Gauss charge and finite-chain flux balance

Let $(M,g,N,o)$ be the smooth oriented and time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions, let $F\in {\mathrm{\Omega }}^2(M)$ and $j\in \mathrm{\Gamma }(TM)$ be smooth, and fix constants $c,{\mu }_0>0$. Assume $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, equivalently $d({\star }_{g}F)={\mu }_0{\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ by Sourced Maxwell sign and current conservation. A finite smooth singular $k$-chain means a finite formal real linear combination of smooth singular $k$-simplices, with the boundary operation of The metric volume form, current flux through 3-faces and finite-chain charges and Stokes theorem for smooth singular chains.

For a finite smooth singular three-chain $\mathrm{\Sigma }$, define its charge by

$Q(\mathrm{\Sigma }):=\frac{1}{c}{\int }_{\mathrm{\Sigma }}{\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

Then:

(a) Gauss charge formula. For every such $\mathrm{\Sigma }$,

$Q(\mathrm{\Sigma })=\frac{1}{{\mu }_{0}c}{\int }_{\mathrm{\partial }\mathrm{\Sigma }}{\star }_{g}F\mathrm{.}$

Consequently, if $\mathrm{\partial }\mathrm{\Sigma }=0$, then $Q(\mathrm{\Sigma })=0$. This is the compact-cycle neutrality constraint for a smooth everywhere-defined sourced field.

(b) Bounding-surface restriction. For every finite smooth singular three-chain $V$, the magnetic flux through its boundary vanishes:

${\int }_{\mathrm{\partial }V}F=0.$

The equation $dF=0$ does not imply that the flux through every closed two-cycle is zero; this item gives a zero-flux conclusion only when the cycle is the boundary of a finite smooth chain in the field domain. Likewise, the electric flux formula in (a) identifies charge with flux only for a bounding surface.

(c) Charge balance. If a finite smooth singular four-chain $C$ satisfies $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$ for three-chains ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2,B$, then

$Q({\mathrm{\Sigma }}_2)-Q({\mathrm{\Sigma }}_1)=-Q(B)\mathrm{.}$

Equal charge on the two slices therefore requires zero net side flux. No asymptotic charge or flux at infinity is asserted; noncompact limits require separate convergence and boundary-flux hypotheses.

The finite-chain hypotheses are essential to the stated argument. On a noncompact spatial domain, charge conservation requires specified convergence of slice integrals and an explicit treatment of limiting side flux; neither follows from the local differential equation alone.