Naked-Singularity Candidates in Collapse

Collapse models can produce singular endpoints from which future-directed causal curves emerge before a trapped region closes around them. Calling such an endpoint a counterexample to cosmic censorship requires more than finding an outgoing radial null geodesic in a symmetric exact solution. One must specify admissible initial data and matter, distinguish local from global visibility, and test whether the behavior persists under generic perturbations.

Helpful background. Trapped surfaces and singularity theorems build on Black-Hole Geometry and Mechanics; perturbative stability uses Black-Hole Perturbation Theory; evolution systems come from Formulations of General Relativity; visibility is defined in Causality and Global Structure; and finite-amplitude evidence requires Computational Relativity.

Candidate classes and visibility tests

In spherical Lemaître–Tolman–Bondi dust collapse, the free radial density profile controls whether outgoing null geodesics can emerge from the central shell-focusing singularity. Some regular profiles yield local or global visibility; shell-crossing singularities are a different, often weak pathology and must not be conflated with shell focusing. Pressure changes the characteristic structure, while self-similar perfect-fluid models make the relevant root conditions analytically accessible. Harada reviews the dust, fluid, and self-similar criteria and their perturbative questions Harada 2004, §§ 2–5.

Massless scalar collapse supplies a different lesson. At the threshold between dispersion and black-hole formation, the critical solution has curvature at arbitrarily small scales and one unstable mode. Tuning one parameter suppresses that mode, yielding a naked critical endpoint in the ideal limit. Choptuik’s simulations establish mass scaling and discrete self-similarity in spherical symmetry Choptuik 1993, pp. 9–12 and Figures 2–4. Codimension-one tuning, however, is evidence against genericity in the ambient data space.

The scalar-field stability question depends decisively on the perturbation space. Christodoulou proves black-hole-forming instability for a class of naked solutions under arbitrarily small characteristic-data perturbations whose derivatives have bounded variation Christodoulou 1999, Theorems 1.1–1.3. Zheng’s current preprint instead proves nonlinear stability of a one-parameter family of continuously self-similar $C^{1,\alpha}$ naked-singularity solutions under small, localized, same-regularity Hölder perturbations and for a restricted small similarity parameter Zheng 2026, Theorems 1.1 and 1.3, pp. 2–3, and Theorem 3.1, pp. 27–29. These are different function-space claims, not contradictory verdicts in one topology.

The finite-regularity continuously self-similar family is also not Choptuik’s smooth discretely self-similar critical solution. Zheng explicitly states that the former is not critical under nearby amplitude scaling in the specified Hölder topology Zheng 2026, discussion following Theorem 1.3, pp. 4–6. Neither result establishes stability beyond spherical symmetry.

Vacuum critical collapse removes any ambiguity about whether a matter model creates the small scales, but its numerical record is less universal than the spherical scalar case. Abrahams and Evans reported mass scaling with exponent near $0.37$ and approximate echoing for a one-parameter family of axisymmetric vacuum gravitational-wave data Abrahams and Evans 1993, pp. 2980–2983. A later comparison of three independent codes obtained excellent agreement on matched families but found no universal threshold solution and only family-dependent approximate self-similarity Baumgarte et al. 2023, Figs. 1–3, pp. 2–5. The original Abrahams–Evans setup remains difficult to reproduce because some data parameters were not recorded and near-threshold evolution is numerically fragile Rostworowski 2025, §§I and V, pp. 1–2, 8–9.

The evidence cutoff includes a June 2026 study of two new Nakamura-wave families. Both display approximate scaling and approximate discrete self-similarity, but their threshold solutions differ from one another and from earlier Teukolsky and Brill families; at the achieved fine tuning there is still no evidence for a unique vacuum critical solution Baumgarte, Gundlach, and Hilditch 2026, §§I and IV–V, pp. 1–13. These computations establish large, increasingly resolved vacuum curvature near a black-hole threshold. They do not yet construct a globally naked vacuum singularity, prove the limiting spacetime exists, or show openness under unrestricted perturbations.

What a dynamical counterexample must establish

The initial hypersurface should be regular, satisfy the constraints and stated energy condition, and belong to the topology in which genericity is claimed. The maximal development must become singular or inextendible in a specified sense. A visibility-based violation of weak censorship requires a causal signal to reach future null infinity, not merely travel a short distance before entering a later horizon. A completeness-based violation instead requires an appropriate obstruction to complete future null infinity. Genericity must be tested in the declared data class, including nonsymmetric perturbations unless the conjecture itself is symmetry restricted.

Matter admissibility also matters. Pressureless dust develops caustics where the continuum model ceases to describe a single-stream medium. A naked singularity driven by an unphysical equation of state or by data outside the chosen regularity class does not settle a formulation about reasonable hyperbolic matter. Conversely, excluding a model only because its result is inconvenient is not legitimate; the admissible class must be fixed first.

Collapse candidates through 2026-08-12

• Existence: naked or locally naked singularities occur in several symmetric dust, fluid, and scalar models; vacuum simulations instead approach ever larger visible curvature without yet constructing the limiting singular spacetime.
• Fine tuning: the scalar critical solution has a measured single unstable mode, placing the exact threshold at codimension one; vacuum results also tune one-parameter thresholds but do not yet identify one universal attractor.
• Stability evidence: some symmetric dust outcomes persist under restricted profile perturbations. Scalar conclusions split by regularity topology: rough bounded-variation perturbations can trigger black-hole formation, while a restricted continuously self-similar family is stable in a localized same-regularity Hölder neighborhood. Independent vacuum codes agree on matched evolutions, while threshold behavior remains family dependent at current fine tuning.
• Unresolved: there is no agreed generic four-dimensional asymptotically flat collapse counterexample with regular unrestricted data and globally visible curvature blow-up.

Comparisons must fix the matter model, regularity, symmetry class, data topology, and local or global visibility. Competing positions often choose different meanings of “generic”: openness within spherical dust profiles is not openness among all Einstein–matter data. This discussion covers results through 2026-08-12. The critical scalar solution demonstrates naked behavior but not open-set genericity. A stability theorem under nonspherical perturbations or a convergent nonsymmetric evolution demonstrating visibility at null infinity would support a broader conclusion.

Check your understanding

A spherical calculation finds an outgoing null ray from a central singularity into a regular neighborhood, but has not established whether any such ray reaches null infinity. What has been shown?

Solution

It shows local nakedness: the singularity can influence a regular neighborhood. It does not show global visibility to distant observers and therefore does not by itself violate a formulation of weak cosmic censorship stated in terms of completeness or visibility of future null infinity. One must establish global escape for at least one causal curve, or an appropriate failure of null-infinity completeness. Failure of one selected ray to escape does not rule out other escaping curves.

References

• Abrahams, Andrew M., and Charles R. Evans. “Critical Behavior and Scaling in Vacuum Axisymmetric Gravitational Collapse.” Physical Review Letters 70 (1993): 2980–2983. doi:10.1103/PhysRevLett.70.2980.
• Baumgarte, Thomas W., Bernd Brügmann, Daniela Cors, Carsten Gundlach, David Hilditch, Anton Khirnov, Tomáš Ledvinka, Sarah Renkhoff, and Isabel Suárez Fernández. “Critical Phenomena in the Collapse of Gravitational Waves.” Physical Review Letters 131 (2023): 181401. doi:10.1103/PhysRevLett.131.181401. Open PDF.
• Baumgarte, Thomas W., Carsten Gundlach, and David Hilditch. “Critical Collapse of Vacuum Spacetimes: Nakamura Wave Initial Data.” arXiv:2606.27431v1 (2026), 15 pp. Abstract. Open PDF.
• Choptuik, Matthew W. “Universality and Scaling in Gravitational Collapse of a Massless Scalar Field.” Physical Review Letters 70 (1993): 9–12. doi:10.1103/PhysRevLett.70.9.
• Christodoulou, Demetrios. “The Instability of Naked Singularities in the Gravitational Collapse of a Scalar Field.” Annals of Mathematics 149 (1999): 183–217. doi:10.2307/121023.
• Harada, Tomohiro. “Gravitational Collapse and Naked Singularities.” Pramana 63 (2004): 741–754. doi:10.1007/BF02705196. Open PDF.
• Rostworowski, Andrzej. “Vacuum Axisymmetric Gravitational Collapse Revisited: Preliminary Investigation.” Physical Review D 111 (2025): 104067. doi:10.1103/PhysRevD.111.104067. Open PDF.
• Zheng, Weihao. “Nonlinear Stability of Continuously Self-Similar Naked Singularities for the Einstein–Scalar Field Equations I: Main Results.” arXiv:2605.16235v1 (15 May 2026). arXiv. Open PDF.