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In flat space time the Lagrangian for the EM potential is (neglecting the source term)

[tex]\mathcal{L}_{flat}=-\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})[/tex]

which is a scalar for flat spacetime. I would have expected the generalization to curved manifolds to be

[tex]\mathcal{L}_{curved}=-\frac{1}{16\pi}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})(\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu})\sqrt{-g}[/tex]

However, the Wikipedia article for [URL [Broken] equations in curved spacetime[/url] gives the Lagrangian still in terms of the regular derivatives, not covariant derivatives. But [tex]\mathcal{L}_{flat}[/tex] is not a scalar in general, is it? I was just reading an article about covariance and Noether's theorem that likewise uses an EM Lagrangian in terms of partial derivatives instead of covariant derivatives.

So which is correct?

Edit: I just checked Gravitation (Misner, Thorne, Wheeler) and eq 22.19a gives the EM tensor as [tex]\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}[/tex]

[tex]\mathcal{L}_{flat}=-\frac{1}{16\pi}(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})[/tex]

which is a scalar for flat spacetime. I would have expected the generalization to curved manifolds to be

[tex]\mathcal{L}_{curved}=-\frac{1}{16\pi}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})(\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu})\sqrt{-g}[/tex]

However, the Wikipedia article for [URL [Broken] equations in curved spacetime[/url] gives the Lagrangian still in terms of the regular derivatives, not covariant derivatives. But [tex]\mathcal{L}_{flat}[/tex] is not a scalar in general, is it? I was just reading an article about covariance and Noether's theorem that likewise uses an EM Lagrangian in terms of partial derivatives instead of covariant derivatives.

So which is correct?

Edit: I just checked Gravitation (Misner, Thorne, Wheeler) and eq 22.19a gives the EM tensor as [tex]\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}[/tex]

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