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Proposition 4.1: positive-expansion spheres of limiting area 4π

PDF p. 21: Proposition 4.1; PDF p. 22: Proof of Proposition 4.1; (4.2)

Statement

Work in ZFC with the regular stationary vacuum data of Regular non-degenerate stationary vacuum spacetime, the model assumptions of Stationary vacuum model and asymptotic normalization, and the mass-one end, pole, normalized Ernst potential and specified Harrison family required by Uniform coordinate-sphere area and null expansion. In the original stationary end coordinates, let $C_{q} = {t = 0, r = R_{0} q^{- 2 / 3}}$ for $0 < q \leq 1$ , as in Comparison spheres and their stationary cylinders. The transformed metric is $g_{q}$ .

For a sufficiently large fixed $R_{0}$ , independent of $q$ , each $C_{q}$ is a smooth embedded $g_{q}$ -spacelike two-sphere. Its stationary projection onto the original section surrounds the bifurcation sphere $S$ in the following intrinsic sense. Put $B = (Σ_{0} \cap E) \cup S$ . For $R = R_{0} q^{- 2 / 3}$ , the truncation $K_{q} = B ∖ {t = 0, r > R}$ is a compact smooth region with boundary $S ⊔ C_{q}$ , where $r$ is the original end radius. The future outward null expansion of $C_{q}$ is strictly positive; its stationary cylinder is timelike, and $t$ is temporal on the exterior side $r \geq R_{0} q^{- 2 / 3}$ . Moreover $∣ C_{q} ∣_{g_{q}} \to 4 π$ as $q ↓ 0$ . Here temporal means that $d t$ is timelike; the null expansion is the trace of the null second fundamental form as defined in Null expansions, screen quotients and focal data.

Facts & Assumptions

[A1]

The construction uses ZFC, including the axiom of choice, and the stated stationary vacuum model. The Axiom of Choice Stationary vacuum model and asymptotic normalization

[F1]

The coordinate definition gives smooth embedded spheres $C_{q}$ , their stationary translates ${t = s, r = R_{0} q^{- 2 / 3}}$ , and $R_{0} q^{- 2 / 3} \to \infty$ with $q R_{0} q^{- 2 / 3} \to 0$ as $q ↓ 0$ . Comparison spheres and their stationary cylinders

[F2]

The original section $Σ_{0}$ agrees with $t = 0$ on the actual stationary end. Its closed exterior portion $B = (Σ_{0} \cap E) \cup S$ is a smooth embedded manifold with compact boundary $S$ . For every sufficiently large original end radius $R$ , the truncation $B ∖ {t = 0, r > R}$ is a compact smooth region with boundary $S ⊔ {t = 0, r = R}$ . This is the compact-truncation conclusion of the existing section supplier under the regular-spacetime hypotheses. Regular non-degenerate stationary vacuum spacetime The exterior Cauchy component has a complete asymptotically Euclidean closure

[F3]

For all sufficiently large $R$ , uniformly in $q \in [0, 1]$ and angle, the supplied estimate gives $∣ {t = 0, r = R} ∣_{g_{q}} = 4 π (1 + q R)^{2} (1 + O (R^{- 1}))$ and $θ_{L_{q}} = 2 q / (1 + q R) + O (R^{- 2})$ . It also gives spacelike coordinate slices, timelike $d t$ , and timelike stationary coordinate cylinders throughout that large-radius region. The future outward normal is normalized by $g_{q} (T, L_{q}) = - d_{q} / 4$ , where $d_{q}$ is the pole denominator. Uniform coordinate-sphere area and null expansion Mass-one convention and Harrison denominator Null expansions, screen quotients and focal data

Proof

Given: The data and parameter range in the Statement, and the uniform estimates in [F3].

Goal: Establish the geometric properties and limiting area for one sufficiently large fixed $R_{0}$ .

1.1givenF1F2F3

Choose $R_{0}$ larger than the end threshold, the compact-truncation threshold in [F2], and the common large-radius threshold in [F3]. Since $q^{- 2 / 3} \geq 1$ , every chosen radius $R = R_{0} q^{- 2 / 3}$ is at least $R_{0}$ . The spheres are smooth and embedded by [F1], and are spacelike because they lie in the spacelike coordinate slices of [F3]. The section agrees with $t = 0$ on the original end, so $C_{q}$ already lies in that section. The stationary end flow is $(t, x) \mapsto (t + s, x)$ ; hence its projection $(t, x) \mapsto (0, x)$ fixes $C_{q}$ . Apply [F2] at $R = R_{0} q^{- 2 / 3}$ : the region $K_{q} = B ∖ {t = 0, r > R}$ is compact and smooth, with boundary $S ⊔ C_{q}$ . This establishes the asserted intrinsic enclosure in the actual end coordinates.

2.1givenF3step 1.1algebra

The uniform expansion estimate has an error bounded in absolute value by $K R^{- 2}$ for some constant $K$ independent of $q$ , angle and sufficiently large $R$ . For $0 < q \leq 1$ , $2 q R^{2} / (1 + q R) = 2 R_{0}^{2} q^{- 1 / 3} / (1 + R_{0} q^{1 / 3}) \geq 2 R_{0}^{2} / (1 + R_{0})$ . Increase $R_{0}$ until the final quantity exceeds $K$ . The leading term then strictly exceeds the absolute error everywhere on every $C_{q}$ , so $θ_{L_{q}} > 0$ . The normal is future outward by [F3], and its expansion has the stated meaning.

3.1F1F3step 1.1step 2.1algebra∎

By [F3], the stationary cylinder at each chosen radius is timelike, and $d t$ is timelike at every point with $r \geq R$ because this entire region lies beyond the common threshold. Finally $q R = R_{0} q^{1 / 3} \to 0$ and $R^{- 1} \to 0$ as $q ↓ 0$ . Applying the uniform area formula in [F3] gives $∣ C_{q} ∣_{g_{q}} \to 4 π$ .

Source notes

The article reference attributes Proposition 4.1, which defines the spheres in the original end coordinates and asserts that their stationary projections surround $S$ . Intrinsic enclosure here follows from the existing compact-truncation supplier. The coordinate definition and uniform-estimate suppliers give the other conclusions; the proposition is not used as a premise.

Depends on

Used by · 5

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Sources

The uniqueness of smooth stationary black holes · PDF pp. 21–22, Proposition 4.1 and its proof