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Advected nonconstant charge ratio without Maxwell closure

Example

Let $M = R^{4}$ with length coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , Minkowski metric $η = diag (- 1, 1, 1, 1)$ , orientation $d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ , future time-orientation $N = \partial_{0}$ , and $x^{0} = c t$ . In SI bookkeeping, assign $[ρ_{0}] = k g m^{- 3}$ , $[L] = m$ , and $[α_{*}] = C k g^{- 1}$ . Choose $ρ_{0} > 0$ , $L > 0$ , and $α_{*} \neq 0$ . Set

$U = c \partial_{0}, ρ = ρ_{0}, α (x) = α_{*} (1 + (x^{1} / L)^{2}), σ = ρ_{0} α, F = 0.$

The fields are smooth and the ratio $α = σ / ρ$ is nonconstant in space. This is a mathematical example of the matter balance and transport equations; the sourced Maxwell equation is not imposed.

Facts & Assumptions

Given: The stated Minkowski coordinates and fields; $ρ_{0}, L > 0$ and $α_{*} \neq 0$ .

[F1]

Smooth formal matter, EM tensors and observer quantities: $g (U, U) = - c^{2}$ , $T_{m}^{a b} = ρ U^{a} U^{b}$ , $j^{a} = σ U^{a}$ , and the sourced Maxwell equation would be $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ with $μ_{0} > 0$ .

[F2]

Charged dust balance equivalence and advected charge ratio: on a positive-density domain, dust exchange is equivalent to continuity and the Lorentz-force equation; under continuity, $\nabla_{a} j^{a} = 0$ is equivalent to $U^{a} \nabla_{a} α = 0$ .

Verification

1.1givenF1F2algebra

The constant Minkowski connection gives $g (U, U) = - c^{2}$ , $\nabla_{a} (ρ U^{a}) = \partial_{0} (ρ_{0} c) = 0$ , and $U^{a} \partial_{a} α = c \partial_{0} α = 0$ . Since $ρ_{0} > 0$ , [F2] yields charge-current conservation $\nabla_{a} (σ U^{a}) = 0$ .

2.1givenF2step 1.1algebra

The acceleration of $U$ is zero, and $F^{b}_{c} U^{c} = 0$ because $F = 0$ . Thus the force equation and mass continuity in [F2] hold, so the matter-exchange equation holds. But $α (0) = α_{*}$ and $α (L) = 2 α_{*}$ , which differ because $α_{*} \neq 0$ ; hence $α$ is not spatially constant.

3.1givenF1algebra∎

The factor $1 + (x^{1} / L)^{2}$ is at least $1$ , so $j = ρ_{0} α U$ is nonzero everywhere. For $F = 0$ , the sourced Maxwell equation would read $0 = - μ_{0} j^{b}$ , which fails. Hence matter balance and charge transport alone do not supply Maxwell closure.

Remarks

The example is not a coupled Einstein–Maxwell solution and makes no claim about physical existence. It shows exactly why constant $σ / ρ$ is not a consequence of the dust exchange identity.

Advected nonconstant charge ratio without Maxwell closure

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources