Schmidt boundaries of foliated space-times

For every $(p + q)$-dimensional foliated Lorentzian manifold $(\mathfrak{M}, g, \mathcal{F})$, where $\mathcal{F}$ is a codimension $q$ space-like foliation, we build its $Q$-completion $\overline{M}$ and $Q$-boundary $\partial_Q \mathfrak{M}$. These are analogs, within transverse Lorentzian geometry of foliated manifolds, to the $b$-completion and $b$-boundary $\dot{\mathfrak M}$ (due to B.G. Schmidt. The bundle morphism $h^\bot : O(\mathfrak{M}, \mathcal{F}, g) \to O(\mathcal{F}, g_Q )$ (mapping the $\mathfrak{o}(p) + \mathfrak{o}(1, q-1)$-component of the Levi-Civita connection $1$-form of $(\mathfrak{M}, g)$ into the unique torsion-free adapted connection on the bundle of Lorentzian transverse orthonormal frames) is shown to induce a surjective continuous map $\partial h^\bot : \partial_{\mathrm{adt}} \mathfrak{M} \to \partial_Q {\mathfrak{M}}$ of the adapted boundary ($\partial_{\mathrm{adt}} \mathfrak{M} \subset \dot{\mathfrak{M}}$) of $\mathfrak{M}$ onto its $Q$-boundary. Map $\partial h^\bot$ is used to characterize $\partial_Q {\mathfrak{M}}$ as the set of end points $\lim_{t \to 1^-} \gamma (t)$, in the topology of $\overline{\mathfrak{M}}$, of all $Q$-incomplete curves $\gamma : [0, 1) \to \mathfrak{M}$. As an application we determine a class $(\partial h^\bot )^{-1} (P) \subset \dot{\mathfrak{M}}$ of $b$-boundary points, where $\mathfrak{M} = \mathbb{R} \times (0, m) \times S^2$, $g$ is Schwartzschild's metric, and $\mathcal{F}$ is the codimension two foliation tangent to the Killing vector fields $\partial /\partial t$ and $\partial /\partial \varphi$.