Electromagnetic tensor in a FRW metric

Question

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!

Answer 1

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.