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.

An exact local Killing field is Killing to every order on a horizon

Statement

Assume the axiom of countable choice ${A C}_{ω}$ . Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold without boundary, with metric $g$ of signature $(-, +, +, +)$ . Let $H \subset M$ be a smooth embedded hypersurface, let $O \subset M$ be an open set containing $H$ , and let $K$ be a smooth Killing field of $g ∣_{O}$ . Write $\nabla$ for the metric-selected connection of $g$ , restricted to $O$ , and $π = L_{K} (g ∣_{O})$ for the deformation tensor. Set $\nabla^{0} π = π$ and $\nabla^{m + 1} π = \nabla (\nabla^{m} π)$ . Every covariant derivative $\nabla^{m} π$ vanishes on $O$ , for every integer $m \geq 0$ , and hence on $H$ . Here restriction to $H$ retains the ambient tensor at each $p \in H$ , with all its ambient tangent arguments. Let $i : H ↪ M$ denote the inclusion, with $H$ carrying the smooth structure from its restricted slice charts. We identify $T_{p} H$ with the subspace $d i_{p} (T_{p} H) \subseteq T_{p} M$ using the injective linear differential $d i_{p} : T_{p} H \to T_{p} M$ . A direction transverse to $H$ at $p$ means a vector $v \in T_{p} M$ with $v \notin T_{p} H$ . Thus an actual smooth Killing field near a smooth horizon is Killing to all orders there, including transverse derivatives; merely specifying a field on $H$ would not by itself define those transverse derivatives.

Statement references

Killing fields of a Lorentzian metric

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

The Axiom of Countable Choice (AC_ω)

Codimension and hypersurfaces

Embedded submanifolds and slice charts

The inclusion of an embedded submanifold is a smooth embedding

Smooth embeddings

Immersions, submersions, and constant-rank maps

The differential of a smooth map

The differential sends derivations to derivations and is linear

Facts & Assumptions

[F1]

A Killing field of $g ∣_{O}$ is defined by $L_{K} (g ∣_{O}) = 0$ on its open domain. Killing fields of a Lorentzian metric

[F2]

On a smooth four-manifold without boundary, a smooth nondegenerate symmetric metric has a unique metric-selected connection, whose induced tensor derivatives are linear and extend to every finite tensor rank. The cited construction assumes ${A C}_{ω}$ for the canonical tangent-bundle smooth structure. Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields

[F3]

The ambient time-oriented Lorentzian data include a smooth symmetric nondegenerate metric and a smooth timelike field $N$ on a Hausdorff second-countable smooth four-manifold without boundary. Countable choice is assumed for the canonical smooth bundle structures used by this definition and [F2]; the vanishing induction makes no further choices. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model The Axiom of Countable Choice (AC_ω)

[F4]

A smooth embedded hypersurface is an embedded submanifold of codimension one, with its smooth structure supplied by restricted slice charts. Transverse directions are understood in the explicit sense given in the Statement, with no spacelike or null requirement on $H$ . Codimension and hypersurfaces Embedded submanifolds and slice charts

[F5]

For the restricted-slice-chart smooth structure, the inclusion $i : H ↪ M$ is a smooth embedding. An embedding is an immersion, and an immersion has injective differential at every point. The differential $d i_{p} : T_{p} H \to T_{p} M$ is well defined and linear, so its image is a vector subspace and it identifies $T_{p} H$ with that image. This is the ambient tangent-subspace identification used here. The inclusion of an embedded submanifold is a smooth embedding Smooth embeddings Immersions, submersions, and constant-rank maps The differential of a smooth map The differential sends derivations to derivations and is linear

Proof

Given: The hypotheses and data in the Statement.

Goal: Prove vanishing of all ambient covariant derivatives on $O$ and on $H$ .

1.1F1F2F3

The ambient data and countable-choice assumption in [F3] meet the hypotheses of [F2], so the metric-selected connection and its induced derivatives are available on $O$ . By [F1], $π$ is the identically zero tensor field on $O$ . Linearity gives $\nabla π = \nabla 0 = 0$ on the same open set.

2.1F2F4F5step 1.1∎

The case $m = 0$ holds by step 1.1. If $\nabla^{m} π = 0$ on $O$ , then [F2] gives $\nabla^{m + 1} π = \nabla 0 = 0$ there. Induction proves the assertion for every $m \geq 0$ . At each $p \in H \subset O$ , evaluating this zero ambient tensor on any ambient tangent arguments still gives zero, including arguments transverse to $T_{p} H$ under the identification in [F5] and the convention in [F4]. These derivatives are defined using the field on $O$ ; data given only on $H$ do not specify its variation off $H$ .

Depends on

Used by · 1

Hawking horizon generator and analytic global rigidity

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.