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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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Local normal collar charts at a smooth spacelike hypersurface

Statement

Let $(M, g, N)$ be a smooth time-oriented Lorentzian four-manifold, let $Σ$ be a smooth three-manifold, and let $i : Σ \to M$ be a smooth spacelike embedding. Suppose a smooth future unit normal field $n$ along $i (Σ)$ is supplied. For every $p \in Σ$ there are an open neighborhood $U_{p} \subseteq Σ$ , a number $ϵ_{p} > 0$ , and an open $V_{p} \subseteq M$ such that the restricted map $Φ_{p} : (- ϵ_{p}, ϵ_{p}) \times U_{p} \to V_{p}$ given by

$Φ_{p} (t, x) = γ_{i (x), n_{i (x)}} (t)$

is a diffeomorphism, where $γ_{q, v}$ is the maximal affinely parameterized metric geodesic with initial data $(q, v)$ . At its center,

$d (Φ_{p})_{(0, p)} (a \partial_{t} + X) = a n_{i (p)} + d i_{p} (X) .$

The neighborhoods and time widths may depend on $p$ ; no globally injective normal exponential map or uniform width is asserted.

Facts & Assumptions

Given: The smooth Lorentzian manifold, the spacelike embedding, and the supplied smooth future unit normal.

[F1]

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model fixes the smooth spacetime structure and time orientation. Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons then makes $i (Σ)$ an embedded three-dimensional submanifold with positive definite induced metric; the supplied future unit normal is timelike and orthogonal to its tangent space.

[F2]

The geodesic initial-value problem and the exponential map of the metric-selected connection: geodesics depend smoothly on time, base point and initial tangent on an open set of admissible data.

[L1]

The smooth inverse function theorem on manifolds: a smooth map whose differential is an isomorphism at a point restricts to a diffeomorphism between neighborhoods of that point and its image.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the tangent-bundle smooth-structure convention used by [F2] is carried under ${A C}_{ω}$ . No stronger choice principle is used in this local construction.

Proof

Given: The geometric data in the Statement and the choice convention [A1].

1.1givenA1F2algebra

The map $(t, x) \mapsto (t, i (x), n_{i (x)})$ is smooth and sends $(0, p)$ to admissible geodesic initial data. By [F2] that datum has an open neighborhood on which the geodesic evaluation map is smooth. Their composition is therefore the normal exponential map $Φ_{p}$ on a neighborhood of $(0, p)$ .

2.1step 1.1F1L1algebraconstruct∎

Differentiating the geodesic at zero gives the displayed differential: the time direction maps to $n_{i (p)}$ , and the slice direction maps to $d i_{p} (X)$ . If $a n_{i (p)} + d i_{p} (X) = 0$ , its inner product with $n_{i (p)}$ is $- a = 0$ because $n$ is orthogonal to $i (Σ)$ ; then $d i_{p} (X) = 0$ and $X = 0$ because an embedding is an immersion. Thus the differential is an isomorphism between four-dimensional tangent spaces. Apply [L1], then shrink its source neighborhood to a product $(- ϵ_{p}, ϵ_{p}) \times U_{p}$ ; the restriction is still a diffeomorphism onto its open image.

Local normal collar charts at a smooth spacelike hypersurface

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources