Collapse Thresholds and Spherical End States

There is no model-independent list of spherical collapse outcomes obtained by inspecting compactness alone. An endpoint statement must specify the matter equations, data class, differentiability, asymptotics, and diagnostic: dispersal, a regular bound object, a black-hole region, shell crossing, a covered singularity, and a locally or globally visible singularity are different conclusions. The most delicate claims also depend on the topology used to call an outcome stable or generic.

Required background. Use The Einstein Equation to identify the continuum system whose maximal development is being classified.

Helpful background. Black-Hole Geometry and Mechanics supplies causal and quasi-local horizon definitions, Black-Hole Perturbation Theory supplies stability concepts, Formulations of General Relativity supplies data and well-posedness language, Relativistic Matter and Compact Objects develops matter models and their closure relations, and Computational Relativity supplies validated endpoint diagnostics.

What an endpoint claim must contain

A collapse problem begins with constraint-satisfying data for a declared Einstein–matter system. Its maximal globally hyperbolic development can then be classified by several independent questions.

1. Does the matter remain regular, disperse to infinity, or approach a regular bound configuration?
2. Does a closed trapped surface or marginally outer trapped surface form?
3. Does the causal definition produce a nonempty black-hole region relative to the chosen future infinity?
4. Does the matter model fail first, as dust does at shell crossing or a smooth fluid does at a shock?
5. If curvature or geodesic incompleteness develops, is it hidden, locally visible along a null geodesic, or visible from the asymptotic region?
6. In which regularity class can the metric, connection, matter, or evolution be extended?

These questions cannot be collapsed into the binary labels “black hole” and “naked singularity.” A shell-crossing density blow-up can be a weak failure of a single-stream dust model. A central shell-focusing singularity is a different geometric event. A marginal surface is slice-dependent, while escape to future null infinity is global. A solution that is stable in one norm can be unstable in a rougher topology because the allowed perturbations differ.

Evidence compared through 12 August 2026

The evidence cutoff for this comparison is 12 August 2026. The table records results inside their actual model and data class; it does not rank the models by astrophysical realism.

System and data	Controlled outcomes	Strongest evidence	Essential limitation
Homogeneous dust ball	Covered focusing singularity and black-hole region	Exact Oppenheimer–Snyder matching and causal diagram Oppenheimer and Snyder 1939, pp. 455–459	Zero pressure, exact homogeneity, spherical symmetry
Inhomogeneous LTB dust	Covered or visible central focusing; shell crossing for other profiles	Exact solutions and radial null-geodesic tests Joshi and Malafarina 2011, §§4.2–4.4	Dust can fail at crossing; visibility and stability depend on profile and topology
Smooth massless scalar families	Dispersal below threshold, black hole above, scale-free limiting behavior at threshold	Convergent numerical critical-collapse evidence across several spherical families Gundlach and Martín-García 2007, §§2–3	The threshold requires one-parameter tuning; nonspherical and rigorous smooth-data conclusions are separate
Christodoulou self-similar scalar solutions	Globally naked singularities in a finite-regularity family	Rigorous construction; black-hole-forming instability for rough BV perturbations Christodoulou 1999, pp. 183–217	Background and perturbation regularity do not coincide
Same self-similar scalar family in a localized Hölder topology	A small open neighborhood retains the naked-singularity outcome for sufficiently small similarity parameter	May 2026 nonlinear-stability preprint Zheng 2026, Theorems 1.1 and 1.3, pp. 2–3	Spherical symmetry, finite regularity, localized topology, restricted parameter range, and preprint status
Barotropic perfect fluids	Dispersal, black-hole formation, critical solutions, shocks, and model-specific visible candidates	Analytic similarity reductions plus numerical evolutions Gundlach and Martín-García 2007, §4.2	Equation of state, sonic regularity, shock treatment, and symmetry determine the conclusion

The scalar rows expose why “naked scalar singularities are unstable” is no longer an adequate unqualified summary. Christodoulou’s theorem produces black-hole formation under generic perturbations whose derivatives lie in bounded variation Christodoulou 1999, pp. 183–217. Zheng’s current preprint instead proves nonlinear stability of a one-parameter family of continuously self-similar $C^{1,\alpha}$ naked-singularity solutions, for small perturbations in a localized same-regularity Hölder neighborhood and for a restricted small similarity parameter Zheng 2026, Theorems 1.1 and 1.3, pp. 2–3. The paper explicitly places the two results in different functional frameworks.

This does not turn the finite-regularity self-similar family into the smooth discretely self-similar solution observed at the Choptuik threshold. The 2026 theorem states that its solutions are not critical with respect to nearby amplitude scaling in the specified Hölder topology Zheng 2026, discussion following Theorem 1.3, pp. 4–6. Conflating these two scalar constructions would erase their different regularity, self-similarity, and phase-space roles.

Competing conclusions and counterexamples

Why the norm changes the question

The topology specifies which disturbances count as small. For example, on a fixed interval the smooth functions $f_n(x)=n^{-1}\sin(nx)$ tend uniformly to zero, but their derivatives $f_n'(x)=\cos(nx)$ do not. Thus small field amplitude is not the same as small field-and-gradient perturbation. A Hölder condition adds control of how rapidly a derivative varies:

$$[f']_\alpha =\sup_{x\ne y}\frac{|f'(x)-f'(y)|}{|x-y|^\alpha}, \qquad 0\lt\alpha\lt1.$$

The scalar theorem uses a specialized localized, weighted version of this idea, rather than an unrestricted textbook $C^{1,\alpha}$ ball. Its background parameter obeys $\alpha=k^2/(1-k^2)$; the construction covers $0\lt k^2\lt1/3$, whereas the stability claim is for sufficiently small positive $k^2$. The initial data are smooth away from the sphere where the initial hypersurface meets the past light cone of the singularity, and have only the specified finite regularity there Zheng 2026, Theorems 1.1 and 1.3 and the following qualifications, p. 2, PDF. Neither an open set in this specialized space nor a BV instability is automatically a statement about generic globally smooth nonspherical data.

Distinct hypotheses, distinct conclusions

One credible position is that weak cosmic censorship should use a rough function space compatible with low-regularity well-posedness; in the spherical Einstein–scalar system, the Christodoulou instability result then supports non-genericity of the known naked solutions. Another is that genericity should be tested in a topology matching the background’s actual regularity; the 2026 Hölder stability theorem then supplies an open naked-singularity set within its restricted framework. The results compete as formulations of “generic,” not as contradictory calculations in one topology.

Exact LTB profiles provide a separate counterexample to the unconditional statement that every regular spherical dust collapse is covered. They do not refute a censorship conjecture posed for generic nonspherical data and an appropriate matter class. Conversely, Oppenheimer–Snyder collapse refutes the claim that finite-time focusing must be visible. The discriminating check is always to write the removed hypothesis: homogeneity, pressure, smoothness, symmetry, asymptotic flatness, or the perturbation norm.

For any proposed endpoint, independently recompute the Misner–Sharp mass and null expansions, verify regular-center data, exclude or identify shell crossing and shocks, and integrate the relevant outgoing null geodesics. Local escape is not global visibility. In numerical scalar or fluid data, the calculation must also demonstrate constraint convergence, boundary control, horizon-finder sensitivity, and stability under a stated family of perturbations.

What remains unresolved

The broad question—what endpoint occurs for generic, smooth, asymptotically flat, nonspherical Einstein–matter data satisfying physically motivated conditions—remains unresolved. Even in spherical symmetry, the answer changes with matter model and functional framework. The intermediate regularity classes between the BV and localized Hölder settings in the self-similar scalar problem are open, as are extensions of the 2026 stability result through its full parameter range and beyond spherical symmetry.

A stronger conclusion would require a theorem with a precisely justified smooth-data topology and broader perturbation class, or an independently converged nonlinear counterexample stable under nonspherical perturbations and robust changes of matter closure. Corrections or withdrawals of the cited results could also change the comparison. Further examples within the same restricted class do not by themselves resolve the remaining symmetry and convergence questions.

Formulation diagnostic

A paper proves that a naked-singularity solution is stable in a small open $C^{1,\alpha}$ neighborhood. Another proves that rough BV perturbations can form trapped surfaces. Is the sentence “the solution is both stable and unstable” a logically sufficient summary? State the missing information.

Solution

No. Stability is a quantified statement about a background, an evolution, a set of allowed perturbations, a norm or topology measuring their size, and an outcome norm. The $C^{1,\alpha}$ neighborhood excludes some rough BV perturbations, while the BV class permits them. A sufficient summary must therefore state both frameworks: stable for small same-regularity localized Hölder perturbations under the theorem’s spherical and parameter hypotheses, and unstable toward trapped-surface formation under the specified rougher BV perturbations. Neither result alone establishes the generic smooth nonspherical outcome.

References

• 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. Open PDF.
• Gundlach, Carsten, and José M. Martín-García. “Critical Phenomena in Gravitational Collapse.” Living Reviews in Relativity 10 (2007): 5. doi:10.12942/lrr-2007-5.
• Joshi, Pankaj S., and Daniele Malafarina. “Recent Developments in Gravitational Collapse and Spacetime Singularities.” International Journal of Modern Physics D 20 (2011): 2641–2729. doi:10.1142/S0218271811020792. Open PDF.
• Oppenheimer, J. Robert, and Hartland Snyder. “On Continued Gravitational Contraction.” Physical Review 56 (1939): 455–459. doi:10.1103/PhysRev.56.455.
• 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.