Kharanshu N. Solanki | Searcharxiv

arXiv2026

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 $γ$ 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.

gr-qc↗

arXiv2025

Tipler Naked Singularities in $N$ Dimensions

A spacetime singularity, identified by the existence of incomplete causal geodesics in the spacetime, is called a (Tipler) strong curvature singularity if the volume form acting on independent Jacobi fields along causal geodesics vanishes in the approach of the singularity. It is called naked if at least one of these causal geodesics is past incomplete. Here, we study the formation of strong curvature naked singularities arising from spherically symmetric gravitational collapse of general type-I matter fields in an arbitrarily finite number of dimensions. In the spirit of Joshi and Dwivedi [26], and Goswami and Joshi [31], we first construct regular initial data in terms of matter variables and geometric quantities, subject to the dominant and null energy conditions. Using this initial data, we derive two distinct (but not mutually exclusive) conditions, which we call the positive root condition (PRC) and the simple positive root condition (SPRC), that serve as necessary and sufficient conditions, respectively, for the existence of naked singularities. In doing so, we generalize the results of [26] and [31]. We further restrict the PRC and the SPRC by imposing the curvature growth condition (CGC) of Clarke and Krolak [24] on all causal curves that satisfy the causal convergence condition. The CGC gives a sufficient condition for the naked singularities implying the PRC and implied by the SPRC, to be of strong curvature type; thereby also implying the $C^2$ inextendibility of the spacetime. Using the CGC, we extend the results of [28] (that hold for dimension $N=4$) to the case $N=5$, showing that strong curvature naked singularities can occur in this case. However, for the case $N\geq6$, we show that past-incomplete causal curves that identify naked singularities do not satisfy the CGC. These results shed light on the validity of the cosmic censorship conjectures in arbitrary dimensions.

gr-qc↗