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Sufficiently far stationary end points can be routed to a chosen cylinder

Statement

Let $E = R_{t} \times {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ , where $R_{0} > 0$ , carry a smooth stationary metric with $T = \partial_{t}$ and $g_{00} = - 1 + 2 M / r + e (x)$ . Let $R_{a} > R_{0}$ and $C \geq 0$ satisfy $∣ e (x) ∣ \leq C r^{- 2}$ for $r \geq R_{a}$ . Choose $R_{f} \geq \max (R_{a}, 8 ∣ M ∣, 2 \sqrt{C})$ and the future orientation for which $T$ is future at large radius. Then $g (T, T) \leq - 1 / 2$ on the far tail $r \geq R_{f}$ . For any fixed cylinder radius $R \geq R_{f}$ , every point of this tail has a smooth future timelike curve in the tail ending on $r = R$ at some finite stationary time. This includes points initially outside that cylinder, for which a crossing argument from the interior would be inapplicable.

Statement references

Stationary Lorentzian data and tensor conventions

Differentiated asymptotic bounds and parameter uniformity

Causal curves, chronological and causal reachability, and inextendibility

Facts & Assumptions

[F1]

The stationary metric and time-direction conventions are those of Stationary Lorentzian data and tensor conventions. The asymptotic hypothesis is exactly the displayed bound $∣ e (x) ∣ \leq C r^{- 2}$ for $r \geq R_{a}$ , the $m = 0$ , $b = 2$ case of the decay convention in Differentiated asymptotic bounds and parameter uniformity.

[F2]

On the stated product, $Φ_{s} (t, x) = (t + s, x)$ is defined for every $s \in R$ . It is smooth, has identity at $s = 0$ , has integral curves of $T = \partial_{t}$ and satisfies the group law, so it is a global flow in the supplied sense. Local and global flows generated by a vector field

[F3]

A smooth curve with everywhere future timelike velocity is a future timelike curve in the supplied curve class, with future direction defined by negative pairing with the time-orientation field $N$ . Causal curves, chronological and causal reachability, and inextendibility

[F4]

For two smooth timelike fields, the sign of their pairing is constant on each nonempty connected subset. Negative pairing means that the two fields determine the same future cone. Two time orientations agree or are opposite on each connected set

[F5]

The unit sphere $S^{2} \subset R^{3}$ is path-connected (the $n = 3$ case), and every path-connected subset is connected in its subspace topology. For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected Every path-connected space is connected, and every path component lies inside a component

[F6]

The nonempty interval $[0, 1]$ is compact, and a continuous real function on a nonempty compact subset of $R$ attains finite maximum and minimum values. Heine-Borel by bisection: every closed bounded interval [a,b] is compact Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value

Proof

Given: The stated data and explicit hypotheses.

Goal: Establish the asserted conditional conclusion.

1.1givenF1F4F5

For $r \geq R_{f}$ , the two terms $2 ∣ M ∣ / r$ and $C / r^{2}$ are at most $1 / 4$ , so $g_{00} \leq - 1 / 2$ . To check connectedness, take any two points $(u, x)$ and $(v, y)$ of the far tail, and put $a = ∣ x ∣$ , $b = ∣ y ∣$ . By [F5], there is a continuous path $ω : [0, 1] \to S^{2}$ from $x / a$ to $y / b$ . The continuous path $s \mapsto ((1 - s) u + s v, ((1 - s) a + s b) ω (s))$ joins these points and stays in the tail because its radius is $(1 - s) a + s b \geq R_{f}$ . Thus the nonempty tail is path-connected and hence connected by [F5]. On the open set where $T$ is timelike, apply [F4] to $N$ and $T$ and this connected subset $R_{t} \times {r \geq R_{f}}$ . The chosen future direction at large radius fixes the sign as $g (T, N) < 0$ on this entire subset; hence $T$ and $N$ determine the same future cone there.

2.1F1F2F3F4F6step 1.1∎

For a starting point $(t_{1}, x_{1})$ with $r_{1} = ∣ x_{1} ∣ \geq R_{f}$ , take $0 \leq λ \leq 1$ and the smooth radial spatial path $x (λ) = (1 - λ) x_{1} + λ (R / r_{1}) x_{1}$ . Its radius stays between $r_{1}$ and $R$ , hence in the far tail. Define $b (λ) = g_{0 i} (t_{1}, x (λ)) x^{' i} (λ)$ and $c (λ) = g_{i j} (t_{1}, x (λ)) x^{' i} (λ) x^{' j} (λ)$ . Smoothness makes $b$ and $c$ continuous on $[0, 1]$ . Applying [F6] to these functions gives finite $B, D \geq 0$ such that $∣ b (λ) ∣ \leq B$ and $∣ c (λ) ∣ \leq D$ throughout this interval; stationarity makes the same bounds valid at every time along this spatial path. Choose a finite $L > \max (1, 8 B, 2 \sqrt{D})$ and set $t (λ) = t_{1} + L λ$ . By step 1.1 its tangent has squared norm $L^{2} g_{00} + 2 L b + c \leq - L^{2} / 2 + 2 L B + D < 0$ , since $2 L B < L^{2} / 4$ and $D < L^{2} / 4$ . Its pairing with $T$ is $L g_{00} + b \leq - L / 2 + B < 0$ , so [F4] and step 1.1 make it future directed relative to $N$ , and [F3] makes the curve future timelike. The entire curve exists in $E$ because its time factor is $R$ , as in [F2], and ends on $r = R$ at the finite time $t_{1} + L$ . The path is smooth, including the constant spatial path if already on the cylinder.

Depends on

Used by · 2

Dependency tree · 2 levels

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