Cofinal stationary end representatives have the same causal exterior

The chart exterior $\{\mathrm{\mid }x\mathrm{\mid }>R_0\}$ is connected (radial paths and paths on a larger sphere join its points), so $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is connected. Let ${\pi }_E:U\to E$ be product projection. Every orbit in $U$ meets $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ by the saturation hypothesis, and ${\pi }_E$ is constant on each orbit. Hence ${\pi }_E(S_{\mathrm{e}\mathrm{x}\mathrm{t}})=E$, so $E$ is connected. In a smooth manifold the points reachable from a fixed point by a finite piecewise $C^1$ path form an open set with open complement: inside a coordinate ball straight coordinate segments extend such paths. Connectedness therefore gives such a path between any two points of $E$. In particular, every $x\in E$ has a finite piecewise $C^1$ path $c:[0,1]\to E$ to some $y\in E_1$. This derives the spatial path property from the declared stationary-end data rather than from connectedness only at large radius.

Fix $p=(t_0,x)\in U$ and a path $c$ from step 1.1. Along $(0,c(s))$, put $V=(0,\dot{c}(s))$ on each of its finitely many $C^1$ pieces. Compactness and timelikeness give constants $\delta >0$ and $B_0,C_0<\mathrm{\infty }$ with $g(T,T)\le -\delta$, $\mathrm{\mid }g(T,V)\mathrm{\mid }\le B_0$ and $\mathrm{\mid }g(V,V)\mathrm{\mid }\le C_0$. Stationarity gives the same bounds after any time translation. For sufficiently large constant $\lambda >0$, both vectors $\lambda T+V$ and $-\lambda T+V$ have squared norm at most $-\delta {\lambda }^2+2B_0\lambda +C_0<0$; their inner products with $T$ are respectively negative and positive. In a Lorentz frame with future timelike $T$ as time axis these signs select the future and past cones. Thus $(t_0+\lambda s,c(s))$ is future timelike and $(t_0-\lambda s,c(s))$ is past timelike, piecewise, with endpoints $q_{+},q_{-}\in U_1$. They lie entirely in $U$, since its time factor is all of $\mathbb{R}$. Reversing the latter gives $q_{-}{\ll }_{M}p{\ll }_M{q}_{+}$. Piecewise timelike curves belong to the supplied locally absolutely continuous curve class; the finite corner set does not affect their almost-everywhere tangent condition.

Since $U_1\subset U$, one has $I_M^{\pm }(U_1)\subset I_M^{\pm }(U)$. If $z\in I_M^{+}(U)$, choose $p\in U$ with $p{\ll }_{M}z$ and prepend the curve $q_{-}{\ll }_{M}p$ of step 2.1. Concatenating the two finite timelike curves gives $q_{-}{\ll }_{M}z$, hence $z\in I_M^{+}(U_1)$. If $z\in I_M^{-}(U)$, choose $p\in U$ with $z{\ll }_{M}p$ and append $p{\ll }_M{q}_{+}$; this gives $z\in I_M^{-}(U_1)$. Concatenation is again in the same curve class and directly verifies transitivity here. Thus both chronological sets agree. Each graph $S_1$ meets every orbit over $E_1$ once, so its full saturation is exactly $U_1$, independently of $b$.

Equality in step 3.1 preserves intersections and boundaries in the fixed topology of $M$, and therefore preserves $D$, $H^{\pm }$, ${\mathcal{E}}^{\pm }$ and their connected components as subsets. The ambient metric and time orientation have not changed. Every inextendibility, causal-diamond and strong-causality condition on the fixed domain $D$ is consequently the same condition, so global hyperbolicity is preserved. On each fixed horizon set the affine null-geodesic equation, maximal continuation within that set, and any connected portion restriction are unchanged. The same parametrized generators therefore have the same maximal affine intervals; unboundedness of either end, and hence the stated completeness, is identical. This argument assumes neither smoothness of the horizon nor existence or completeness of generators where those were not already supplied.

For fixed $\mathrm{\Sigma }$ its ambient closure and the supplied manifold boundary $B$ are unchanged. Since the horizon set and its generator images are unchanged by step 4.1, the predicates $B\subset H$ and $\mathrm{\mid }B\cap G\mathrm{\mid }=1$ are identical before and after the replacement. This proves invariance of the specified completed section and its cross-section condition. It proves no inclusion $S_1\subset \mathrm{\Sigma }$, nor existence of a deformed section containing $S_1$ with prescribed completion; those require a separate applicable input.

At fixed $x$, the map $t\mapsto at+f(x)$ is a bijection of $\mathbb{R}$ and ${\mathrm{\partial }}_{t^{\mathrm{\prime }}}=a^{-1}T$. Thus its level graphs have the same orbit saturation over their spatial domain, and every cofinal or cutoff replacement already falls under step 3.1. Positivity of $a$ preserves future direction. Rewriting coordinates changes neither the metric as a tensor nor the affine geodesic equation and its geometric solutions. Even a positive constant rescaling of an affine parameter preserves unboundedness of its ends. Consequently the coordinate change has all the claimed invariances, with spacelikeness of any chosen level imposed separately as stated.