Null generators and once-only cross-sections

Assume Countable Choice for the general construction, as required by the supplied smooth tangent-bundle and vector-field interfaces. Let $H\subset M$ be a specified horizon subset of a time-oriented smooth Lorentzian four-manifold $(M,g,N)$. An affinely parametrized null geodesic is a smooth curve $c:I\to M$, where $I\subseteq \mathbb{R}$ is an interval with nonempty interior, with nonzero future-directed tangent, $g(\dot{c},\dot{c})=0$ and $D_s\dot{c}=0$. Here $D_s$ is the covariant derivative along $c$ induced by the Levi–Civita connection ${\mathrm{\nabla }}^g$; the notation ${\mathrm{\nabla }}_{\dot{c}}^g\dot{c}$ means this derivative along the curve, without requiring an ambient extension of $\dot{c}$. In chart coordinates $c^a=x^a\circ c$, the equivalent equation is ${\ddot{c}}^a+{\mathrm{\Gamma }}^a{}_{bc}(c){\dot{c}}^b{\dot{c}}^c=0$. At an included endpoint, smoothness and these derivatives have the one-sided interpretation.

A generator of $H$ means the image of such a null geodesic maximally continued while remaining in $H$; two affine parametrizations with the same image describe the same generator. A finite endpoint belonging to $H$ may be included, with the preceding endpoint convention. Maximal continuation here is relative to $H$, and is not a claim of affine geodesic completeness in $M$.

For the family $\mathcal{G}(H)$ of these generator images, a once-only cross-section is a subset $B\subset H$ such that $B\cap G$ has exactly one point for every $G\in \mathcal{G}(H)$. This counts points, not affine parameter values. It imposes neither smoothness nor a normalization of tangent fields. Coverage of an arbitrary $H$ by generators, future inextendibility, and regularity of a particular horizon are separate geometric assertions; they are not proved merely by this terminology.

For the stationary application, take an asymptotically flat spacelike end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and a complete Killing field $T$ future timelike on that end, and use its flow saturation $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and chronological exterior $D=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. In this causal-boundary application assume also Dependent Choice, as in the chronology-openness supplier. Define the full future event horizon by $H^{+}={\mathrm{\partial }}_M{I}_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and its portion by ${\mathcal{E}}^{+}={\mathrm{\partial }}_{M}D\cap I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. Then ${\mathcal{E}}^{+}=H^{+}\cap I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. For the cited causal-boundary assertion use also the source's asymptotic convention: $g(T,T)\le -ϵ<0$ for some $ϵ>0$ at sufficiently large radius on $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$. This uniformity is not needed for the portion identity. The cited input then provides, from each point of $H^{+}$, a future-inextendible null geodesic ray staying in $H^{+}$; its past continuation can exit $H^{+}$. Inextendibility here means absence of a future endpoint in $M$, not infinite affine parameter length. For a portion ${\mathcal{E}}^{+}\subset H^{+}$, use the maximal connected pieces of the horizon generator curves lying in that portion, and then their images. The once-only condition uses those images; it does not assert existence of a section.

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

Metric compatibility, torsion freedom and the metric-selected connection class

Causal curves, chronological and causal reachability, and inextendibility

Covariant derivative along a curve

Geodesic of an affine connection

Coordinate geodesic equation

Asymptotically flat spacelike end and derivative decay

Complete stationary flow and its exterior region

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

Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets

Given: The objects and data of the Definition, with Countable Choice for the general construction and Dependent Choice for its stationary application.

Goal: Verify the curve and cross-section predicates and the horizon-portion identity, distinguishing the imported ray assertion.

1.1 Under Countable Choice, the supplied Lorentzian and metric-selected-connection interfaces apply to the general construction. By the along-curve derivative interface, $D_s\dot{c}$ is intrinsic, including at one-sided endpoints. Applying the coordinate-geodesic proposition to ${\mathrm{\nabla }}^g$ gives exactly the displayed equation. The additional null and future predicates exclude constant curves. Maximal continuation within $H$ specifies the absence of a larger connected geodesic extension in $H$; defining this predicate asserts no existence. [F1, F2]

1.2 In the stationary application put $U=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and $P=I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$, so $D=U\cap P$. For any $p\in U$, openness supplies a neighborhood $V\subset U$; on $V$ the sets $D$ and $P$ agree. Consequently $p\in {\mathrm{\partial }}_{M}D$ if and only if $p\in {\mathrm{\partial }}_{M}P$. Intersecting with $U$ gives ${\mathcal{E}}^{+}=H^{+}\cap U$, defining the required relationship without identifying the full horizon with its portion. [F3, F4]

2.1 Passing to images removes parametrization from the once-only condition: it is precisely $\mathrm{\mid }B\cap G\mathrm{\mid }=1$ for each specified generator image. Included endpoints are points of the image, so require no special multiplicity convention. No smooth horizon tangent bundle or existence of a section is needed to formulate this predicate. [F1, F2, step 1.1]

3.1 With the stated uniform timelikeness convention, the exterior data match the imported stationary causal-boundary premise, so its future rays are available on $H^{+}$. Restricting each such generator curve to maximal connected parameter pieces in ${\mathcal{E}}^{+}$ defines the portion convention; step 2.1 then supplies its once-only intersection predicate. The future-ray assertion is imported, not established by the terminology or by this local justification, and implies no affine completeness or smoothness of the horizon. [F1, F3, F5, step 2.1, step 1.2] ∎

The ray assertion [F5] is imported from Chruściel–Costa §4.1, opening paragraph; the exterior construction is §2.2. Its precise statement, hypotheses, locator and reading limits are in the source record. This local justification only applies that premise and verifies the definitions and the portion identity.