LemmaStatement: AI-generatedProof: AI-generated

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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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Killing-horizon derivative contracted with the binormal

Statement

For the nonrotating normalized horizon data of Normalized Komar data on a nonrotating horizon section, one has $(\nabla^{a} T^{b}) b_{a b} = - 2 κ$ pointwise on $C$ . The sign is fixed by its specified binormal, which is unchanged by the boosts in The normal binormal is invariant under oriented boosts. Use the antisymmetric Killing equation in Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor.

Facts & Assumptions

[F1]

Along the section $T = c (n + s)$ with $c > 0$ , $\nabla_{T} T = κ T$ , and the bilinear form $A (U, V) = g (\nabla_{U} T, V)$ is antisymmetric. The binormal convention is fixed by the normalized-data item and boost lemma. Normalized Komar data on a nonrotating horizon section The normal binormal is invariant under oriented boosts Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Proof

**Given:**The section and Killing generator in the Statement.

Goal: Compute the binormal contraction.

1.1givenF1

Contract $\nabla_{T} T = κ T$ with $n$ . Since $T = c (n + s)$ and $A (n, n) = 0$ , this gives $c A (s, n) = - κ c$ , so $A (s, n) = - κ$ . Antisymmetry gives $A (n, s) = κ$ . No derivative of $c$ is taken: the first derivative slot is pointwise linear in its vector argument.

2.1F1step 1.1∎

By the definition of $b$ , $(\nabla^{a} T^{b}) b_{a b} = A (s, n) - A (n, s) = - 2 κ$ . The boost lemma makes the computation independent of the chosen oriented orthonormal normal pair.

Depends on

Used by · 2

Dependency tree · 2 levels

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