LemmaStatement: AI-generatedProof: AI-generated

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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.

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General curvature coefficient in the outward hypersurface gauge

Statement

In Outward unit variation in the normalized null basis and Agreement of expansion and normal-connection conventions, write $G = Ric - \frac{1}{2} {Scal}_{g} g$ for the Einstein tensor, and define the scalar $Y : = V ∣ χ_{l} ∣_{q}^{2} + G (l, u)$ . For the unit outward variation of Surface measure, flux densities and null normal gauge, $Y = \frac{1}{2} ∣ χ_{L} ∣_{q}^{2} + G (L, n) = \frac{1}{2} ∣ χ_{L} ∣_{q}^{2} + \frac{1}{2} G (L, L) + G (L, \underline{L})$ . In particular, if $χ_{L} = 0$ and $G (L, L) = 0$ , then $Y = G (L, \underline{L})$ . Vanishing expansion alone is not substituted for these two conditions.

Facts & Assumptions

[F1]

The normalized null decomposition and second-form identifications hold under the declared surface hypotheses. Outward unit variation in the normalized null basis Agreement of expansion and normal-connection conventions Surface measure, flux densities and null normal gauge

[F2]

Ricci curvature, scalar curvature and the Einstein tensor are the smooth metric-derived tensors of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Definition (c), on the smooth time-oriented Lorentzian four-manifold of [F1]. In particular, $G$ is a covariant two-tensor and is bilinear on each tangent space.

Proof

Given: The data and explicit hypotheses of the Statement.

Goal: Establish its identities with the indicated normalizations.

1.1givenF1algebra

The normalization lemma gives $V = 1 / 2$ , $u = n$ and $l = L$ . Substitution into the defining scalar gives the first formula, using the same second fundamental form as the expansion.

2.1step 1.1F1F2algebra∎

The gauge identity $n = L / 2 + \underline{L}$ and bilinearity give the second formula. Setting the explicitly assumed $χ_{L}$ and $G (L, L)$ to zero yields the final equality, without discarding either term otherwise.

Depends on

Used by · 1

General MOTS first variation before electrovac specialization

Dependency tree · 2 levels

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