Restricted smooth charged-dust model

Let $U$ be an open subset of the smooth oriented, time-oriented Lorentzian four-manifold and use the metric, units, and electromagnetic conventions of Einstein–Maxwell primitive quantities and SI conventions. A restricted smooth charged-dust configuration consists of constants $m>0$ in kilograms and $q\in \mathbb{R}$ in coulombs, a smooth future-directed vector field $U^a$ with $g_{ab}{U}^a{U}^b=-c^2$, and a smooth scalar field $\nu :U\to (0,\mathrm{\infty })$. Put $u^a:=U^a\mathrm{/}c$, so $g_{ab}{u}^a{u}^b=-1$, and at each $x\in U$ define its local rest space by $H_x:=\{v\in T_{x}M:g_x(v,u_x)=0\}$. The restriction ${\gamma }_x:=g_x{\mathrm{\mid }}_{H_x}$ is a positive-definite inner product. In any basis $(e_1,e_2,e_3)$ of $H_x$ with linear coordinates $(y^1,y^2,y^3)$, its proper 3-volume density is $d{V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}=\sqrt{\det ({\gamma }_x(e_i,e_j))}\mathrm{\mid }d{y}^{1}d{y}^{2}d{y}^3\mathrm{\mid }$; this density is independent of the chosen basis. The scalar $\nu (x)$ is the proper number density measured in this rest space: the local particle-number measure has density $\nu d{V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$. Thus $[\nu ]={\mathrm{m}}^{-3}$. Define the rest-space projector $h^a{}_b={\delta }^a{}_b+U^a{U}_b\mathrm{/}{c}^2$. The associated four-current and matter stress tensor are

$j^a:=q\nu U^a,T_m^{ab}:=m\nu U^a{U}^b\mathrm{.}$

Their length-coordinate SI units are $[U^a]=\mathrm{m}{\mathrm{s}}^{-1}$, $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, and $[T_m^{ab}]=\mathrm{J}{\mathrm{m}}^{-3}$. The single-velocity condition means that at every event in the model domain all dust matter there is assigned the same vector $U_x$; a state with distinct future unit timelike velocities at one event is outside this model. In this item, a caustic or multistream breakdown is a point where the material description can no longer be represented by smooth fields $\nu ,U$ with $0<\nu <\mathrm{\infty }$ and one such velocity. The model domain is restricted to open regions excluding those breakdown points. This is a scope convention, not a claim that caustics are avoided or that the fields exist globally. As the continuum closure of this model, adopt on the smooth solution domain

${\mathrm{\nabla }}_a(\nu U^a)=0,m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U),$

where ${\mathrm{\nabla }}_U{U}^a:=U^b{\mathrm{\nabla }}_b{U}^a$ and $g(F^{\mathrm{♯}}(v),w)=F(w,v)$, as in the external-charge convention of Stationary charged action and Lorentz force. The latter supplies test-body sign and normalization context only; it proves neither the continuum number equation nor a continuum or coupled PDE evolution law.

This definition applies only while $\nu ,U,F,g$ remain smooth and the single-velocity dust description is valid. It makes no claim that such solutions exist globally or avoid caustics, and it does not assign a distributional stress tensor to point particles.