Complete stationary flow and its exterior region

Let $(M,g,N)$ be a time-oriented Lorentzian four-manifold and $T$ a smooth Killing field. Completeness means that the flow $\varphi :\mathbb{R}\times M\to M$ of $T$ is defined for every real time and every point, with ${\varphi }_0$ the identity and ${\varphi }_{s+t}={\varphi }_s\circ {\varphi }_t$.

A specified spacelike end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ has the end-chart data of the spacelike-end interface: a spacelike embedded hypersurface without manifold boundary and a specified diffeomorphism $\chi :\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R_0\}\to S_{\mathrm{e}\mathrm{x}\mathrm{t}}$, where $R_0>0$. Its radius is $r(\chi (x))=\mathrm{\mid }x\mathrm{\mid }$. Only these end-chart data are used here; no asymptotic decay condition is required. Require a specified $R_1>R_0$ such that $T$ is future timelike on $\chi (\{x:\mathrm{\mid }x\mathrm{\mid }\ge R_1\})$. The set $\{x:\mathrm{\mid }x\mathrm{\mid }\ge R_1\}$, including its finite-radius sphere, lies inside the original open chart domain. Restrict the chart to this tail and denote its image again by $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$; only this subset is used in the flow saturation $M_{\mathrm{e}\mathrm{x}\mathrm{t}}={\bigcup }_{t\in \mathbb{R}}{\varphi }_t(S_{\mathrm{e}\mathrm{x}\mathrm{t}})$. All subsequent sets are relative to this specified restricted end.

All chronological relations here are computed in $M$: $I^{+}(A)=\{q:\mathrm{\exists }p\in A,p{\ll }_{M}q\}$ and $I^{-}(A)=\{p:\mathrm{\exists }q\in A,p{\ll }_{M}q\}$. Define $D=I^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$, and define ${\mathcal{E}}^{+}={\mathrm{\partial }}_{M}D\cap I^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and ${\mathcal{E}}^{-}={\mathrm{\partial }}_{M}D\cap I^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. The subscript on ${\mathrm{\partial }}_M$ specifies the ambient topological boundary: for $A\subseteq M$, ${\mathrm{\partial }}_{M}A={\bar{A}}^M\setminus {\mathrm{int}}_M(A)$, with closure and interior taken in the manifold topology of $M$. These are the future and past portions of the boundary of the domain of outer communications; the full future event boundary is instead ${\mathrm{\partial }}_M{I}^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. These expressions define separate sets; no equality between them is built into their names.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Killing fields of a Lorentzian metric

Local and global flows generated by a vector field

Causal curves, chronological and causal reachability, and inextendibility

Asymptotically flat spacelike end and derivative decay

Interior, closure, boundary, exterior, derived set and isolated point in a topological space

**Given:**The objects and specified data in the Definition.

Goal: Verify that the displayed conditions define unambiguous objects and predicates on these data.

1.1 Since $R_1>R_0$, every point with $\mathrm{\mid }x\mathrm{\mid }\ge R_1$ belongs to the original open chart domain. Restricting the specified chart therefore determines the closed tail, including its sphere, as a subset of $M$ without any extension of the chart. Globality of the supplied flow makes ${\varphi }_t(S_{\mathrm{e}\mathrm{x}\mathrm{t}})$ defined for every real $t$, so its union specifies a subset of $M$. The flow law gives ${\varphi }_s(M_{\mathrm{e}\mathrm{x}\mathrm{t}})=M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, since $t\mapsto t+s$ permutes $\mathbb{R}$. [F1, F3]

2.1 The chronological predicates then specify the two subsets $I^{\pm }(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. Their intersection, its boundary in $M$, and the indicated intersections determine all the named sets. This construction needs no theorem identifying their topology or smoothness. [F2, F4, step 1.1] ∎