Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Charged-dust current and stress balance

Physical scope

A local deduction for smooth charged-dust fields satisfying the independently adopted number-continuity and Lorentz-force equations. It does not construct dust solutions, prove avoidance of caustics, or extend to singular particles.

Statement

For the smooth charged-dust model of Restricted smooth charged-dust model, with constant $m > 0$ and $q$ , current $j^{a} = q ν U^{a}$ , and stress $T_{m}^{a b} = m ν U^{a} U^{b}$ , every configuration satisfying its adopted closure obeys

$\nabla_{a} j^{a} = 0, \nabla_{a} T_{m}^{a b} = F^{b c} j_{c} .$

This is a model deduction from the continuum equations, not a consequence of the external test-particle theorem and not a claim that a smooth continuum solution exists.

Facts & Assumptions

Given: A smooth metric-compatible connection, smooth fields $ν > 0$ , $U^{a}$ , and $F_{a b}$ , constants $m > 0$ and $q$ , and the dust equations of Restricted smooth charged-dust model.

[F1]

Restricted smooth charged-dust model: $j^{a} = q ν U^{a}$ , $T_{m}^{a b} = m ν U^{a} U^{b}$ , $\nabla_{a} (ν U^{a}) = 0$ , and $m \nabla_{U} U^{b} = q F^{b}_{c} U^{c}$ .

[A1]

The fields are smooth on the open domain and the charge and mass parameters are constant, so the covariant product rule applies without distributional terms.

Proof

Proof technique: expand the two divergences by the product rule and substitute the adopted dust equations.

1.1A1F1algebra

Since $q$ is constant, the product rule and [F1] give $\nabla_{a} j^{a} = q \nabla_{a} (ν U^{a}) = 0$ .

2.1A1F1step 1.1algebra

Differentiate the stress tensor in [F1]: $\nabla_{a} T_{m}^{a b} = m (\nabla_{a} (ν U^{a})) U^{b} + m ν U^{a} \nabla_{a} U^{b}$ . The first term is zero by number continuity, and $U^{a} \nabla_{a} U^{b} = \nabla_{U} U^{b}$ by definition.

3.1F1step 2.1algebra∎

Use the adopted force equation in [F1] in step 2.1: $\nabla_{a} T_{m}^{a b} = q ν F^{b}_{c} U^{c} = F^{b c} (q ν U_{c}) = F^{b c} j_{c}$ , because $F^{b}_{c} = g_{c a} F^{b a}$ and $j_{c} = q ν U_{c}$ . This is the claimed matter stress exchange.

Depends on

Restricted smooth charged-dust model

Charged-dust current and stress balance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources