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Lemma 4.1: Cauchy sections and global hyperbolicity

PDF p. 23: Lemma 4.1 and proof; PDF p. 24: Lemma 4.1 and proof

Statement

Adopt the regular mass-one stationary vacuum construction, with maximal original exterior section and $T ∣_{S} \equiv̸ 0$ , and assume the Axiom of Choice.

Choose the single smooth stationary primitive $W_{0}$ on $E$ and a neighborhood of $S$ supplied by the global one-form construction, with its normalized Ernst potential $σ = f + i ψ$ , $f = - g (T, T)$ , and supplied pole $p \in S$ where $T_{p} = 0$ . Put $T^{♭} = g (T, \cdot)$ , $D_{q} = ∣ q (1 + σ) + (1 - σ) ∣^{2}$ , $d_{q} = D_{q} (p)$ , and $W_{q} = (1 - q)^{2} W_{0} - 4 q (1 - q) T^{♭}$ . Use the Harrison metric

$g_{q} = \frac{1}{4} (D_{q} g + T^{♭} \otimes W_{q} + W_{q} \otimes T^{♭} - f D_{q}^{- 1} W_{q} \otimes W_{q}) .$

For each $q \in (0, 1]$ , choose a complete smooth spacelike maximal stationary graph section $Σ_{q}$ with boundary $S$ as supplied by the maximal-slice result. Its stationary section time $v_{q}$ vanishes on its interior and satisfies $T v_{q} = 1$ ; write $Σ_{q, a}^{\circ} = {p \in E : v_{q} (p) = a}$ for $a \in R$ .

Each $Σ_{q, a}^{\circ}$ is a spacelike Cauchy hypersurface of $(E, g_{q})$ . Every inextendible causal curve has section-time image all of $R$ and meets each such section exactly once. Consequently $(E, g_{q})$ is globally hyperbolic. No corresponding global causal assertion for $g_{0}$ is claimed.

A causal curve is locally absolutely continuous with nonzero causal velocity almost everywhere, and all curves and causal relations are relative to $E$ . A Cauchy hypersurface meets every inextendible causal curve exactly once. Global hyperbolicity means strong causality and compact causal diamonds in $E$ .

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Mass-one convention and Harrison denominator

Stationary Harrison one-form and metric

Single-potential construction of the Harrison one-form family

Closedness, periods and smooth global choice of W0

Positivity of the Harrison denominator and horizon bound

Theorem 3.1: complete maximal slices for positive q

Stationary section time and translated maximal sections

Causal exterior, chronological boundaries and Cauchy data

Facts & Assumptions

[A1]

The regular stationary model and mass-one normalization supply the background hypotheses stated above. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Mass-one convention and Harrison denominator

[A2]

Work in ZFC, including countable and dependent choice. The Axiom of Choice

[F1]

For each fixed $q \in (0, 1]$ , the maximal-slice result supplies the stated complete smooth spacelike maximal graph section with boundary $S$ . The section-time construction identifies the exterior with the stationary product, defines $v_{q}$ , and identifies its level sets with the stationary translates of the section interior. Theorem 3.1: complete maximal slices for positive q Stationary section time and translated maximal sections

[F2]

The positive- $q$ ADM result gives a smooth time-oriented Lorentzian metric on the exterior, a positive lapse, and $g_{q}^{- 1} (d v_{q}, d v_{q}) < 0$ ; the translated level sets are spacelike. ADM form and stable causality of positive-q metrics

[F3]

Under the same construction and for the chosen section, the finite-section-time escape result says that every inextendible causal curve has section-time image $R$ . Its causal curves and endpoints are relative to the exterior. Causal speed and finite-section-time escape estimates Causal exterior, chronological boundaries and Cauchy data

[F4]

For a time-oriented smooth Lorentzian four-manifold, the existence of a Cauchy set implies strong causality and compact causal diamonds, assuming countable and dependent choice. The existence of a Cauchy set implies strong causality and compact causal diamonds

[F5]

The global one-form result applies to the regular mass-one vacuum data with original maximal exterior section and $T ∣_{S} \equiv̸ 0$ . It supplies the normalized smooth Ernst potential, the pole $p \in S$ with $T_{p} = 0$ , and one smooth stationary $W_{0}$ on the exterior and a neighborhood of $S$ , with $d W_{0} = 4 ((f - 1) F + ψ ⋆_{g} F)$ , $F = \frac{1}{2} d T^{♭}$ , and $W_{0} (T) = D_{0} (p) - D_{0}$ . Closedness, periods and smooth global choice of W0

[F6]

For these normalized data the denominator-positivity result gives $D_{q} > 0$ on $E$ for every $q \in [0, 1]$ . The one-form family defines the displayed $W_{q}$ and gives $W_{q} (T) = d_{q} - D_{q}$ . The Harrison metric definition supplies the displayed tensor formula and its Lorentzian signature whenever the one-form conditions and $D_{q} > 0$ hold. Positivity of the Harrison denominator and horizon bound Single-potential construction of the Harrison one-form family Stationary Harrison one-form and metric Mass-one convention and Harrison denominator

Proof

Given: The construction, choice convention, positive parameter $q$ , and chosen section in the Statement.

Goal: Establish the Cauchy and global-hyperbolicity conclusions for every real section time.

1.1givenA1F5F6algebra

Fix $q \in (0, 1]$ . The hypotheses of the global construction in [F5] are exactly [A1] and the original-section conditions in the Statement, so its normalized potential and $W_{0}$ exist on the required domain. The family in [F6] is smooth and stationary there. Differentiating its formula using [F5] gives $d W_{q} = 4 (1 - q) (((1 - q) f - (1 + q)) F + (1 - q) ψ ⋆_{g} F)$ , while [F6] gives $W_{q} (T) = d_{q} - D_{q}$ . At the supplied pole, $f (p) = 0$ , hence $d_{q} = (1 + q)^{2} + (1 - q)^{2} ψ (p)^{2} > 0$ . The positivity result applies to the same normalized potential and original maximal mass-one data, giving $D_{q} > 0$ throughout $E$ . Thus all one-form and domain conditions of the metric definition hold, and its formula defines the smooth stationary Lorentzian $g_{q}$ on the entire exterior. This is the fixed family used by the section, ADM and escape suppliers.

2.1givenA1F1F2step 1.1

Fix $a \in R$ . The supplied section and its stationary translates define the smooth level sets of $v_{q}$ throughout the exterior. The ADM result gives a positive lapse and shows that each such level set is spacelike. In particular, $Σ_{q, a}^{\circ}$ is a spacelike hypersurface.

3.1F2F3step 2.1

Let $γ$ be an inextendible future-directed causal curve in the exterior. The negative squared norm in [F2], with the section-time orientation, makes $v_{q}$ strictly increase along $γ$ ; equivalently, the finite-escape result uses its section-time parametrization. By [F3] its image under $v_{q}$ is all of $R$ . Therefore it takes the value $a$ exactly once. Reversing orientation gives the same intersection conclusion for a past-directed curve. By the stated definition, $Σ_{q, a}^{\circ}$ is a Cauchy hypersurface.

4.1A2F2F4step 3.1∎

The exterior is a time-oriented smooth Lorentzian four-manifold, and step 3.1 supplies a Cauchy set. The choice assumptions of [F4] follow from [A2]. Hence [F4] gives strong causality and compact causal diamonds, which is global hyperbolicity in the stated convention. The argument applies to each fixed positive $q$ and makes no assertion at $q = 0$ .

Source notes

The article reference attributes the original claim. The proof fixes the metric through the global primitive, one-form family, denominator positivity and metric definition, then uses the completed section, temporality and escape results and the mathematical Cauchy-set theorem; the article is not used as a prerequisite proof.

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Sources

The uniqueness of smooth stationary black holes · PDF pages 23–24, Lemma 4.1 and its proof