Large Data, Thresholds, and Open Regimes

Small-data stability and trapped-surface formation establish rigorous open regimes on opposite sides of gravitational dynamics, but they do not meet at a universal threshold. Between them lie large vacuum data with no initially trapped surface, symmetry-reduced matter models, and numerically observed critical solutions. “Amplitude” is gauge dependent and cannot by itself define a geometric boundary between dispersion and collapse.

This comparison includes theorem versions available through 2026-08-12. An open-set large-data global theorem, a codimension-one critical theorem, or a rigorous threshold counterexample outside symmetry could sharpen the boundary between dispersion and collapse.

Required background. Analysis for Mathematical Relativity supplies continuation and concentration compactness; The Einstein Equation supplies constraints and trapped geometry; PDEs and Green Functions for Gravitation supplies critical-scaling language.

Helpful background. Causality and Global Structure clarifies horizons, completeness, and singularity conclusions.

Large data is not one regime

The size of an initial data set must be measured after solving the constraints and quotienting gauge. Scaling a coordinate component can change lapse or coordinates rather than curvature. Different norms also emphasize different effects: a high Sobolev norm can be large because of harmless high-frequency oscillation, while a scale-invariant curvature concentration can be dynamically decisive.

At least four questions are distinct:

1. Do specified data have a complete global development?
2. Does a trapped surface form from initially untrapped data?
3. Is there a codimension-one boundary between dispersion and collapse in a named one-parameter family?
4. Is that boundary universal under nonsymmetric perturbations and changes of matter model?

A theorem answering one does not answer the others.

What is known on common axes

Regime	Geometry, matter, and data	Conclusion	Status and limitation
Small asymptotically flat vacuum data	$3+1$ vacuum, no symmetry; constrained weighted data near Minkowski	Complete global development with decay	Peer-reviewed theorem; perturbative neighborhood only.
Vacuum short-pulse data	$3+1$ vacuum, no symmetry; large incoming curvature flux concentrated on a characteristic hypersurface	A trapped surface forms in the future development	Peer-reviewed theorem; formation, not a classification of the final singular spacetime.
Spherical Einstein–scalar black-hole formation	$3+1$, spherical symmetry, massless scalar field; large characteristic data	Rigorous black-hole formation in the model	Peer-reviewed theorem; gravitational radiation degrees of freedom are removed by symmetry.
Christodoulou naked-singularity instability	$3+1$ spherical Einstein–massless-scalar; continuously self-similar naked-singularity backgrounds whose initial data have localized finite Hölder regularity; perturbations with BV derivatives	Generic perturbations in this rougher class produce black-hole regions	Peer-reviewed 1999 theorem; “generic” is tied to the BV-derivative topology and does not imply instability under perturbations at the background’s own Hölder regularity.
Zheng and Singh–Zheng stability claims, versions 1 of 2026-05-15	The same spherical model; a one-parameter subfamily with sufficiently small nonzero self-similarity parameter; perturbations in a small open neighborhood in the localized $C^{1,\alpha}$ topology of the background	Linear and nonlinear stability of the corresponding naked singularities	Two linked 2026 preprints, not yet peer-reviewed at the cutoff; restricted to spherical symmetry, this family, and this same-regularity topology.
Type-II critical collapse	Principally spherical Einstein–massless-scalar one-parameter families	Numerical evidence for a universal discretely self-similar critical solution and mass scaling	Reproducible numerical evidence, not a general Einstein theorem or a gauge-invariant universal amplitude.
Unrestricted large vacuum data without trapped surfaces initially	$3+1$ vacuum, no symmetry; broad asymptotically flat class	Global dispersion, later collapse, and final-state alternatives are not classified	Open.

The perturbative cell is the Minkowski stability theorem. Christodoulou proves symmetry-free vacuum trapped-surface formation from regular characteristic data in Christodoulou 2009, main theorem and Chapters 1–2. In the different spherical Einstein–scalar model, Christodoulou 1991, main theorem, pp. 339–373 proves black-hole formation for suitable large data. Choptuik’s critical scaling is numerical evidence from Choptuik 1993, pp. 9–12.

Regularity changes the naked-singularity verdict

The apparently competing instability and stability statements use different neighborhoods of the same rough backgrounds. Christodoulou’s result makes the relevant naked singularities non-generic when perturbation derivatives are measured in the BV class; see Christodoulou 1999, main theorem, pp. 183–217. The perturbations allowed there are rougher at the distinguished sphere where the background itself has only localized Hölder regularity.

The two version-1 preprints posted on 2026-05-15 instead fix a sufficiently small nonzero self-similarity parameter and perturb at the regularity of the background. Zheng 2026, Theorem 1.3 and precise Theorem 3.1 claims nonlinear stability for scalar-field data in a small open neighborhood in the corresponding localized $C^{1,\alpha}$ topology. Its companion Singh–Zheng 2026, Theorem 1.2 and precise Theorem 3.1 supplies the linearized stability analysis used in the nonlinear argument.

There is therefore no theorem-level contradiction: a family can be unstable in a topology admitting rougher BV perturbations and stable in a smaller same-regularity Hölder neighborhood. At the evidence cutoff, the newer stability result is a linked preprint claim, is confined to spherical Einstein–scalar evolution and a particular one-parameter family, and has not established a corresponding stable set in unrestricted vacuum gravity. The correct frontier statement must name the function space before using “generic” or “stable.”

Trapped-surface criteria are not amplitude thresholds

A trapped surface is a closed spacelike two-surface whose two future null expansions are negative. Its existence is geometric, though finding one can depend on a foliation. A short-pulse theorem places sufficient incoming shear energy into an angular and affine interval so that Raychaudhuri focusing makes both expansions negative.

The sufficient inequality is not a necessary-and-sufficient scalar threshold on all initial data. Angular distribution, pulse width, background geometry, and constraint coupling matter. Two data sets with the same norm can focus differently. Moreover, formation of a trapped surface feeds into a singularity theorem under global hypotheses, but it does not alone identify the event horizon, prove curvature blow-up, or determine the final state.

Conversely, “no trapped surface initially” is weak information. Trapped surfaces can form later, and their absence on one initial slice does not make the data perturbatively small.

Critical collapse and codimension

For a chosen smooth one-parameter family of spherical scalar data $D(p)$, numerical evolution finds a critical parameter $p_*$ separating dispersion from black-hole formation. On the supercritical side,

$$M_{\mathrm{BH}}\sim C(p-p_*)^\gamma$$

over a scaling window, with fine structure associated with discrete self-similarity. This is a powerful organizing phenomenon. It does not by itself establish that the critical set is a smooth codimension-one hypersurface in the full constraint manifold, that $\gamma$ is unchanged by all matter models, or that nonsymmetric gravitational modes are stable.

The most credible conjectural picture treats the critical solution as having one unstable mode in the symmetry-reduced phase space. A rigorous general result would need a well-posed function space, construction of the critical solution, spectral control of all perturbations, nonlinear stable/unstable manifolds, and a gauge-invariant description of the two outcomes. Those ingredients are not yet assembled for unrestricted $3+1$ vacuum gravity.

Continuation criteria do not solve the classification

A continuation theorem says a solution extends while a listed norm stays finite. It converts global existence into an a priori-estimate problem; it does not supply that estimate for arbitrary large data. Likewise, numerical evolutions may strongly support a phase diagram without proving that all constraint-satisfying data fall into its observed basins.

Symmetry can make large-data global theorems possible by eliminating gravitational degrees of freedom or reducing the equations to $1+1$ dimensions. Such results are essential evidence, but removing symmetry can introduce angular concentration, trapped surfaces, and new instabilities. The unrestricted question remains genuinely different.

Solved diagnostic: why amplitude is gauge dependent

Compute the metric after the time-coordinate change below. Is the map a valid chart for every $\lambda$, and are its constant-time surfaces always spacelike? Explain why these are different questions.

Solution

Start with Minkowski space and make the coordinate change

$$t'=t+\lambda f(x),$$

where $f$ is smooth and compactly supported, with $\lambda f$ having units of time. Its inverse is $t=t'-\lambda f(x)$, so this is a valid global chart for every finite $\lambda$. The metric becomes

$$g=-dt'^2+2\lambda\partial_i f\,dt'\,dx^i +(\delta_{ij}-\lambda^2\partial_i f\partial_j f)dx^i dx^j.$$

Its components can be large while the curvature remains zero. However, $t'=\mathrm{constant}$ is spacelike only where $\lambda^2\lvert\nabla f\rvert^2\lt1$. Thus large components in a spacetime chart do not necessarily define admissible data on spacelike slices. For large components while keeping the usual spacelike foliation, a spatial rescaling $x'=\epsilon x$ instead gives $g=-dt^2+\epsilon^{-2}dx'^2+dy^2+dz^2$. This chart is no longer normalized to Euclidean components at infinity: the asymptotic coordinate normalization is part of a meaningful small-data norm.

By contrast, curvature fluxes, null expansions of a geometrically specified surface, and scale-invariant norms tied to a normalized foliation carry geometric information. Even these do not generally collapse the phase space to one number, which is why no universal amplitude threshold is presently justified.

The next chapter studies one major non-dispersive background where the global problem becomes decay toward a modulated black hole: Black-Hole Stability, Decay, and Scattering.

References

• Choptuik, Matthew W. “Universality and Scaling in Gravitational Collapse of a Massless Scalar Field.” Physical Review Letters 70 (1993): 9–12. DOI.
• Christodoulou, Demetrios. “The Formation of Black Holes and Singularities in Spherically Symmetric Gravitational Collapse.” Communications on Pure and Applied Mathematics 44 (1991): 339–373. DOI.
• Christodoulou, Demetrios. “The Instability of Naked Singularities in the Gravitational Collapse of a Scalar Field.” Annals of Mathematics 149 (1999): 183–217. DOI.
• Christodoulou, Demetrios. The Formation of Black Holes in General Relativity. EMS Monographs in Mathematics. European Mathematical Society, 2009. DOI.
• Singh, Jaydeep, and Weihao Zheng. “Nonlinear Stability of Continuously Self-Similar Naked Singularities for the Einstein–Scalar Field Equations II: Linearized Stability.” arXiv preprint, version 1, May 15, 2026. arXiv:2605.16095v1.
• Zheng, Weihao. “Nonlinear Stability of Continuously Self-Similar Naked Singularities for the Einstein–Scalar Field Equations I: Main Results.” arXiv preprint, version 1, May 15, 2026. arXiv:2605.16235v1.

Further reading

• Lindblad, Hans, and Igor Rodnianski. “The Global Stability of Minkowski Space-Time in Harmonic Gauge.” Annals of Mathematics 171 (2010): 1401–1477. arXiv:math/0411109.