Observer Maxwell equations and conserved source charge

Let $(M,g)$ be a smooth ($C^{\mathrm{\infty }}$), Hausdorff, second-countable, boundaryless, oriented, time-oriented four-manifold with Lorentz metric of signature $(-,+,+,+)$ and oriented volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Use SI units: $c>0$ has exact defining value $299792458\mathrm{m}{\mathrm{s}}^{-1}$; its identification as the vacuum light/clock conversion is adopted in this framework, not derived here. The constant ${\mu }_0>0$ has units $\mathrm{N}{\mathrm{A}}^{-2}$ and is experimentally determined in the revised SI; set ${ϵ}_0:=({\mu }_0{c}^2{)}^{-1}$ in $\mathrm{F}{\mathrm{m}}^{-1}$. The manifold orientation is independent of the time orientation. 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 vector field whose components have units $\mathrm{A}{\mathrm{m}}^{-2}$, both on $\mathcal{D}$. An observer field is a smooth future-directed vector field $n$ with $g(n,n)=-1$; no hypersurface-orthogonality or global inertial chart is assumed. On decomposable $k$-covectors define $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k{⟩}_g:=\det (g^{-1}({\alpha }_i,{\beta }_j))$ and extend bilinearly. Define the Hodge star by $\alpha \wedge \ast \beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ and define insertion by $(i_X\omega )(Y_1,\dots ,Y_{k-1})=\omega (X,Y_1,\dots ,Y_{k-1})$. The observer fields are the spatial covectors $E_a:=c{F}_{ab}{n}^b=-c(i_{n}F{)}_a$ and $B_a:=-(\ast F{)}_{ab}{n}^b=(i_n\ast F{)}_a$. Antisymmetry makes both covectors spatial, that is, they annihilate $n$. In a positively oriented orthonormal tetrad with time axis $n$ (an ordered basis with metric matrix $\mathrm{diag}(-1,1,1,1)$), let ${\delta }_{ij}$ be the Euclidean spatial metric and let ${\epsilon }_{ijk}$ be the alternating symbol with ${\epsilon }_{123}=+1$; at an event write $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={\epsilon }_{ijk}{B}_k$ for the tetrad components at that event. Define the local charge and current densities by $\rho :=j^0\mathrm{/}c$ and $J^i:=j^i$, so $[\rho ]=\mathrm{C}{\mathrm{m}}^{-3}$ and $[J^i]=\mathrm{A}{\mathrm{m}}^{-2}$. For spatial components define $\mathrm{\nabla }\cdot X:={\sum }_i{\mathrm{\partial }}_i{X}_i$ and $(\mathrm{\nabla }\times X{)}_k:={\epsilon }_{kij}{\mathrm{\partial }}_i{X}_j$. Coordinates are lengths, with $x^0=ct$ and ${\mathrm{\partial }}_0=c^{-1}{\mathrm{\partial }}_t$; $E_i$ has units $\mathrm{V}{\mathrm{m}}^{-1}$ and $B_i$ has units tesla.

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as stated in [A2] and the adopted curved vacuum Maxwell equations in [A1] on $\mathcal{D}$, with $\mathrm{\nabla }$ the Levi-Civita connection of $g$. A point-normal Lorentz chart at $p\in \mathcal{D}$ means a positively oriented chart in $\mathcal{D}$ with $g_{ab}(p)=\mathrm{diag}(-1,1,1,1)$, ${\mathrm{\partial }}_c{g}_{ab}(p)=0$, and coordinate time axis $n(p)$. In this chart define ${\widehat{E}}_i(x):=-c{F}_{0i}(x)$ and ${\widehat{B}}_i(x):=\frac{1}{2}{\epsilon }_{ijk}{F}_{jk}(x)$; at $p$ these are the components measured by $n(p)$. Derivatives in the pointwise formulas below act on these chart-component functions, not on components of a varying observer congruence. Then ${\mathrm{\nabla }}_a{j}^a=0$. At each event $p$, in a positively oriented point-normal Lorentz chart whose time axis is $n(p)$, the component equations at $p$ are

$\mathrm{\nabla }\cdot \widehat{\mathbf{E}}=\rho \mathrm{/}{ϵ}_0,\mathrm{\nabla }\times \widehat{\mathbf{B}}-c^{-2}{\mathrm{\partial }}_t\widehat{\mathbf{E}}={\mu }_0\mathbf{J},\mathrm{\nabla }\cdot \widehat{\mathbf{B}}=0,{\mathrm{\partial }}_t\widehat{\mathbf{B}}+\mathrm{\nabla }\times \widehat{\mathbf{E}}=0.$

These four component equations hold throughout a fixed Minkowski inertial chart with constant observer $n={\mathrm{\partial }}_0$. 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 ${\mathrm{\Delta }}^k\to \mathcal{D}$; a finite smooth singular $k$-chain is a finite real linear combination of such simplices, with boundary the alternating sum of their oriented faces. A smooth singular three-simplex is spacelike when its differential has rank three and its pullback metric is positive definite on its interior; a spacelike three-chain has this property on each simplex and is oriented by its future-directed unit normal. For a finite smooth oriented spacelike three-chain $\mathrm{\Sigma }$ in the field domain, define its source charge by $Q(\mathrm{\Sigma }):=c^{-1}{\int }_{\mathrm{\Sigma }}{i}_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ (in coulombs). If $\mathrm{\partial }\mathrm{\Sigma }$ is its finite smooth boundary chain, then

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

The component identities at a general event are local observer measurements; they do not assert that an arbitrary observer field supplies a global inertial coordinate system or a preferred foliation.

The postulate and its current are adopted model inputs. The equations and charge identity are conditional deductions; they are not experimental observations or evidence that the adopted model describes nature. The current supplier statements and proofs were inspected; readiness is bound to the current physical adoption [A1] and supplier dependency closure.

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.