Strongly causal complete stationary ambient gluing around the exterior

First orient $E$, without using the boost collar or doubled realization. Apply the finite-density, volume-time and smooth-temporal inputs successively to $E$: it is a nonempty smooth time-oriented globally hyperbolic four-manifold, [A1] supplies their choice hypotheses, and the density satisfies the three measure conditions required by the volume-time theorem. The resulting complete future timelike field $X$ has temporal derivative one. Its flow lines have no endpoint in $E$, since the temporal function tends to the two infinities on them, so each meets the original Cauchy section $C={\mathrm{\Sigma }}_0\cap E$ exactly once. Transversality, flow uniqueness and completeness make the flowout map $\mathbb{R}\times C\to E$ a bijective local diffeomorphism, hence a diffeomorphism. Orient $C$ by the original radial chart of ${\mathrm{\Sigma }}_0$ and give $E$ the transported product orientation, with the future time direction first. Along $C$ this is precisely its future-normal orientation: any future timelike transverse vector differs from a positive multiple of the future normal by a tangent vector.

Use the concrete working construction in [F7]. Choose a smaller open annular collar $A\subset {\mathrm{\Sigma }}_0$ of $S$, with compact closure inside the original annular collar, and its future timelike flowout $H(s,a)$ with $\mathrm{\mid }s\mathrm{\mid }<ϵ(a)$ and image $O\subset O_0$, where $ϵ$ is positive and smooth. This is the transverse flowout theorem applied to the same future timelike field, restricted inside $M_0^{\mathrm{\prime }}$. Orient $O$ by the radial orientation on $A$ and the increasing future time coordinate. For $H(s,a)\in O\cap E$, [F7] forces $a$ to be on the outer part $A\cap C$: an inner or $S$ base point would be causally related to $E$. The whole flow segment between $a$ and $H(s,a)$ lies in $E$. Indeed both endpoints have timelike connections to and from the end, so concatenating those connections with subsegments of this timelike flow gives both end connections at each intermediate point. At $s=0$ the two orientations agree by step 1.1; the sign comparing two smooth orientations is constant along this connected segment in $O\cap E$. They therefore agree on every component of the overlap, and define an orientation on $M^{\mathrm{\prime }}=E\cup O$.

Verify that this restriction still supplies the working data. Its section is ${\mathrm{\Sigma }}^{\mathrm{\prime }}=C\cup A={\mathrm{\Sigma }}_0^{\mathrm{\prime }}\cap M^{\mathrm{\prime }}$, so smoothness, spacelikeness, closedness and acausality are inherited from the original working section, and edgelessness is local and persists on this open restriction. The inner annulus has the same open radial topology after reparametrization. No new chronological connections can appear because $M^{\mathrm{\prime }}\subset M_0^{\mathrm{\prime }}$. Every point of $E$ retains both end connections inside $E$: given end paths to and from that point, each intermediate point on either path has both end connections by timelike concatenation. Thus $E$ is still the exterior in $M^{\mathrm{\prime }}$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cap I_{M^{\mathrm{\prime }}}^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}^{\mathrm{\prime }}\cap I_{M^{\mathrm{\prime }}}^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})=C$. The open set $O$ contains a neighborhood of all of $S$. The local one-sided timelike access through $E$ used in [F7] remains available after this restriction, by taking its short local segments inside $O$ and its end paths inside $E$. Hence the original two horizon germs and their transverse nonexpanding intersection are retained. The restricted metric and field retain the vacuum equation, end decay and completeness of stationary orbits in the whole $E$. Together with the orientation in step 2.1 and the positive mass $M_0>0$ from [F4], these are the required GR and SBS data. Only now invoke the local boost input and the compatible canonical splice, doubled realization and band for these same data.

The right wedge and the source collar have the same complete stationary coordinates, so the gluing map is a diffeomorphism and the metrics and fields agree. To check Hausdorffness, only a sequence in the common wedge approaching a non-right-wedge band point could have a second limit in $E$. Such an approach has $r=\sqrt{UV}\to 0$. If its exterior images converged in $E$, their quotient images would lie in a compact set, contradicting the quotient-escape assertion. The outer cutoff $\mathrm{\mid }UV\mathrm{\mid }={\delta }^2$ is excluded from the band, so it introduces no second limit point. The graph of the identification therefore has no pair of distinct limits in the two pieces, and the gluing is Hausdorff. The images of countable bases of the two open pieces form a countable base. The ordinary gluing charts give the smooth manifold and the common vacuum metric.

The maps to $\widehat{M}$ agree on the overlap by equivariance and their common local germ. They define a smooth global local isometry $F_{\ast }:M_{\ast }\to \widehat{M}$ and the pulled-back time orientation. Completeness of the stationary field follows from the compatible complete actions on the two invariant open pieces: a point in either piece has its entire real-parameter orbit there, and the actions agree on the overlap. The band supplier already provides a smooth Cauchy temporal function $\widehat{\tau }$ on $\widehat{M}$; use that same ambient function on both pieces. Its construction is the finite-density, volume-time and smooth-temporal chain in [F6], with nonemptiness supplied by $S$ and choice supplied by [A1]. Its pullback $\tau =\widehat{\tau }\circ F_{\ast }$ has past timelike gradient because $F_{\ast }$ is a time-orientation-preserving local isometry.

Write $g_{\ast }$ for the glued metric and set $q=g_{\ast }^{-1}(d\tau ,d\tau )<0$ and $g^{\mathrm{\prime }}=g_{\ast }-d\tau \otimes d\tau$. In the $g_{\ast }$-orthogonal decomposition $\mathbb{R}\mathrm{\nabla }\tau \oplus \ker d\tau$, the second summand stays positive definite and the value on $\mathrm{\nabla }\tau$ changes from $q$ to $q-q^2<0$; thus $g^{\mathrm{\prime }}$ is smooth Lorentzian. For every nonzero $g_{\ast }$-causal vector $v$, the temporal pairing result gives $d\tau (v)\mathrm{\ne }0$, so $g^{\mathrm{\prime }}(v,v)=g_{\ast }(v,v)-d\tau (v{)}^2<0$. Any future $g_{\ast }$-timelike orientation field remains $g^{\mathrm{\prime }}$-timelike. Also $g^{\mathrm{\prime }-1}(d\tau ,d\tau )=q\mathrm{/}(1-q)<0$, and $d\tau$ is positive on that orientation field, so $\tau$ is temporal for $g^{\mathrm{\prime }}$ as well. Apply the temporal monotonicity lemma to $g^{\mathrm{\prime }}$ to exclude all closed $g^{\mathrm{\prime }}$-causal curves. This is exactly the wider-metric certificate for stable causality.

Check strong causality directly, including possible multiple sheets of $F_{\ast }$. Given $x\in M_{\ast }$ and an open neighborhood $O$ of $x$, choose an open $W\subset O$ on which $F_{\ast }$ is a diffeomorphism to an open neighborhood of $F_{\ast }(x)$. Global hyperbolicity of $\widehat{M}$ includes strong causality, so choose an open causally convex $V$ with $F_{\ast }(x)\in V\subset F_{\ast }(W)$. Put $U=(F_{\ast }{\mathrm{\mid }}_W{)}^{-1}(V)$. If a future causal curve has two points in $U$, its intervening image under $F_{\ast }$ is a future causal curve by the smooth-composition chain rule and local isometry, and lies entirely in $V$. The inverse in $W$ lifts that image on the whole intervening interval. It agrees with the original curve at the initial point; their equality set is open by local injectivity of $F_{\ast }$ and closed by Hausdorffness, hence is the entire interval. The curve therefore stays in $U$. This gives arbitrarily small open causally convex neighborhoods as required.

The restriction of $F_{\ast }$ to $C_0\cup S$ is injective and has image in the doubled Cauchy hypersurface. That hypersurface is acausal by the Cauchy-set input, so any causal curve between distinct section points would give a causal curve between distinct points of it, a contradiction. The right collar has no extra intersections with $C_0$: a nonzero stationary translate of its canonical section is a timelike pure-boost translate of a circle translate in the same canonical section, contradicting acausality of $C_0$ in $E$; omit the circle in the nonrotating case. Indeed the pure boost is timelike near the outward canonical section and remains so on its full band orbit by boost invariance. Thus in the band the section has exactly the equation $U=V\ge 0$. It is closed there, while $C_0$ is closed in $E$; closedness in this open cover proves closedness in $M_{\ast }$. Finally $C_0$ is connected and $S$ is in its closure, so $C_0\cup S$ is connected. This retains the claimed section and all ambient conclusions without asserting global injectivity of $F_{\ast }$.