Stationary vacuum normal form and compatible regularity section

On the supplied end decompose $T=Nn+Y$. Future timelikeness gives $N>0$ and $h(Y,Y)<N^2$. The source equations in [F2] apply to this vacuum Killing data, with $\rho ={\tau }_{ij}=0$; weakening the stated decay exponent to $\beta$ verifies all hypotheses of Proposition 2.2. Its affine expansion gives $N=-{\mathrm{\Lambda }}_{0i}{y}^i+O(r^{1-\beta })$. If the boost coefficient is nonzero, a ray on the full exterior sphere makes the linear term negative, contradicting $N>0$. Thus ${\mathrm{\Lambda }}_{0i}=0$ and $N=o(r)$. If a rotational coefficient remains, choose a ray with ${\mathrm{\Lambda }}_{ij}{y}^j\mathrm{/}r\mathrm{\ne }0$; then $h(Y,Y)$ grows quadratically while $N^2=o(r^2)$, contradicting timelikeness. Hence $\mathrm{\Lambda }=0$. The improved source estimate gives $N=A^0+O_k(r^{-\beta })$ and $Y^i=A^i+O_k(r^{-\beta })$. Taking the limit in $g(T,T)=-N^2+h(Y,Y)\to -1$ gives $(A^0{)}^2-\mathrm{\mid }A{\mathrm{\mid }}^2=1$, and $N>0$ forces $A^0>0$. This establishes the limiting unit timelike translation without a positive-energy premise. The argument applies to any other asymptotically flat spacelike end in the same saturation on which the same norm limit holds.

Apply [F3] to that translation, or to the translation explicitly assumed in the first paragraph of the Statement. The imported expansion has the displayed Schwarzschild leading terms; subsequent inverse-power/logarithmic terms and their differentiated remainders give the stated $o_3(r^{-1})$ estimates. Write its time as $t$ with $Tt=1$. This establishes the local end conclusion. For the remaining argument fix the $I^{+}$-regular data and keep the underlying geometry and stationary field fixed. In this setting the far-out stationary orbit product is defined for the complete real flow: the local stationary construction extends by time translation, and the original acausal end intersects each timelike orbit at most once.

Apply [F4] to obtain $u,{\mathrm{\Sigma }}_0$ and its compact boundary $B$ on $C$. Use end coordinates $y$ on ${\mathrm{\Sigma }}_0$, extended by $T$, and write $r=\mathrm{\mid }y\mathrm{\mid }$. The end is asymptotically flat in the same saturation, so step 1.1 applies to its lapse and shift as well. They converge to finite constants $B^0>0,B^i$ with $(B^0{)}^2-\mathrm{\mid }B{\mathrm{\mid }}^2=1$. In the coordinates $(u,y)$, $g_{ij}\to {\delta }_{ij}$, $g_{ui}\to B^i$ and $g_{uu}\to -1$; both metric and inverse therefore are uniformly bounded. Moreover $g^{-1}(du,du)=-N_0^{-2}\to -(B^0{)}^{-2}$. The normal-form end of step 2.1 is a cofinal coordinate description on the same orbit space. Since $Tt=Tu=1$, its time has the form $t=u+f(y)$ there, and $g^{-1}(dt,dt)\to -1$ by its metric expansion.

Here is the uniform cone and height control needed for gluing. Put $V^2=-g(T,T)$ and $a_{\ast }=-T^{\mathrm{♭}}\mathrm{/}{V}^2$. Every covector $a$ with $a(T)=1$ decomposes uniquely as $a=a_{\ast }+z$, $z(T)=0$, and $g^{-1}(a,a)=-V^{-2}+g^{-1}(z,z)$. On $z(T)=0$, the inverse form is positive definite, with coefficients and inverse uniformly bounded in the $y$ coordinates by the limiting metric in step 3.1. Choose $\delta >0$ and $R$ large enough that both $a_0=du$ and $a_1=dt$ have squared norm at most $-2\delta$ for $r>R$. Their $z$ parts then have norm at most $\sqrt{V^{-2}-2\delta }$. The Euclidean triangle inequality in this positive-definite form shows that every $a_{\theta }=(1-\theta )du+\theta dt$, $0\le \theta \le 1$, has squared norm at most $-2\delta$. These covectors have uniformly bounded coordinate coefficients. In particular $df=dt-du$ is bounded. Integrating along radial rays from a fixed sphere, on which $f$ is bounded, gives $\mathrm{\mid }f(y)\mathrm{\mid }\le C_{f}r$ after increasing $R$. No bounded-height or asymptotic identification of the two slices has been assumed.

Choose a smooth nondecreasing function $\eta :\mathbb{R}\to [0,1]$ equal to zero on $(-\mathrm{\infty },0]$ and one on $[1,\mathrm{\infty })$, with bounded derivative. For example use $p(s)=e^{-1\mathrm{/}s}$ for $s>0$, $p(s)=0$ otherwise, and $\eta (s)=p(s)\mathrm{/}(p(s)+p(1-s))$. For a constant $L>0$ put $\chi (r)=\eta (\log (r\mathrm{/}R)\mathrm{/}L)$ and extend $\chi f$ by zero off the selected far-out stationary end. It is smooth because the cutoff is flat at its inner edge. For $0\le s\le 1$ define ${\tau }_s=u+s\chi f$. Then $T{\tau }_s=1$ and $d{\tau }_s=(1-s\chi )du+s\chi dt+sfd\chi$. The last term has coordinate norm at most $C_f\mathrm{\parallel }{\eta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}L$. The bounds in step 4.1 on the inverse metric and on $a_{\theta }$ make its change to the squared norm at most $C_1\mathrm{/}L+C_2\mathrm{/}{L}^2$, for constants independent of $r,s,L$. Choose $L$ so that this is less than $\delta$. Thus $g^{-1}(d{\tau }_s,d{\tau }_s)<-\delta$ on the modified end; elsewhere ${\tau }_s=u$. Its gradient is past timelike: on the end its covector is positive on future timelike $T$, and off the end it equals $du$. Near the boundary of $C$ it also equals $du$, which has timelike differential up to that boundary by [F4]. This gives a smooth family of temporal functions on $C$.

The acausality conclusion is ambient, not merely a local spacelikeness test. Let $U=M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, $F=I_M^{+}(U)$ and $P_{-}=I_M^{-}(U)$. Since $F$ is open, $C=F\cap \overline{P_{-}}$. The push-up rule makes $F$ closed under taking causal futures. If $q\le b$ with $b\in \overline{P_{-}}$, every $a\ll q$ satisfies $a\ll b$ by push-up. The closure-of-past-set assertion in [F5] therefore gives $a\in P_{-}$. Such points $a$ approach $q$ along a local timelike curve, so $q\in \overline{P_{-}}$. Thus $\overline{P_{-}}$ is closed under causal pasts. Every ambient causal curve whose endpoints lie in $C$ stays in $C$. Along it, ${\tau }_s$ is locally absolutely continuous and its derivative is positive almost everywhere by step 5.1 and the chain rule in [F5]. In a Lorentz frame this positivity is the inequality $a_0{v}^0+\mathbf{a}\cdot \mathbf{v}>0$ for a past-gradient timelike covector, whose positive time coefficient exceeds its spatial norm, and a nonzero future causal vector with $v^0\ge \mathrm{\mid }\mathbf{v}\mathrm{\mid }$. Integration on any nontrivial parameter subinterval makes ${\tau }_s$ strictly increasing. Consequently $\{{\tau }_s=0\}$, including its boundary in $C$, is acausal in $M$.

Write $x$ for a point of the whole completed starting section ${\bar{\mathrm{\Sigma }}}_0$, extending the function $F_0=\chi f$ by zero outside its selected end. In the product from [F4] the completed new section is the graph $x\mapsto (-s{F}_0(x),x)$. It is smoothly diffeomorphic to ${\bar{\mathrm{\Sigma }}}_0$, equals it near $B$, and has positive-definite tangent metric by step 5.1. Its compact core and finite-end topology are therefore retained. There is no extra ambient closure from spatial infinity: if a sequence on this graph escaping along the changed end converged to a point $p\in M$, the norm limit would give $g(T,T)(p)=-1$. For small fixed $ϵ>0$, the translated points ${\mathrm{\Phi }}_{-ϵ}(p_n),{\mathrm{\Phi }}_{ϵ}(p_n)\in U$ approach the respective past and future timelike translates of $p$. Openness of chronology would place $p$ both in $I_M^{+}(U)$ and $I_M^{-}(U)$, hence in $D$. The product chart at $p$ would then make the spatial projections of $p_n$ converge, contradicting escape along the end. Any unmodified ends have their original completion. Thus the graph has precisely boundary $B$ in the ambient completion, and its once-only intersections with the unchanged horizon generators are the same as before.

At $s=1$ and $r\ge R{e}^L$ one has ${\tau }_1=t$, so the final regularity section contains exactly the desired far-out normal-form time level. The family of graphs in step 7.1 deforms only the end and fixes $B$. The metric, field, causal domain and horizons were not changed. This proves the simultaneous choice asserted in the Statement, with the original unit normalization, full end expansion and regularity properties. The expansion and structure theorem are the imported premises; the compatibility construction is the local argument above.