# What is the EM Lagrangian in curved spacetime?

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

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

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

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

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 $$\mathcal{L}_{flat}$$ 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 $$\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$$

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bcrowell
Staff Emeritus
Gold Member
Wald also uses covariant derivatives on p. 455.

But did you see the following text in the WP article? "Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out."

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But did you see the following text in the WP article? "Despite the use of partial derivatives, these equations are invariant under arbitrary curvilinear coordinate transformations. Thus if one replaced the partial derivatives with covariant derivatives, the extra terms thereby introduced would cancel out."

No. I didn't read that . Thanks. I thought that might be it though and attempted to work it out myself earlier. I must have made a mistake. They didn't cancel for me. I'll give it another shot.

No. I didn't read that . Thanks. I thought that might be it though and attempted to work it out myself earlier. I must have made a mistake. They didn't cancel for me. I'll give it another shot.

They must cancel because of the symmetry of the Christoffel symbols.

$$A_{m;n} - A_{n;m} = A_{m,n}- A_{n,m} - {\Gamma^a}_{mn} A_a + {\Gamma^a}_{nm} A_a$$

dextercioby