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Identification of the spherical exterior and its positive stationary lapse

Statement

Work in ZFC. Let $(D, g, T)$ be smooth time-oriented stationary Lorentzian data, and let $U \subset D$ be a nonempty connected open stationary region. Require that the flow of $T$ restricts to a smooth global flow $Φ : R \times U \to U$ . No causal character of $T$ on all of $U$ is required. Take all chronological relations and boundaries in $D$ , and set $A = I_{D}^{-} (U)$ , $B = I_{D}^{+} (U)$ and $E = A \cap B$ .

Suppose $E$ is globally hyperbolic and $C = Σ \cap E$ is a smooth spacelike Cauchy hypersurface of $E$ , where $Σ \subset D$ is smooth and spacelike with a specified diffeomorphism $Σ ≅ {x \in R^{3} : r = ∣ x ∣ > 1 / 2}$ . Assume $\emptyset \neq U \cap Σ \subset {r > 1}$ and $S = \partial_{D} A \cap \partial_{D} B = {r = 1} \subset Σ$ . Then $C = {r > 1}$ .

Suppose additionally that the smooth outer portion ${r \geq 1} \subset Σ$ is maximal, $Ric (g) = 0$ along it, and $T$ is tangent to $S$ . On an open neighborhood of its full tail ${r > R}$ for some $R > 1$ , supply coordinates $(t, x^{1}, x^{2}, x^{3})$ with $T = \partial_{t}$ and $Σ = {t = 0}$ there, whose spatial coordinates agree with the specified identification of $Σ$ . Require constants $a > 0$ and $0 < α \leq 1$ such that $∣ g_{μ ν} - η_{μ ν} ∣ \leq a r^{- α}$ for $μ, ν \in {0, 1, 2, 3}$ , uniformly on this coordinate neighborhood as $r \to \infty$ , where $η = diag (- 1, 1, 1, 1)$ . Require $T$ to be future timelike sufficiently far out in this chart. With future unit normal $n$ , $h = g ∣_{T Σ}$ , $k (V, W) = - g (\nabla_{V} n, W)$ and $T = N n + X$ , the lapse satisfies $Δ_{h} N = ∣ k ∣_{h}^{2} N$ , $N ∣_{S} = 0$ , $N \to 1$ uniformly at infinity and $0 < N < 1$ on $C$ .

Statement references

Causal exterior, chronological boundaries and Cauchy data

Stationary Lorentzian data and tensor conventions

Spacelike sections, maximality and lapse–shift data

Exact asymptotically stationary end contract

Connectedness and causal convexity of a chronological exterior

A Cauchy section cannot end on only one chronological horizon

A smooth Cauchy hypersurface gives a connected timelike-flow product

Mean-curvature variation and the stationary lapse equation

Exterior Schrödinger maximum principles, Hopf growth and Harnack

The Axiom of Choice

Facts & Assumptions

[A1]

Work in ZFC, including the Axiom of Choice and its countable/dependent-choice consequences. The choice use is inherited through the causal suppliers and the density and temporal-flow construction in the Cauchy-product supplier. The Axiom of Choice

[F1]

The chronological exterior, ambient boundaries and Cauchy condition have the causal interface's meanings; the stationary and hypersurface interfaces supply the Killing field, Ricci convention, future unit normal, induced metric, second fundamental form and lapse decomposition. Causal exterior, chronological boundaries and Cauchy data Stationary Lorentzian data and tensor conventions Spacelike sections, maximality and lapse–shift data

[F2]

For a nonempty connected open stationary region with a smooth global stationary flow, without a causal-character requirement on that flow, the connectivity supplier gives $U \subset E$ and connectedness of $E$ . For the stated Cauchy section, boundary exclusion gives $\partial_{Σ} C \subset \partial_{D} A \cap \partial_{D} B$ , and the globally hyperbolic Cauchy-product supplier gives connectedness of $C$ . Connectedness and causal convexity of a chronological exterior A Cauchy section cannot end on only one chronological horizon A smooth Cauchy hypersurface gives a connected timelike-flow product

[F3]

The metric bound in the Statement is the zero-order component estimate in clause (1) of the asymptotically stationary end definition. Only that estimate is used; no derivative or Hölder estimate or coordinate existence theorem is invoked. The matching of the spatial coordinates, the full-tail domain, $T = \partial_{t}$ and $Σ = {t = 0}$ and the future direction of $T$ sufficiently far out are explicit hypotheses here, rather than consequences of the stationary-data definition. Exact asymptotically stationary end contract Stationary Lorentzian data and tensor conventions

[F4]

On a smooth maximal spacelike hypersurface, a Killing field's lapse obeys $Δ_{h} N = (∣ k ∣_{h}^{2} + Ric (n, n)) N$ with the stated hypersurface and curvature conventions. This is a local identity, imposing no boundary data or end normalization. Mean-curvature variation and the stationary lapse equation

[F5]

On a connected smooth Riemannian exterior with compact nonempty smooth boundary, a smooth solution of $(Δ_{h} - V) u = 0$ with $V \geq 0$ , zero boundary values and uniform limit $1$ at infinity satisfies $0 < u < 1$ in the interior. Exterior Schrödinger maximum principles, Hopf growth and Harnack The exterior, compact-core, end-chart and Laplacian conventions are those of Weighted exterior elliptic domains and admissible weights; no weighted norm estimate is needed here.

[F6]

A component is the largest connected subset through a point. In a connected space the only clopen subsets are the empty set and the whole space. The unit sphere in $R^{3}$ is path connected, and a path-connected space is connected. Connected components, quasicomponents, and totally disconnected spaces For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected Every path-connected space is connected, and every path component lies inside a component

[F7]

For a square matrix $M$ over a field with $\det M \neq 0$ , the inverse is $M^{- 1} = adj (M) / \det M$ . Here the field is $R$ and the matrix size is four. The adjugate gives an explicit inverse formula over a field

[F8]

In $R^{3}$ with its Euclidean topology, every closed bounded subset is compact. A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology

[F9]

On a time-oriented smooth Lorentzian four-manifold, the chronological future $I_{D}^{+} (u)$ and past $I_{D}^{-} (u)$ of every point $u$ are open. Clause (a) of the chronological-openness supplier applies with the locally absolutely continuous timelike-curve convention used here; its Countable Choice and Dependent Choice hypotheses hold in ZFC by [A1]. Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

Proof

Given: The stationary-region, Cauchy-section, coordinate and maximality hypotheses in the Statement, in ZFC.

Goal: Identify the outer component and prove its lapse equation, boundary value, normalization and strict bounds.

1.1givenA1F1F2F9

Apply the chronological-exterior supplier with the nonempty connected open region $U$ and its supplied global flow; no timelikeness of $T$ throughout $U$ is needed. It gives $U \subset E$ and connectedness of $E$ . Apply [F9] in $D$ : for every $u \in U$ , the sets $I_{D}^{-} (u)$ and $I_{D}^{+} (u)$ are open. By the chronological definitions, $A = ⋃_{u \in U} I_{D}^{-} (u)$ and $B = ⋃_{u \in U} I_{D}^{+} (u)$ , so $A$ and $B$ are open as unions of open subsets of $D$ , and $E = A \cap B$ is open. In particular $U \cap Σ \subset C$ , and this section is nonempty and lies in ${r > 1}$ by hypothesis. The boundary-exclusion supplier applies to the stated Cauchy hypersurface and gives $\partial_{Σ} C \subset S$ . Since $A$ and $B$ are open, their boundaries are disjoint from them, so $S \cap C = \emptyset$ .

2.1givenA1F1F2F6step 1.1

The set $C$ is open in $Σ$ and has no boundary in $Σ ∖ S$ , hence is both open and closed in $Σ ∖ S$ . Under the specified diffeomorphism, that space is the disjoint union of the clopen shells ${1 / 2 < r < 1}$ and ${r > 1}$ . Each shell is path connected: join two points radially to a common radius in its radial interval, join their unit directions by a path in the unit sphere from [F6] at that radius, and concatenate with the reversed radial segment. Each segment stays in the shell, so [F6] makes each shell connected; being disjoint clopen connected sets covering the space, they are exactly its components. Connectedness of $E$ and its global hyperbolicity allow the Cauchy-product supplier to conclude that $C$ is connected. It meets the outer shell through the nonempty set $U \cap Σ$ ; connectedness prevents it from meeting both shells. Its intersection with the outer shell is nonempty and clopen in that connected shell, so [F6] makes it the whole shell. Therefore $C = {r > 1}$ .

3.1givenF1F3F7F8step 2.1algebra

The normal and lapse are smooth on ${r \geq 1}$ , and tangency of $T$ to $S \subset Σ$ gives $N = - g (T, n) = 0$ on $S$ . On the specified end chart the bound in [F3] gives $g_{μ ν} \to η_{μ ν}$ uniformly. The determinant and cofactors are finite polynomials in the matrix entries, so their limits are uniform as well. Thus $\det (g_{μ ν}) \to \det η = - 1$ uniformly, and its absolute value is at least $1 / 2$ sufficiently far out. Applying [F7] to the real $4 \times 4$ matrix gives $g^{00} \to - 1$ uniformly. Since $Σ = {t = 0}$ , its unit normal is $n = - \nabla t / \sqrt{- g^{00}}$ on the tail: this sign is future-directed because $T = \partial_{t}$ is future timelike and $g (- \nabla t, T) = - 1$ . Consequently $N = d t (T) / \sqrt{- g^{00}} = (- g^{00})^{- 1 / 2} \to 1$ uniformly. Coordinate matching makes this the full spatial infinity of $C$ ; for finite $b \geq 1$ , the Euclidean annuli ${x \in R^{3} : 1 \leq ∣ x ∣ \leq b}$ are closed and bounded, hence compact by [F8]. Their images ${1 \leq r \leq b} \subset Σ$ under the specified diffeomorphism are compact: pulling any open cover back to the compact annulus and taking a finite subcover proves this directly.

4.1givenF1F4F5step 2.1step 3.1∎

The local maximal lapse identity in [F4] applies on exactly the outer portion assumed maximal in the Statement. With $Ric (g) = 0$ there it yields $Δ_{h} N = ∣ k ∣_{h}^{2} N$ . By step 2.1 its closure in $Σ$ is the smooth connected Riemannian exterior ${r \geq 1}$ , with smooth metric and compact nonempty boundary $S$ . To check the exterior convention in [F5], choose $r_{0} > R$ : the complement of the compact set ${1 \leq r \leq r_{0}}$ is the specified end ${r > r_{0}}$ , and its required core $K = {1 \leq r \leq 2 r_{0}}$ is compact by step 3.1. Its lapse is smooth up to that boundary, its potential $V = ∣ k ∣_{h}^{2}$ is smooth and nonnegative, and step 3.1 supplies zero boundary values and the uniform end limit $1$ . Thus every hypothesis of [F5] is satisfied, and the exterior maximum principle gives $0 < N < 1$ on $C$ , completing all conclusions.

Depends on

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