Primitives and postulates

Let $(A,V,\eta ,C^{+})$ be an oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with inertial chart $x:A\to {\mathbb{R}}^4$ (Inertial frames and inertial charts of the affine Minkowski model), let $c>0$ be the framework's conversion constant, and let $\delta$ be the Euclidean spatial form of that frame on $n^{\perp }$, $n:=e_0$, with norm $\mathrm{\mid }\cdot \mathrm{\mid }$ and positive spatial orientation (The Euclidean spatial structure and cross product of an inertial frame). Fix a nonempty open interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and a nonempty open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, and put

$U:=\{p\in A:\text{ there are }t\in I,y\in \mathrm{\Omega }\text{ with }{x}^0(p)=ct,(x^1,x^2,x^3)(p)=y\}\mathrm{.}$

Then:

(a) [Chart restriction] $x(U)=\{(ct,y):t\in I,y\in \mathrm{\Omega }\}$ and $\mathrm{\Phi }:U\to I\times \mathrm{\Omega }$, $\mathrm{\Phi }(p):=(x^0(p)\mathrm{/}c,(x^1,x^2,x^3)(p))$, is a bijection, because the inertial chart $x:A\to {\mathbb{R}}^4$ is a bijection and $c>0$; its components are the laboratory time coordinate $t(p):=x^0(p)\mathrm{/}c$ and the laboratory position $r(p):=(x^1(p),x^2(p),x^3(p))$. In the chart coordinates the pair $(t(p),r(p))$ is the reading of the laboratory decomposition of $p$. Writing $r(p)=(r^1(p),r^2(p),r^3(p))\in {\mathbb{R}}^3$, the standard basis of ${\mathbb{R}}^3$ (The standard list $e:n\to F^n$ with $e_i(i)=1_F$ and $e_i(j)=0_F$ for $j\mathrm{\ne }i$ is an ordered basis of $F^n$; hence ${\dim }_F{F}^n=n$, and $F^0$ is the zero space with basis $\mathrm{\varnothing }$ and dimension $0$) labels the three coordinate directions, and the identification ${\mathbb{R}}^3\cong n^{\perp }$ is the linear map ${\sum }_{i=1}^3{r}^i{e}_i$ into the three-dimensional real vector space $n^{\perp }$ (Vector space over a field, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

(b) [Slices and spatial structure] For each $t\in I$ the slice $U_t:=\{p\in U:t(p)=t\}$ is carried bijectively onto $\mathrm{\Omega }$ by $r$; the constant vector $n=e_0$ is future unit timelike, and $(e_1,e_2,e_3)$ is the $\delta$-orthonormal positively oriented basis of $n^{\perp }$ in which the coordinates of $r$ are expressed.

(c) [Laboratory fields] For a nonnegative integer $m$, a laboratory field of rank $m$ is a map $F:\mathrm{\Omega }\times I\to {\mathbb{R}}^m$; it is identified with the map $U\to {\mathbb{R}}^m$, $p\mapsto F(r(p),t(p))$, and with the family $(F(\cdot ,t){)}_{t\in I}$ of maps $\mathrm{\Omega }\to {\mathbb{R}}^m$, its values at fixed time. Such a field is of class $C^k$, for $m\ge 1$, when the map $(y,t)\mapsto F(y,t)$ is $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms. For $m=0$, set ${\mathbb{R}}^0:=\{()\}$; the unique field into this singleton is declared $C^{\mathrm{\infty }}$ (and $C^k$ for every finite $k$), and each of its partial derivatives is the same singleton-valued zero map. For $m\ge 1$ and $k\ge 1$, ${\mathrm{\partial }}_{y_i}F$ (at fixed $t$) and ${\mathrm{\partial }}_{t}F$ (at fixed $y$) are the partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. No regularity of a field is asserted beyond the class stated for it.

(d) [Conventions] The product decomposition is a coordinate identification inside the single inertial frame $(o;e_0,e_1,e_2,e_3)$; it is not a Galilean transformation, it does not compare two inertial frames, and it does not assert absolute simultaneity. The objects $I,\mathrm{\Omega },t,r,\delta$ and the identification ${\mathbb{R}}^3\cong n^{\perp }$ belong to the named frame; changes of inertial description are the proper orthochronous maps adopted in Adopted inertial covariance of the model equations. The frame is an origin together with an ordered basis; it is not a congruence of observers, and no moving observer field is defined here.

Adopt the SI system of units for this framework, with the following quantity table and conventions; every symbol is a quantity of this framework and the laboratory structure is that of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields.

(a) [Coordinates and constants] $x^0=ct$ and each spatial coordinate $x^i$ carries the metre (m); the time coordinate $t$ carries the second (s); the conversion constant $c>0$ carries m s${}^{-1}$ and has the exact SI defining value $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$; the elementary charge $e$ has the exact SI defining value $e=1.602176634\times {10}^{-19}\mathrm{C}$, and this item does not assert that physical charge is quantised; the magnetic constant ${\mu }_0>0$ carries N A${}^{-2}$ and is measured, so no exact numerical value of ${\mu }_0$ is asserted. The pre-2019 SI fixed the historical nominal ${\mu }_{0,\mathrm{n}\mathrm{o}\mathrm{m}}:=4\pi \times {10}^{-7}\mathrm{H}{\mathrm{m}}^{-1}=4\pi \times {10}^{-7}\mathrm{N}{\mathrm{A}}^{-2}$ exactly; since the revised SI took effect on 20 May 2019, ${\mu }_0$ is experimentally determined. The SI brochure quotes the nominal ${\mu }_{0,\mathrm{n}\mathrm{o}\mathrm{m}}$ as the estimate at the 2019 adoption with relative standard uncertainty $2.3\times {10}^{-10}$; this is historical estimate and uncertainty context, not an exact equality for the actual ${\mu }_0$ or a current measured value. The electric constant is defined within the framework by

${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$

in F m${}^{-1}$, consistent with the exact SI relation ${\epsilon }_0{\mu }_0=1\mathrm{/}{c}^2$, so that $c^2{\epsilon }_0{\mu }_0=1$ and ${\epsilon }_0$ inherits the relative standard uncertainty of ${\mu }_0$.

(b) [Electromagnetic quantities] Charge $q$ carries the coulomb (C); the charge density $\rho$ of the framework's source definition carries C m${}^{-3}$ and the current density $J$ carries A m${}^{-2}$, so that the four-current components $(c\rho ,J)$ carry A m${}^{-2}$; the electric field $E$ carries V m${}^{-1}$; the magnetic field $B$ carries the tesla (T), where $1\mathrm{T}=1\mathrm{N}{\mathrm{A}}^{-1}{\mathrm{m}}^{-1}$; the scalar potential $\phi$ carries V and the vector potential $A$ carries T m, so that the four-potential components $(\phi \mathrm{/}c,A)$ carry T m; the components of the electromagnetic two-form of the framework's tensor definition carry T; a force carries the newton (N), $1\mathrm{N}=1\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}$.

(c) [Mechanical quantities] A rest mass parameter $m$ carries the kilogram (kg) and a four-momentum $P=mU$ of the relativistic particle model carries kg m s${}^{-1}$; field energy density carries J m${}^{-3}$, energy flux W m${}^{-2}$, so that $1\mathrm{W}=1\mathrm{J}{\mathrm{s}}^{-1}=1\mathrm{V}\mathrm{A}$.

(d) [Bookkeeping only] The unit of a quantity records the adopted system of units; it does not show that a quantity is primitive in this or any other framework, it contains no measurement, and it does not identify a quantity with a similarly named quantity of another framework. The exact defining values of $c$ and $e$ fix numerical conventions of the SI; they do not by themselves identify $c$ with the vacuum propagation constant of ideal light, which is the physical adoption of the framework's Minkowski postulate.

Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its spatial Euclidean structure and its units be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values, and let $k\ge 1$. An electric field of class $C^k$ on the region is a map $E:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ that is of class $C^k$ in the sense of $C^k$ Euclidean maps and diffeomorphisms; a magnetic field of class $C^k$ is a map $B:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ of the same class. At fixed $t\in I$ the maps $E(\cdot ,t),B(\cdot ,t):\mathrm{\Omega }\to {\mathbb{R}}^3$ are real spatial vector fields on the region $\mathrm{\Omega }$, read in the $\delta$-orthonormal basis $(e_1,e_2,e_3)$ of the frame. The framework uses: $C^1$ fields in the classical Maxwell branch, $C^2$ fields where second or mixed time-space derivatives of the fields appear, and $C^{\mathrm{\infty }}$ fields in the smooth existence branches; every statement carries its own regularity and no regularity beyond it is asserted.

Conventions:

(a) The polar-vector and axial-vector conventions mean the following component rules under a spatial orthogonal coordinate change $y^{\mathrm{\prime }}=Qy$, where $Q$ is a real $3\times 3$ matrix with $Q^{\mathsf{T}}Q=I_3$. Transpose, matrix product and identity are those of The transpose $A^{\mathsf{T}}$ of a matrix and Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, and $\det Q$ is the determinant of For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix. At corresponding points $E^{\mathrm{\prime }}(y^{\mathrm{\prime }},t)=QE(y,t)$ and $B^{\mathrm{\prime }}(y^{\mathrm{\prime }},t)=(\det Q)QB(y,t)$. Thus polar and axial components agree in their transformation rule when $\det Q=1$ and differ by a sign when $\det Q=-1$. These rules specify the orientation convention; the adopted inertial descriptions have positive orientation, and no physical orientation-reversing change of description is adopted here.

(b) The fields are defined by their values on $\mathrm{\Omega }\times I$, and no value at a point of singular source support is asserted.

(c) $E$ carries V m${}^{-1}$ and $B$ carries T.

(d) The identification of the pair $(E,B)$ with the components of an alternating two-form, including its transformation behaviour, is made in the framework's electromagnetic two-form definition on this page and is not presupposed here.

Adopt as physical assumptions of this framework:

(i) The oriented time-oriented affine Minkowski space $(A,V,\eta ,C^{+})$ used by Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, together with the conversion constant $c>0$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values, is the spacetime of the framework: its points represent idealized physical events, displacements represent idealized spacetime separations, and the causal classification of the model is the framework's causal structure. In the concrete coordinate realization The Minkowski affine four-space: the form $\eta$, causal cones, frames and Lorentz maps, $A=V={\mathbb{R}}^4$ and $\eta =\mathrm{diag}(-1,1,1,1)$. That supplier uses the symbol $C^{+}$ for the nonzero future causal cone, including null vectors; here, as in the special-relativity model, $C^{+}$ denotes its future timelike subset. The future null boundary vectors retain their separate causal classification.

(ii) The inertial frames and charts of Inertial frames and inertial charts of the affine Minkowski model are the inertial descriptions of the framework, and changes of description are the proper orthochronous isometries adopted in Adopted inertial covariance of the model equations.

(iii) The ideal clock reading rule along future timelike curves and the identification of ideal vacuum light rays with future null straight lines, both with the same constant $c$, are those adopted in Adopted ideal-clock and ideal-vacuum-light interpretation.

(iv) The laboratory description of fields and sources is the restricted inertial chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the spatial Euclidean structure and the SI conventions of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values.

Status and scope: the mathematical model defines the geometry and its theorems prove geometric facts only; that physical events are described by the model, that ideal clocks read the stated proper time and that ideal vacuum light follows null straight lines are adopted assumptions, not deductions from experiment or from the mathematical axioms. The special-relativity postulates adopted here do not by themselves select the Maxwell equations, a force law or any matter model: the electromagnetic equations and couplings of this framework are further postulates of it. The framework is classical, flat and non-quantum; gravitational curvature, quantum fields and measurement postulates are outside its scope.

The particle and light objects of the framework are the following curves and vector fields of the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model), with the physical interpretation of Adopted Minkowski physical model of the electromagnetic framework and the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Worldlines] A worldline is a regular causal curve $x:J\to A$ in one of the regularity classes of Curves, regular causal curves and future timelike worldlines in affine Minkowski space ($J$ an order-convex subset of $\mathbb{R}$ with at least two elements). A massive worldline is a $C^k$ future timelike worldline, $k\ge 1$, with everywhere nonzero future timelike tangent. A future null curve is a $C^1$ curve with everywhere nonzero future null tangent. An ideal vacuum light ray of the adopted physical model follows a null straight line: on its parameter interval it has the form $x(s)=p+\lambda (s)v$, where $p\in A$, $v\in V$ is nonzero future null, and $\lambda$ is $C^1$ with ${\lambda }^{\mathrm{\prime }}(s)>0$. This is the straight-ray adoption of Adopted Minkowski physical model of the electromagnetic framework; arbitrary future null curves are retained as kinematic objects without asserting that they describe vacuum light propagation. Along a null curve the integrand $-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})$ of the proper-time construction (Proper time along a future timelike worldline) vanishes identically, so that construction yields no proper-time parametrization of it, and a light ray is never treated as a massive worldline. A worldline is a curve in spacetime; its reading $r\circ x$ in a named laboratory chart is a coordinate description of it, not an object of another observer.

(b) [Four-velocity and four-acceleration] For a massive worldline the four-velocity $U$ and, when it is $C^2$, the four-acceleration $A$ are those of Four-velocity and four-acceleration of a future timelike worldline; wherever defined they satisfy $\eta (U,U)=-c^2$ and $\eta (U,A)=0$, by Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.

(c) [Chart reading] If a massive $C^2$ worldline is described in a laboratory chart of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields by $t\mapsto (ct,r(t))$ on an open interval of laboratory times, then $\mathrm{\mid }{r}^{\mathrm{\prime }}(t)\mathrm{\mid }<c$ for every $t$, with four-velocity $U=\gamma (c,r^{\mathrm{\prime }})$ and Lorentz factor $\gamma =(1-\mathrm{\mid }{r}^{\mathrm{\prime }}{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$, by clause (e) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition.

(d) [Frames and observers] An inertial frame in this framework is the origin-with-basis object of Inertial frames and inertial charts of the affine Minkowski model; a point observer is a future unit timelike vector $n\in V$ with rest space $n^{\perp }$ and rest projection $h$, as in clauses (c)-(d) of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. On an open spacetime region $W\subseteq A$, an observer congruence field means a $C^1$ map $n:W\to V$ with $n(p)$ future timelike and $\eta (n(p),n(p))=-1$ at every $p$; its observer histories, when given, are integral curves satisfying $dx\mathrm{/}d\tau =cn(x)$. A tetrad field is a tuple of $C^1$ maps $(f_0,f_1,f_2,f_3):W\to V^4$ whose values form a positively oriented $\eta$-orthonormal basis at every event, with $f_0$ future timelike. Regularity here means componentwise $C^1$ in inertial coordinates. These fields need not be constant and are distinct from an inertial origin-with-basis frame, its coordinate chart and a single point observer; no global congruence or tetrad is asserted to exist.

(e) [Scope] No force law, no self-field and no extended-body structure is supplied here; an accelerating massive worldline requires a force model stated separately, and an ideal light ray carries no dispersive, quantum or polarisation structure in this definition.

Let the laboratory region and its chart be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Charge parameter] The framework takes a real charge parameter $q\in \mathbb{R}$, measured in coulombs, attached to a source; its sign is the sign of the source and it is a parameter of the model. Charge quantisation is not asserted.

(b) [Smooth source fields] A smooth (volume) source field on $\mathrm{\Omega }\times I$ is a pair $(\rho ,J)$ of maps $\rho :\mathrm{\Omega }\times I\to \mathbb{R}$, $J:\mathrm{\Omega }\times I\to {\mathbb{R}}^3$ that is continuous on $\mathrm{\Omega }\times I$ and of class $C^1$ wherever a displayed equation differentiates it; $\rho$ is the frame charge density (C m${}^{-3}$) and $J$ the frame current density (A m${}^{-2}$). For each fixed $t\in I$ retain the compact Jordan branch: if $V\subseteq \mathrm{\Omega }$ is compact and Jordan measurable, define

$Q_R(V,t):={\int }_V\rho (y,t)dy$

by The Riemann integral of a bounded function over a bounded Jordan measurable set, with integrability supplied by A continuous real function on a compact Jordan measurable set is Riemann integrable over that set. For the full measurable-volume branch assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the published Lebesgue-measure interface. If $V$ is any bounded Lebesgue-measurable subset with compact closure $\bar{V}\subseteq \mathrm{\Omega }$, define

$Q(V,t):={\int }_V\rho (y,t)d{\lambda }_3(y)={\int }_{{\mathbb{R}}^3}{F}_t(y){\mathbf{1}}_V(y)d{\lambda }_3(y),$

where $F_t=\rho (\cdot ,t)$ on $\mathrm{\Omega }$ and $0$ elsewhere; the signed integral is defined through its positive and negative parts (Integral over a measurable subset, Integrable real and complex functions, and their integrals). The function $F_t$ is Borel, and hence Lebesgue measurable: for every open $O\subseteq \mathbb{R}$ its preimage inside the open set $\mathrm{\Omega }$ is relatively open in $\mathrm{\Omega }$, hence open in ${\mathbb{R}}^3$, and the closed complement of $\mathrm{\Omega }$ is added precisely when $0\in O$ (A continuous map has Borel preimages of Borel sets, Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable). This integral is absolutely finite: $\rho (\cdot ,t)$ is bounded on the compact $\bar{V}$ by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, $V$ has finite Lebesgue measure by Lebesgue measure is sigma-finite, and every metrically bounded subset of ${\mathbb{R}}^n$ has finite outer measure, and the monotonicity and homogeneity of the nonnegative integral (Monotonicity and nonnegative homogeneity of the nonnegative integral) bound $\int \mathrm{\mid }{F}_t\mathrm{\mid }{\mathbf{1}}_{V}d{\lambda }_3\le M_t{\lambda }_3(V)<\mathrm{\infty }$, the indicator integral being its simple-function value (The nonnegative integral agrees with the simple integral on simple functions); the positive and negative parts are measurable by Closure properties of measurable functions used by the integral. On the compact Jordan branch, under the same Countable Choice assumption, $Q=Q_R$ by Riemann–Lebesgue comparison for distribution test integrands. The empty region has charge zero in both branches. The current flux remains defined independently: for each oriented surface patch in $\mathrm{\Omega }$ the current flux at time $t$ is the surface integral of $J(\cdot ,t)$ in the induced orientation, as defined in Unit normal fields, orientations, and flux through a regular surface patch. Point sources and sheet sources are not source fields in this sense; they are different objects, defined by their own pairings when the framework uses them.

(c) [Covariant identification] Write $j:=(c\rho ,J)$, so $j^0=c\rho$ and $j^i=J^i$. The framework adopts as a model premise the identification of the charge-current of a source with a vector field of the affine Minkowski model, and hence the component transformation law of Transformation of tensor, covector, derivative and current components under affine Lorentz maps for $j$ between inertial charts. This is a model premise, not a theorem about physical currents.

(d) [Frame density, scalar multiplier and the normalized material branch] Suppose additionally that the identified $j$ is $C^1$ and $j={\rho }_{0}U$, with a $C^1$ future timelike vector field $U$ and a $C^1$ real scalar multiplier ${\rho }_0$. Since $U^0>0$, define the normalized chart direction $u:=U\mathrm{/}{U}^0$, so $u^0=1$, $u^i=U^i\mathrm{/}{U}^0$. Then $j=({\rho }_0{U}^0)u$ and, because the electromagnetic convention is $j^0=c\rho$,

$\rho ={\rho }_0{U}^0\mathrm{/}c,J^i={\rho }_0{U}^i=c\rho u^i\mathrm{.}$

Pointwise equality $\rho ={\rho }_0$ holds if and only if ${\rho }_0(U^0-c)=0$, including the zero-multiplier case. Where ${\rho }_0\mathrm{\ne }0$, equality is equivalent to $U^0=c$, which does not alone imply $U^i=0$ for an arbitrary future timelike $U$. Thus a scalar multiplier is not automatically a Lorentz rest charge density. The direction $u$ is chart normalized, with no unit or proper-four-velocity normalization imposed in this general branch; if its coordinate velocity interpretation is adopted, the spatial velocity is $v^i:=c{u}^i$.

For the separate convective source model, explicitly assume that $U$ is the material proper-time four-velocity, normalized by $\eta (U,U)=-c^2$ as in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition. This is an additional model condition, not a consequence of the factorization $j={\rho }_{0}U$. In that branch, $U=\gamma (c,v)$, $\mathrm{\mid }v\mathrm{\mid }<c$, $\gamma =(1-\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}{c}^2{)}^{-1\mathrm{/}2}$, and

$\rho =\gamma {\rho }_0,J=\rho v\mathrm{.}$

Here ${\rho }_0$ is the rest charge density of this stipulated convective model: in a comoving inertial frame $U=(c,0,0,0)$ and $j=(c{\rho }_0,0,0,0)$. The equality $\rho ={\rho }_0$ then holds exactly where ${\rho }_0=0$ or $v=0$ (equivalently $\gamma =1$). Both signs of charge and the zero-density case are retained.

(e) [Scope] This definition fixes the source objects of the smooth branch and their transformation interface. It asserts no conservation law (that is proved for stated source classes) and no regularity of singular sources.

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.

Adopt as a physical postulate of this framework the following four equations for the fields of The electric and magnetic fields as fixed-time spatial vector fields and the sources of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, with ${\mu }_0>0$, $c>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and the SI units of Framework-local SI units, the vacuum constants and the exact/measured status of their values.

(a) [Smooth branch] Let $E,B$ be jointly $C^1$ fields with the additional derivatives required by each equation existing and continuous, and let $\rho ,J$ be continuous. Then, with the divergence and curl of Divergence and curl of a $C^1$ vector field and the partial derivative ${\mathrm{\partial }}_t$ at fixed position of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$,

$\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E\mathrm{.}$

(b) [Weak branch] Let instead $E,B$ be locally integrable ${\mathbb{R}}^3$-valued fields and $\rho ,J$ locally integrable scalar and vector fields on $I\times \mathrm{\Omega }$, identified with their regular distributions (Regular distribution from a locally integrable function, Locally integrable functions embed in distributions). Let $\mathrm{div}$, $\mathrm{curl}$ and ${\mathrm{\partial }}_t$ act as distributional derivatives, componentwise, and let $\psi$ range over $C_c^{\mathrm{\infty }}(I\times \mathrm{\Omega })$ (Test function space d of an open set). Then the same four identities hold in the sense of Distribution: the pairings

$⟨\mathrm{div}E-\rho \mathrm{/}{\epsilon }_0,\psi ⟩,⟨\mathrm{div}B,\psi ⟩,⟨\mathrm{curl}E+{\mathrm{\partial }}_{t}B,\psi ⟩,⟨\mathrm{curl}B-{\mu }_{0}J-c^{-2}{\mathrm{\partial }}_{t}E,\psi ⟩$

all vanish for every such $\psi$, all derivatives being distributional derivatives (Distributional derivative).

(c) [Scope] The postulate contains no magnetic charge: $\mathrm{div}B=0$ and no magnetic-source term appears in $\mathrm{curl}E$. Sources that are not locally integrable functions (point or sheet sources) are covered by neither branch and require their own definition. The equations are adopted classical vacuum physics in the named inertial chart; they are not deduced from charge conservation, from empirical agreement or from the special-relativity postulates, and this item by itself asserts no covariance under changes of inertial frame.

Assume the vacuum Maxwell postulate The microscopic vacuum Maxwell equations: smooth and weak branches on the laboratory region $U\cong I\times \mathrm{\Omega }$ of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields with the fields of The electric and magnetic fields as fixed-time spatial vector fields and the sources of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current, and fix the constants ${\mu }_0>0$, $c>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ of Framework-local SI units, the vacuum constants and the exact/measured status of their values. Let $n\ge 1$, let $(E_1,B_1,{\rho }_1,J_1),\dots ,(E_n,B_n,{\rho }_n,J_n)$ be $n$ field-source pairs each satisfying one and the same branch of the postulate on the same region (all in the smooth branch, or all in the weak branch), and let $a_1,\dots ,a_n\in \mathbb{R}$. Put $E:={\sum }_{j=1}^n{a}_j{E}_j$, $B:={\sum }_{j=1}^n{a}_j{B}_j$, $\rho :={\sum }_{j=1}^n{a}_j{\rho }_j$ and $J:={\sum }_{j=1}^n{a}_j{J}_j$. Then:

(a) $(E,B,\rho ,J)$ satisfies the same branch of the postulate on the same region with the same constants: pointwise in the smooth branch and in the sense of the pairings of clause (b) of The microscopic vacuum Maxwell equations: smooth and weak branches in the weak branch.

(b) Nothing beyond this linearity is asserted: the result does not apply to a source whose density depends nonlinearly on the fields (a nonlinear constitutive law), and no convergence statement is made for infinite series or integrals of solutions or for infinite linear combinations.

(c) The theorem is conditional on the adopted postulate and is not an empirical claim about nature.

Adopt as physical postulates of this framework, on the spacetime and particle model of Adopted Minkowski physical model of the electromagnetic framework and Worldlines, four-velocity, chart readings and the frame/observer distinction and on the special-relativity particle model Adopted massive-particle momentum, observer measurement and isolated-collision conservation:

(i) [Test charge in a prescribed external field] Let $q\in \mathbb{R}$ be a constant charge parameter (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current), $m>0$ an externally supplied rest mass, $x:J\to A$ a future timelike $C^2$ massive worldline with four-velocity $U$, and let $F$ be a prescribed field of The electromagnetic two-form, its observer decomposition and the four-current that is $C^1$ on a neighbourhood of the graph of $x$ and is generated externally to the particle. Adopt the equation of motion

$m\frac{dU}{d\tau }=q\mathrm{\Phi }(U),$

where, at each original parameter $s\in J$, $\mathrm{\Phi }={\mathrm{\Phi }}_{x(s)}$ is the raised-index operator of the field at $x(s)$, defined by $\eta ({\mathrm{\Phi }}_{x(s)}(u),w)=F(x(s))(w,u)$ (Antisymmetric contractions vanish and $m{U}^{\mathrm{\prime }}=q\mathrm{\Phi }(U)$ preserves $\eta (U,U)$); equivalently $md{U}^{\mu }\mathrm{/}d\tau =q{F}^{\mu \nu }{U}_{\nu }$. Here $dU\mathrm{/}d\tau$ is the derivative of $U$ with respect to its proper-time function $\tau (s)$: explicitly $dU\mathrm{/}d\tau =U^{\mathrm{\prime }}(s)\mathrm{/}{\tau }^{\mathrm{\prime }}(s)$, with ${\tau }^{\mathrm{\prime }}(s)=c^{-1}\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}>0$. It is the four-acceleration of the worldline (Worldlines, four-velocity, chart readings and the frame/observer distinction, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition). The particle's own field is excluded from $F$: this is the external-field coupling, and it defines no self-force.

(ii) [Continuum matter coupling] For a smooth source $(\rho ,J)$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current interacting with the field through an explicitly stated matter model to which this coupling applies, adopt the force density $\rho E+J\times B$ (unit N m${}^{-3}$) and the power density $J\cdot E$ (unit W m${}^{-3}$) as the electromagnetic source terms of that matter model, with the spatial cross product of The cross product in ${\mathbb{R}}^3$; they describe the local interaction of the field with matter and are not a property of a prescribed current alone.

(iii) [Scope] The rest mass and the charge are inputs; $m$ is not derived within electromagnetism. The coupling is adopted for constant $q$ and $m>0$; no magnetic-monopole coupling, no radiation reaction and no self-force are included. In particular the nonrelativistic coordinate statement $m{r}^{\mathrm{\prime }\mathrm{\prime }}=q(E+r^{\mathrm{\prime }}\times B)$ is not adopted as exact: the adopted statement is the covariant one of clause (i), and any low-speed statement requires its own approximation hypotheses and error bounds.