LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For a Killing field the acceleration and surface-gravity equations agree

Statement

Let $K$ be a smooth Killing field of a smooth Lorentz metric $g$ (a smooth symmetric nondegenerate covariant $2$ -tensor of Lorentz signature) on a smooth manifold of any finite dimension, with metric-compatible Levi–Civita connection $\nabla$ . Here Killing means $L_{K} g = 0$ , and the connection is a torsion-free affine connection satisfying $X (g (Y, Z)) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ for smooth local vector fields. Then the ambient covector identity $d (g (K, K)) = - 2 g (\nabla_{K} K, \cdot)$ holds, without orientation or time-orientation assumptions.

Consequently, on any nonzero-generator part $H_{*}$ of a null Killing horizon, $\nabla_{K} K = κ K$ is equivalent to $d (g (K, K)) = - 2 κ K^{♭}$ for any smooth scalar $κ$ , where $K^{♭} = g (K, \cdot)$ . Here $H_{*}$ is a portion of a smooth null hypersurface on which $K$ is nonzero, tangent and normal. Whenever these equivalent equations hold, a rescaling $K \mapsto a K$ by a nonzero real constant changes the unique signed coefficient to $a κ$ . For arbitrary real constants, including zero, the algebraic identity $\nabla_{a K} (a K) = a^{2} \nabla_{K} K$ still holds; at $a = 0$ it defines no surface-gravity coefficient. In the four-dimensional time-oriented setting, the signed surface-gravity terminology is that of the bifurcate-horizon convention.

Statement references

Bifurcate Killing horizons, rotation and normalization

Affine connection on a smooth manifold

Torsion tensor of an affine connection

The Lie derivative is a derivation of the tensor algebra

Tensor Lie derivative agrees with X on functions and bracket on vector fields

The exterior derivative of a function is its differential

Facts & Assumptions

[F1]

The affine-connection rules include function-linearity in the differentiating field and the Leibniz rule in the differentiated field. Torsion freedom means $[X, Y] = \nabla_{X} Y - \nabla_{Y} X$ . Metric compatibility has the explicit meaning given in the Statement. Affine connection on a smooth manifold Torsion tensor of an affine connection

[F2]

The Lie derivative preserves tensor products and contractions, acts on functions by $L_{K} f = K f$ , and acts on vector fields by $L_{K} X = [K, X]$ . These interfaces apply on smooth manifolds without dimension, metric-signature or orientation restrictions. The Lie derivative is a derivation of the tensor algebra Tensor Lie derivative agrees with X on functions and bracket on vector fields

[F3]

For every smooth function $f$ and smooth local vector field $X$ , $d f (X) = X f$ . The exterior derivative of a function is its differential

[F4]

In the four-dimensional time-oriented horizon convention, signed surface gravity is specified on the nonzero-generator part by the ambient covector equation. Its normalization uses a nonzero constant rescaling. Only this naming convention is used here; the equivalence and scale calculation are proved below. Bifurcate Killing horizons, rotation and normalization

Proof

Given: The hypotheses and data in the Statement.

Goal: Prove the ambient identity, its equivalence to the acceleration equation, and the precise rescaling rule.

1.1F1F2given

Apply the Lie derivative to the scalar contraction $g (X, Y)$ . The product and contraction rules give $(L_{K} g) (X, Y) = K (g (X, Y)) - g ([K, X], Y) - g (X, [K, Y])$ . By metric compatibility and torsion freedom this equals $g (\nabla_{X} K, Y) + g (X, \nabla_{Y} K)$ : expand $K (g (X, Y))$ as $g (\nabla_{K} X, Y) + g (X, \nabla_{K} Y)$ and substitute $[K, X] = \nabla_{K} X - \nabla_{X} K$ and $[K, Y] = \nabla_{K} Y - \nabla_{Y} K$ . Since $K$ is Killing, the resulting sum vanishes.

2.1F1F3step 1.1

For every smooth local vector field $X$ , metric compatibility and the preceding identity with $Y = K$ imply $X (g (K, K)) = 2 g (\nabla_{X} K, K) = - 2 g (X, \nabla_{K} K)$ . By $d f (X) = X f$ this says $d (g (K, K)) (X) = - 2 g (\nabla_{K} K, X)$ . Local vector fields realize every tangent vector, so the identity holds as an equality of ambient covectors.

3.1F4step 2.1given

Substituting $\nabla_{K} K = κ K$ gives the stated covector equation. Conversely, that equation and step 2.1 give $g (\nabla_{K} K - κ K, X) = 0$ for every tangent vector $X$ , hence $\nabla_{K} K = κ K$ by nondegeneracy of $g$ . On $H_{*}$ the coefficient is unique because $K \neq 0$ . This proves the equivalence used for the signed coefficient, including the four-dimensional terminology of [F4], without using that convention as a proof of the identity.

4.1F1F2F4step 1.1step 3.1∎

For any real constant $a$ , the affine-connection rules give $\nabla_{a K} (a K) = a^{2} \nabla_{K} K$ . If $a \neq 0$ , then step 1.1 applied to $a K$ gives $L_{a K} g = a L_{K} g = 0$ , and $a K$ has the same nonzero-generator part as $K$ . When the equations in step 3.1 hold, $\nabla_{a K} (a K) = (a κ) (a K)$ , so uniqueness and step 3.1 identify its signed coefficient as $a κ$ . If $a = 0$ , the algebraic identity remains true, but the nonzero-generator part is empty and the equation does not select a coefficient.

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