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Piecewise timelike transfer yields arrival on the cylinder under far-tail access

Statement

Let $E = R_{t} \times B$ be a connected smooth four-dimensional stationary product with two smooth time-oriented Lorentzian metrics $g$ and $\hat{g}$ and common complete stationary flow $Φ_{s} (t, b) = (t + s, b)$ generated by $T = \partial_{t}$ . Use the inherited ZFC convention. Supply a coordinate end $B_{\infty} \subset B$ identified with ${x \in R^{3} : r = ∣ x ∣ > a}$ , where $a > 0$ . On $R_{t} \times B_{\infty}$ , assume the routing metric bounds $g_{00} = - 1 + 2 M / r + e (x)$ , $e = O_{6} (r^{- 2})$ , $g_{0 i} = O_{6} (r^{- 2})$ and $g_{i j} - δ_{i j} = O_{6} (r^{- 1})$ , for $M \in R$ . Fix $R_{a} > a$ and $C \geq 0$ such that $∣ e (x) ∣ \leq C r^{- 2}$ for $r \geq R_{a}$ , choose $R_{f} \geq \max (R_{a}, 8 ∣ M ∣, 2 \sqrt{C})$ , and use the $g$ time orientation for which $T$ is future at large radius. Fix any $R \geq R_{f}$ . In this end define $C_{R} (s) = {(s, x) : ∣ x ∣ = R}$ and its comparison cylinder $C_{R} = ⋃_{s \in R} C_{R} (s) = {(t, x) : t \in R, ∣ x ∣ = R}$ . This is a coordinate definition; no causal character of the cylinder for $\hat{g}$ is assumed.

Assume a transfer operation sends each smooth future $g$ -timelike curve on a compact interval in $E$ to a finite piecewise smooth future $\hat{g}$ -timelike curve on a compact interval in $E$ , with the same initial point and the same ordered spatial path in $B$ (allowing an increasing reparametrization and preserving both spatial endpoints). Assume every $Φ_{s}$ preserves the future cone of $\hat{g}$ . If $p \in E$ has future $g$ -timelike access inside $E$ to some specified $z \in U_{f} = R_{t} \times {r > R_{f}}$ , then a finite piecewise smooth future $\hat{g}$ -timelike curve from $p$ ends on $C_{R}$ , hence on some $C_{R} (s)$ at finite stationary time. No assumption that $p$ starts inside that cylinder is needed.

Statement references

Stationary Lorentzian data and tensor conventions

Local and global flows generated by a vector field

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

An open chronological exterior is causally convex without a timelike-end assumption

Differentiated asymptotic bounds and parameter uniformity

Causal curves, chronological and causal reachability, and inextendibility

Chart independence, concatenation, reparametrization and transitivity for causal curves

The Axiom of Choice

Facts & Assumptions

[F1]

The routing supplier applies to the original metric on the supplied product end with the metric bounds, constants, radius and future orientation stated above. It gives a smooth future $g$ -timelike route from $z$ to $r = R$ staying in $r \geq R_{f}$ . The differentiated bounds and the radius $r = ∣ x ∣$ use the asymptotic-data convention. Sufficiently far stationary end points can be routed to a chosen cylinder Differentiated asymptotic bounds and parameter uniformity

[F2]

The Lorentzian, time-direction and inherited ZFC conventions are those of the stationary-data interface. The supplied global flow has the group law, is smooth, preserves both metrics by stationarity, and preserves $E$ for all real times. Its explicit formula fixes the spatial projection $π_{B} (t, b) = b$ and gives $Φ_{s} (C_{R} (u)) = C_{R} (u + s)$ . Stationary Lorentzian data and tensor conventions Local and global flows generated by a vector field

[F3]

Apply the open-set/geodesic-piece supplier with ambient spacetime $(E, g)$ and open set $U = E$ . Its conclusion $U \subset I_{g}^{-} (U) \cap I_{g}^{+} (U)$ makes that chronological exterior equal to $E$ . Since $U_{f} \subset U$ and the access curve lies in $E$ , its finite smooth geodesic replacement from $p$ to $z$ lies in $E$ . Chronological access and future timelike curves use the stated reachability convention. An open chronological exterior is causally convex without a timelike-end assumption Causal curves, chronological and causal reachability, and inextendibility

[F4]

Finite concatenations of future timelike curves with matching endpoints remain future timelike; increasing affine changes of compact parameter intervals preserve them. A finite piecewise smooth curve is locally absolutely continuous. These are clauses (c), (d) and (e) of the causal-curve calculus, used under the inherited ZFC convention, which supplies its Countable Choice and Dependent Choice hypotheses. Chart independence, concatenation, reparametrization and transitivity for causal curves Stationary Lorentzian data and tensor conventions The Axiom of Choice

Proof

Given: The stationary product, coordinate end, routing bounds, explicit access point $z$ , and regularity-preserving transfer operation in the Statement.

Goal: Produce a finite piecewise smooth future $\hat{g}$ -timelike curve from $p$ to $C_{R}$ .

1.1givenF1F3F4

Apply [F3] to the supplied future $g$ -timelike access curve from $p$ to $z$ . Append the far-tail route of [F1], which is applicable since $r (z) > R_{f}$ and $R \geq R_{f}$ . This gives a finite chain of smooth future $g$ -timelike pieces $γ_{1}, \dots, γ_{m}$ in $E$ , with successive matching endpoints, initial point $p$ , and final point on $C_{R}$ . Parametrize each piece on $[0, 1]$ using [F4]. This construction allows the access point and the original point to lie outside the target cylinder.

2.1givenF2step 1.1

Transfer $γ_{1}$ using the stated operation. Suppose a finite piecewise smooth future $\hat{g}$ -timelike chain has been obtained through the transfer of $γ_{j}$ , with endpoint $y_{j}$ . Spatial endpoint preservation gives $π_{B} (y_{j}) = π_{B} (γ_{j + 1} (0))$ . Write $y_{j} = (u_{j}, b_{j})$ and $γ_{j + 1} (0) = (v_{j}, b_{j})$ , and set $s_{j} = u_{j} - v_{j}$ . Transfer $γ_{j + 1}$ at its original initial point and apply the fixed map $Φ_{s_{j}}$ to its entire output. By [F2] and future-cone preservation this translated output is defined throughout its compact parameter interval, lies in $E$ , is future $\hat{g}$ -timelike, and starts at $y_{j}$ . Smoothness of $Φ_{s_{j}}$ preserves each smooth piece, so the output still has only finitely many smooth pieces. The translation also preserves its spatial endpoints.

3.1F2F4step 1.1step 2.1∎

Iterate step 2.1 over the finite original chain. Use increasing affine reparametrizations to place the transferred pieces on consecutive compact intervals and concatenate them by [F4]. The total number of smooth subpieces is a finite sum of finite numbers, so the resulting curve is finite piecewise smooth and future $\hat{g}$ -timelike, with initial point $p$ . Its final spatial point is that of $γ_{m} (1)$ , whose radius is $R$ ; hence its endpoint lies on $C_{R}$ , or equivalently on $C_{R} (s)$ for its finite real time coordinate $s$ . This is arrival on the cylinder and requires no initial interior position.

Depends on

Used by · 1

Eventual chronological access to translated comparison spheres

Dependency tree · 2 levels

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