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

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Selected total-stress/Killing charges

Physical scope

Conditional total-stress conservation and finite-domain Killing flux for the selected smooth charged-dust Einstein–Maxwell branch, plus the local gauge-compensated first integral for a separate prescribed-field charged test body. Finite-chain side flux and local-potential conditions are retained; no gravitational-field energy, global charge, self-field, or global trajectory claim is made. The probe branch carries ACω through the published RPM action-equivalence result.

Statement

Let $Ω_{+}$ 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 $T_{t o t}^{a b} : = T_{m}^{a b} + T_{E M}^{a b}$ . Let $ξ$ be a smooth Killing field on $Ω_{+}$ , meaning $L_{ξ} g = 0$ , and set $J_{ξ}^{a} : = T_{t o t}^{a b} ξ_{b}$ . For any finite smooth singular $4$ -chain $C$ in $Ω_{+}$ with boundary $\partial C = Σ_{2} - Σ_{1} + B$ , define $Q_{ξ} (Σ) : = \int_{Σ} ι_{J_{ξ}} {v o l}_{g}$ . Then

$\nabla_{a} T_{t o t}^{a b} = 0, \nabla_{a} J_{ξ}^{a} = 0, Q_{ξ} (Σ_{2}) - Q_{ξ} (Σ_{1}) = - Q_{ξ} (B) .$

The two slice charges are equal only when the side flux $Q_{ξ} (B)$ 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 ${A C}_{ω}$ for the canonical tangent-bundle smooth structure. On a local potential domain $V \subseteq Ω_{+}$ , let $A$ be a supplied smooth one-form with $F = d A$ and let $χ$ be a smooth scalar such that $L_{ξ} A = d χ$ . Let $z : [a, b] \to V$ be a finite $C^{2}$ future-timelike curve segment, parameterized by proper time $τ$ , 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 $m \nabla_{U} U = q F^{♯} (U)$ for its constant $m > 0$ , $q \in R$ , and $U = d z / d τ$ . Consequently

$C_{ξ} : = m g (U, ξ) + q (A (ξ) - χ)$

is constant along the segment and is invariant under $A \mapsto A + d λ$ , $χ \mapsto χ + L_{ξ} λ$ . This conditional test-body identity uses no self-field, global potential, trajectory-existence, or Einstein-solution claim.

Facts & Assumptions

Given: The smooth Einstein–Maxwell dust branch, positive-density domain, Killing field, finite-chain boundary and side-flux hypotheses in the first clause; and, separately, the local potential, symmetry compensator, finite fixed-endpoint test-body segment and ${A C}_{ω}$ hypothesis in the second clause.

[A1]

Selected transported charged-dust Einstein–Maxwell model: the selected branch adopts $T_{m}^{a b} = ρ U^{a} U^{b}$ , $j^{a} = σ U^{a}$ , smooth Maxwell fields, and the undivided dust equations on a positive-density region.

[A2]

Smooth curved vacuum Maxwell constitutive adoption: on the smooth oriented Lorentzian domain the adopted field law is $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ with the selected SI sign.

[L1]

Dust continuity, transport and Lorentz acceleration: on $ρ > 0$ , the adopted dust equations give the exact matter-exchange identity $\nabla_{a} T_{m}^{a b} = F^{b}_{c} j^{c}$ .

[F1]

Curved electromagnetic stress divergence: for smooth $F, j$ , $μ_{0} > 0$ , $d F = 0$ , and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , the electromagnetic tensor obeys $\nabla_{a} T_{E M}^{a b} = - F^{b c} j_{c}$ . Its current proof and mathematical dependencies remain draft material, so this use is conditional.

[F2]

Killing exchange current and gauge-compensated charged first integral: under the smooth Maxwell equations and matter exchange, a symmetric total stress gives a divergence-free Killing current, the finite-chain side-flux balance, and, for a local potential and Lorentz-force test curve with $L_{ξ} A = d χ$ , the stated gauge-compensated constant.

[F3]

Charged and spinning test-body quantities: the test body has constant $m > 0$ , constant charge $q$ , a $C^{2}$ future-timelike worldline parameterized by proper time, and a prescribed field with a local potential where used.

[A3]

External-field charged-particle action: on a finite local-potential segment, the adopted model trajectories are stationary points of the displayed fixed-endpoint external-field action.

[F4]

Stationary charged action and Lorentz force: under that postulate, the charged-body definitions, finite-variation hypotheses and ${A C}_{ω}$ , stationarity is equivalent to $m \nabla_{U} U = q F^{♯} (U)$ . This published supplier is used only at its stated finite-segment interface.

[A4]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the Axiom of Countable Choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure; the test-body clause uses it through [F4].

Proof

1.1A1A2L1F1algebra

By [A1], [A2] and the local result [L1], the dust model gives $\nabla_{a} T_{m}^{a b} = F^{b}_{c} j^{c}$ on $Ω_{+}$ . The adopted Maxwell equations in [A1] match the smoothness and source-sign hypotheses of [F1], which gives $\nabla_{a} T_{E M}^{a b} = - F^{b c} j_{c}$ . Raising/lowering with $g$ identifies these force-density terms with opposite signs, so adding the two identities yields $\nabla_{a} T_{t o t}^{a b} = 0$ .

1.2A3F2F3F4A4given

Under [A3], the finite proper-time segment and charged-body quantities satisfy [F3] and [A3]; therefore [F4] supplies the Lorentz-force equation required by [F2]. No new trajectory is constructed.

2.1F2step 1.1given

The total stress is symmetric as the sum of symmetric matter and electromagnetic tensors. The Killing-field and finite-chain hypotheses in the first clause match [F2], so its current conclusion gives $\nabla_{a} J_{ξ}^{a} = 0$ and its boundary formula gives $Q_{ξ} (Σ_{2}) - Q_{ξ} (Σ_{1}) = - Q_{ξ} (B)$ . Equality follows only when the specified side flux is zero.

3.1F2step 1.2given∎

Apply the local test-body clause of [F2] to the equation from step 1.2 and the supplied $F = d A$ , $L_{ξ} A = d χ$ . It gives constancy and gauge invariance of $C_{ξ}$ on the segment. The result remains conditional on the local potential, symmetry and finite-segment hypotheses.

Selected total-stress/Killing charges

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources