Nonlinear Stability of Kerr

Nonlinear Kerr stability asks whether every sufficiently small vacuum perturbation of a subextremal Kerr initial data set has a future exterior that remains regular and settles, after gauge and parameter modulation, to another Kerr metric. The phrase “sufficiently small” is inseparable from spin range, regularity, asymptotic expansion, gauge, and the norm used for convergence. This page compares the established slow-rotation theorem program with Peter Hintz’s recent full-subextremal preprint claim. Evidence and versions were checked through 2026-08-12; the latter is presented as a preprint theorem claim awaiting independent scrutiny, not as settled consensus.

Required background. Analysis for Mathematical Relativity supplies tame estimates and nonlinear iteration, while Partial Differential Equations and Green Functions supplies gauge-fixed hyperbolic evolution.

Helpful background. Einstein Equations fixes the vacuum constraints, Causality and Global Structure fixes the maximal exterior, and Black-Hole Geometry and Mechanics together with Black-Hole Perturbation Theory supplies Kerr parameters and linear decay mechanisms.

The nonlinear stability statement

Let $g_{M,\mathbf a}$ be a four-dimensional asymptotically flat vacuum Kerr metric, with physical mass $M>0$, angular-momentum vector $\mathbf J$, specific-angular-momentum vector $\mathbf a=\mathbf J/M$, and $\lvert\mathbf a\rvert<GM$. Choosing an axis can turn $\mathbf a$ into the usual signed scalar, but the full theorem permits its direction to modulate. Initial data $(\Sigma,\gamma,K)$ satisfy the vacuum constraints. With the site’s extrinsic-curvature sign,

$$K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu,$$

the constraints are

$$R(\gamma)+(\operatorname{tr}_\gamma K)^2-|K|_\gamma^2=0, \qquad D^j\bigl(K_{ij}-(\operatorname{tr}_\gamma K)\gamma_{ij}\bigr)=0.$$

A stability theorem must specify a weighted high-regularity neighborhood of the Kerr data, construct a future development containing the black-hole exterior, and identify final parameters $(M_f,\mathbf a_f)$ together with any asymptotic Poincaré modulation, including a small Lorentz boost. “Settles to Kerr” means decay of the modulated metric in a stated gauge and norm, not convergence of the original coordinate components without adjustment. Completeness toward future null infinity, regularity across the future horizon, and control in the intermediate region are distinct parts of the conclusion.

Why the final parameters must vary

An exact Kerr metric with mass $M+\delta M$ is itself a small stationary perturbation of Kerr with mass $M$. It cannot decay back to the original metric: its conserved stationary parameter differs. To first order,

$$g_{M+\delta M,\mathbf a+\delta\mathbf a}-g_{M,\mathbf a} =\delta M\,\partial_M g +\sum_{i=1}^3\delta a^i\,\partial_{a^i}g +O(\lvert\delta p\rvert^2),$$

where $\delta p$ denotes the parameter displacement in fixed reference units. The nondecaying terms are tangent directions to the Kerr family, not radiative instabilities. A stability proof chooses evolving parameters to remove these directions from the decaying remainder and proves that the parameters themselves tend to limits. Radiation can carry away energy and angular momentum, so the final black-hole parameters are not simply every initial asymptotic charge relabeled.

Slow-rotation theorem program

The Klainerman–Szeftel theorem and the Giorgi–Klainerman–Szeftel wave-estimate installment treat sufficiently small dimensionless spin $\lvert\mathbf a\rvert/(GM)\ll1$. They use a geometric foliation and generalized covariant modulation conditions to identify a nearby Kerr limit, coupled Regge–Wheeler-type equations for curvature, transport estimates for Ricci coefficients, and a bootstrap hierarchy with many derivatives. The wave- estimate installment proves the M1, M2, and M8 components needed by the larger program. The installment’s role and hypotheses are stated in Giorgi, Klainerman, and Szeftel 2024, §1.5.4, Theorems 1.5.2–1.5.4, while the full slow-rotation theorem is Klainerman and Szeftel 2023, Main Theorem, §3.4.

The small-spin assumption is structural in that theorem program. It lets one compare key operators and trapped geometry with Schwarzschild while retaining Kerr modulation. It is not permission to extrapolate the estimates uniformly to near-extremal spin. The conclusion also belongs to the program’s particular asymptotic data class and gauge construction; “slow rotation” is only one axis of its hypotheses.

The full-subextremal v2 preprint claim

Hintz’s arXiv:2606.28253v2, revised 2026-08-03, claims nonlinear stability for every fixed subextremal background. Its simple theorem assumes smooth vacuum data that equal Kerr data outside a compact set and are small in a sufficiently high Sobolev norm. The more general theorem permits finite partial polyhomogeneous expansions: the metric perturbation may start at $r^{-1-\epsilon_0}$ and the second-fundamental-form perturbation one inverse power faster, with a remainder decaying at least as $r^{-3-\epsilon_0}$ in the specified weighted norm.

The claimed solution is constructed in a generalized wave-map gauge modified by finite-dimensional gauge sources determined by the asymptotic expansion. Final mass, spin vector, a small Lorentz boost $S$, gauge sources, and radiation tail enter a Nash–Moser iteration as unknown modulation parameters. In spatially compact regions, the metric is claimed to approach a nearby Kerr metric at $O(t_*^{-2-\epsilon_{\mathcal K}})$, with separate quantitative descriptions at null infinity and the horizon. The restricted statement and the precise version appear in Hintz 2026, arXiv:2606.28253v2, Theorem 1.1, pp. 2–3, and Theorem 13.1, pp. 318–319; remarks and construction, pp. 319–323.

Comparison on fixed axes

The two results should not be arranged on a single “stronger/weaker” line.

• Spin. The GKS theorem assumes slow rotation; the Hintz preprint claims every fixed subextremal spin, still excluding extremality.
• Data class. The GKS program and related earlier results admit their own structureless weighted asymptotics. Hintz admits weaker leading decay when a finite polyhomogeneous structure is supplied, but requires stronger $r^{-3-\epsilon_0}$ remainder decay for structureless data. The v2 paper explicitly notes that the theorem classes do not contain one another.
• Gauge and mechanism. GKS uses its geometric foliation, curvature system, and modulation hierarchy. Hintz uses a modified generalized wave-map gauge, constraint damping, tame linear estimates, asymptotic expansions, and Nash–Moser iteration.
• Status. The slow-rotation program is a multiwork proof with substantial established literature behind its components. The full-subextremal claim is a 339-page v2 preprint relying on two companion papers as black-box inputs. Length and internal detail are not independent verification.

Adversarial checks and unresolved scope

The v2 claim should be tested at the interfaces most likely to hide a gap: uniformity as spin approaches extremality is not claimed and must not be inferred; the constraint-damped linear operator must propagate genuine vacuum solutions; derivative loss must be compatible with Nash–Moser tame bounds; the finite-dimensional gauge sources must cover all asymptotic obstructions; and the constructed exterior must match the stated maximal-development conclusion. Its mode-stability and forward-estimate inputs also require independent checking in exactly the tensorial gauge used by the nonlinear iteration.

The open question is therefore no longer stated simply as “is slow Kerr stable?” The unresolved assessment, as of the cutoff, is whether the full-subextremal preprint and companion estimates withstand line-by-line independent review and whether comparable conclusions hold for broader, non-polyhomogeneous data. Extremal Kerr is outside both theorems. Matter, charge, and non-asymptotically-flat boundary conditions are also outside the vacuum data class. A future peer-reviewed full-subextremal theorem, substantive correction, independent verification, or counterexample is a trigger to update this comparison.

Compare the stability theorems

Does the spin inclusion $\{\lvert a\rvert/(GM)\ll1\}\subset\{\lvert a\rvert/(GM)\lt1\}$ imply that the Hintz v2 theorem subsumes the GKS theorem for all data covered by GKS?

Solution

No. Spin is only one hypothesis. The theorems use different weighted regularity, asymptotic data classes, gauges, and modulation schemes. Hintz’s partial polyhomogeneity allows some weak leading tails, while its unstructured remainder condition is stronger than the decay allowed in parts of the earlier program. Thus the domains are not nested merely because one spin interval is larger. A valid implication would require matching every hypothesis and every conclusion, not only $\lvert\mathbf a\rvert/(GM)$.

References

• Giorgi, Elena, Sergiu Klainerman, and Jérémie Szeftel. “Wave Equations Estimates and the Nonlinear Stability of Slowly Rotating Kerr Black Holes.” Pure and Applied Mathematics Quarterly 20 (2024): 2865–3849. DOI.
• Hintz, Peter. “Nonlinear Stability of Subextremal Kerr Black Holes.” arXiv:2606.28253v2 (revised 2026-08-03). Versioned arXiv.
• Klainerman, Sergiu, and Jérémie Szeftel. “Kerr Stability for Small Angular Momentum.” Pure and Applied Mathematics Quarterly 19 (2023): 791–1678. DOI.