LemmaStatement: AI-adaptedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Killing derivative is an exact closed two-form

Statement

For the stationary Lorentzian data, put $α = T^{♭}$ and $F_{a b} = \nabla_{a} T_{b}$ . Then $F_{a b} = - F_{b a}$ , $F = \frac{1}{2} d α$ and $d F = 0$ . Here a two-form has coordinate expansion $F = \frac{1}{2} F_{a b} d x^{a} \land d x^{b}$ .

Facts & Assumptions

[F1]

The metric, Killing field, local flow and torsion-free metric connection are those of Stationary Lorentzian data and tensor conventions.

[F2]

Exterior differentiation has its coordinate formula and satisfies $d^{2} = 0$ . Lorentzian exterior calculus and Hodge star

Proof

Given: The data and hypotheses in the Statement.

Goal: Prove all asserted identities under these hypotheses.

1.1F1algebra

In a coordinate chart, the derivative of the flow pullback is $(L_{T} g)_{a b} = T^{c} \partial_{c} g_{a b} + g_{c b} \partial_{a} T^{c} + g_{a c} \partial_{b} T^{c}$ . Metric compatibility and symmetry of the Christoffel symbols rewrite this as $\nabla_{a} T_{b} + \nabla_{b} T_{a}$ . The Killing condition makes this sum zero, so $F$ is alternating.

2.1step 1.1F1F2algebra∎

Torsion freedom gives $(d α)_{a b} = \partial_{a} T_{b} - \partial_{b} T_{a} = \nabla_{a} T_{b} - \nabla_{b} T_{a} = 2 F_{a b}$ . Applying $d$ once more gives $d F = \frac{1}{2} d^{2} α = 0$ .

External input

The cited Killing equation supplies background for the local exactness calculation.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Lecture Notes on General Relativity · §5, equations (5.41)–(5.42), PDF p. 147

The vanishing Lie derivative of the metric is equivalent to the Killing equation.

Hypotheses: A smooth metric and its Levi–Civita connection.

Reading limits: Read the cited Killing equations. The exactness and closure consequences are proved locally in this item.

Sources

Lecture Notes on General Relativity · §5, equations (5.41)–(5.42), PDF p. 147