I have been thinking about the Maxwell equations lately and was wondering about their "natural" differential form formulation to get some nice geometric interpretation. This post mainly concerns the inhomogenous microscopic Maxwell equations on some spacetime ##(M,g)##, as the homogenous ones ##d F = 0## just mean that the total electromagnetic flux through a ##2##-surface ##\Omega## does not change under a smooth homotopy of ##\Omega##, i.e.(adsbygoogle = window.adsbygoogle || []).push({});

$$\int\limits_{\Omega} F = \int\limits_{\Omega'} F \, .$$

When one checks out the literature, one often finds the following vector field:

$$ j := j^\mu \, \partial _\mu =

\begin{pmatrix}

\frac{1}{c} \rho\\

\vec{j}

\end{pmatrix} \, .$$

My first question is: What is the natural interpretation of the flow of this vector field?

There are a lot of different formulations one finds in the literature, but I have thought about it for a while and it seems like defining a ##3##-form ##J## out of ##j## to get the total charge in some spacetime region ##\Omega \subseteq M##

$$Q := \int\limits_{\Omega} \, J$$

is most convenient as ##J## can then naturally be interpreted as a charge density in our spacetime. My final questions are: Using this definition, how does one get ##J## from ##j## coordinate independently in SI-units s.t. the above equation holds (using the Cartan derivative, Hodge operator, metric, etc.)? How do the inhomogenous Maxwell equations look like in this form language in SI units?

I tried doing the calculation, but for some reason I always get a wrong factor in the ##J## and the continuity equation ##d J = 0## has the wrong sign.

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# Current 3-form, current density, current vector, etc.

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