Charge parameter, frame charge and current densities, volume charge and flux, and the four-current

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.