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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