Obstructions to global visibility of singularities in asymptotically flat spacetimes

TL;DR

This paper investigates conditions under which singularities in asymptotically flat spacetimes are hidden from view, developing mathematical criteria based on null geodesic behavior and applying them to scalar field collapse models.

Contribution

It introduces new generator-wise criteria for obstructing the global visibility of singularities using Sturm-type ODE analysis and applies these to collapse scenarios.

Findings

Criteria prevent null generators from reaching infinity, hiding singularities.

First zero of the Jacobi map indicates conjugate points and geodesic focusing.

Application to scalar field collapse demonstrates practical relevance.

Abstract

Consider an $(N+1)$-dimensional asymptotically flat spacetime and a future-directed, affinely parametrized outgoing null generator $\gamma$ of an achronal boundary $\partial J^+(S_\varepsilon)$, where $\{S_\varepsilon\}$ 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, $$ \theta'=-\frac1m\theta^2-\|\sigma\|^2-\mathrm{Ric}(k,k), $$ where $k$ is the tangent to $\gamma$. This equation linearizes, via the rescaling $u:=A^{1/m}$ with $A := |\det D|$ the Jacobi-map $m$-volume, to the Sturm-type ODE $$ u''+\frac1m f\,u=0,\qquad f:=\|\sigma\|^2+\mathrm{Ric}(k,k)\ge 0. $$ 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 Pr\"ufer-angle estimate yields failure of disconjugacy whenever $\int_c^d \sqrt{f/m}\,d\lambda>\pi$ on a subinterval. We prove that $u(\lambda_\ast)=0$ is equivalent to the existence of a focal (conjugate) point and implies $\theta= m u'/u\to-\infty$ at $\lambda_\ast$. Using the standard structure of achronal boundaries, this yields a geodesic-wise obstruction: if every generator that could reach $\mathscr I^+$ satisfies one of the above conditions in the regular spacetime region, then $J^+(S_\varepsilon)\cap \mathscr I^+=\emptyset$, 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.