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A finite temporal target cannot be reached from a sufficiently small collar

Statement

Assume the Axioms of Countable Choice and Dependent Choice. Let $(Σ, h)$ be a smooth Riemannian three-manifold with boundary $S$ and interior $Σ^{\circ}$ . Let $E = R_{v} \times Σ^{\circ}$ carry a smooth stationary time-oriented Lorentzian metric for which $v$ is temporal.

Use intrinsic length distance $d_{h}$ on $Σ$ , componentwise with value $+ \infty$ between distinct components, and define the boundary distance by $ρ (x) = \inf_{s \in S} d_{h} (x, s)$ , with $\inf \emptyset = + \infty$ . Extend it stationarily by $ρ (v, x) = ρ (x)$ . Let $a > 0$ be such that ${ρ < a}$ lies in a supplied smooth boundary collar. Components not meeting $S$ have $ρ = + \infty$ and contribute no points to this collar sublevel.

Suppose that, along the temporal parametrization of every future causal curve, $∣ d ρ / d v ∣ \leq C ρ$ almost everywhere where $0 < ρ < a$ , for a fixed $C \geq 0$ . The absolute continuity needed to interpret and integrate this inequality follows from the stated smooth geometry, as proved below. Let a nonempty compact target $Q \subset E$ project outside ${ρ < a}$ and satisfy $0 < \min_{Q} v \leq \max_{Q} v = V < \infty$ . Then no point of ${v = 0, 0 < ρ < a e^{- C V}}$ reaches $Q$ by a future causal curve in $E$ . The boundary $S$ is excluded from $E$ and is separately regarded as inaccessible.

Facts & Assumptions

[F1]

Causal curves have locally absolutely continuous coordinates and nonzero future causal velocity almost everywhere. Temporal means that the smooth gradient of $v$ is past timelike. The stationary product and stationary extension use the section-time conventions, without requiring the additional maximal-graph hypotheses of that interface. Causal exterior, chronological boundaries and Cauchy data Stationary section time and translated maximal sections

[F2]

Boundary charts and the supplied smooth collar have their usual half-space and embedding meanings. A smooth Riemannian metric up to $S$ is a positive-definite smooth tensor in those charts; no collar-existence theorem is invoked. Smooth charts, atlases, and structures with boundary Smooth collars of a manifold boundary

[F3]

Intrinsic lengths, componentwise distance and distance to a subset have the supplied definitions. On each component meeting $S$ , distance to $S$ is finite and $1$ -Lipschitz for $d_{h}$ . The distance topology is the manifold topology, including at the boundary; the metric and topology suppliers explicitly treat half-balls in their proofs. Riemannian speed and length Riemannian distance on a connected manifold Extended riemannian distance on a disconnected manifold Distance from a point to a subset Distance to a nonempty subset is one lipschitz The riemannian distance topology is the manifold topology

[F4]

On compact coordinate neighborhoods, a smooth positive-definite metric is bounded above and below by positive Euclidean multiples. Absolute continuity is the small-total-interval-length condition for sums of endpoint increments. Local comparison of a riemannian metric with the euclidean metric Absolute continuity on a compact interval

[F5]

Under the stated choice assumptions, $v \circ γ$ is locally absolutely continuous and strictly increasing on every future causal curve, and $d v (γ^{'}) > 0$ almost everywhere. For an absolutely continuous scalar function $f$ on a compact interval, $f^{'} \in L^{1}$ and $f (t) - f (0) = \int_{0}^{t} f^{'} (s) d s$ after translating the interval if necessary. Temporal functions are time functions, and time functions forbid closed causal curves Fundamental theorem of calculus for absolutely continuous functions The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain

[F6]

If a continuous nonnegative $r$ obeys $r (t) \leq A + C \int_{0}^{t} r (s) d s$ on a compact interval, where $A, C \geq 0$ , the constant-coefficient case of the supplied integral Gronwall theorem gives $r (t) \leq A e^{C t}$ . Gronwall's integral inequality with variable and constant coefficients

[F7]

The inverse metric and musical isomorphisms satisfy $\nabla v = (d v)^{♯}$ , $g (\nabla v, Z) = d v (Z)$ and $g^{- 1} (d v, d v) = g (\nabla v, \nabla v)$ ; the inverse metric is smooth. For a future unit timelike vector $n$ , its orthogonal complement $n^{⊥}$ is three-dimensional and positive definite, every vector $Z$ decomposes as $Z = - g (Z, n) n + π_{n} Z$ , and $g (π_{n} Z, π_{n} Z) = g (Z, Z) + g (Z, n)^{2}$ , where $π_{n} Z = Z + g (n, Z) n$ . The inverse metric, musical isomorphisms and index raising The observer rest-space decomposition and the spatial projection

Proof

Given: The smooth product, boundary metric, collar, choice assumptions, causal differential inequality and compact target in the Statement.

Goal: Prove the finite-time collar exclusion with the stated radius.

1.1givenF2F3

Work in a connected component $H$ of $Σ$ meeting $S$ . By [F3], $ρ$ is finite and continuous there, and $ρ = 0$ on $S \cap H$ . At an interior point $x$ , the interior is open, so the distance-topology assertion supplies $ε > 0$ with $B_{d_{h}} (x, ε) \subset Σ^{\circ} \cap H$ . Thus $d_{h} (x, s) \geq ε$ for every $s \in S \cap H$ , and $ρ (x) \geq ε > 0$ . Every continuous curve starting in $H$ stays in $H$ , since components are open and a parameter interval is connected. Components missing $S$ have $ρ = + \infty$ and cannot contain an allowed starting point.

1.2givenF1F4F5F7

Let $γ (s) = (v (s), y (s))$ be a future causal curve on a compact parameter interval. By [F5], $v (s)$ is absolutely continuous and strictly increasing, so it is a homeomorphism onto its time interval. Using the inverse metric of [F7], put $N = (- g^{- 1} (d v, d v))^{- 1 / 2}$ , $n = - N \nabla v$ and $X = \partial_{v} - N n$ . Temporality gives $N > 0$ , $g (n, n) = - 1$ , future-directed $n$ and $d v (n) = 1 / N$ . Since $g (n, Z) = - N d v (Z)$ , the tangent space of a $v$ -slice is $\ker d v = n^{⊥}$ ; [F7] therefore makes its induced metric $k$ positive definite. Also $d v (X) = 0$ , so $X$ is slice-tangent and $γ^{'} = N v^{'} n + (y^{'} + X v^{'})$ is its orthogonal normal–tangent decomposition. The projection identity in [F7] now gives $k (y^{'} + X v^{'}, y^{'} + X v^{'}) = g (γ^{'}, γ^{'}) + N^{2} (v^{'})^{2} \leq N^{2} (v^{'})^{2}$ almost everywhere. On a compact product-coordinate neighborhood inside $E$ , smoothness, $N > 0$ and the lower metric bound of [F4] therefore give $∣ y^{'} (s) ∣ \leq L v^{'} (s)$ for a finite $L$ ; enlarge $L$ to bound all coordinate components. Coordinatewise application of the absolutely continuous fundamental theorem in [F5] yields $∣ y (s_{2}) - y (s_{1}) ∣ \leq L (v (s_{2}) - v (s_{1}))$ , after enlarging $L$ once more for the Euclidean norm. Thus the spatial time graph $y (v)$ is locally Lipschitz. This uses only the temporal condition and the mathematical decomposition in [F7], with no maximality or field-equation assumption.

2.1F2F3F4step 1.1

In a sufficiently small convex coordinate ball, or half-ball at the boundary, the coordinate segment from $x$ to $y$ has $h$ -length at most $K ∣ x - y ∣$ by the upper metric bound in [F4] and the length definition in [F3]. Hence $d_{h} (x, y) \leq K ∣ x - y ∣$ . Combining this with the distance-to-set bound gives $∣ ρ (x) - ρ (y) ∣ \leq K ∣ x - y ∣$ on every such neighborhood in a component meeting $S$ . In particular $ρ$ is locally Lipschitz in spatial coordinates, without an assumption that distance is smooth.

3.1F3F4F5step 1.1step 2.1step 1.2

If the connected component of $Σ$ containing the spatial image of $γ$ meets $S$ , steps 2.1 and 1.2 show that $r (v) = ρ (y (v))$ is locally Lipschitz. On any compact time interval, choose a finite subdivision subordinate to these local neighborhoods. Taking the maximum of the finitely many Lipschitz constants and splitting endpoint increments at subdivision points makes $r$ Lipschitz on that whole interval. For any disjoint finite family of subintervals, the sum of its absolute $r$ -increments is then bounded by this constant times their total length; this is absolute continuity by [F4]. Thus [F5] applies to $r$ , rather than absolute continuity being an extra assumption on causal curves.

4.1givenF5F6step 1.1step 3.1

Suppose a curve from $(0, x_{0})$ with $0 < ρ_{0} = ρ (x_{0}) < a$ reaches a point of $Q$ at time $v_{Q} \leq V$ . Its spatial component remains in a component meeting $S$ , and $r$ is positive and continuous throughout $[0, v_{Q}]$ by steps 1.1 and 3.1. Since $r (0) = ρ_{0} < a$ and $r (v_{Q}) \geq a$ , the nonempty closed set ${v \in [0, v_{Q}] : r (v) \geq a}$ has a least element $v_{*} > 0$ . Continuity gives $r (v_{*}) = a$ and $0 < r (v) < a$ for $0 \leq v < v_{*}$ . The assumed derivative inequality therefore holds almost everywhere before $v_{*}$ ; absolute continuity and [F5] give $r (t) \leq ρ_{0} + C \int_{0}^{t} r (s) d s$ for every $0 \leq t \leq v_{*}$ . All hypotheses of [F6] hold with $A = ρ_{0}$ , so $a = r (v_{*}) \leq ρ_{0} e^{C v_{*}} \leq ρ_{0} e^{C V}$ .

5.1givenF3step 1.1step 4.1∎

Every reachable starting point with $0 < ρ_{0} < a$ consequently satisfies $ρ_{0} \geq a e^{- C V}$ . This excludes the asserted strict sublevel at $v = 0$ . Since $ρ = 0$ on $S$ and is continuous in each boundary-meeting component, $S \cup {0 < ρ < a e^{- C V}}$ is a neighborhood of $S$ in the section. Its interior part has just been excluded from causal access; $S$ itself remains outside $E$ by the separate domain convention.

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