Eventual chronological access to translated comparison spheres

The original lapse/product supplier gives $\mathcal{E}=\mathbb{R}\times C$, $N>0$ on $C$, $N{\mathrm{\mid }}_S=0$ and $f=-g(T,T)<1$. Choose the adapted spherical-exterior coordinates $y$ supplied by the original lapse/product result, so that $\bar{C}=\{\mathrm{\mid }y\mathrm{\mid }\ge 1\}$ and $S=\{\mathrm{\mid }y\mathrm{\mid }=1\}$. Write $x$ for the separate asymptotic end chart on an open subset $\mathcal{A}\subset C$, and $r=\mathrm{\mid }x\mathrm{\mid }$ only there. No equality of $x$ and $y$ is assumed. The regular data supply a smooth spacelike section across compact $S$, hence a smooth positive induced metric $h$ there. The transverse-intersection result and tangency to both horizons give $T{\mathrm{\mid }}_S\in TS$. The original outer section is maximal. On the stationary end $h_{ij}={\delta }_{ij}+O_6(r^{-1})$ and the shift covector is $g_{0i}=O_6(r^{-2})$. In particular choose $a$ larger than the chart threshold so that $h(v,v)\ge \frac{1}{2}\mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}\mathrm{u}\mathrm{c}\mathrm{l}}^2$ for all end tangent vectors at $r\ge a$.

We first prove that the closed chart tail $H_a=x^{-1}(\{r\ge a\})$ is closed in $\bar{C}$. Its inner sphere $B_a=x^{-1}(\{r=a\})$ and every finite chart annulus $x^{-1}(\{a\le r\le b\})$ are compact and closed in the Hausdorff manifold $\bar{C}$. Suppose $z\in {\overline{H_a}}^{\bar{C}}\setminus H_a$. Choose a relatively compact section coordinate ball, or boundary half-ball, $V$ about $z$ disjoint from $B_a$, small enough that coordinate straight segments remain in $V$. Choose $w\in H_a\cap V$, necessarily with $r(w)>a$, and take the coordinate segment $\lambda :[0,1]\to V$ from $w$ to $z$. Its $h$-length is finite because the smooth positive metric has bounded coefficients on the closure of $V$. While $\lambda$ stays in the open chart tail $r>a$, step 1.1 and the length formula give $\mathrm{\mid }\frac{d}{ds}(r\circ \lambda )\mathrm{\mid }\le \sqrt{2}\mathrm{\mid }\dot{\lambda }{\mathrm{\mid }}_h$, and hence $r(\lambda (s))\le r(w)+\sqrt{2}{L}_h(\lambda )=b<\mathrm{\infty }$. At the first exit time, including a possible exit at $s=1$, all earlier points lie in the closed compact chart annulus $a\le r\le b$. Its closedness puts the exit point in that annulus. Since $V$ misses $B_a$, the exit point has $r>a$, an open condition, contradicting the exit. Thus such a $z$ cannot exist. This proves closedness without an assumed compatibility of the two charts. The same local chart description shows ${\mathrm{\partial }}_{\bar{C}}{H}_a=B_a$.

Supply the standing Harrison data from existing results. The regular vacuum hypotheses give the unique normalized stationary potential $\sigma =f+i\psi$ on $\mathcal{E}\cup O$ by F18. The original outer section is maximal and mass one, and $T{\mathrm{\mid }}_S\equiv ̸0$, so F19 supplies a pole $p_{\ast }\in S\cap O$, $T_{p_{\ast }}=0$, and a smooth stationary normalized primitive $W_0$. Shrink $O$ if necessary to use their common neighborhood. Take the construction's family $W_q=(1-q{)}^2{W}_0-4q(1-q)T^{\mathrm{♭}}$ from F20. It is smooth and stationary on $\mathcal{E}$. Using $d{T}^{\mathrm{♭}}=2F$ and the primitive equation gives $d{W}_q=4(1-q)(((1-q)f-(1+q))F+(1-q)\psi {\star }_{g}F)$. The family contraction identity gives $W_q(T)=d_q-D_q$, with the same pole used to define every $d_q=D_q(p_{\ast })$. Thus the required one-form data are furnished by the primitive result, not asserted to exist by their definition.

The set $H_a$ is noncompact: its continuous chart radius takes every value in $[a,\mathrm{\infty })$, whereas a continuous real function on a nonempty compact set is bounded. Choose $L>1$ so large that the compact sphere $B_a$ lies in $\{\mathrm{\mid }y\mathrm{\mid }<L\}$. The closed spherical shell $Q_L=\{1\le \mathrm{\mid }y\mathrm{\mid }\le L\}$ is compact. Since $H_a$ is closed in $\bar{C}$ and noncompact, it cannot be contained in $Q_L$. The outer shell $V_L=\{\mathrm{\mid }y\mathrm{\mid }>L\}$ is path connected: move radially to a common radius greater than $L$, join the directions by a path in $S^2$, and reverse the second radial segment. It is therefore connected. The nonempty subset $H_a\cap V_L$ is closed in $V_L$; it is also open there, since the only boundary of $H_a$ is $B_a$, which misses $V_L$. It equals $V_L$ by connectedness. Moreover $V_L$ misses $B_a$, so it lies in the open chart tail $r>a$. Consequently $K_a=\bar{C}\setminus x^{-1}(\{r>a\})$ is a closed subset of $Q_L$ and is compact. Repeating the same argument for any larger chart radius proves the exact compact-complement hypothesis required by the far-tail supplier.

The mass-one normalization uses the original open maximal outer section $C$ and its given mass limit. With the potential and pole of step 2.2, put $B_q=q(1+\sigma )+(1-\sigma )$ and $D_q=\mathrm{\mid }{B}_q{\mathrm{\mid }}^2$. Since $f<1$ on $\mathcal{E}$, $\mathrm{Re}{B}_q=2q+(1-q)(1-f)>0$ for $0<q\le 1$, so $D_q>0$ throughout the exterior. At the pole $f(p_{\ast })=0$, hence $d_q=(1+q{)}^2+(1-q{)}^2\psi (p_{\ast }{)}^2>0$. Apply the one-form and Lorentzian-algebra interfaces with $U=\mathcal{E}\cup O$ and $V=\mathcal{E}$: the potential and pole are on $U$, and the one-form and positive denominator are on $V$. They define the standing metric $g_q=(D_q\mathrm{/}4)A_q^{\ast }g$ on the whole exterior. Give it the transported future orientation for which $A_q$ maps future $g_q$-timelike vectors to future $g$-timelike vectors. Its stationary flow is complete by the original product and preserves $g_q$. For each initial future timelike vector its images under the flow remain timelike and vary continuously with flow time from that vector; hence they stay in the future cone. All the Harrison hypotheses of that result are now met on $\mathcal{E}$.

Use the explicit outgoing-access premise in the Statement. Choose the physical radius $R_f$ beyond its threshold $R_a$, beyond the chart threshold, and large enough for the routing estimates used below. With $U_f=\mathbb{R}\times \{r>R_f\}$, every point of $\mathcal{E}$ reaches $U_f$ by a future original-metric timelike curve contained in $\mathcal{E}$. The original designated end, metric and exterior are unchanged. The compact-complement arguments of steps 1.1 and 3.1 remain geometric conclusions, but no inference from those conclusions to the assumed access property is needed here.

Apply the far-end routing lemma with $M=M_0$, the coefficient in the regular-data expansion $g_{00}=-1+2M_0\mathrm{/}r+O_6(r^{-2})$. The remainder bound gives finite $C\ge 0$ and $R_a$ with $\mathrm{\mid }{g}_{00}+1-2M_0\mathrm{/}r\mathrm{\mid }\le C{r}^{-2}$ for $r\ge R_a$. Choose $R_f\ge \max (R_a,8\mathrm{\mid }{M}_0\mathrm{\mid },2\sqrt{C})$, also sufficiently large for step 4.1. Increase the freely chosen $R_0$ to be at least $R_f$ and satisfy the comparison-sphere conditions. Since $0<q\le 1$, the comparison radius $R=R_0{q}^{-2\mathrm{/}3}$ is at least $R_f$. Let $p\in \mathcal{E}$ and let $z\in U_f$ be the endpoint of its access curve from step 4.1. Apply the finite smooth-piece lemma with ambient spacetime $(\mathcal{E},g)$ and open set $U=\mathcal{E}$. Its inclusion $U\subset I_{\mathcal{E}}^{-}(U)\cap I_{\mathcal{E}}^{+}(U)$ makes the chronological exterior in that application equal to $\mathcal{E}$; also $U_f\subset U$ is open and $p{\ll }_{\mathcal{E}}z$. The lemma therefore replaces the access curve from $p$ to $z$, with the same endpoints, by finitely many smooth future timelike geodesic pieces in $\mathcal{E}$. Now append the routing lemma's smooth future timelike radial curve from $z$ to radius $R$, which lies in $r\ge R_f$. This yields a finite concatenation of smooth future timelike pieces from $p$ to the cylinder, including when $R=R_f$ and when $z$ is outside the cylinder.

Apply Stationary transfer of timelike curves to each smooth piece from step 5.1, affinely parametrized on $[0,1]$. Each transferred piece has the same initial point and spatial path as that piece. Translate each subsequent output by the stationary time difference between its initial point and the preceding endpoint. Completeness and stationarity preserve the exterior and future $g_q$-timelikeness. The conditional piecewise-transfer lemma therefore glues a finite future timelike path from the given point to the cylinder, ending at some $C_a$. This is chronological reachability by a piecewise smooth timelike path; no globally smooth concatenation is claimed.

On the far cylinder $T$ is future $g$-timelike by the routing bound. The Harrison algebra gives $A_{q}T=(d_q\mathrm{/}{D}_q)T$ with $d_q,D_q>0$, and $g_q(T,T)=d_q^{2}g(T,T)\mathrm{/}(4D_q)<0$. Under the transported time orientation this same positive multiple makes $T$ future $g_q$-timelike. Its complete stationary orbit stays on the cylinder. Thus the translated-cylinder-pasts lemma, applied in the open spacetime $\mathcal{E}$ with metric $g_q$, gives the arrival point access to every $C_t$ with $t>a$, and the pasts $I_{g_q}^{-}(C_t)$ are open and increasing.

The preceding argument applies to every point of $\mathcal{E}$, hence every point of the specified ${\mathrm{\Sigma }}^{\circ }$, without needing that section to be the original one. Apply the compact eventual-entry lemma to the increasing open pasts: every compact $L\subset {\mathrm{\Sigma }}^{\circ }$ has a finite common threshold. This proves the claimed pointwise and compact-uniform conclusions for the fixed parameter, without a uniform-in-$q$ assertion.