From Linear Decay to Nonlinear Stability

A decaying linear perturbation is only the first ingredient of nonlinear black-hole stability. The perturbation also changes the geometry through which it propagates. A successful proof must control that feedback, preserve the Einstein constraints, choose a regular gauge, and identify the final Kerr parameters. This page derives a small energy model that explains the required estimates, then compares the published slow-rotation program with the full-subextremal preprint. The model is not a proof of either Einstein theorem.

Evidence cutoff: 2026-09-12. The comparison below distinguishes theorem scope from the public publication record.

Required background. PDEs and Green Functions for Gravitation supplies wave operators, energy estimates, and causal evolution. Bootstrap and tame estimates are introduced below.

Helpful background. Black-Hole Geometry and Mechanics describes the Kerr family; Formulations of General Relativity distinguishes gauge and constraints; Mathematical Relativity develops global existence; Black-Hole Perturbation Theory provides the linear-field setting.

Nonlinear stability of a family of metrics

Work in four-dimensional asymptotically flat vacuum general relativity, with zero cosmological constant. We use geometrized units $G=c=1$ in this calculation: the Kerr mass $M$ and specific angular momentum $a=J/M$ are lengths. For physical mass $M_{\rm phys}$, restore $M=GM_{\rm phys}$; subextremality is $\lvert a\rvert\lt GM_{\rm phys}$.

Initial data consist of a spatial metric $\gamma$ and extrinsic curvature $K$ on a spacelike hypersurface $\Sigma$. With $K_{ij}=\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$ and future unit normal $n$, the vacuum constraints are

$$R[\gamma]+(\operatorname{tr}_\gamma K)^2-K_{ij}K^{ij}=0, \qquad D^jK_{ij}-D_i\operatorname{tr}_\gamma K=0.$$

Here $D$ and $R[\gamma]$ are the spatial covariant derivative and scalar curvature. These are restrictions on admissible initial perturbations, not optional errors to be made small. Their geometric origin and propagation are explained in Wald 1984, §10.2, equations (10.2.28)–(10.2.36), pp. 259–262.

Orbital stability means remaining near the Kerr family, allowing the admissible coordinate and reference-frame identifications. Asymptotic stability additionally requires convergence of the intrinsic parameters and decay of a radiative remainder in specified norms and regions. A finite comparison takes the form

$$g=\Phi^*g_b+h_{\rm rad}, \qquad b=(M,\boldsymbol a),$$

where $\Phi$ is a controlled diffeomorphism and $\boldsymbol a$ specifies the spin axis as well as its magnitude. This is a decomposition to construct, not a gauge-invariant definition of the remainder. A Lie derivative is only the first variation of the pullback; it is not the full nonlinear coordinate change.

Mass and spin are intrinsic Kerr parameters. A boost, center trajectory, or asymptotic frame is different, and need not be a gauge transformation trivial at the boundaries. In particular, a moving center need not converge in the original frame. Radiation can change the remnant parameters without changing the conserved total ADM charges on complete asymptotically flat Cauchy slices. Even an exact nearby Kerr solution need not approach the original reference metric: no radiation is needed for that simple obstruction.

A theorem must also specify its future region, norm, regularity and completeness conclusions. Existence for all coordinate time does not establish infinite affine length. Exterior stability does not assert regularity of the entire black-hole interior.

An evolving wave operator

A quasilinear equation is linear in its highest derivatives, but the coefficients of those derivatives depend on the unknown solution. In generalized harmonic evolution, the reduced metric equations have principal part $g^{\alpha\beta}\partial_\alpha\partial_\beta g_{\mu\nu}$. Thus the wave operator itself changes when $g=g_0+h$ changes.

For small $h$ in a fixed regular chart, raise its indices with $g_0$. Matrix inversion gives

$$g^{\alpha\beta}=g_0^{\alpha\beta}-h^{\alpha\beta}+O(h^2).$$

Consequently a schematic reduced equation, normalized to have that wave principal part and written with a frozen linear operator $L_0$, contains

$$L_0h =h^{\alpha\beta}\partial_\alpha\partial_\beta h +Q(h, \partial h) +O(h^2)\partial^2h +\text{gauge and parameter terms}.$$

Here $L_0$ includes the background curvature and linear lower-order terms; $Q$ contains terms at least quadratic with no second derivatives of $h$. This formula identifies the derivative structure, not the complete tensor Einstein equation. The positive sign of the displayed quasilinear term comes from moving the inverse-metric correction to the right-hand side.

Treating $h\,\partial^2h$ as an arbitrary source can demand an uncontrolled extra derivative. Energy arguments instead retain the evolving principal operator and use integration by parts, commutators, and its geometric structure. They must preserve hyperbolicity and control coefficient norms. The local reduced-equation construction alone supplies no global decay estimate Wald 1984, §10.2, pp. 260–262.

The required linear input is therefore a robust inhomogeneous estimate: it controls the response to a source on every sufficiently close geometry encountered in the argument, with constants and derivative requirements that the same argument can bound. A homogeneous estimate on one fixed Kerr metric does not automatically provide this.

A bootstrap estimate that actually improves

A bootstrap starts with a local solution, assumes deliberately generous bounds on an interval, and uses the equations to improve them strictly. Continuity and an appropriate continuation theorem then exclude a finite endpoint. The following scalar inequality isolates the energy step.

Use dimensionless time $\tau=t/M_0$, where $M_0\gt0$ is a fixed background length. Let $E_N(\tau)\ge0$ be an absolutely continuous, dimensionless squared high-order norm, with $E_N(0)\le\varepsilon^2$. For application to a metric, such a norm must fix the foliation, positive tensor norm, spatial weights and derivatives through order $N$. Here $E_N$ is abstract: it is not identified with a particular theorem’s energy or with signed Killing energy.

Suppose the estimates, on the bootstrap interval, yield

$$E_N'(\tau)+D_N(\tau) \le C B(\tau)E_N(\tau)+R_N(\tau), \qquad D_N(\tau)\ge0,$$

and, separately,

$$0\le B(\tau)\le A\varepsilon(1+\tau)^{-p}, \qquad 0\le R_N(\tau)\le C\varepsilon^3(1+\tau)^{-1-q}, \qquad p\gt1,\quad q\gt0.$$

The constants $A,C\gt0$ do not depend on the interval length or $\varepsilon$. The low-order coefficient $B$ represents, for example, a weighted sup norm of first derivatives; its decay must be proved separately. The nonnegative $D_N$ represents controlled spacetime bulk or outgoing flux. Uncontrolled terms, including any indefinite flux, belong on the right. The residual $R_N$ represents all remaining source pairings after their own bounds; calling a term “quadratic” does not prove this cubic estimate.

Define the integrating-factor exponent

$$\Lambda(\tau)=C\int_0^\tau B(s)\,ds \le\frac{CA\varepsilon}{p-1}.$$

Multiplying by $e^{-\Lambda}$ and integrating gives

$$e^{-\Lambda(\tau)}E_N(\tau) +\int_0^\tau e^{-\Lambda(s)}D_N(s)\,ds \le \varepsilon^2+\int_0^\tau e^{-\Lambda(s)}R_N(s)\,ds.$$

Since $\Lambda$ is nondecreasing and $\int_0^\infty(1+s)^{-1-q}\,ds=1/q$, both the energy and accumulated bulk obey

$$E_N(\tau)+\int_0^\tau D_N(s)\,ds \le\varepsilon^2 \left(1+\frac{C\varepsilon}{q}\right) \exp\!\left(\frac{CA\varepsilon}{p-1}\right).$$

For sufficiently small $\varepsilon$, the last factor is at most $3/2$. This improves an assumed $E_N\le2\varepsilon^2$ to $E_N\le3\varepsilon^2/2$, uniformly on the interval. An explicit sufficient smallness condition is derived in the first exercise.

This closes only the displayed energy estimate. The argument still has to improve the assumed bound for $B$, control the gauge and all continuation quantities, and establish the geometric and asymptotic conclusions. See Bootstrap and Geometric Continuation for that final logical step.

At the borderline $p=1$, the same comparison instead allows

$$E_N(\tau)\le \varepsilon^2\left(1+\frac{C\varepsilon}{q}\right) (1+\tau)^{CA\varepsilon}.$$

This can fit a weak-growth top-order bootstrap when lower derivatives still decay. It is not a uniform bound, and a nonintegrable upper bound does not itself prove that the actual solution grows.

Nonlinear error control

The table connects the abstract inequality to the work a black-hole proof must perform. Each row is a separate requirement; a successful estimate in one row cannot supply the others.

Nonlinear difficulty	Required estimate or construction	What would be insufficient
Changing principal coefficients	Inhomogeneous energy and decay estimates stable under the controlled metric perturbation	Homogeneous decay on one frozen background
Quadratic source and slow tail	Low-order sup-norm decay, product estimates and an integrable residual, or a justified weak-growth hierarchy	A small source at each fixed time with no time-integral bound
Trapping and derivative loss	A quantified loss compatible with the high/low hierarchy or tame iteration	Requiring one extra uncontrolled derivative at every step
Gauge and constraints	Regular gauge, homogeneous subsidiary evolution with compatible zero data, and no incoming constraint violation	Damping alone or only the initial value of the gauge constraint
Stationary and resonant directions	Modulated Kerr parameters, a controlled frame and treatment of leading nondecaying terms	Demanding that every metric component decay to the original Kerr metric

A tame estimate places the highest regularity linearly in the bound, while allowing nonlinear dependence on controlled lower norms. A Nash–Moser iteration combines such estimates with smoothing to compensate for a quantified loss of derivatives. It is not a remedy for an arbitrary loss: the smoothing, regularity reserve and linearized inverse must be proved compatible.

Constraint propagation is a separate uniqueness argument. A harmonic reduction produces a homogeneous subsidiary wave equation for its gauge constraint once the reduced metric is known. Both the constraint and its normal derivative must vanish initially; the Einstein constraints and compatible gauge data supply the latter in the standard construction. A timelike artificial boundary also needs constraint-preserving incoming data. Lower-order damping can help estimates, but does not turn arbitrary nonzero constraint data into an exact vacuum solution.

Finally, “every fixed subextremal Kerr background” does not mean uniformity as $\lvert a\rvert/M\to1$. The allowed perturbation size and constants may depend on the background. The extremal-horizon result has its own charge, geometry and decay assumptions; it cannot be bypassed by taking an uncontrolled extremal limit.

A decaying linearization with nonlinear blow-up

Consider an elementary dimensionless model, not an Einstein or black-hole equation:

$$u'=-\frac{p}{1+\tau}u+u^2, \qquad u(0)=\varepsilon\gt0,\quad p\gt0.$$

The linearized solution is $u_{\rm lin}=\varepsilon(1+\tau)^{-p}$, which decays for every $p\gt0$. To solve the nonlinear equation, set $w=(1+\tau)^p u$. Then

$$w'=(1+\tau)^{-p}w^2, \qquad \frac1{w(\tau)}=\frac1\varepsilon-I_p(\tau), \qquad I_p(\tau)=\int_0^\tau(1+s)^{-p}\,ds.$$

Hence, until its denominator first vanishes,

$$u(\tau)= \frac{\varepsilon(1+\tau)^{-p}} {1-\varepsilon I_p(\tau)}.$$

For $p\gt1$, the integral is bounded by $1/(p-1)$, so $0\lt\varepsilon\lt p-1$ ensures global decay. For $0\lt p\le1$, the integral diverges and every positive $\varepsilon$ produces finite-time blow-up. In particular, $p=1$ gives $\tau_*=\exp(1/\varepsilon)-1$. The quadratic term may be tiny initially and still accumulate over a very long interval.

The lesson is not that Einstein perturbations obey this equation. It is that even decay of the entire linearized solution does not control nonlinear accumulation without a quantitative source estimate. The actual tensor structure, constraints and spatial dispersion decide whether this mechanism is present or excluded.

Stability results through 2026-09-12

The following comparison is current through 2026-09-12. The first three rows link to the detailed theorem treatments in this chapter; they are different analytic inputs, not successive names for one theorem.

Result	Scope and data	Conclusion	Limitation
Whiting mode stability	Massless separated Teukolsky modes; fixed subextremal Kerr; continued physical endpoint branches	Excludes exponentially growing separated modes	Not an arbitrary-data evolution estimate
Scalar boundedness and decay	Neutral massless scalar on fixed subextremal Kerr; the stated weighted data and foliation	Boundedness, integrated decay and quantitative decay	Not the coupled linearized or nonlinear Einstein equation
Linearized gravitational stability	Schwarzschild linearized vacuum gravity; controlled characteristic data and gauge	Boundedness and decay to linearized Kerr modulo controlled gauge	Not nonlinear error control
Klainerman–Szeftel, published 2023 program	Asymptotically flat vacuum; sufficiently small rotation and sufficiently small admissible high-regularity initial layer	Nonlinear stability with complete future null infinity and convergence toward nearby Kerr	Preserves the source’s weighted norms and initial-layer geometry
Hintz, arXiv:2606.28253v2	Asymptotically flat vacuum; each fixed subextremal background; restricted data example below	Preprint theorem of nonlinear settling to nearby Kerr	Public record checked lists a preprint; no uniform extremal estimate or arbitrary-data assertion

Small-spin theorem

The exact smallness condition in the Klainerman–Szeftel author-final text is $I_{k_{\rm large}+10}\le\epsilon_0^2$, with sufficiently large $k_{\rm large}$, sufficiently small $\epsilon_0$, and sufficiently small $\lvert a_0\rvert/M_0$. The initial layer is a vacuum evolution of asymptotically flat Cauchy data, with the specified outgoing/ingoing frames and their transition. Its norm $I_k$ controls weighted connection and curvature components and frame transition coefficients through $k$ derivatives. It is not an unspecified norm of metric components.

The result constructs a future development with complete null infinity and quantitative convergence to nearby Kerr in the source’s boundedness/decay norms. Its generally covariant modulated (GCM) spheres fix geometric normalizations; GCM is part of the construction, not an extra symmetry assumption on perturbations. The proof uses companion curvature and GCM estimates, not the short energy model above. The precise layer, norms and statement are in Klainerman and Szeftel 2023, author-final PDF, §§3.1, 3.3.6 and 3.4, pp. 117–121 and 137–144. These locators are not the journal’s pagination.

Full-subextremal preprint

A concrete sufficient data class in Hintz’s Theorem 1.1 is smooth constrained data on its regular horizon-crossing slice, agreeing exactly with a fixed subextremal Kerr data set outside a compact region and sufficiently close in $H^d$, a Sobolev norm controlling square-integrable derivatives through order $d$, for sufficiently large $d$. The maximal globally hyperbolic development contains the constructed future region, extending across the horizon, and its metric approaches nearby Kerr there. On fixed compact spatial sets the component difference decays as $O((1+\tau)^{-2-\epsilon_{\mathcal K}})$ for some $\epsilon_{\mathcal K}\gt0$, in the constructed gauge. Here $\tau=\tilde t/M_0$ uses the theorem’s horizon-regular time $\tilde t$, not an arbitrary time reparameterization. The theorem specifies different estimates near infinity. This is not a uniform global-in-space rate Hintz 2026, v2, Theorem 1.1, pp. 1–3, PDF.

The broader data class has finite inverse-power/logarithmic expansions plus a faster-decaying weighted remainder. It is not nested with all data classes of the slow-rotation program; the source explicitly warns against that inference. Its tensorial wave-map/Nash–Moser construction uses companion constraint-damping and tame-wave estimates Hintz 2026, v2, p. 2, footnote 1, and §1.1, pp. 3–4, PDF. These companions are Hintz 2026, Constraint Damping, v1, abstract and Hintz 2026, (Non-)Linear Waves II, v1, abstract; their role here is a declared proof dependency, not a separate proof reproduced on this page.

What the publication record does and does not say

The journal record for the slow-rotation work and the arXiv record for Hintz v2 have different publication status. The latter still lists the 3 August 2026 version, with no journal reference in the record checked at the cutoff. This does not reveal a private referee report, establish community consensus, or constitute evidence against the mathematics. Full independent verification of the nonlinear proof is not supplied by this page. The evidence does not establish a uniform extremal limit or settle every admissible data class.

The meaningful comparison is between different formulations and data requirements, not two invented competing verdicts. A technical objection would need to identify a failure in an estimate, constraint argument, iteration or theorem hypothesis. The present source comparison reports no such established objection, but it is not an exhaustive certification that none exists.

The theorem’s version matters: a revised preprint, substantive technical critique, erratum, journal publication, or extension of the admissible data may change its assumptions or parameter-uniform estimates. Compare the precise statements, companion results, norms, and rates when using a later version. For detailed nonlinear proof methods, continue to Mathematical Relativity; finite-amplitude evolution and endpoints belong in Black-Hole Dynamics.

Exercises

Make the bootstrap margin explicit

Find a sufficient upper bound on $\varepsilon$ ensuring that the energy factor above is at most $3/2$. State what this does not yet prove.

Solution

It suffices to impose

$$\varepsilon\le \min\!\left\{\frac{q}{4C}, \frac{(p-1)\log(6/5)}{CA}\right\}.$$

Then $1+C\varepsilon/q\le5/4$ and the exponential is at most $6/5$, whose product is $3/2$. This strictly improves the factor $2$ but does not derive the low-order decay, preserve the gauge, verify a continuation criterion, or prove completeness and convergence.

Test a borderline decay rate

Solve the model at $p=1$. For $p\gt1$, examine the threshold $\varepsilon=p-1$: does the solution decay?

Solution

At $p=1$,

$$u(\tau)= \frac{\varepsilon}{(1+\tau)[1-\varepsilon\log(1+\tau)]}, \qquad \tau_*=\exp(1/\varepsilon)-1.$$

For $p\gt1$,

$$I_p(\tau)=\frac{1-(1+\tau)^{1-p}}{p-1}.$$

At $\varepsilon=p-1$, cancellation gives $u(\tau)=(p-1)/(1+\tau)$, still global and decaying. Thus the strict inequality in the sufficient small-data condition is not necessary for global existence in this particular model. If $\varepsilon\gt p-1$, the denominator vanishes at finite time. This distinction guards against promoting a sufficient bootstrap bound to a sharp instability threshold for Einstein’s equation.

Diagnose a derivative deficit

Suppose a proposed iteration requires $\lVert h\rVert_{X^N}\le C\lVert F\rVert_{Y^{N+1}}$, but its nonlinear source is controlled only in $Y^N$ from an $X^N$ bound. Why is simply iterating not enough? Would a small residual or a nearby stationary Kerr solution repair the problem?

Solution

The next iterate demands an uncontrolled derivative of the source. Smallness in $Y^N$ is not a bound in $Y^{N+1}$, so these estimates do not establish an iteration preserving the chosen space. Remedies require additional structure: a sharper estimate, a closed high/low hierarchy, or a tame smoothing iteration with a quantified regularity reserve.

A nearby stationary Kerr solution illustrates a different issue: its parameter difference can persist forever while the geometry remains stable. Absorbing that finite-dimensional component does not produce a missing derivative estimate for the radiative remainder. Both the parameter identification and the derivative bounds are needed.

References

• Hintz, Peter. “(Non-)Linear Waves on Asymptotically Flat Spacetimes. II: Trapping, Bound States, Nonlinear Applications.” arXiv:2606.28008v1, 2026. Version record.
• Hintz, Peter. “Constraint Damping on Subextremal Kerr Spacetimes.” arXiv:2606.27658v1, 2026. Version record.
• Hintz, Peter. “Nonlinear Stability of Subextremal Kerr Black Holes.” arXiv:2606.28253v2, submitted 3 August 2026. Version record. Open PDF.
• Klainerman, Sergiu, and Jérémie Szeftel. “Kerr Stability for Small Angular Momentum.” Pure and Applied Mathematics Quarterly 19(3) (2023): 791–1678. DOI. Open PDF, author-final text dated 14 July 2023.
• Wald, Robert M. General Relativity. University of Chicago Press, 1984. Publisher.