Global Gauss flux and compact-slice neutrality

Let $(M,g)$ be a smooth ($C^{\mathrm{\infty }}$), Hausdorff, second-countable, boundaryless, oriented, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with oriented metric volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Use SI constants $c>0$ with exact defining value $299792458\mathrm{m}{\mathrm{s}}^{-1}$ and ${\mu }_0>0$ in $\mathrm{N}{\mathrm{A}}^{-2}$ (experimentally determined in the revised SI); define ${ϵ}_0:=({\mu }_0{c}^2{)}^{-1}$ in $\mathrm{F}{\mathrm{m}}^{-1}$. Let $\mathcal{D}\subseteq M$ be the open field domain. Let $F$ be a smooth real two-form with components in tesla and let $j$ be a smooth current vector field with components in $\mathrm{A}{\mathrm{m}}^{-2}$ on $\mathcal{D}$. Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as stated in [A2] and the adopted Maxwell equations in [A1].

Use the Lorentzian Hodge star ${\star }_g$, the unique smooth bundle map defined by Lorentzian exterior calculus and Hodge star: on decomposable $k$-covectors its induced pairing is $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k{⟩}_g:=\det (g^{-1}({\alpha }_i,{\beta }_j))$, extended bilinearly, and $\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. In four dimensions with signature $(-,+,+,+)$, that supplier also proves ${\star }_g^2=-\mathrm{i}\mathrm{d}$ on two-forms. Write $\ast$ for ${\star }_g$ when the metric is fixed.

For $k\ge 0$, let ${\mathrm{\Delta }}^k:=\{(t_0,\dots ,t_k)\in {\mathbb{R}}^{k+1}:t_i\ge 0,{\sum }_i{t}_i=1\}$. A smooth singular $k$-simplex is a smooth map ${\mathrm{\Delta }}^k\to \mathcal{D}$, and a finite smooth singular $k$-chain is a finite real linear combination of such simplices; $\mathrm{\partial }$ is the alternating sum of their oriented faces. A smooth singular three-simplex is spacelike when its differential has rank three and its pulled-back metric is positive definite on its interior; a spacelike three-chain has this property on each simplex. Orient spacelike simplices using their future-directed unit normal. Define insertion by $(i_j\omega )(X_1,\dots ,X_{k-1})=\omega (j,X_1,\dots ,X_{k-1})$. For every finite smooth singular three-chain $C$, set ${\mathrm{\Phi }}_j(C):=c^{-1}{\int }_C{i}_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$; for a spacelike slice $\mathrm{\Sigma }$ write $Q(\mathrm{\Sigma }):={\mathrm{\Phi }}_j(\mathrm{\Sigma })$, in coulombs.

For every finite smooth oriented four-chain $V$ whose boundary decomposes as $\mathrm{\partial }V={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, where ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ are finite spacelike three-chains and $B$ is the finite side chain, the source-current flux satisfies

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

For every compact boundaryless oriented smooth spacelike slice $\mathrm{\Sigma }$ in the domain, represented by a finite smooth three-cycle, $Q(\mathrm{\Sigma })=0$. For every finite smooth three-chain $W$, ${\int }_{\mathrm{\partial }W}F=0$; where $j=0$ throughout $W$, ${\int }_{\mathrm{\partial }W}\ast F=0$. These conclusions concern bounding surfaces. A closed surface is nonbounding in the field domain when it is a finite smooth two-cycle $C$ with no finite smooth three-chain $W$ in that domain such that $C=\mathrm{\partial }W$; no zero-flux conclusion is made for such a surface. The field topology is the global real closed-form model of [F2]; an optional normalized $U(1)$ connection sector with chosen flux normalization ${\kappa }_0$ is extra structure, where $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ under multiplication, and no electric-charge quantization is inferred.

Punctured-Minkowski example (a separate specialization of the general setup). For $b\mathrm{\ne }0$ with $b$ measured in webers, let $M_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}:={\mathbb{R}}_{x^0}\times ({\mathbb{R}}^3\setminus \{0\})$ be the open punctured spacetime, with metric $g_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}:=-(d{x}^0{)}^2+d{x}^2+d{y}^2+d{z}^2$, standard orientation $d{x}^0\wedge dx\wedge dy\wedge dz$, future time orientation given by ${\mathrm{\partial }}_{x^0}$, and field domain ${\mathcal{D}}_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}:=M_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}$. Take $j_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}=0$; in this example, “source-free” means that the adopted equations [A1] reduce to $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=0$. Write $r_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}:=(x^2+y^2+z^2{)}^{1\mathrm{/}2}$, and define on this domain

$F_b:=\frac{b}{r_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}^3}(xdy\wedge dz+ydz\wedge dx+zdx\wedge dy)\mathrm{.}$

The coordinates $x^0=ct,x,y,z$ and $r_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}$ are in metres. This is one particular instance of the arbitrary $M,g,\mathcal{D},F,j$ fixed above; the following example claims concern this instance only. For every radius $R>0$, the sphere $S_R^2$ at fixed $x^0$, with outward orientation, has ${\int }_{S_R^2}{F}_b=4\pi b\mathrm{\ne }0$. Its Hodge-dual source-free field $-{\star }_{g_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}}{F}_b$ has ${\int }_{S_R^2}{\star }_{g_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}}(-{\star }_{g_{\mathrm{p}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}}}{F}_b)=4\pi b\mathrm{\ne }0$ as well. The sphere is nonbounding in this punctured domain; this example makes no claim about a source or field value at the excluded origin.

Charge conservation in an open region requires zero side flux or inclusion of that side flux, as the first conclusion states. Noncompact slices require a separately specified exhaustion and convergence/decay hypotheses, which are not present here. The actual supplier statements and proofs have been inspected. This item remains conditional: the punctured-sphere result [F3] remains draft and unaccepted despite the now-present prerequisite carriers, and [A1] is a draft physical adoption.