Smooth matter quantities, current and observer observables

Let $\mathrm{\Omega }$ be an open subset of the smooth, oriented, time-oriented Lorentzian four-manifold $(M,g,N,o)$ of Einstein–Maxwell primitive quantities and SI conventions, with signature $(-,+,+,+)$ and metric-selected Levi–Civita connection $\mathrm{\nabla }$. The exact tensor types, formal matter systems, stress convention, observer conventions and smoothness levels below are those of Smooth formal matter, EM tensors and observer quantities. Use SI units with length coordinates $x^0=ct$ as in Einstein–Maxwell primitive quantities and SI conventions; no numerical value of the experimentally determined ${\mu }_0$ is treated as exact.

At a point $p\in \mathrm{\Omega }$, an observer is a future-directed unit timelike vector $n\in T_{p}M$ satisfying $g(n,n)=-1$. A smooth field of such vectors is an observer congruence. A local tetrad on an open set $V\subseteq \mathrm{\Omega }$ is a tuple of smooth vector fields $(e_0,e_1,e_2,e_3)$ forming a basis of $T_{x}M$ at every $x\in V$ and satisfying $g(e_{\alpha },e_{\beta })={\eta }_{\alpha \beta }$, where $\eta =\mathrm{diag}(-1,1,1,1)$, as in Einstein–Maxwell primitive quantities and SI conventions; its value at a point is an orthonormal tangent-space basis. An observer, a congruence, a tetrad and a coordinate chart are different objects. A four-velocity field $U$ is a $C^2$ future-timelike vector field with $g(U,U)=-c^2$; $n=U\mathrm{/}c$ is its unit observer at each point. A proper-time worldline segment is a $C^2$ future-directed timelike curve $z:I\to \mathrm{\Omega }$ on an interval $I$ of proper time $\tau$ in seconds, with $dz\mathrm{/}d\tau =U$ and $g(U,U)=-c^2$ along the segment. These are local model fields and curves; no global flow or completeness is included.

The matter variables are constitutive inputs of a selected model: rest-mass density $\rho$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, rest-charge density $\sigma$ in $\mathrm{C}{\mathrm{m}}^{-3}$, rest-energy density $ϵ$ in $\mathrm{J}{\mathrm{m}}^{-3}$, and isotropic pressure $p$ in $\mathrm{J}{\mathrm{m}}^{-3}$. A smooth matter current $j^a$ has units $\mathrm{A}{\mathrm{m}}^{-2}$, and a symmetric matter stress tensor $T_m^{ab}$ has units $\mathrm{J}{\mathrm{m}}^{-3}$ in the stated length-coordinate convention. Their physical meanings are fixed only after a constitutive branch is selected:

• For charged dust, $T_m^{ab}=\rho U^a{U}^b$ and $j^a=\sigma U^a$. Here $\rho \ge 0$ is rest-mass density, not energy density.
• For a perfect fluid, $T_m^{ab}=(ϵ+p)U^a{U}^b\mathrm{/}{c}^2+p{g}^{ab}$. Its rest-frame entries are $(ϵ,p,p,p)$. No equation of state is included by this tensor definition.
• An optional conducting-fluid current is $j^a=\sigma U^a+q^a$, where $q^a{U}_a=0$ and $q^a$ is a rest-space conduction-current density in $\mathrm{A}{\mathrm{m}}^{-2}$. The separate constitutive choice $q^a=\chi E_U^a$ uses a scalar conductivity $\chi \ge 0$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and is not part of the general definition.

For a smooth real electromagnetic two-form $F$ and positive ${\mu }_0$, define the electromagnetic stress by

$T_{\mathrm{E}\mathrm{M}}^{ab}:=\frac{1}{{\mu }_0}(F^{ac}{F}^b{}_c-\frac{1}{4}{g}^{ab}{F}_{cd}{F}^{cd}),T_{ab}^{\mathrm{E}\mathrm{M}}:=g_{ac}{g}_{bd}{T}_{\mathrm{E}\mathrm{M}}^{cd}\mathrm{.}$

The components of $F$ in an orthonormal tetrad have tesla units. For an observer $n$, the observer electric field is $E_n^a:=c{F}^a{}_b{n}^b$ in $\mathrm{V}{\mathrm{m}}^{-1}$; with $U=cn$, write $E_U^a:=F^a{}_b{U}^b=E_n^a$. The observer magnetic field is the spatial covector $B_{n,a}:=-({\star }_{g}F{)}_{ab}{n}^b$ in tesla, with the orientation and Hodge convention of Einstein–Maxwell primitive quantities and SI conventions. These observer fields depend on $n$; they are not global observer-independent three-vectors.

Define the matter-plus-field force density by $f^a:=F^a{}_b{j}^b$, in $\mathrm{N}{\mathrm{m}}^{-3}$, and the matter exchange equation by ${\mathrm{\nabla }}_a{T}_m^{ab}=f^b$. This equation is a condition of a selected coupled model, not a consequence of the tensor definitions alone. For any unit observer $n$, define the measured energy density of a symmetric stress tensor $T_{ab}$ by ${\mathcal{E}}_n:=T_{ab}{n}^a{n}^b$ in $\mathrm{J}{\mathrm{m}}^{-3}$. Define its spatial energy-current vector by ${\mathcal{S}}_n^a:=-h_n{}^a{}_c{T}^c{}_b{n}^b$, where $h_n{}^a{}_b={\delta }^a{}_b+n^a{n}_b$ projects to $n^{\perp }$; its orthonormal-frame physical energy flux is $c{\mathcal{S}}_n^i$ in $\mathrm{W}{\mathrm{m}}^{-2}$. These contractions are derived, observer-dependent quantities.

Within the selected Einstein–Maxwell formulation, the fields $g$ and $F$ may be dynamical unknowns in a coupled branch or prescribed data in a test branch. The matter variables $\rho ,\sigma ,ϵ,p$ and any chosen conductivity or equation of state are independent constitutive input. The current, stress, force density, electromagnetic stress, observer fields and observer contractions are derived from those inputs by the displayed definitions. This definition makes no identification with Newtonian or thermodynamic quantities solely because units agree; it records no measurement and asserts no matter evolution, entropy law, or coupled Einstein–Maxwell solution.

On an open region $\mathrm{\Omega }$ of the smooth oriented, time-oriented Lorentzian spacetime and SI conventions of Smooth matter quantities, current and observer observables, adopt the equation-based Einstein and smooth curved Maxwell systems of Equation-based Einstein–Maxwell model and Smooth curved vacuum Maxwell constitutive adoption, with the smooth matter quantities and electromagnetic stress of Smooth matter quantities, current and observer observables. Thus the coupled branch includes the adopted Einstein equation $G_{ab}+\mathrm{\Lambda }{g}_{ab}=(8\pi G_N\mathrm{/}{c}^4)(T_{m,ab}+T_{ab}^{\mathrm{E}\mathrm{M}})$ and Maxwell equations $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, using the fixed local SI signs, units and curvature convention of those suppliers. Their adopted equations remain physical assumptions; neither is deduced here from flat spacetime, variational stationarity, or mathematical structure.

Choose smooth $\rho ,\sigma ,U$ on $\mathrm{\Omega }$, with $\rho \ge 0$, $U$ future timelike, $g(U,U)=-c^2$, and $\sigma =0$ wherever $\rho =0$. Select the pressureless dust closure

$T_m^{ab}=\rho U^a{U}^b,j^a=\sigma U^a\mathrm{.}$

As the matter model, adopt the undivided smooth mass and Lorentz-force equations

${\mathrm{\nabla }}_a(\rho U^a)=0,\rho U^a{\mathrm{\nabla }}_a{U}^b=\sigma F^b{}_c{U}^c,$

and charge-current conservation ${\mathrm{\nabla }}_a{j}^a=0$. On each open region ${\mathrm{\Omega }}_{+}:=\{p\in \mathrm{\Omega }:\rho (p)>0\}$ define $\alpha :=\sigma \mathrm{/}\rho$. Under the adopted mass continuity and charged-dust forms, Charged dust balance equivalence and advected charge ratio gives the exact equivalence between charge-current conservation and transport $U^a{\mathrm{\nabla }}_a\alpha =0$. The ratio is not defined and no division is made on $\rho =0$. This ratio statement is restricted to ${\mathrm{\Omega }}_{+}$; no global coupled evolution theorem is asserted.

This is a selected classical pressureless continuum model with one four-velocity field $U$ on the stated smooth domain. It excludes viscosity, conduction, multistream averaging, spin, point sources, self-field products and a general dust action. The finely fixed charge-to-mass ratio used in the specific ECD solutions of Meinel and Hütten is not a law of this general model. The postulate asserts neither a global smooth continuation nor validity of a single-velocity description beyond the stated domain; it makes no coupled-solution existence, uniqueness, stability, or empirical claim.

On a smooth open state domain $\mathrm{\Omega }$ in the spacetime and SI conventions of Smooth matter quantities, current and observer observables, choose $C^1$ rest-energy density $ϵ>0$, pressure $p$, rest-charge density $\sigma$, and a $C^2$ future-timelike field $U$ with $g(U,U)=-c^2$. The equation-based coupled branch uses Equation-based Einstein–Maxwell model and Smooth curved vacuum Maxwell constitutive adoption with a smooth symmetric matter tensor and current supplied by this selected fluid model. Adopt the perfect-fluid tensor

$T_m^{ab}=\frac{ϵ+p}{c^2}{U}^a{U}^b+p{g}^{ab}$

and either the convective current $j^a=\sigma U^a$ or a separately specified conduction current $j^a=\sigma U^a+q^a$ with $q^a{U}_a=0$. Adopt charge continuity ${\mathrm{\nabla }}_a{j}^a=0$ and the local matter-exchange balance ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$. The fields and derivatives are smooth enough for these equations to hold pointwise.

For an optional Ohmic branch, choose a smooth scalar conductivity $\chi \ge 0$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and impose $q^a=\chi E_U^a$, where $E_U^a=F^a{}_b{U}^b$ is the rest-frame electric field in $\mathrm{V}{\mathrm{m}}^{-1}$. This is a separate constitutive adoption, not a consequence of covariance or of the Maxwell equations. A smooth equation of state may be supplied on the chosen local state domain; no particular function, relativistic entropy law, causality assertion, or thermodynamic interpretation of $ϵ,p$ is included here. The stress-projection and comoving work identities follow conditionally from Perfect-fluid exchange projections and conduction heating, but no global fluid or conducting-fluid solution theory is asserted.

This postulate adopts a local classical matter branch. It does not presume an unspecified matter action or infer empirical validity from the source literature.

Let ${\mathrm{\Omega }}_{+}\subseteq \mathrm{\Omega }$ be an open positive-density region of the charged-dust model in Selected transported charged-dust Einstein–Maxwell model. Let $\rho ,\sigma \in C^1({\mathrm{\Omega }}_{+},\mathbb{R})$ with $\rho >0$, let $U\in C^2({\mathrm{\Omega }}_{+},T{\mathrm{\Omega }}_{+})$ be future timelike with $g(U,U)=-c^2$, and let the smooth electromagnetic two-form and metric be those of that model. Set $a^b:=U^a{\mathrm{\nabla }}_a{U}^b$, $T_m^{ab}:=\rho U^a{U}^b$, and $j^a:=\sigma U^a$.

On ${\mathrm{\Omega }}_{+}$ the adopted charged-dust matter-exchange equation ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$ is equivalent to

${\mathrm{\nabla }}_a(\rho U^a)=0,\rho a^b=\sigma F^b{}_c{U}^c\mathrm{.}$

Under the mass-continuity equation, ${\mathrm{\nabla }}_a{j}^a=0$ is equivalent to transport of the charge-to-mass ratio $\alpha :=\sigma \mathrm{/}\rho$ along the flow, $U^a{\mathrm{\nabla }}_a\alpha =0$. These are model deductions on ${\mathrm{\Omega }}_{+}$; the ratio and the divided force law are not defined at zero density. The result is local to its stated smooth positive-density domain and asserts neither a global smooth continuation nor validity of the single-velocity model beyond that domain. It supplies no Maxwell or Einstein solution or coupled-field existence theorem.

Let $\mathrm{\Omega }$ be a smooth open state domain of Selected charged perfect-fluid exchange/Ohmic branch. Assume $ϵ,p,\sigma \in C^1(\mathrm{\Omega },\mathbb{R})$, $U\in C^2(\mathrm{\Omega },T\mathrm{\Omega })$ is future timelike with $g(U,U)=-c^2$, $F$ is a smooth electromagnetic two-form, and the matter stress and current satisfy the selected fluid closure and exchange equation. Define

$D:=U^a{\mathrm{\nabla }}_a,\theta :={\mathrm{\nabla }}_a{U}^a,a^b:=D{U}^b,h^b{}_c:={\delta }^b{}_c+\frac{U^b{U}_c}{c^2},f^b:=F^b{}_c{j}^c,E_U^a:=F^a{}_b{U}^b\mathrm{.}$

Then the longitudinal work and rest-space force equations are

$Dϵ+(ϵ+p)\theta =-U_b{f}^b,\frac{ϵ+p}{c^2}{a}^b+h^{ba}{\mathrm{\nabla }}_{a}p=h^b{}_c{f}^c\mathrm{.}$

For the convective branch $j^a=\sigma U^a$, the comoving electromagnetic work $-U_b{f}^b$ is zero. For the optional smooth Ohmic branch $j^a=\sigma U^a+q^a$, $q^a=\chi E_U^a$, with $\chi \ge 0$, it is

$-U_b{f}^b=\chi g(E_U,E_U)\ge 0.$

Here $E_U$ is the rest-frame electric field and the induced metric on $U^{\perp }$ is positive definite. This is a conditional mechanical energy identity; it does not assert an entropy-production law, equation-of-state evolution, causal constitutive theory, or fluid existence.

Let ${\mathrm{\Omega }}_{+}$ be a smooth positive-density region of the selected charged-dust model, and require $F,j,T_m$ to satisfy the smoothness hypotheses of the Maxwell-stress and Killing-current suppliers. Assume the dust fields satisfy the model equations and the conditional matter-exchange identity of Dust continuity, transport and Lorentz acceleration, and assume the smooth Maxwell equations of Smooth curved vacuum Maxwell constitutive adoption. Define $T_{\mathrm{t}\mathrm{o}\mathrm{t}}^{ab}:=T_m^{ab}+T_{\mathrm{E}\mathrm{M}}^{ab}$. Let $\xi$ be a smooth Killing field on ${\mathrm{\Omega }}_{+}$, meaning ${\mathcal{L}}_{\xi }g=0$, and set $J_{\xi }^a:=T_{\mathrm{t}\mathrm{o}\mathrm{t}}^{ab}{\xi }_b$. For any finite smooth singular $4$-chain $C$ in ${\mathrm{\Omega }}_{+}$ with boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, define ${\mathcal{Q}}_{\xi }(\mathrm{\Sigma }):={\int }_{\mathrm{\Sigma }}{\iota }_{J_{\xi }}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Then

${\mathrm{\nabla }}_a{T}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{ab}=0,{\mathrm{\nabla }}_a{J}_{\xi }^a=0,{\mathcal{Q}}_{\xi }({\mathrm{\Sigma }}_2)-{\mathcal{Q}}_{\xi }({\mathrm{\Sigma }}_1)=-{\mathcal{Q}}_{\xi }(B)\mathrm{.}$

The two slice charges are equal only when the side flux ${\mathcal{Q}}_{\xi }(B)$ vanishes. This is a matter-plus-electromagnetic stress current, not a local gravitational-energy tensor; no asymptotic or universal global energy is asserted.

There is also the following separate test-body clause. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the canonical tangent-bundle smooth structure. On a local potential domain $V\subseteq {\mathrm{\Omega }}_{+}$, let $A$ be a supplied smooth one-form with $F=dA$ and let $\chi$ be a smooth scalar such that ${\mathcal{L}}_{\xi }A=d\chi$. Let $z:[a,b]\to V$ be a finite $C^2$ future-timelike curve segment, parameterized by proper time $\tau$, with fixed-endpoint variations as in External-field charged-particle action. If the segment is stationary for that adopted action, then Stationary charged action and Lorentz force gives $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$ for its constant $m>0$, $q\in \mathbb{R}$, and $U=dz\mathrm{/}d\tau$. Consequently

$C_{\xi }:=mg(U,\xi )+q(A(\xi )-\chi )$

is constant along the segment and is invariant under $A\mapsto A+d\lambda$, $\chi \mapsto \chi +{\mathcal{L}}_{\xi }\lambda$. This conditional test-body identity uses no self-field, global potential, trajectory-existence, or Einstein-solution claim.

Use the smooth electromagnetic stress, observer contractions and selected matter tensors of Smooth matter quantities, current and observer observables, with ${\mu }_0>0$ and a future unit observer $n$. The electromagnetic tensor is symmetric and trace-free, and it satisfies WEC, NEC, SEC and DEC. In a positively oriented orthonormal rest tetrad, with the cross product taken in the oriented Euclidean rest space, its energy density and spatial energy-current components are

$u=\frac{{ϵ}_0\mathrm{\mid }{E}_n{\mathrm{\mid }}^2+\mathrm{\mid }{B}_n{\mathrm{\mid }}^2\mathrm{/}{\mu }_0}{2},{\mathcal{S}}_n=\frac{E_n\times B_n}{{\mu }_{0}c}\mathrm{.}$

For charged dust with $\rho \ge 0$, the matter tensor and its sum with the electromagnetic tensor satisfy all four conditions. For a selected perfect-fluid tensor with rest energy density $ϵ$ and pressure $p$, the exact pointwise conditions on the matter tensor are

$\mathrm{W}\mathrm{E}\mathrm{C}⟺ϵ\ge 0\text{ and }ϵ+p\ge 0,\mathrm{N}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0,$

$\mathrm{S}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0\text{ and }ϵ+3p\ge 0,\mathrm{D}\mathrm{E}\mathrm{C}⟺ϵ\ge \mathrm{\mid }p\mathrm{\mid }\mathrm{.}$

If the perfect-fluid tensor satisfies any one of these conditions, adding the electromagnetic tensor preserves that condition. The inequalities are not assumed universally, and adding the electromagnetic tensor to arbitrary matter need not repair a violated condition.