Gravitational collapse of a spherically symmetric inhomogeneous cloud with non-zero isotropic pressure can end in a locally naked singularity that is strong in the sense of Tipler: Mosani, Dey and Joshi (Phys. Rev. D 101, 044052, 2020) exhibited a non-zero-measure set of initial data, with outgoing-geodesic exponent α = 3, for which outgoing radial null geodesics leave a Tipler-strong central singularity. Later work corrected the admissible exponent set (in four dimensions nakedness requires α ∈ {5/3, 7/3, 3}, so strength pins α = 3) and established global visibility only for pressureless dust. Three interlocked questions remain open: whether the pressure-bearing strong singularities are globally visible, in the sense that the escaping geodesics reach distant observers rather than being trapped at the cloud boundary; whether such outcomes persist for generic mass profiles rather than the specially tuned perturbations that decouple the unknown collapse dynamics; and how imposing an equation of state p(ρ) changes existence, strength, and visibility. Together these decide whether the constructed objects are genuine counterexamples to weak cosmic censorship or nongeneric artifacts of fine-tuned data.

Background

When a sufficiently massive body undergoes unhindered gravitational collapse, general relativity predicts a spacetime singularity, and Penrose's cosmic censorship hypothesis conjectures that such singularities are hidden from distant observers inside black holes; the weak form asks that no signal from the singularity ever reach far-away observers. The question has been studied most sharply for spherically symmetric collapse of matter of type I (matter whose stress-energy tensor has one timelike and three spacelike eigenvectors; a perfect fluid with density ρ and isotropic pressure p is the special case with equal spatial eigenvalues), written in comoving coordinates (t, r) with areal radius R(t, r) and Misner–Sharp mass F(t, r) — the mass contained inside the shell of areal radius R, with F = r³M near the regular center, where M(r, v) is the mass profile; the field equations likewise introduce the velocity function b(r), an integration constant related to the speed with which each matter shell falls in, with b(r) = 1 + r²b₀(r) near the regular center. Writing R = r·v(t, r) introduces the scaling function v, and the singularity curve t_s(r) records when each shell reaches R = 0; the central shell-focusing singularity is the point r = 0, t = t_s(0). Its visibility is decided by outgoing radial null geodesics (ORNGs): if a family of future-directed ORNGs terminates at the central singularity in the past with finite positive tangent, the singularity is locally naked; if such geodesics also escape the boundary of a finite cloud and reach distant observers before the trapped region (the region where light cannot move outward, bounded by the apparent horizon) closes, the singularity is globally naked — the case relevant to the weak censorship conjecture. Near the singularity the escaping geodesic obeys R = X₀r^α with X₀ > 0, where the exponent α takes discrete values fixed by which coefficient χᵢ(0) of the Taylor expansion of the singularity curve in powers of r first fails to vanish: α = 5/3 when χ₁(0) ≠ 0, α = 7/3 when χ₂(0) ≠ 0, α = 3 when χ₃(0) ≠ 0. A naked singularity is only a serious censorship counterexample if it is physically strong: strong in the sense of Tipler means the volume element spanned by independent Jacobi fields (solutions of the geodesic-deviation equation, encoding tidal deformation of infalling objects) is crushed to zero volume at the singularity, and the Clarke–Krolak criterion gives a sufficient condition — along at least one incomplete causal geodesic with affine parameter λ, the limit of λ² times the Ricci contraction of the tangent stays positive. For inhomogeneous dust (Lemaître–Tolman–Bondi collapse), Joshi and Dwivedi (1993) showed that locally naked singularities arise for generic initial data with α ≤ 3, and Mosani, Dey and Joshi later constructed Tipler-strong singularities that are globally naked in non-marginally bound dust (2020) and in marginally bound dust with astrophysical parameter estimates (2021) — always, however, with zero pressure. For non-zero isotropic pressure, Mosani, Dey and Joshi (2020) bypassed the fact that the end-state analysis requires the collapse dynamics v(t, r), which is unknown in general, by choosing mass-profile perturbations of the dust profile whose fourth-order term δ(v)r⁴ both generates pressure and annihilates the terms coupled to the derivatives of v; for a non-zero-measure set of initial data the outcome is a Tipler-strong locally naked singularity with α = 3. No equation of state was imposed (the pressure arises only through the time dependence of the perturbation), and the authors explicitly left open whether the singularities are globally visible and whether α > 3 can give naked singularities, while the genericity of the construction and what an equation of state would change are implicit in its acknowledged limitation — the specially tuned mass profile that bypasses the unknown collapse dynamics — and were confirmed as open by the subsequent literature. Numerically, globally naked shell-focusing singularities were found by Harada (1998) for a γ-law equation of state only in the near-dust regime γ ≲ 1.01, and Giambò (2006) proved global visibility for anisotropic elastic matter with tangential pressure — but neither covers the isotropic-pressure, Tipler-strong class constructed above. Later work of Solanki, Mosani, Deshpande and Joshi (2024) corrected the admissible exponent set: for spherically symmetric type-I collapse in four dimensions, nakedness necessarily requires α ∈ {5/3, 7/3, 3}, so together with the Clarke–Krolak strength criterion α = 3 is the unique Tipler-strong locally naked value with respect to that criterion, and the source paper's question whether α > 3 can give naked singularities is closed in the negative.

Problem Statement

For a spherically symmetric collapsing cloud of perfect fluid with non-zero isotropic pressure (type-I matter, regular C² initial data, weak energy condition), in the collapse formalism where the areal radius is R = r·v(t, r), the Misner–Sharp mass near the center is specified by the mass profile M(r, v) with the velocity function b(r), and the central shell-focusing singularity is Tipler strong and locally naked — which in four dimensions pins the outgoing-geodesic exponent to α = 3 with χ₃(0) > 0, since nakedness requires α ∈ {5/3, 7/3, 3} and the Clarke–Krolak strength criterion holds at α = 3 — the following questions, posed by the source work and left open by the subsequent literature, remain: (1) Global visibility: determine conditions on the initial mass profile M(r, v), the velocity function b(r), and the size of the collapsing cloud under which the outgoing radial null geodesics that leave the central singularity escape the boundary of the cloud to reach distant observers before the trapped region closes; in particular, determine whether a non-zero-measure set of such initial data yields a Tipler-strong, globally naked singularity with non-zero pressure, or establish that the escaping family is always trapped for pressure-bearing data of this class. (2) Genericity and stability: the closed-form construction works only for mass profiles whose special perturbation form annihilates the terms coupled to the unknown collapse dynamics w(r, v) = v′(t, r); determine whether Tipler-strong locally naked outcomes persist for general mass profiles in which those coupled terms are present, and under perturbations beyond the constructed family, so that the result is not an artifact of the tuned choice. (3) Role of an equation of state: determine how imposing a physically motivated relation p(ρ) modifies the existence, strength, and visibility of these outcomes. These three questions share a single scientific objective: to decide whether pressure-bearing, physically strong, naked singularities in spherical general-relativistic collapse can be globally visible and generic — genuine counterexamples to the weak cosmic censorship hypothesis — or whether the known constructions are nongeneric artifacts of fine-tuned initial data.

References

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