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A variable-time flow through a fixed boundary has the boosted unit normal

Statement

Let a smooth Killing field $K$ vanish on a compact spacelike boundary surface $S$ of a reference spacelike section in a smooth time-oriented Lorentzian four-manifold $(M, g)$ of signature $(-, +, +, +)$ . Let $\overset{ˉ}{n}$ be the future unit reference normal and $\overset{ˉ}{ν}$ the exterior unit boundary normal. Fix $κ > 0$ and impose the signed normal-boost convention $\nabla_{\overset{ˉ}{n}} K = κ \overset{ˉ}{ν}$ and $\nabla_{\overset{ˉ}{ν}} K = κ \overset{ˉ}{n}$ on $S$ , where $\nabla$ is the Levi–Civita connection. Let $ϕ^{K} : D \to M$ be a local flow of $K$ . On a reference neighborhood of $S$ , let $u$ be smooth down to $S$ , with $(u (x), x) \in D$ , and suppose the variable-time map $Φ (x) = ϕ_{u (x)}^{K} (x)$ is a smooth embedding with spacelike image, the candidate section. Then on $S$ its future unit normal is $n = \cosh (κ u) \overset{ˉ}{n} + \sinh (κ u) \overset{ˉ}{ν}$ . In particular $- g (n, \overset{ˉ}{n}) = \cosh (κ u)$ . The derivative of $u$ contributes nothing to the boundary tangent map because $K = 0$ there.

Statement references

Spacelike sections, maximality and lapse–shift data

Killing fields of a Lorentzian metric

Normal boost action and invariant two-forms at a fixed surface

Local and global flows generated by a vector field

The chain rule for differentials of smooth maps

Immersions and embeddings for manifolds with boundary

Facts & Assumptions

[F1]

A smooth embedded spacelike section has its unique future unit normal, including at its smooth spacelike boundary. Spacelike sections, maximality and lapse–shift data

[F2]

With the stated signed infinitesimal boost in the ordered normal basis $(\overset{ˉ}{n}, \overset{ˉ}{ν})$ , the Killing flow fixes $S$ pointwise, its differential is the identity on $T S$ , and its normal-plane differential is the boost with off-diagonal entries $\sinh (κ t)$ . Normal boost action and invariant two-forms at a fixed surface

[F3]

The derivative of a smooth flow with respect to its time is its generating field at the image point. Local and global flows generated by a vector field

[F4]

For composable smooth maps $F, G$ , the differential satisfies $d (G \circ F)_{p} = d G_{F (p)} \circ d F_{p}$ . The chain rule for differentials of smooth maps

[F5]

A smooth embedding between manifolds with boundary is an immersion and a homeomorphism onto its image; its differential on the full tangent spaces is injective at every point, including boundary points. Immersions and embeddings for manifolds with boundary

Proof

Given: The reference section, its boundary $S$ , the signed boost hypotheses, and the smooth variable-time embedding with spacelike image in the Statement.

Goal: Establish the boundary normal formula, the angle identity, and independence of height derivatives in the boundary tangent map.

1.1F1F2F3F4F5

At $p \in S$ , the flow fixes $p$ . Apply the chain rule to $x \mapsto (u (x), x) \mapsto ϕ_{u (x)}^{K} (x)$ . For every vector $X$ in the reference tangent space at $p$ , this gives $d Φ_{p} (X) = d ϕ_{u (p)}^{K} ∣_{p} (X) + d u_{p} (X) K_{ϕ_{u (p)}^{K} (p)} = d ϕ_{u (p)}^{K} ∣_{p} (X)$ , since $ϕ_{u (p)}^{K} (p) = p$ and $K_{p} = 0$ . At the boundary this calculation is made using smooth local extensions in boundary charts; smoothness down to $S$ ensures that the resulting differentials agree with those of the given maps. Thus the candidate tangent hyperplane is the image of the reference tangent hyperplane under $d ϕ_{u (p)}^{K} ∣_{p}$ , independently of all height derivatives.

2.1F1F2step 1.1algebra∎

The identity action on $T_{p} S$ and the normal-plane boost preserve $g_{p}$ , so $d ϕ_{u (p)}^{K} ∣_{p}$ is a linear isometry. Its image of ${\overset{ˉ}{n}}_{p}$ is $\cosh (κ u (p)) {\overset{ˉ}{n}}_{p} + \sinh (κ u (p)) {\overset{ˉ}{ν}}_{p}$ , a future unit timelike vector because $\cosh (κ u (p)) > 0$ and $\cosh^{2} (κ u (p)) - \sinh^{2} (κ u (p)) = 1$ . By step 1.1 this vector is orthogonal to the candidate tangent hyperplane, hence is its unique future unit normal. Pairing with ${\overset{ˉ}{n}}_{p}$ and using $g ({\overset{ˉ}{n}}_{p}, {\overset{ˉ}{n}}_{p}) = - 1$ and $g ({\overset{ˉ}{ν}}_{p}, {\overset{ˉ}{n}}_{p}) = 0$ gives $- g (n_{p}, {\overset{ˉ}{n}}_{p}) = \cosh (κ u (p))$ .

Depends on

Used by · 3

Dependency tree · 2 levels

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