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I have difficulties to understand how to solve the Maxwell equations on curved spacetime. I want to solve the equations in the weak regime $g_{\mu\nu}=\eta_{\mu\nu}+h{\mu\nu},~ h_{\mu\nu}\ll 1$ without sources $J_{\mu}=0$. I use $\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$.

The Maxwell equations in curved spacetime are $$F^{\mu\nu}_{~~~~~||\mu}=0 \tag{$*$}$$ and $$\epsilon^{\mu\nu\lambda\kappa}F_{\lambda\kappa||\nu}=0 \tag{$**$}$$ where $||$ means covariant derivative. In the following I focus on $(*)$. I can rewrite $(*)$ as $$0=F^{\mu\nu}{}_{||\mu}=F^{\mu\nu}{}_{|\mu}+\Gamma^{\mu}{}_{\mu\rho}F^{\rho\nu}=\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}F^{\mu\nu}),$$ where I have used the definition of the Christoffel symbols $\Gamma^{\mu}{}_{\mu\rho}=\frac{1}{\sqrt{g}}\partial_{\rho}\sqrt{g}$ and $g=-\text{det}(g^{\mu\nu})$. In summary, I get $$\partial_{\mu}(\sqrt{g}F^{\mu\nu})=0, \tag{$***$}$$ and I furthermore use the transverse traceless gauge $$h_{\mu\nu}=\begin{pmatrix}0&0&0&0\\0&h_+&h_\times&0\\0&h_\times&-h_+&0\\0&0&0&0\end{pmatrix}$$ and assume that $h_{\mu\nu}$ is a background field. The coupling to the Einstein Equations is neglected. I can expand $$g=1-h_+^2-h_\times^2+\mathcal{O}(h^4)$$

I now can split $F^{\mu\nu}$ in powers of $h$: $F^{\mu\nu}=F_{(0)}^{\mu\nu}+F_{(1)}^{\mu\nu}+\cdots$ and expand all fields in equation $(***)$

The zeroth order yields normal electrodynamics, $$\partial_{\mu}F^{\mu\nu}_{(0)}=0.$$ However the first order field strength tensor also yields normal electrodynamics, $$\partial_{\mu}F^{\mu\nu}_{(1)}=0$$

Now comes my question:

If I rewrite $(***)$ to $$\partial_{\mu}(\sqrt{g}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta})=0, \tag{$****$}$$ and do the expansions I get non-trivial first-order equations for $F^{\mu\nu}_{(1)}$, $$\partial_{\beta}F^{\mu\beta}_{(1)}=-(\partial_{\nu}h^{\mu}{}_{\alpha})F^{\alpha\nu}_{(0)}-\partial_\nu(h^{\nu}_{~\beta}F^{\mu\beta}_{(0)})-h^{\mu}{}_{\alpha}\partial_{\beta}F^{\alpha\beta}_{(0)}$$

How can this be? Does it have to do something with the choice of the system where we do the calculation? I am interested in the case that a gravitational wave hits the Earth. What would be the physical electromagnetic fields? Would they be components of $F^{\mu\nu}_{(1)}$ or components of $F_{(1)\mu\nu}$? I think this should not matter since we have to raise and lower the indices with $\eta$ in order not to change the order of the tensor!? My question also arises due to this paper. They solve my equation $(****)$ what is their equation (9) in the paper. However to get to the physical electromagnetic fields they say that one has to do another transformation, c.f., their equation (A.9)-(A.11). I simply do not understand this.

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    $\begingroup$ Where you have defined the expansion of $F^{\mu \nu}$ as an expansion in powers of h, I don't quite understand what you mean, since there is no $h$ multiplying the $F^{\mu \nu}_{(1)}$ term. Is it a power series in $det(h_{\mu \nu})$? $\endgroup$
    – Stratiev
    Commented Jun 5, 2020 at 19:13
  • $\begingroup$ $F^{\mu\nu}_{(1)}$ has no definition when I do the expansion. It is the function that I am looking for. At the stage of the expansion just consider it as an expersiion which contains one $h$. If you look at the end result you can see that this makes sense $\endgroup$
    – user255856
    Commented Jun 6, 2020 at 20:06
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    $\begingroup$ Before equation (48) they pointed out that "Any physical measurement by an observer in curved spacetime should be carried out in a local inertial frame, i.e., the observable quantities are the projections of the physical quantities on to the four orthonormal bases $e^{\mu}_0, e^{\mu}_1, e^{\mu}_2,e^{\mu}_3$ carried by the observer. Therefore, the observable EM fields are $F_{\alpha \beta}=F_{\mu \nu}e^{\mu}_{\alpha}e^{\nu}_{\beta}$" $\endgroup$
    – Alex Trounev
    Commented Jun 6, 2020 at 20:23
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    $\begingroup$ @jojo123456 Probably they just try to solve equation $g_{ik}=\eta _{\mu \nu}e^{\mu}_ie^{\nu}_k$. $\endgroup$
    – Alex Trounev
    Commented Jun 7, 2020 at 23:45
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    $\begingroup$ Have you looked at hep.princeton.edu/~mcdonald/examples/rotatingEM.pdf? $\endgroup$
    – honeste_vivere
    Commented Jun 10, 2020 at 14:53

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