nu_1\dots\nu_k}{}_{\mu_1\dots\mu_{d-k}}=\sqrt{|g|}g^{\alpha_1 \nu_1}\dots g^{\alpha_k\nu_k}\varepsilon_{\alpha_1\dots\alpha_k~\mu_1\dots\mu_{d-k}} $$

The expression $\sqrt{-g}$ is more generally $\sqrt{|\det(g)|}$. In Lorentz spacetime the determinant is necessarily negative so there is no harm (and no imaginary units) in writing $\sqrt{-g}$.

References: Spacetime and geometry by S. Carrol.