A smooth spacelike reference section is a bounded rapidity graph

Project the normal tube to $(v,y)$. At $S$ its kernel is the timelike line spanned by $e_0$, which is transverse to the spacelike tangent plane of $\mathrm{\Sigma }$. Thus [F3] supplies local graphs $u=w(v,y)$ with $w(0,y)=0$; local uniqueness makes them agree on overlaps after shrinking. Write the exterior-pointing unit normal to $S$ within $\mathrm{\Sigma }$ as $\nu =\beta e_0+\alpha e_1$. The stated side hypothesis gives $\alpha >0$, while unit length gives ${\alpha }^2-{\beta }^2=1$. Thus the exterior side corresponds to increasing $v$, and ${\mathrm{\partial }}_{v}w(0,y)=\beta \mathrm{/}\alpha$ has absolute value less than one; no equality $\nu =e_1$ is used. Compactness of $S$ and the product-neighborhood assertion in [F6] give a common exterior graph collar $0\le v<\epsilon$. Define $q(v,y)={\int }_0^1{\mathrm{\partial }}_{v}w(tv,y)dt$. The integrand is smooth in each chart, and [F4], applied to $t\mapsto w(tv,y)$, gives $w(v,y)=vq(v,y)$, including at $v=0$; hence $q(0,y)={\mathrm{\partial }}_{v}w(0,y)$. To check joint smoothness, differentiate successively in the chart variables $(v,y)$ using [F5] on closed coordinate intervals. Every resulting derivative is the integral of the corresponding continuous derivative of the integrand. These integrals are jointly continuous: for any such integrand $H$ and fixed parameter $p_0$, the open set $\mathrm{\mid }H(t,p)-H(t,p_0)\mathrm{\mid }<\eta$ contains $[0,1]\times \{p_0\}$, so [F6] gives a neighborhood of $p_0$ where the inequality holds uniformly in $t$; the integral-error bound in [F5] then gives continuity. This also proves smoothness up to $v=0$ using its one-sided derivatives (or a local smooth extension of the graph there).

Apply [F6] on $S$ to obtain $m={\max }_y\mathrm{\mid }q(0,y)\mathrm{\mid }<1$, and choose $m<\theta <1$. The open set $\mathrm{\mid }q(v,y)\mathrm{\mid }<\theta$ contains $\{0\}\times S$, so the tube lemma in [F6] gives this bound on a common smaller collar. Put $R(v,y)=v\sqrt{1-q(v,y{)}^2}$. Then ${\mathrm{\partial }}_{v}R(0,y)=\sqrt{1-q(0,y{)}^2}\ge \sqrt{1-m^2}>0$. Continuity and the same tube lemma allow a further shrinking with ${\mathrm{\partial }}_{v}R>c=\frac{1}{2}\sqrt{1-m^2}$ throughout. Fix $0<{\epsilon }_1<\epsilon$ within that collar. By [F7], $R({\epsilon }_1,y)>c{\epsilon }_1$ and $v\mapsto R(v,y)$ is strictly increasing; the intermediate value theorem gives a unique inverse $v=b(r,y)$ for $0\le r<r_0=c{\epsilon }_1$. The inverses are smooth by [F3], including at zero, and their uniqueness makes the chartwise inverses agree. Substituting into [F2]'s boost coordinates gives $a(r,y)={\kappa }^{-1}\mathrm{arctanh}q(b(r,y),y)$. It is smooth down to zero, and $\mathrm{\mid }a\mathrm{\mid }\le A={\kappa }^{-1}\mathrm{arctanh}\theta$, so the entire exterior collar has the asserted bounded-rapidity graph form.

On the spacelike graph, the defining function $f=\tau -a(r,y)$ has $df(K)=1$ and timelike normal covector $df$. If $n$ is its future unit normal, write $df=\lambda g(n,\cdot )$ there. Since $K$ is future timelike, $g(n,K)<0$, so $\lambda <0$. The positive lapse $N=-g(T,n)>0$ from [F1] therefore gives $df(T)=-\lambda N=1+\mathrm{\Omega }{Z}_{S}a>0$. For any fixed formal $(r,\tau ,y)$ with $0<r<r_0$, set $h(t)=\tau +t-a(r,{\psi }_{-\mathrm{\Omega }t}y)$. The circle action in [F2] is defined for all real $t$, and $h^{\mathrm{\prime }}(t)=1+\mathrm{\Omega }{Z}_{S}a(r,{\psi }_{-\mathrm{\Omega }t}y)>0$, using the graph inequality at that base point. Thus [F7] applied on every finite interval proves strict increase. Step 2.1 gives $\mathrm{\mid }h(t)-(\tau +t)\mathrm{\mid }\le A$, so $h(-\tau -A-1)\le -1$ and $h(-\tau +A+1)\ge 1$. The intermediate value theorem in [F7] gives a zero between these endpoints, strict increase gives uniqueness, and at the zero $\tau +t=a(r,{\psi }_{-\mathrm{\Omega }t}y)$ gives $\mathrm{\mid }\tau +t\mathrm{\mid }\le A$.