Acausal hypersurface with compact core and finitely many ends

Let $S\subset M$ be a connected spacelike embedded hypersurface containing a specified asymptotically flat end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ with the decay data of Asymptotically flat spacelike end and derivative decay. Acausal means that no distinct points of $S$ are joined, in either direction, by a future causal curve in the ambient $M$. It is an additional global condition, not part of positive definiteness of the induced metric.

Use the completion data of Ambient closure and intrinsic boundary of a hypersurface: ${\bar{S}}^M$ is a topological three-manifold with manifold interior $S$ and boundary $B={\bar{S}}^M\setminus S$, and $B$ is a topological two-manifold. A compact core with finitely many asymptotic ends means that ${\bar{S}}^M=C_0\cup S_1\cup \cdots \cup S_N$ for a compact subset $C_0$ and finitely many end regions $S_j$ with asymptotically flat end charts as above. The chosen end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is one of the specified asymptotic regions, up to truncation at a larger radius. This is a finite-end condition, not compactness of ${\bar{S}}^M$.

For a specified future horizon portion ${\mathcal{E}}^{+}$, the boundary section condition means $B\subset {\mathcal{E}}^{+}$ and $\mathrm{\mid }B\cap G\mathrm{\mid }=1$ for every generator image $G$ of ${\mathcal{E}}^{+}$, in the sense of Null generators and once-only cross-sections. It is a topological once-only cross-section condition; a smooth embedding or a transverse differentiable intersection is not added to it.

Asymptotically flat spacelike end and derivative decay

Ambient closure and intrinsic boundary of a hypersurface

Null generators and once-only cross-sections

Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

**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 The induced metric tests spacelikeness pointwise, while causal reachability tests acausality using curves in $M$. The completion and its manifold boundary are specified independently. The finite union condition then quantifies over a compact set, a finite integer and end charts of the already defined type. [F1, F2]

2.1 With those objects fixed, boundary inclusion and once-only intersection are the subset and cardinality predicates of the generator interface. Their conjunction with step 1.1 gives a definite section-data predicate. No existence, compactification by points at infinity, or smoothness theorem is inferred. [F1, F2, step 1.1] ∎