Obstructions to global visibility of singularities in asymptotically flat spacetimes

Consider an $(N+1)$-dimensional asymptotically flat spacetime and a future-directed, affinely parametrized outgoing null generator $\gamma$ of an achronal boundary $\mathrm{\partial }{J}^{+}(S_{\epsilon })$, where $\{S_{\epsilon }\}$ is a nested family of smooth compact codimension $2$ surfaces approaching a singular boundary set $S$ in the past. In the twist-free case and under the null energy condition, the Raychaudhuri equation on the $m:=N-1$ dimensional screen bundle reads, $${\theta }^{\mathrm{\prime }}=-\frac{1}{m}{\theta }^2-\mathrm{\parallel }\sigma {\mathrm{\parallel }}^2-\mathrm{R}\mathrm{i}\mathrm{c}(k,k),$$ where $k$ is the tangent to $\gamma$. This equation linearizes, via the rescaling $u:=A^{1\mathrm{/}m}$ with $A:=\mathrm{\mid }\det D\mathrm{\mid }$ the Jacobi-map $m$-volume, to the Sturm-type ODE $$u^{\mathrm{\prime }\mathrm{\prime }}+\frac{1}{m}fu=0,f:=\mathrm{\parallel }\sigma {\mathrm{\parallel }}^2+\mathrm{R}\mathrm{i}\mathrm{c}(k,k)\ge 0.$$ We develop two purely generator-wise criteria forcing a first zero of $u$: (i) an exact Volterra identity combined with concavity leads to a barrier-weighted integral inequality, and (ii) Sturm comparison and a Prüfer-angle estimate yields failure of disconjugacy whenever ${\int }_c^d\sqrt{f\mathrm{/}m}d\lambda >\pi$ on a subinterval. We prove that $u({\lambda }_{\ast })=0$ is equivalent to the existence of a focal (conjugate) point and implies $\theta =m{u}^{\mathrm{\prime }}\mathrm{/}u\to -\mathrm{\infty }$ at ${\lambda }_{\ast }$. Using the standard structure of achronal boundaries, this yields a geodesic-wise obstruction: if every generator that could reach ${\mathcal{I}}^{+}$ satisfies one of the above conditions in the regular spacetime region, then $J^{+}(S_{\epsilon })\cap {\mathcal{I}}^{+}=\mathrm{\varnothing }$, and hence $S$ is not globally visible. As an application, we illustrate one of these criteria in the Einstein-massless scalar field collapse model of Christodoulou.