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.

Zero boundary expansions without maximality

Statement

Let $\overline{C} = C \cup S$ be the smooth exterior section with boundary supplied under the regular stationary-vacuum hypotheses. Let $n$ be its future unit normal and $ν$ the unit normal to $S$ pointing into $\overline{C}$ (toward the end), and put $k (X, Y) = - g (\nabla_{X} n, Y)$ . With $γ = g ∣_{T S}$ , define the unaveraged boundary mean curvature by $H_{S} = {tr}_{γ} [(X, Y) \mapsto g (\nabla_{X} ν, Y)]$ for $X, Y \in T S$ , using covariant derivatives along $S$ . If the two horizon null expansions vanish at $S$ , then $H_{S} = 0$ and ${tr}_{S} k = 0$ , where ${tr}_{S} k = {tr}_{γ} (k ∣_{T S \times T S})$ . In particular both expressions $θ_{\pm} = \pm H_{S} + {tr}_{S} k$ vanish, independently of the sign convention for $k$ . No maximality is assumed.

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: The two zero boundary traces and their stated sign independence.

[F1]

The regular-spacetime hypotheses are formulated in ZFC. The exterior section is a smooth spacelike manifold with smooth compact boundary $S \subset Σ_{0}$ . The exterior Cauchy component has a complete asymptotically Euclidean closure Regular non-degenerate stationary vacuum spacetime

[F2]

The two smooth null horizons $H^{+}$ and $H^{-}$ contain $S$ , intersect transversely there, and have zero screen expansion. Their screen expansion is the trace of $([X], [Y]) \mapsto g (\nabla_{X} L, Y)$ for a local nonzero radical field $L$ ; zero expansion is unchanged by nonzero smooth rescaling. Regular non-degenerate stationary vacuum spacetime

[F3]

The induced data have a smooth future unit normal $n$ , induced metric $h = g ∣_{T \overline{C}}$ , and second fundamental form $k (X, Y) = - g (\nabla_{X} n, Y)$ . The surface expansion of a null normal $ℓ$ is ${tr}_{γ} [(X, Y) \mapsto g (\nabla_{X} ℓ, Y)]$ . Spacelike sections, maximality and lapse–shift data Null expansions, screen quotients and focal data

[F4]

For a spacelike two-surface in a time-oriented Lorentzian four-manifold, the normal plane is Lorentzian with two future null rays. A null hypersurface with radical $R L$ has tangent space $L^{⊥}$ . On a spacelike section its quotient screen is isometric to the section tangent plane, and surface null second fundamental forms scale by $χ_{e^{f} L} = e^{f} χ_{L}$ . These assertions do not require an affine generator. Local null normal pairs and the positive screen quotient

Proof

Given: The data of the Statement and [F1]–[F4].

Goal: Identify the two horizon traces with the boundary trace formulas and solve them.

1.1F1F2F4

At $p \in S$ , the inclusion $T_{p} S \subset T_{p} \overline{C}$ makes $γ$ positive definite. For either horizon let $R L$ be its radical line. Since $T_{p} S \subset T_{p} H^{\pm} = L^{⊥}$ , $L$ is normal to $S$ . The line $R L$ cannot intersect the positive-definite plane $T_{p} S$ nontrivially. Their dimensions are one and two, so $T_{p} H^{\pm} = T_{p} S \oplus R L$ . Thus $S$ is a spacelike section of each horizon, and [F4] identifies its tangent metric with each screen metric.

2.1F1F2F3F4step 1.1algebra

The fields $n, ν$ are orthogonal unit normals to $S$ , with squared lengths $- 1, 1$ . They span its normal plane, whose null lines are exactly $R (n + ν)$ and $R (n - ν)$ , because $g (a n + b ν, a n + b ν) = - a^{2} + b^{2}$ . Put $ℓ_{\pm} = n \pm ν$ . They are future null, since $g (ℓ_{\pm}, n) = - 1$ , and satisfy $g (ℓ_{+}, ℓ_{-}) = - 2$ . The two horizon radical lines are distinct: if they agreed, [F4] would give equal tangent hyperplanes, contrary to their transverse intersection. Hence, after a local interchange of horizon labels, each $ℓ_{\pm}$ is a smooth positive multiple $a_{\pm} L_{\pm} ∣_{S}$ of a smooth future horizon radical field.

3.1givenF2F3F4step 1.1step 2.1algebra

Under the screen isometries of step 1.1, the horizon form restricts to $χ_{L_{\pm} ∣_{S}} (X, Y) = g (\nabla_{X} (L_{\pm} ∣_{S}), Y)$ on $T S$ ; differentiation in a tangent direction depends only on the restriction to $S$ . Its $γ$ -trace is therefore zero by [F2]. Write $a_{\pm} = e^{f_{\pm}}$ locally and apply the surface rescaling law in [F4] to obtain ${tr}_{γ} χ_{ℓ_{\pm}} = 0$ . Linearity of the connection and the definitions in the Statement and [F3] give $χ_{ℓ_{\pm}} (X, Y) = - k (X, Y) \pm g (\nabla_{X} ν, Y)$ and consequently $0 = - {tr}_{S} k \pm H_{S}$ . No affine parametrization is used.

4.1givenstep 3.1algebra∎

Adding the two equalities of step 3.1 gives $- 2 {tr}_{S} k = 0$ ; subtracting them gives $2 H_{S} = 0$ . Thus both $θ_{\pm} = \pm H_{S} + {tr}_{S} k$ vanish for the normal pointing into the exterior. Replacing $k$ by $- k$ changes the sign of its trace and preserves both zero values. The argument has imposed no maximality condition.

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