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.