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Canonical outward normals and null gauge on a graph

Statement

Use an admissible normal graph $F_{u}$ in the tube of the preceding graph-domain lemma, and let $n$ be the smooth future unit normal to the fixed hypersurface $Σ$ . In tube coordinates $(x^{1}, x^{2}, r)$ the covector $α_{u} = d r - (\partial_{a} u) d x^{a}$ , evaluated at $(x, u (x))$ , annihilates the graph tangent plane. Write $h$ also for its pullback $T^{*} h$ to the tube. For a tube covector $α$ , let $α^{♯_{h}}$ be the unique vector satisfying $h (α^{♯_{h}}, V) = α (V)$ for every tube tangent vector $V$ , and define $∣ α ∣_{h} = \sqrt{h (α^{♯_{h}}, α^{♯_{h}})}$ . Set $ν_{u} = α_{u}^{♯_{h}} / ∣ α_{u} ∣_{h}$ , and push this vector to $Σ$ . These definitions agree on overlaps and give the outward smooth $h$ -unit normal along the graph, with $ν_{0} = ν$ . Here outward means positive pairing with $T_{*} \partial_{r}$ .

The fields $L_{u} = n + ν_{u}$ and ${\underline{L}}_{u} = (n - ν_{u}) / 2$ along the graph are future null normals satisfying $g (L_{u}, {\underline{L}}_{u}) = - 1$ . They depend smoothly on the first graph data $(x, u, \partial u)$ and fix the normal gauge throughout the deformation, rather than choosing a new arbitrary boost at each parameter.

Facts & Assumptions

[F1]

Admissible domains for outward normal graphs supplies $T$ , the admissible graph and its embedding into $Σ$ .

[F2]

The future unit normal, induced metric and second fundamental form of a spacelike hypersurface supplies the smooth future unit normal $n$ of this fixed hypersurface and its smooth induced positive metric $h$ .

[F3]

Null normal pairs and screen geometry defines null expansion and the effect of changing a null normal's positive scale; its normalized second null normal may be rescaled from pairing $- 2$ to pairing $- 1$ .

[F4]

Pullback of a riemannian metric as a tensor defines $(T^{*} h)_{z} (V, W) = h_{T (z)} (d T_{z} V, d T_{z} W)$ . Since $T$ is the diffeomorphism supplied by [F1], this tensor is a smooth positive definite metric on the tube.

[F5]

Musical isomorphisms defines $α^{♯_{h}}$ by $h (α^{♯_{h}}, V) = α (V)$ . The covector norm in the Statement is defined from this vector and the positive metric $h$ .

Proof

**Given:**An admissible graph in the supplied tube, with $h, n, ν$ as above.

Goal: Justify the smooth, normalized outward graph gauge.

1.1F1F2F4F5givenalgebra

The scalar function $r - u (p)$ is defined on the part of the tube over the domain of $u$ , independently of a coordinate chart on $S$ . Its differential is $α_{u}$ and takes value one on $\partial_{r}$ . It is nonzero and annihilates each tangent vector $\partial_{a} + (\partial_{a} u) \partial_{r}$ to the graph. The tube metric is smooth and positive definite by [F4]. The defining identity in [F5] gives $h (α_{u}^{♯_{h}}, V) = α_{u} (V)$ , so $α_{u}^{♯_{h}} \neq 0$ and $∣ α_{u} ∣_{h} > 0$ . Dividing by this norm and pushing forward by $T$ gives a unique smooth unit normal with positive pairing with $T_{*} \partial_{r}$ , since $h (α_{u}^{♯_{h}} / ∣ α_{u} ∣_{h}, \partial_{r}) = 1 / ∣ α_{u} ∣_{h} > 0$ . At $u = 0$ , $d T (X, 0) = X$ and $T_{*} \partial_{r} = ν$ by [F1], so this normal is $ν$ . The scalar construction makes the local formulas agree. The inverse-matrix and square-root formulas show smooth dependence on $(x, u, \partial u)$ .

2.1F2F3step 1.1algebra∎

Orthogonality gives $g (n, n) = - 1$ , $g (ν_{u}, ν_{u}) = 1$ , $g (n, ν_{u}) = 0$ . Direct multiplication gives $g (L_{u}, L_{u}) = g ({\underline{L}}_{u}, {\underline{L}}_{u}) = 0$ and $g (L_{u}, {\underline{L}}_{u}) = - 1$ . Both are nonzero and future-directed because their scalar product with the future timelike $n$ is negative. They are normal to the graph. The factors used to normalize the second null vector do not affect the expansion of $L_{u}$ , defined by [F3]. This proves the stated gauge with its smoothness and orientation.

Depends on

Used by · 9

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