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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Metric transport along a null generator

Statement

On the null branch with generator $L$ in Null second fundamental form descends to the screen quotient, write $q = g ∣_{T H}$ . For tangent fields $X, Y$ , $(L_{L} q) (X, Y) = 2 B (X, Y)$ . Thus vanishing $B$ means preservation of the degenerate tangent metric along the local generator flow. The zero-trace condition remains invariant under the rescalings in Rescaling a null generator preserves zero expansion; no implication from zero trace to vanishing $B$ is asserted here. The Lie derivative and connection conventions are those of Stationary Lorentzian data and tensor conventions.

Facts & Assumptions

[F1]

The descended null form is symmetric; the ambient connection is torsion-free and metric-compatible. Positive rescaling multiplies the expansion. Null second fundamental form descends to the screen quotient Rescaling a null generator preserves zero expansion Stationary Lorentzian data and tensor conventions

[F2]

For any smooth tensor on a smooth manifold, its Lie derivative along a smooth vector field is the derivative at zero of pullback by its local flow. It obeys the tensor-product rule, commutes with contractions, acts on functions by directional differentiation, and acts on vector fields by the Lie bracket. Apply these facts on $H$ to the smooth covariant two-tensor $q$ and the tangent generator $L$ ; no nondegeneracy of $q$ is required. The Lie derivative of a tensor field The Lie derivative is a derivation of the tensor algebra Tensor Lie derivative agrees with X on functions and bracket on vector fields

[F3]

On every common local flow domain, a smooth tensor has zero Lie derivative if and only if its pullback by each defined flow map equals the tensor. Apply this to $q$ and the local flow of $L$ on $H$ ; completeness of $L$ is not assumed. A tensor field is flow-invariant exactly when its Lie derivative vanishes

Proof

Given: The branch, tangent metric and generator in the Statement.

Goal: Compute the Lie derivative and its stated consequence.

1.1givenF1F2

Regard $q (X, Y)$ as the contraction of $q \otimes X \otimes Y$ . The derivation and contraction rules in F2 give $L (q (X, Y)) = (L_{L} q) (X, Y) + q ([L, X], Y) + q (X, [L, Y])$ . Thus $(L_{L} q) (X, Y) = L (g (X, Y)) - g ([L, X], Y) - g (X, [L, Y])$ . The brackets here are intrinsic to $H$ ; ambient extensions of tangent fields give the same brackets on $H$ . Replacing each bracket by its torsion-free connection expression and expanding the first term by metric compatibility leaves $g (\nabla_{X} L, Y) + g (X, \nabla_{Y} L)$ .

2.1F1F3step 1.1∎

Symmetry of $B$ makes this sum $2 B (X, Y)$ . Hence $B = 0$ gives $L_{L} q = 0$ . By F3, for the local generator flow $Φ_{t}$ this implies $Φ_{t}^{*} q = q$ on every common local flow domain, proving preservation of the degenerate tangent metric. The rescaling statement follows from F1 and asserts only invariance of zero trace.

Depends on

Used by · 1

Hawking and rotational fields near a bifurcate sphere

Dependency tree · 2 levels

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