Maxwell's equations with differential forms

[addaF]

Hello!

I was not quite sure about posting in this category, but I think my question fits here.

I am wondering about Maxwell equations in vacuum written with differential forms, namely:
\begin{equation} \label{pippo}
dF = 0 \qquad d \star F = 0
\end{equation}
I know $F$ is a 2-form, and It can be written by using the 1-form $F = d A$. I am not now interested in the derivation (even if the first is straightforward), but I want to see the equivalence with the Maxwell equation written in a covariant way:
\begin{equation} \label{pluto}
\varepsilon^{\mu \nu \rho \sigma} \partial_\nu F_{\rho \sigma} = 0 \qquad \partial^\mu F_{\mu \nu} = 0
\end{equation}

If I explicit everything, knowing the components of $F_{\mu \nu}$, I can recover both \ref{pluto} equations starting from \ref{pippo}. I'm interested in a more "direct" derivation, but I cannot find any reference in textbooks.
Thank you in advance,

Francesco

NB: I am not a native english speaker, sorry for that.

 

[jvicens]

Hello Francesco,

Thank you for your question. It is great to see someone interested in the mathematical formulation of Maxwell's equations.

To see the equivalence between the two forms of Maxwell's equations, we can start by considering the definition of the exterior derivative of a 1-form. We know that for a 1-form $\omega$, the exterior derivative is defined as $d\omega = (\partial_\mu \omega_\nu - \partial_\nu \omega_\mu) dx^\mu \wedge dx^\nu$. In the case of the 2-form $F$, we can write it as $F = F_{\mu \nu} dx^\mu \wedge dx^\nu$.

Applying the definition of the exterior derivative, we have $dF = (\partial_\mu F_{\nu \rho} - \partial_\nu F_{\mu \rho}) dx^\mu \wedge dx^\nu \wedge dx^\rho$. Notice that this is the same as the first equation in \ref{pluto}, with the indices $\mu,\nu,\rho,\sigma$ being replaced by $\nu,\rho,\mu,\sigma$ respectively.

To see the equivalence of the second equations, we can use the Hodge dual operator $\star$. Recall that for a p-form $\omega$, the Hodge dual is defined as $\star \omega = \frac{1}{p!} \varepsilon_{\mu_1 \mu_2 ... \mu_p} \omega^{\mu_1 \mu_2 ... \mu_p}$, where $\varepsilon_{\mu_1 \mu_2 ... \mu_p}$ is the Levi-Civita symbol. Applying the Hodge dual to the first equation in \ref{pippo}, we have $\star dF = 0$. Using the definition of the Hodge dual, we can rewrite this as $d\star F = 0$, which is the second equation in \ref{pippo}.

Finally, we can use the fact that for a 2-form $F$, we have $\star F = -F$. Applying this to the second equation in \ref{pippo}, we have $d\star F = -dF = 0$, which is the second equation in \ref{pluto}