Collar barrier against finite section-time boundary arrival

Let $U$ be the collar intersected with $S$, an open subset of $S$. The inverse collar map is smooth in boundary charts; taking its first coordinate makes $\rho$ smooth on $U$, so the map $F(v,p)=\rho (p)$ below is smooth. On every section-time subarc whose spatial image lies in $U$, apply smooth composition to $F:\mathbb{R}\times U\to \mathbb{R}$, $F(v,p)=\rho (p)$, and the causal curve $\gamma (v)=(v,x(v))$. Then $r=\rho \circ x$ is locally absolutely continuous and $r^{\mathrm{\prime }}=d\rho (x^{\mathrm{\prime }})=d\rho (x^{\mathrm{\prime }}+X)-X(\rho )$ almost everywhere. The pointwise Cauchy–Schwarz inequality and the ADM vector-speed bound therefore give $\mathrm{\mid }{r}^{\mathrm{\prime }}\mathrm{\mid }\le \mathrm{\mid }d\rho {\mathrm{\mid }}_h\mathrm{\mid }{x}^{\mathrm{\prime }}+X{\mathrm{\mid }}_h+\mathrm{\mid }X(\rho )\mathrm{\mid }\le N\mathrm{\mid }d\rho {\mathrm{\mid }}_h+\mathrm{\mid }X(\rho )\mathrm{\mid }$. This uses no global extension of $\rho$.

Within a connected time interval on which $\rho <a$, the speed estimate gives $\mathrm{\mid }\rho \mathrm{\prime }\mathrm{\mid }\le C\rho$ almost everywhere. On any compact subinterval $[s,t]$ of it, the fundamental theorem gives $\rho (u)\le \rho (t)+C{\int }_u^t\rho (w)dw$ for $s\le u\le t$. The reflected Gronwall inequality yields $\rho (s)\le \rho (t)e^{C(t-s)}$, or $\rho (t)\ge \rho (s)e^{-C(t-s)}$. Applying the forward version gives the same lower bound in the reverse orientation.

For $v\ge v_0$, if $\rho (x(v))\ge a$ or $x(v)$ is outside the collar, the asserted bound is immediate since $b(x(v))=a\ge b(x(v_0))$. Otherwise use the connected component of $\{u:\rho (x(u))<a\}$ containing $v$. If the component does not contain $v_0$, its left endpoint $\mathrm{\ell }$ satisfies $v_0\le \mathrm{\ell }<v$, including the case $\mathrm{\ell }=v_0$. Continuity gives $\rho (x(\mathrm{\ell }))=a$; pass to that endpoint in the previous lower bound. If the component contains $v_0$, use the value $\rho (x(v_0))$ instead. Both cases give the stated estimate with elapsed time at most $v-v_0$. Reversing the time interval proves the other orientation. The positive resulting lower bound for bounded elapsed time excludes all boundary approaches, even with repeated collar entries.