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Causal speed and finite-section-time escape estimates

PDF p. 23: Proof of Lemma 4.1; (4.3); PDF p. 24: Proof of Lemma 4.1; (4.3)

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. Fix $q \in (0, 1]$ and a complete smooth spacelike maximal graph section $Σ_{q}$ supplied by Theorem 3.1: complete maximal slices for positive q. Its compact boundary is the bifurcation sphere $S$ . Let $v = v_{q}$ be stationary section time, and write the stationary metric as $g_{q} = - N^{2} d v^{2} + h_{i j} (d x^{i} + X^{i} d v) (d x^{j} + X^{j} d v)$ , with positive lapse $N$ , induced Riemannian metric $h$ , and shift $X$ .

A causal curve is locally absolutely continuous with nonzero causal velocity almost everywhere. A future causal curve admits section-time parameterization and obeys $h_{i j} ((x^{i})^{'} + X^{i}) ((x^{j})^{'} + X^{j}) \leq N^{2}$ almost everywhere. Near $S$ , let $ρ$ be the $h$ -distance to $S$ , extended constantly along the stationary flow. In a sufficiently small collar, $N = O (ρ)$ and $X (ρ) = O (ρ)$ , and hence $∣ d ρ / d v ∣ \leq C ρ$ almost everywhere and $ρ (v) \geq ρ (v_{0}) e^{- C ∣ v - v_{0} ∣}$ while the curve remains in the collar. On the asymptotic end, for the original radial coordinate $r$ , $∣ r^{'} ∣ \leq ∣ d r ∣_{h} (N + ∣ X ∣_{h}) \leq C$ sufficiently far out. Constants may depend on the fixed $q$ .

A causal curve cannot reach $S$ or escape to infinity in finite section time. If its spatial projection remains in an interior compact set and its section time approaches a finite endpoint, the curve has an endpoint in the exterior. Thus an inextendible causal curve has section-time image $R$ . These assertions concern section time, not geodesic affine completeness.

Facts & Assumptions

Given: The model, choice convention, positive parameter, supplied section and causal curve in the Statement.

Goal: Prove the speed, collar, end and endpoint assertions.

[A1]

The regular data supply the compact sphere $S$ , the stationary field tangent to both transverse horizon branches, and the one-ended exterior. The mass normalization and stationary model specify the transformed family. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Mass-one convention and Harrison denominator

[A2]

Work in ZFC, including the choice assumptions inherited by the mathematical suppliers. The Axiom of Choice

[F1]

The section theorem supplies a smooth spacelike graph with boundary $S$ , complete up to that boundary, and normalized end derivatives tending to zero. The graph and section-time definitions identify the exterior with the stationary product and define $v$ . Theorem 3.1: complete maximal slices for positive q Stationary maximal graphs and normalized graph heights Stationary section time and translated maximal sections

[F2]

The lapse theorem gives $N > 0$ in the interior and $N = 0$ on $S$ . The ADM theorem gives the displayed stationary metric; the shift is the section-tangent part of $T = N n + X$ . Transformed lapse equation and bounds ADM form and stable causality of positive-q metrics Spacelike sections, maximality and lapse–shift data

[F3]

Apply the additional positive-parameter conclusion of Asymptotically flat positive-q ends and cylindrical limit for the actual smooth primitive of Closedness, periods and smooth global choice of W0 in the standing regular mass-one construction, with maximal original exterior section and $T ∣_{S} \equiv̸ 0$ . The pole, normalized contraction and single family are unchanged; only $(W_{0})_{i} = O_{5} (r^{- 3})$ is required, not equality with the exact radial integral. For this fixed $q > 0$ , the normalized metric is $- 1 + O_{5} (r_{q}^{- 1})$ in its time-time component, $O_{5} (r_{q}^{- 2})$ in its mixed components and $δ_{i j} + O_{5} (r_{q}^{- 1})$ spatially. These are the end bounds used below.

[F4]

The causal section-time speed lemma supplies the causal reparameterization, the displayed speed inequality, and $∣ (f \circ x)^{'} ∣ \leq N ∣ d f ∣_{h} + ∣ X (f) ∣$ for smooth $f$ . Causal relations and endpoints are relative to the exterior. ADM section time and causal spatial speed Causal exterior, chronological boundaries and Cauchy data

[F5]

The normal-tube, distance-gradient and boundary-normal-coordinate results of Lee are recorded in External input below. Together they supply the geometry used to construct the intrinsic boundary-distance collar in step 1.2. Lee, Theorem 5.25, Corollary 6.39, Theorem 6.40 and Example 6.44.

[F6]

The compact-confinement lemma combines a boundary collar barrier and a proper-radius bound; the compact-endpoint lemma supplies a limit at a finite time endpoint for a trajectory confined to an interior compact set. Combining the collar and end barriers into interior compact confinement An interior compact causal trajectory has a finite-time endpoint

[F7]

Given a smooth compact-boundary collar with coordinate $ρ$ , smooth positive-definite $h$ up to its boundary, and $N \leq A ρ$ , $∣ X (ρ) ∣ \leq A ρ$ , $∣ d ρ ∣_{h} \leq A$ , the collar-barrier lemma gives $b (x (v)) \geq b (x (v_{0})) e^{- (A^{2} + A) ∣ v - v_{0} ∣}$ for $b = \min (ρ, a)$ in the collar, extended by $a$ outside it, where $0 < a$ is smaller than the collar width. Its proof uses the absolute-continuity fundamental theorem and both orientations of Gronwall, including curves that leave and re-enter the collar. Collar barrier against finite section-time boundary arrival

[F8]

In the stationary ADM setting, suppose the connected section has compact boundary, $h$ extends smoothly and positively to that boundary, $N$ and $X$ extend smoothly, and a supplied smooth nonnegative radius $R$ has compact sublevels. If $∣ d R ∣_{h} (N + ∣ X ∣_{h})$ is bounded outside a compact set, the proper-radius lemma gives $∣ R (x (v)) - R (x (v_{0})) ∣ \leq C_{R} ∣ v - v_{0} ∣$ for all section times. Its sufficient end realization is a compact core with finitely many product ends having compact cross sections, with $R$ equal to the unbounded end coordinate outside a larger compact core. Proper end radius and exclusion of finite-time spatial infinity

[F9]

The smooth-cutoff lemma supplies $η \in C_{c}^{\infty} (R)$ with $0 \leq η \leq 1$ , $η = 1$ on $∣ s ∣ \leq 1$ , and $η = 0$ on $∣ s ∣ \geq 2$ , by taking $n = 1$ . Explicit compactly supported smooth cutoffs

External input

Lee's normal-tube theorem gives a uniform normal-exponential diffeomorphism around a compact embedded submanifold in a smooth Riemannian manifold. In that tube the normal-length function equals ambient distance and has unit gradient away from the submanifold. Example 6.44 supplies smooth extension of a boundary metric to the double and restriction to inward boundary normal coordinates. These are imported premises; the intrinsic-distance comparison and the causal deductions below are local applications. The exact hypotheses, locators and reading limits are in the Lee source entry.

Proof

1.1A1A2F1F2F4

Fix the parameter and section. The graph supplies the global stationary product, while the positive lapse and ADM form satisfy the hypotheses of the section-time speed lemma. It follows that future causal curves have strictly increasing $v$ and that their section-time parameterizations satisfy the displayed quadratic speed inequality almost everywhere.

1.2F1F5A2

The induced Riemannian metric $h$ is smooth and positive definite up to the compact boundary $S$ . Use the boundary construction in [F5] to extend it smoothly to the connected double, with extended metric $\hat{h}$ . The compact normal-tube theorem gives a common $ϵ > 0$ for which $F (p, s) = \exp_{p}^{\hat{h}} (s ν_{p})$ , $p \in S$ , $∣ s ∣ < ϵ$ , is a diffeomorphism onto a normal tube, where $ν$ is the inward unit normal. Shrink $ϵ$ so that its inward half is a collar in the original section. The distance and gradient results in [F5] give $d_{\hat{h}} (F (p, s), S) = s$ and $∣ d s ∣_{h} = 1$ for $0 < s < ϵ$ . Every curve confined to the section is also an ambient curve, so $d_{h} (F (p, s), S) \geq s$ ; the inward normal segment lies in the section and has length $s$ , proving the reverse inequality. Thus the intrinsic boundary distance $ρ$ is exactly the smooth inward coordinate $s$ , including at $S$ .

2.1A1A2F1F2F4F7step 1.1step 1.2

The unit normal and the lapse and shift are smooth up to $S$ . Since $T$ is tangent to both transverse horizon branches, $T ∣_{S} \in T S$ . The lapse vanishes on $S$ , so $X ∣_{S} = T ∣_{S} \in T S$ and $X (ρ) ∣_{S} = 0$ . In the collar of step 1.2, smoothness and compactness give $N \leq A ρ$ and $∣ X (ρ) ∣ \leq A ρ$ on a smaller collar, with $A \geq 1$ . The scalar speed inequality and $∣ d ρ ∣_{h} = 1$ give $∣ ρ^{'} ∣ \leq 2 A ρ$ almost everywhere. All hypotheses of [F7] now hold. Choose $a$ smaller than this collar's width and apply that lemma to the truncated coordinate $b$ . On the inner collar $ρ < a$ , where $b = ρ$ , its estimate is the stated exponential lower bound, with $C = A^{2} + A \geq 2 A$ , on every interval contained in that collar. The global bound for $b$ also handles arbitrary departures and returns.

2.2F1F2F3F4step 1.1algebra

In the normalized end chart, the metric coefficients tend to Minkowski values and the section graph derivatives tend to zero. Pulling the metric back to the graph shows that $h$ and $h^{- 1}$ are uniformly bounded there. The lapse and shift, computed from the stationary metric and graph normal, are likewise bounded: the normalizing timelike covector has squared norm bounded away from zero for sufficiently large radius. Since $r$ differs from normalized radius by the fixed positive factor $q^{- 1}$ , $∣ d r ∣_{h}$ is bounded. The speed inequality consequently gives $∣ r^{'} ∣ \leq ∣ d r ∣_{h} (N + ∣ X ∣_{h}) \leq C$ on the end.

3.1A1A2F1F2F8F9step 1.1step 1.2step 2.1step 2.2construct

Use the compact-core one-ended section data in [A1, F1, F2]. On its product end choose radii $0 < r_{1} < r_{2}$ beyond the compact core and put $δ = (r_{2} - r_{1}) / 2 > 0$ . For the one-dimensional cutoff $η$ in [F9], define $χ (r) = 0$ for $r \leq r_{1}$ and $χ (r) = 1 - η ((r - r_{1}) / δ)$ for $r > r_{1}$ . On a neighborhood of $r_{1}$ both pieces are identically zero, so they join smoothly. Thus $0 \leq χ \leq 1$ , $χ = 0$ for $r \leq r_{1} + δ$ , and $χ = 1$ for $r \geq r_{2}$ . Define $R = r_{1} + χ (r) (r - r_{1})$ on the end and $R = r_{1}$ on the remaining core. These formulas agree near the gluing region, so $R$ is smooth, nonnegative and equals $r$ for $r \geq r_{2}$ . The compact core and compact spherical cross sections give the sufficient end realization of [F8], hence compact sublevels. The smooth positive-definite extension of $h$ and the smooth extensions of $N, X$ were established in steps 1.2 and 2.1. Outside a larger compact set, $R = r$ and step 2.2 gives $∣ d R ∣_{h} (N + ∣ X ∣_{h}) \leq C$ . Thus every hypothesis of [F8] holds, and its conclusion supplies the global estimate $∣ R (x (v)) - R (x (v_{0})) ∣ \leq C_{R} ∣ v - v_{0} ∣$ , including portions of the curve in the compact remainder.

4.1A1A2F1F4F6F7F8step 1.1step 1.2step 2.1step 2.2step 3.1∎

Steps 1.2 and 2.1 give the smooth collar data and global truncated-coordinate barrier required by [F6], and step 3.1 gives its proper-radius data and estimate. The compact-confinement lemma therefore places any bounded section-time segment in a compact subset of the section interior; in particular it cannot reach $S$ or escape along the end at finite section time. If a finite endpoint of the section-time interval existed for an inextendible curve, the compact-endpoint lemma would give a limit in the exterior, contradicting inextendibility. Reversing the time direction treats the other endpoint, so the image is all of $R$ .

Source notes

The article reference attributes the original estimate. The argument uses the completed section, lapse, end and mathematical causal suppliers; the article's claim is not used as its own proof.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

John M. Lee, Introduction to Riemannian Manifolds, second edition (2018) · Theorem 5.25 and normal-tube definitions, printed p. 133 (PDF p. 145); Corollary 6.39, p. 180 (PDF p. 192); Theorem 6.40, p. 181 (PDF p. 193); Example 6.44, p. 183 (PDF p. 195).

The normal exponential map gives a uniform tubular neighborhood for a compact embedded submanifold. Its normal-length coordinate equals ambient distance there and has unit gradient off the submanifold. Boundary normal coordinates follow by smoothly extending the metric to the double and restricting Fermi coordinates to the inward side.

Hypotheses: The tubular and distance results concern an embedded submanifold of a smooth Riemannian manifold without boundary; compactness gives uniform radius, and Theorem 6.40 assumes the ambient manifold connected. Example 6.44 assumes a smooth Riemannian manifold with nonempty boundary and supplies the double/metric-extension construction. Here the section is smooth and spacelike up to its compact boundary S; its connected double with the extended induced Riemannian metric meets those hypotheses. No spacetime causality or field equation is assumed in this input.

Reading limits: Retained prior source-input or attribution record: Consulted the PDF extracted text at the stated definitions, theorem statements, Corollary 6.39 and Theorem 6.40 distance arguments, and Example 6.44. Used these established geometric inputs without auditing their earlier supporting results or reconstructing the smooth metric-extension theorem. The intrinsic-distance comparison and causal application are proved locally below. Fresh maintenance limits: current item and supplier contracts and the local applications were checked; no new reading of the external proof or PDF images is claimed.

Sources

The uniqueness of smooth stationary black holes · PDF pages 23–24, proof of Lemma 4.1 and equation (4.3)
John M. Lee, Introduction to Riemannian Manifolds, second edition (2018) · Theorem 5.25 and normal-tube definitions, printed p. 133 (PDF p. 145); Corollary 6.39, p. 180 (PDF p. 192); Theorem 6.40, p. 181 (PDF p. 193); Example 6.44, p. 183 (PDF p. 195).