The electromagnetic two-form, its observer decomposition and the four-current

Let the laboratory region, its inertial chart and the fields and sources be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields and Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values; fix the ordered basis $(e_0,e_1,e_2,e_3)$ of the chart and the oriented volume form $\epsilon$ of The oriented volume form epsilon and the permutation symbols of an inertial frame, with ${\epsilon }_{0123}=+1$ and $\eta =\mathrm{diag}(-1,1,1,1)$.

(a) [Components] For a field pair $(E,B)$ of class $C^k$, $k\ge 1$, define the covariant components of the electromagnetic two-form $F$ in the chart by $F_{00}=0$, $F_{0i}:=-E_i\mathrm{/}c$, $F_{ij}:={ϵ}_{ijk}{B}_k$ ($i,j,k\in \{1,2,3\}$) and $F_{ab}:=-F_{ba}$, the index-raised components being $F^{0i}=E_i\mathrm{/}c$, $F^{i0}=-E_i\mathrm{/}c$ and $F^{ij}={ϵ}_{ijk}{B}_k$. Here ${ϵ}_{ijk}$ is the spatial permutation symbol of The oriented volume form epsilon and the permutation symbols of an inertial frame. Then $F$ is a $C^k$ type-$(0,2)$ tensor field on $U$ in the sense of C^k tensor fields on affine Minkowski space and their inertial components, alternating at every event, and the components above are exactly its component functions: at $p\in U$ the tensor $F(p)$ is the bilinear form $F(p)(u,w)=F_{ab}(p)u^a{w}^b$ on $V$ in the coordinates $u^a,w^b$ of $u,w$ in $(e_0,e_1,e_2,e_3)$. Indeed each $F_{ab}$ is a finite sum of products of the $C^k$ functions $E_i,B_k$ of the chart coordinates with the constants $c,{ϵ}_{ijk}$, hence a $C^k$ function of $x(p)$, and the component criterion of C^k tensor fields on affine Minkowski space and their inertial components applies.

(b) [Observer decomposition] Let $n:=e_0$ be the future unit timelike vector of the chart and let $e_a:=F_{ab}{n}^b$, $b_a:=(\ast F{)}_{ab}{n}^b$ be the observer components of Hodge decomposition of an alternating two-form for an observer, invariants and the optional field identification; then $e_i=+E_i\mathrm{/}c$, $b_i=-B_i$ and $e_0=b_0=0$, and the invariants and square wedge of that theorem read

$\frac{1}{2}{F}_{ab}{F}^{ab}=\mathrm{\mid }B{\mathrm{\mid }}^2-\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}{c}^2,F_{ab}(\ast F{)}^{ab}=+4(E\cdot B)\mathrm{/}c,F\wedge F=-2(E\cdot B)\epsilon \mathrm{/}c,$

where $\mathrm{\mid }\cdot \mathrm{\mid }$ and $\cdot$ are the spatial Euclidean structure of the frame. (For the same pair $(E,B)$, that theorem's optional clause (d) uses $F_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}^{i0}=E_i\mathrm{/}c$ and $F_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}^{ij}=-{ϵ}_{ijk}{B}_k$, so $F_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}=-F$ in all six components, not only the magnetic block. This framework fixes clause (a) and uses the theorem's general algebra; its optional SI dictionary must be converted by negating the entire tensor.)

(c) [Potentials when they exist] If $\phi$ and $A$ are $C^1$ with $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $B=\mathrm{curl}A$, and $A^{\mu }:=(\phi \mathrm{/}c,A)$, $A_{\mu }:={\eta }_{\mu \nu }{A}^{\nu }$, then $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$; conversely, if a $C^1$ covector field $A_{\mu }$ on $U$ satisfies $F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu }$, then $E,B$ recovered from $F$ by clause (a) obey $E=-\mathrm{\nabla }\phi -{\mathrm{\partial }}_{t}A$ and $B=\mathrm{curl}A$ with $\phi :=-c{A}_0$ and the spatial potential vector $A:=(A_1,A_2,A_3)$. Here $\mathrm{\nabla }$ and $\mathrm{curl}$ are those of Divergence and curl of a $C^1$ vector field in the spatial coordinates, and the spatial cross-product conventions of The cross product in ${\mathbb{R}}^3$ are used. The definition does not assert that every field pair admits such a potential; where a $C^1$ potential represents the $C^1$-or-better field of clause (a), the field is closed in the explicit sense ${\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=0$. This conclusion does not require second classical derivatives of the potential: put $H_{\lambda \mu \nu }:={\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }$, which is continuous because $F$ is $C^1$; for every smooth compactly supported test $\psi$, substituting $F=dA$ and integrating by parts twice, moving the derivatives first onto $\psi$ (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign (b)) and then off the potential factors after commuting only the smooth test derivatives (Clairaut--Schwarz theorem for continuous second partial derivatives), gives $\int H\psi =0$; the fundamental lemma (a) of Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign then gives $H=0$. The framework's potential theorems establish existence only under their stated hypotheses; Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ supplies the coordinate derivatives ${\mathrm{\partial }}_{\mu }$ used throughout.

(d) [Four-current and index-raising operator] With the adopted four-current $j=(c\rho ,J)$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, the raised-index operator $\mathrm{\Phi }$ of Antisymmetric contractions vanish and $m{U}^{\mathrm{\prime }}=q\mathrm{\Phi }(U)$ preserves $\eta (U,U)$, defined pointwise on $V$ by $\eta (\mathrm{\Phi }(u),w)=F(w,u)$, satisfies $\mathrm{\Phi }(u{)}^{\mu }={\eta }^{\mu \alpha }{F}_{\alpha \beta }{u}^{\beta }$; the components of $F$ carry the tesla (T), those of $j$ carry A m${}^{-2}$, and those of a four-potential $A^{\mu }$ carry T m.