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Zero Killing contraction makes the sourced Killing current divergence-free

Statement

Use the time-oriented Lorentzian open set, smooth metric, metric-selected connection, Killing field, positively oriented finite coordinate slab with compact closure, and finite-slab conventions of Finite Killing energy current and reference normalization, including its definitions of $h_{i j}, n_{a}, w, j_{ξ}, H_{ξ}$ and the net outward side flux $Φ_{j_{ξ}} (B)$ . Assume ${A C}_{ω}$ as required by its connection construction. Let $T^{a b}$ be smooth and symmetric and let $f^{b}$ be a smooth vector field such that $\nabla_{a} T^{a b} = f^{b}$ ; this replaces the conserved-stress hypothesis for this corollary. If $f^{b} ξ_{b} = 0$ , then $\nabla_{a} j_{ξ}^{a} = 0$ , and, conditional on the finite-slab balance theorem Complete finite-slab production and outward-flux balance at its actual interface, the finite slice charges satisfy $H_{ξ} (x_{0}^{+}) - H_{ξ} (x_{0}^{-}) + Φ_{j_{ξ}} (B) = 0$ . Thus the endpoint charges are equal exactly when the net side flux is zero. In the static subcase, assume $ξ = N_{ξ} u$ with $N_{ξ} > 0$ ; the algebraic zero-work condition $u_{b} f^{b} = 0$ then implies $f^{b} ξ_{b} = 0$ . No infinite-domain or exhaustion limit is asserted.

Facts & Assumptions

Given: The geometric setup, current definition, and finite-slab convention of Finite Killing energy current and reference normalization, with a smooth symmetric tensor $T^{a b}$ , a smooth vector field $f^{b}$ , $\nabla_{a} T^{a b} = f^{b}$ , and $f^{b} ξ_{b} = 0$ . The conserved-stress equation in the lemma is not assumed here.

[A1]

Assume ${A C}_{ω}$ , the Axiom of Countable Choice, to carry the exact hypothesis of the lemma and its Levi-Civita supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Finite Killing energy current and reference normalization defines $j_{ξ}^{a} = - T^{a b} ξ_{b}$ , $H_{ξ}$ as the finite proper-normal flux charge, and the exact finite-slab geometric notation; its MI01 balance input is the theorem statement recorded in [L2].

[F2]

Killing fields of a Lorentzian metric defines a Killing field by $L_{ξ} g = 0$ .

[F3]

Under [A1], Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields supplies the induced tensor derivative product rule.

[F4]

On the oriented chart domain inherited from the lemma, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (a) proves $L_{ξ} g = 0 ⟺ \nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0$ for every smooth vector field $ξ$ ; its other clauses are not used.

[L1]

Finite spacelike slab and normal flux charge defines a positively oriented product slab with open spatial rectangle and compact closure, future unit slice normal, proper normal-flux charge, and upper-minus-lower coordinate side flux, with $C^{1}$ current regularity and mathematical real coordinates.

[L2]

Complete finite-slab production and outward-flux balance assumes the ${A C}_{ω}$ background, the slab data and a $C^{1}$ current $J$ ; with $Π = \nabla_{a} J^{a}$ , it states $Q_{J} (t_{1}) - Q_{J} (t_{0}) + Φ_{J} (B) = \int_{K} Π {v o l}_{g}$ .

Proof

technique · direct; compute the source term in the Killing-current divergence and apply the finite-slab balance to the resulting current

1.1A1F1F3givenalgebra

By [A1] and [F3], the induced tensor product rule and the supplied source equation give $\nabla_{a} j_{ξ}^{a} = - (\nabla_{a} T^{a b}) ξ_{b} - T^{a b} \nabla_{a} ξ_{b} = - f^{b} ξ_{b} - T^{a b} \nabla_{a} ξ_{b}$ .

2.1F2F4step 1.1givenalgebra

Symmetry of $T^{a b}$ , the Killing definition [F2], and its exact characterization [F4] give $T^{a b} \nabla_{a} ξ_{b} = \frac{1}{2} T^{a b} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) = 0$ . The assumed zero Killing contraction therefore gives $\nabla_{a} j_{ξ}^{a} = 0$ .

3.1F1L1L2step 2.1algebra

The smooth current is $C^{1}$ , so the balance [L2] applies at its stated interface. Substituting step 2.1 into its production integral gives zero, while [F1] identifies the normal current flux with $H_{ξ}$ ; hence the conditional endpoint-plus-side-flux identity follows.

4.1F1L2givenstep 2.1step 3.1algebra∎

In the balance from step 3.1, zero net side flux implies equal endpoint charges, and equal endpoint charges imply zero net side flux. In the static case, $f^{b} ξ_{b} = N_{ξ} f^{b} u_{b} = 0$ whenever $u_{b} f^{b} = 0$ ; thus zero local work suffices for current conservation.

Zero Killing contraction makes the sourced Killing current divergence-free

Statement

Facts & Assumptions

Proof

Depends on

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