LemmaStatement: AI-adaptedProof: 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.

Outward unit variation in the normalized null basis

Statement

In the normal plane of the surface data Surface measure, flux densities and null normal gauge, let $l = L = n + ν$ , $k = 2 \underline{L} = n - ν$ and $v = ν$ . Then $g (l, k) = - 2$ , $g (l, v) = 1$ , $v = - k / 2 + V l$ with $V = 1 / 2$ , and the accompanying normal $u = k / 2 + V l$ is $n$ . The future normal $n$ is timelike unit and $ν$ is spacelike unit, with $g (n, ν) = 0$ .

Facts & Assumptions

[F1]

The supplied surface gauge fixes the orthonormal normal pair and its null combinations. Surface measure, flux densities and null normal gauge

Proof

Given: The data and explicit hypotheses of the Statement.

Goal: Establish its identities with the indicated normalizations.

1.1givenF1algebra

Expand the pairings using $g (n, n) = - 1$ , $g (ν, ν) = 1$ and $g (n, ν) = 0$ ; they give $g (n + ν, n - ν) = - 2$ and $g (n + ν, ν) = 1$ .

2.1step 1.1F1algebra∎

The identity $- k / 2 + V l = (- 1 / 2 + V) n + (1 / 2 + V) ν$ equals $ν$ exactly for $V = 1 / 2$ . Substituting this value in $k / 2 + V l$ gives $n$ .

Depends on

Surface measure, flux densities and null normal gauge

Used by · 3

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.

Sources

Andersson–Mars–Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · Sections 2–3, printed pages 5–8