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Remark 1.3: lapse bounds and stationary product

PDF p. 4: Remark 1.3 and (1.3); PDF p. 5: Remark 1.3 and (1.3)

Statement

Work in ZFC, including the Axiom of Choice. Assume the regular stationary vacuum data and the stationary vacuum model. Write $j : Σ_{0} \to {y \in R^{3} : ∣ y ∣ > 1 / 2}$ for the specified hypersurface identification, $s = ∣ j (\cdot) ∣$ , $S = {s = 1}$ and $C = Σ_{0} \cap E$ . Write $r = ∣ x ∣$ for the physical end radius in the stationary end coordinates $(t, x)$ ; no agreement between these separately supplied coordinates is assumed.

Then $C$ is exactly the component of $Σ_{0} ∖ S$ containing the stationary $t = 0$ end. Its ambient closure $B = C \cup S$ is a connected smooth embedded spherical exterior with compact boundary $S$ , diffeomorphic to ${y \in R^{3} : ∣ y ∣ \geq 1}$ and complete in its intrinsic length metric with the boundary included. In the physical end coordinates, $h_{i j} = δ_{i j} + O_{6} (r^{- 1})$ and $B ∖ {t = 0, r > R}$ is compact for every sufficiently large $R$ ; these truncations exhaust $B$ . Thus it has one asymptotically Euclidean end. In the specified $j$ coordinates, $C$ may be either ${s > 1}$ or ${1 / 2 < s < 1}$ . An adapted topological identification, obtained from $j$ by a radial inversion if necessary, places $C$ on the outer side without asserting agreement with the physical end coordinates. There is an auxiliary complete future timelike-flow diffeomorphism $R \times C \to E$ .

On $Σ_{0}$ use the future unit normal $n$ , $h = g ∣_{T Σ_{0}}$ , $k (U, V) = - g (\nabla_{U} n, V)$ , $H = {div}_{g} n = - {tr}_{h} k$ and $T = N n + X$ , with $N = - g (T, n)$ and $X$ tangent. Without a maximality assumption, the lapse on $C$ satisfies $Δ_{h} N = ∣ k ∣_{h}^{2} N - X (H)$ , $N ∣_{S} = 0$ and $N \to 1$ uniformly on its physical end, where $Δ_{h} = {div}_{h} {grad}_{h}$ .

If the actual exterior section $C$ is maximal, meaning $H = 0$ on $C$ , then $Δ_{h} N = ∣ k ∣_{h}^{2} N$ and $0 < N < 1$ on $C$ . The stationary-flow map $(a, c) \mapsto ϕ_{a}^{T} (c)$ is a diffeomorphism $R \times C \to E$ , and $f = - g (T, T) = N^{2} - ∣ X ∣_{h}^{2} < 1$ throughout $E$ after stationary extension. The horizon boundary is excluded from both open products. In particular these conditional conclusions hold if the physical end lies on the specified side ${s > 1}$ and the portion ${s \geq 1}$ is maximal: then $C = {s > 1}$ . Maximality of that specified side alone does not identify it with $C$ .

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Lapse, shift, induced metric and second fundamental form

Mean-curvature variation and the stationary lapse equation

Exterior Schrödinger maximum principles, Hopf growth and Harnack

Spacelike sections, maximality and lapse–shift data

The Axiom of Choice

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

The exterior Cauchy component has a complete asymptotically Euclidean closure

A complete stationary flow transverse positively to a Cauchy section is a product

Weighted exterior elliptic domains and admissible weights

The adjugate gives an explicit inverse formula over a field

Facts & Assumptions

[A1]

Work in ZFC, including the Axiom of Choice and the inherited countable/dependent-choice hypotheses of the causal and timelike-flow suppliers. The Axiom of Choice

[F1]

The adopted physical assumptions are the vacuum equation, stationarity and future-timelike unit normalization of $T$ sufficiently far out. These are model assumptions rather than observations. Stationary vacuum model and asymptotic normalization

[F2]

Regularity supplies the smooth spacelike section, Cauchy portion, globally hyperbolic exterior, complete stationary orbits meeting the section, transverse horizon branches and tangency of $T$ to both branches. On the separate stationary end, $T = \partial_{t}$ , the section is $t = 0$ , and the metric has the stated angularly uniform differentiated decay. Regular non-degenerate stationary vacuum spacetime

[F3]

Use the future-normal, induced-metric, second-fundamental-form, lapse, shift and mean-curvature conventions in the Statement; in particular $H = - {tr}_{h} k$ , maximality means $H = 0$ , and $- g (T, T) = N^{2} - ∣ X ∣_{h}^{2}$ . Lapse, shift, induced metric and second fundamental form Spacelike sections, maximality and lapse–shift data

[F4]

For a spacelike variation with vector $a n + X$ , the pulled-back mean curvature has derivative $Δ_{h} a - (∣ k ∣_{h}^{2} + Ric (n, n)) a + X (H)$ . A local Killing flow preserves the pulled-back mean curvature. 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 end limit $1$ satisfies $0 < u < 1$ in the interior. The exterior convention requires a physical end chart and a compact core; no weighted norm is used here. Exterior Schrödinger maximum principles, Hopf growth and Harnack Weighted exterior elliptic domains and admissible weights

[F6]

The chronological-exterior lemma gives inclusion of the connected open stationary end in $E$ , openness and connectedness of $E$ , and causal convexity. For the supplied Cauchy section, boundary exclusion gives $\partial_{Σ_{0}} C \subset S$ . Connectedness and causal convexity of a chronological exterior A Cauchy section cannot end on only one chronological horizon

[F7]

A connected globally hyperbolic smooth time-oriented spacetime with a smooth spacelike Cauchy hypersurface has a connected section and a product based on that section given by an auxiliary complete future timelike flow. A smooth Cauchy hypersurface gives a connected timelike-flow product

[F8]

Under the regular-spacetime data, the actual exterior component has the smooth embedded complete closure $B = C \cup S$ , an adapted topological spherical-exterior identification and a single physical asymptotically Euclidean end with compact physical truncations. It requires neither maximality nor matching of the specified hypersurface chart to the end chart. The exterior Cauchy component has a complete asymptotically Euclidean closure

[F9]

On a connected globally hyperbolic spacetime with smooth spacelike Cauchy section $C$ , a complete smooth vector field whose every orbit meets $C$ and whose lapse relative to its future normal is positive gives the stationary-flow product map as a global diffeomorphism. No causal-character assumption on the field is needed. A complete stationary flow transverse positively to a Cauchy section is a product

[F10]

For an invertible real square matrix, its inverse is its adjugate divided by its nonzero determinant. Apply this formula to the $3 \times 3$ induced metric matrix on the physical end. The adjugate gives an explicit inverse formula over a field

Proof

Given: The regular stationary vacuum data and ZFC; actual-exterior maximality is assumed only for the conditional conclusions.

Goal: Establish the intrinsic exterior and general lapse identity, then the maximal-exterior lapse bounds and stationary product.

1.1givenA1F2F6F7

Apply [F6] with the designated stationary end. It is nonempty and connected, its coordinate translations give a global flow, and [F2] supplies the Cauchy and global-hyperbolicity hypotheses. Thus $E$ is open and connected, contains that end, and $\partial_{Σ_{0}} C \subset S$ . Apply [F7] to this smooth Cauchy portion to obtain connectedness of $C$ and the auxiliary timelike-flow product.

1.2givenF1F2F3F4

The local Killing flow transports the induced geometry and future normal of the section isometrically, so its pulled-back mean curvature has zero derivative by [F4]. Apply the full variation formula with $a = N$ and tangential part $X$ : $0 = Δ_{h} N - (∣ k ∣_{h}^{2} + Ric (n, n)) N + X (H)$ . The vacuum equation sets the Ricci term to zero and gives $Δ_{h} N = ∣ k ∣_{h}^{2} N - X (H)$ without assuming maximality.

2.1givenA1F2F3F8step 1.1

Apply [F8] to the same regular data, with the same specified section and physical end. It identifies $C$ as the component containing that end and proves all the asserted closure, smoothness, completeness, compact-truncation and differentiated end properties. The adapted topological identification can interchange the specified sides; it does not change $C$ or the physical end chart. No conclusion here selects the specified outer side without an end-side premise.

3.1givenF1F2F3F10step 2.1algebra

At $S$ , tangency to both transverse horizons gives $T_{p} \in T_{p} H^{+} \cap T_{p} H^{-} = T_{p} S \subset T_{p} Σ_{0}$ . Hence $N = - g (T, n) = 0$ there. On the physical $t = 0$ end, $h_{i j} = g_{i j} \to δ_{i j}$ uniformly. Its determinant tends to $1$ , since it is a polynomial in the entries, so [F10] gives $h^{i j} \to δ^{i j}$ uniformly. Orthogonality of $n$ to the section gives $g_{0 i} = h_{i j} X^{j}$ and hence $∣ X ∣_{h}^{2} = h^{i j} g_{0 i} g_{0 j}$ . By [F3], $N^{2} = - g_{00} + h^{i j} g_{0 i} g_{0 j} \to 1$ uniformly. The future timelike field $T$ has positive lapse on this tail, so $N \to 1$ . Step 2.1 ensures that this physical tail is the whole infinity needed for an exterior end limit.

4.1givenF3F5step 2.1step 3.1step 1.2

Now suppose that the actual $C$ is maximal. Then $H = 0$ and $X (H) = 0$ on $C$ , so step 1.2 gives $(Δ_{h} - ∣ k ∣_{h}^{2}) N = 0$ . The metric and lapse are smooth up to $S$ , since they restrict smooth data on the spacelike section through $S$ . Step 2.1 supplies a connected Riemannian exterior with nonempty compact smooth boundary. In the physical chart choose any sufficiently large $r_{0}$ ; both $B ∖ {r > r_{0}}$ and the required core $B ∖ {r > 2 r_{0}}$ are compact by step 2.1, meeting [F5]'s precise exterior convention. The potential $V = ∣ k ∣_{h}^{2}$ is smooth and nonnegative, and step 3.1 supplies the zero boundary value and uniform unit end limit. Therefore [F5] gives $0 < N < 1$ on $C$ .

5.1givenF2F3F9step 1.1step 2.1step 4.1∎

Under this maximality hypothesis, every complete stationary orbit in $E$ meets $Σ_{0}$ by [F2], and that intersection belongs to $C$ because the orbit stays in $E$ . Step 4.1 supplies positive lapse on $C$ , so [F9] gives the stationary-flow diffeomorphism $R \times C \to E$ . Its injectivity uses positive crossings of a Cauchy defining coordinate, without assuming stationary trajectories are causal. On $C$ , [F3] gives $f = N^{2} - ∣ X ∣_{h}^{2} \leq N^{2} < 1$ . The Killing flow preserves $g$ and $T$ , hence $f$ , and orbit coverage extends this bound throughout $E$ . Finally, if the physical end is in the specified outer side, the component identification in step 2.1 gives $C = {s > 1}$ ; maximality of ${s \geq 1}$ then implies maximality of this $C$ , so all the conditional conclusions apply. Neither product includes $S$ .

Source notes

Remark 1.3 and equation (1.3), PDF pp. 4–5, attribute the original lapse equation, zero boundary value, unit end normalization, strict lapse bound, stationary product and inequality $f < 1$ . Theorem 1.1 on p. 4 states maximality of the specified portion $∣ x ∣ \geq 1$ and also assumes $T ∣_{S} \equiv̸ 0$ . The article uses the same spatial notation for its end and section descriptions. This citation records its claim and scope; it is not a proof of the corrected result above.

The generated regularity definition supplies separate end and hypersurface identifications. Its specified-side maximality does not by itself imply maximality of the actual $C$ . The corrected Statement retains the intrinsic exterior, complete physical end and auxiliary product unconditionally, and states the lapse bounds, stationary product and $f < 1$ under the explicit actual-exterior maximality hypothesis. The specified-coordinate conclusion is retained with its end-side premise. This corrects the generated formulation; it establishes no defect in the original article or its intended coordinate convention. The ADM causal-completeness supplier presupposes a product and is not used to construct either product here.

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Sources

The uniqueness of smooth stationary black holes · PDF pp. 4,5; Remark 1.3 and (1.3)