length. Recall that untrapped means that $\sigma$ eventually escapes every compact set. Eventually $\sigma$ remains inside ${M \backslash K}$ and we take $\sigma(0)$ to be any point inside ${M \backslash K}$. In this parametrization, the growth of $\sigma_r$ is bounded below by a linear function.

Let $\sigma = (\sigma_Y, \sigma_r)$ be an untrapped geodesic ray in ${M \backslash K}$ parametrized by arc length. Then there exist constants $C, B, t_0 > 0$ such that $$\sigma_r(t)\geq C t + B, \mbox{ for all } t > t_0.$$ \[lemma:geodesicrgrowth\]

As a geodesic, the normal component of $\sigma$ satisfies $$\ddot\sigma_r + \dot\sigma^{\alpha} \dot\sigma^{\beta} \Gamma^0_{\alpha \beta} = 0,$$ where we have used the index conventions from Section \[sec:notation\] and the decomposition of the metric $g = dr^2 + g_Y(r)$ to obtain the vanishing of the $\Gamma^0_{00}$ and $\Gamma^{0}_{\alpha 0}$-Christoffel symbols. The $\Gamma^0_{\alpha \beta}$-Christoffel symbol is the scalar second fundamental form and so by our comparison results of Theorem \[lemma:metriccomp\] we have: $$\begin{split}
\label{eqn:sigmar}
\ddot\sigma_r & = - \dot\sigma^{\alpha} \dot\sigma^{\beta} \Gamma^0_{\alpha \beta}  \\
 & \geq a \tanh(a(r + L_1) ) |\dot\sigma_Y|^2_{g} \\
 & \geq 0. 
\end{split}$$

Thus $\dot\sigma_r$ is nondecreasing. Since $\sigma_r$ is eventually unbounded, there is a $t_0$ where $\dot\sigma_r(t_0) > 0$ and so 