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In some papers [like https://arxiv.org/pdf/2204.06883.pdf, eq. (31) ], I see that the Electromagnetic tensor field, for a FRW metric (written in a conformal way) \begin{equation} ds^{2} = a^{2}(\tau) \Big( -d\tau^{2} + dx^{2} + dy^{2} + dz^{2} \Big) \end{equation} is

\begin{equation} F_{\mu \nu}=a^{2}(\eta)\left(\begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} & -B_{y} \\ E_{y} & -B_{z} & 0 & B_{x} \\ E_{z} & B_{y} & -B_{x} & 0 \end{array}\right) \end{equation}

i.e. the Minkowski field tensor multiplied by $a^{2}(\tau)$. However, I'm having a hard time understanding where this formula comes from, even if it intuitively works.

I mean, in other papers [https://arxiv.org/pdf/1905.09968.pdf, pag.4] I read that if my metric is conformal to another metric, $g_{\mu\nu}=\Omega^{2}\tilde{g}_{\mu\nu} $, then $F_{\mu\nu}= \tilde{F}_{\mu\nu}$. In my case, $\tilde{g}_{\mu\nu}$ is the Minkowski metric and $\Omega^{2}=a^{2}$. So, I would expect that the e.m. tensor in the metric above is simply the Minkowski one, without the $a^{2}$ factor. Where does it come from? Or, in other words, given a generic metric, how can I get the e.m. tensor $F_{\mu\nu}$ ?

Thank you in advance!

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  • $\begingroup$ As a general rule we prefer links to the abstract pages for papers where possible. $\endgroup$ Aug 13, 2022 at 23:29

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OP misunderstood the equation $(31)$ of the first paper. Component $E_i$ and $B_i$ are meant to be the components of electric and magnetic fields that would be measured by a comoving observer at that spacetime point and the equation $(31)$ just explains how that components form a tensor without any reference to another metric. The fields $\mathbf{E}(t,\mathbf{x})$ and $\mathbf{B}(t,\mathbf{x})$ are not meant to be the solution of the Maxwell's equations for flat space.

On the other hand the second paper deals with transformation of Faraday tensor under conformal transformation of the metric. Both $F_{μν}$ and $\tilde{F}_{μν}$ are solutions of Maxwell's equations for their respective metrics.

Or, in other words, given a generic metric, how can I get the e.m. tensor $F_{μν}$ ?

By solving the Maxwell's equations (in curved spacetime) or, if the EM field exerts measurable influence on the metric, by solving Einstein–Maxwell-(whatever) equations.

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  • $\begingroup$ @AVS Ok, thank you for your reply. But, anyway, if the second paper is right, the relation " 𝐹𝜇𝜈=𝐹̃ 𝜇𝜈 "(where r.h.s. is F in a FRW metric, while l.h.s. is F in Minkowski metric ) is still true, or not ? $\endgroup$ Aug 17, 2022 at 23:26
  • $\begingroup$ @gravitone123: the relation "$𝐹_{𝜇𝜈}=𝐹̃_{𝜇𝜈}$ … is still true, or not? Locally true, at least. Globally, there may be some subtleties (i.e. solution that is acceptable in one case would lead to globally unacceptable behavior (e.g. incompatible with assumptions which allow us to ignore backreaction) in another). $\endgroup$
    – A.V.S.
    Aug 18, 2022 at 6:52

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