Criteria for Nonlinear Black-Hole Stability

The central small-data vacuum exterior problem changed materially in 2026: a current preprint proves nonlinear stability for perturbations of every fixed subextremal Kerr black hole. That result answers a precise theorem, not every stability question one might attach to Kerr. As of the evidence cutoff 2026-08-12, the remaining task is to map exactly which data, regularity, asymptotics, regions, matter models, and parameter limits are controlled—and which independent nonlinear tests can discriminate physical decay from gauge relaxation.

Required background. PDEs and Green Functions for Gravitation supplies stability norms and nonlinear closure. Helpful background. Black-Hole Geometry and Mechanics supplies Kerr parameters; Black-Hole Perturbation Theory supplies linear decay; Formulations of General Relativity supplies constraints and gauges; Computational Relativity supplies independent nonlinear evolutions.

Kerr stability across parameter and data classes

Regime	Analytic status	Boundary of the result
Minkowski, small asymptotically flat vacuum data	global nonlinear stability and completeness	no black-hole horizon or trapping
Schwarzschild and restricted near-Schwarzschild classes	nonlinear exterior stability with differing gauges and data restrictions	not by itself arbitrary Kerr spin
slowly rotating Kerr	nonlinear stability established by a series of works	spin restricted in those formulations
any fixed subextremal Kerr background	Hintz v2 proves small-data vacuum exterior relaxation to nearby Kerr	not extremal, matter, large data, or complete interior
extremal Kerr	nonlinear stability not settled	degenerate horizon redshift and additional conserved behavior
Kerr with massive or self-interacting fields	model-dependent stable and unstable sectors	outside vacuum theorem
arbitrary finite-amplitude vacuum data	final-state problem remains open	no small-data neighborhood assumption

The simple full-subextremal statement assumes smooth constraint data equal to the chosen Kerr data outside a compact set and sufficiently small in a high Sobolev norm. The precise theorem allows specified partially polyhomogeneous asymptotics. The developed exterior, including its event horizon and asymptotic regions, settles to a nearby Kerr member with quantified decay Hintz 2026, Theorem 1.1 and Remark 1.2, pp. 2–3, and Theorem 13.1, pp. 318–323, PDF.

The proof relies on companion analytic inputs for constraint damping and tame forward estimates and is a current preprint at this cutoff. That publication status does not weaken its mathematical claim by decree, but it makes independent expert verification and updates especially important. Earlier slowly rotating and Schwarzschild-family results remain valuable because they use different geometric structures and expose distinct mechanisms Klainerman and Szeftel 2022, Theorem 1.1 and §1.3.2, pp. 4–6, PDF, Dafermos et al. 2021, Theorem I.3.1, pp. 5–8, PDF.

Spin range and data range are separate axes. The 2026 theorem requires a finite asymptotic expansion for its slowly decaying terms; it does not subsume every less-structured data class of earlier theorems Hintz 2026, p. 2, footnote 1, PDF. Comparing theorems by the largest allowed spin alone misses this distinction.

Analytic acceptance criteria

A theorem establishing relaxation throughout a declared parameter regime must provide:

1. constraint-satisfying initial data with quantitative differentiability, asymptotic expansion, weighted norm, and smallness;
2. a well-posed gauge-fixed system and propagation or damping of the constraints;
3. nondegenerate horizon energy, trapping-aware integrated decay, null-infinity estimates, and low-frequency analysis;
4. separation of radiation, pure gauge, and physical Kerr parameter modes;
5. tame estimates on the dynamical background strong enough to control nonlinear errors;
6. convergence to a nearby Kerr member with stated regional decay, and completeness of the relevant exterior boundaries;
7. constants whose dependence on spin is explicit enough to say whether the result is pointwise subextremal or uniform toward extremality.

The 2026 theorem supplies this exterior control for every fixed subextremal background. Completeness here means, for example, that future null infinity does not end after a finite affine interval; it is not automatically the stronger scattering statement called asymptotic completeness. Here the source uses $G=c=1$, so $M$ and $a=J/M$ are lengths and $|a|/M=c|J|/(GM_{\rm phys}^2)$. “Full subextremal range” means no slow-rotation restriction on the fixed background; it does not mean estimates remain uniform as $|a|/M\to1$.

Independent nonlinear evolution criteria

Numerical evolution can test the geometric interpretation and explore regimes beyond theorem hypotheses, but coordinate settling is not enough. A convincing calculation should specify the continuum vacuum system, initial-data family, constraint solver, gauge, domain, and extraction prescription. It should report:

• constraint and balance residuals over at least three effective resolutions or an equivalent spectral tolerance sequence;
• horizon mass and spin from quasi-local invariants;
• asymptotic Bondi flux and closure with final mass and angular momentum;
• curvature or waveform invariants after a declared asymptotic-frame alignment;
• convergence of the inferred final Kerr parameters and radiative decay rates;
• sensitivity to gauge, outer boundary, excision, and finite extraction.

A linearized solution and a full nonlinear evolution should agree in a controlled overlap region after matching invariant charges. An independent formulation or code is stronger than more resolution in one implementation. No simulation was rerun for this page; these are criteria for evaluating published and future calculations.

Competing positions and open scope

Within the theorem’s precise vacuum small-data exterior class, the credible current position is stability. Unresolved questions concern extension of that conclusion to extremal backgrounds, weaker or different asymptotics, coupled matter, larger perturbations, the complete interior, and uniform estimates across a closed parameter domain.

A purported instability can coexist with the theorem only if it identifies such an axis. Massive-field superradiance changes the equations; inner-horizon mass inflation changes the region; critical collapse changes perturbation size; extremal horizon instabilities change the parameter boundary. None refutes the small-data subextremal vacuum exterior result.

This assessment should be revisited after a correction or withdrawal, peer-reviewed verification of the full-range theorem, a result uniform to extremality, a theorem with matter or large data, or an independently converged counterexample in the same stated data class.

Formulation-choice diagnostic

A simulation starts from small vacuum data near Kerr with dimensionless spin $\chi=cJ/(GM^2)=0.98$. Its coordinate metric approaches a stationary-looking state, but the final mass and spin are inferred only from coordinate parameters. Does it independently support nonlinear stability?

Solution

Not yet. The spin lies within the fixed subextremal theorem regime, but the simulation must demonstrate constraint and resolution convergence, identify final mass and spin quasi-locally or asymptotically, close radiative budgets, align the asymptotic frame, and show decay of invariant radiative quantities. Coordinate stationarity can be produced by gauge. These checks would make the evolution an independent physical test rather than a coordinate comparison.

References

• Dafermos, Mihalis, Gustav Holzegel, Igor Rodnianski, and Martin Taylor, “The Nonlinear Stability of the Schwarzschild Family of Black Holes”, arXiv:2104.08222v1, 2021. Open PDF.
• Hintz, Peter, “Nonlinear Stability of Subextremal Kerr Black Holes”, arXiv:2606.28253v2, 2026. Open PDF.
• Klainerman, Sergiu, and Jérémie Szeftel, “Brief Introduction to the Nonlinear Stability of Kerr”, arXiv:2210.14400v2, 2022. Open PDF.