Obstructions to global visibility of singularities in asymptotically flat spacetimes

TL;DR

This work links the global visibility of singularities in asymptotically flat spacetimes to null-geodesic focusing along a twist-free, NEC-compliant congruence. By reducing the Raychaudhuri equation to the linear Sturm-type ODE $u'' + \frac{1}{m} f\,u = 0$ with $f = \|\sigma\|^2 + \mathrm{Ric}(k,k) \ge 0$ via the Jacobi-map rescaling, the authors derive two generator-wise obstructions: a barrier-based integral inequality and a Sturm/Prüfer criterion that guarantees a first zero of $u$ (i.e., a conjugate point). They prove that the occurrence of such a zero along generators that could reach future null infinity prevents global visibility of the singular boundary, under suitable AF boundary conditions. The framework is illustrated through the Einstein–massless scalar field collapse in spherical symmetry, where $u$ corresponds to the area radius and the integral criteria translate into flux bounds along null generators, highlighting a concrete connection between curvature, shear, and naked-singularity visibility in gravitational collapse scenarios.

Abstract

Consider an $(N+1)$-dimensional asymptotically flat spacetime and a future-directed, affinely parametrized outgoing null generator $γ$ 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, $$ θ'=-\frac1mθ^2-\|σ\|^2-\mathrm{Ric}(k,k), $$ where $k$ is the tangent to $γ$. This equation linearizes, via the rescaling $u:=A^{1/m}$ with $A := |\det D|$ the Jacobi-map $m$-volume, to the Sturm-type ODE $$ u''+\frac1m f\,u=0,\qquad f:=\|σ\|^2+\mathrm{Ric}(k,k)\ge 0. $$ 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 Prüfer-angle estimate yields failure of disconjugacy whenever $\int_c^d \sqrt{f/m}\,dλ>π$ on a subinterval. We prove that $u(λ_\ast)=0$ is equivalent to the existence of a focal (conjugate) point and implies $θ= m u'/u\to-\infty$ at $λ_\ast$. 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.

Figures (1)

• Figure 1: Conformal extensions for the spherically symmetric spacetimes arising from the gravitational collapse of type-I matter fields. The extension on the left admits a locally visible singularity, whereas the extension on the right admits a globally visible singularity. The interior of the collapsing matter could be matched with a Schwarzschild exterior which is globally hyperbolic, and hence in particular $\tilde{U}$ is a causally simple neighbourhood of $\mathscr I^+$.

Theorems & Definitions (70)

• Definition 2.1: Optical map and scalars
• Remark 2.2: Energy condition
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Definition 2.5: Jacobi map and beam $m$-volume
• Lemma 2.6
• proof
• Proposition 2.7