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arxiv: 2606.28253 · v1 · pith:4EXB32URnew · submitted 2026-06-26 · 🌀 gr-qc · math.AP

Nonlinear stability of subextremal Kerr black holes

Peter Hintz This is my paper

Pith reviewed 2026-06-29 02:52 UTC · model grok-4.3

classification 🌀 gr-qc math.AP
keywords nonlinear stabilityKerr black holesEinstein vacuum equationswave map gaugeNash-Moser iterationconstraint dampingTeukolsky equationblack hole stability
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The pith

Spacetimes from initial data near subextremal Kerr black holes settle to a nearby Kerr solution at rate O(t_*^{-2-ε_K}) in spatially compact regions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes global nonlinear stability of the full subextremal Kerr family under the Einstein vacuum equations. Solutions starting from initial data with O(r^{-1-ε0}) decay, including finite expansions with logarithmic terms, approach a nearby Kerr black hole. The proof uses a generalized wave map gauge whose source terms are unknowns in a Nash-Moser iteration, together with black-box inputs from companion papers on constraint damping and tame estimates for the linearized equations. The argument works directly with the tensorial Einstein equation and invokes the Teukolsky equation only for linear mode stability.

Core claim

We settle the global nonlinear stability problem for the family of Kerr black holes in the full subextremal range: spacetimes evolving from initial data close to those of a subextremal Kerr black hole as solutions of the Einstein vacuum equation Ric(g)=0 settle down to a nearby member of the Kerr family at the rate O(t_*^{-2-ε_K}) in spatially compact regions. For the initial data we require O(r^{-1-ε0})-decay for ε0>0, more precisely an arbitrary but finite expansion into terms r^{-z}(log r)^k where z>1, k in N_0, plus a remainder with O(r^{-3-ε0})-decay. The gauge source terms, final black hole parameters, and gravitational wave tail are treated as unknowns in the nonlinear iteration.

What carries the argument

Generalized wave map gauge modified by gauge source terms lying in a finite-dimensional space determined by the initial-data expansion, inside a nonlinear Nash-Moser iteration scheme that solves the gauge-fixed Einstein equation directly.

If this is right

  • The final mass and angular momentum are recovered as part of the solution rather than prescribed in advance.
  • The same gauge and iteration framework controls the gravitational wave tail at the stated rate.
  • The argument applies to initial data whose expansion may contain arbitrarily many logarithmic terms.
  • No reduction of the Einstein equation to scalar wave equations is required except for the linear mode-stability step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decay rate O(t_*^{-2-ε}) is slow enough that late-time tails remain visible in spatially compact observations but fast enough to guarantee asymptotic flatness at future null infinity.
  • The method separates the determination of the final Kerr parameters from the dynamical decay, which may allow similar iteration schemes when matter fields or cosmological constants are added.
  • Numerical evolutions of near-Kerr data could directly measure the predicted power-law index in the approach to stationarity.

Load-bearing premise

The two companion papers supply a strong form of constraint damping valid throughout the subextremal range together with tame estimates for the relevant linearized wave equations.

What would settle it

An explicit initial-data set with the stated decay whose evolution under Ric(g)=0 either fails to approach any Kerr solution or decays slower than O(t_*^{-2-ε_K}) in a fixed compact region.

Figures

Figures reproduced from arXiv: 2606.28253 by Peter Hintz.

Figure 1.1
Figure 1.1. Figure 1.1: On the left: the domain Ω, drawn in a product fashion. The dashed line labeled H+ is the event horizon for gb. On the right: the domain Ω, drawn as a subset of the Penrose diagram of gb. The Einstein vacuum equations (1.1) can be cast as a quasilinear wave equation for g after gauge fixing. The initial data are the first and second fundamental form γ and k of Σ inside of (Ω, g). The constraint equations … view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Illustration (and construction) of the compactified spacetime manifold M (without the factor of S 2 ω). (In order to include points in I + where t∗ = 0, one simply replaces ρ0 and ρI near I 0 ∩ I + by ρ0 = 1 1−t∗ and ρI = 1−t∗ r .) If one defines M as a compactification of {r ≥ m0, t ≥ −1 2 r}, it contains the domain Ω from (1.2). and so on, with the same bounds holding also for all b-derivatives of u. (… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: This diagram summarizes what influences decay and asymptotics in the four asymptotic regimes of spacetime. The arrow labeled “initial data” is dis￾cussed in §1.3.1, the first “transport” arrow and the “radiation” arrow in §1.3.2, the second “transport” arrow and the “resonances” arrow in §1.3.3, the “indicial roots” arrows in §§1.3.4–1.3.5, and the “0-energy bound states” arrow in §1.3.6. This diagram wi… view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: The compact manifold X + sc-b with corners suitable for low-energy spectral theory on asymptotically flat spaces X◦ ⊂ R 3 x . Here ρ = r −1 = |x| −1 , and σ is the spectral parameter (here restricted to [0, 1]). · · · E+, α+ EK, αK M F F −1 · · · E+−1, α+−1 ((0, 0) ∪ (EK−1), αK−1 X + sc-b [PITH_FULL_IMAGE:figures/full_fig_p024_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Relationships between the ι +- and K+-orders of functions on space￾time M and the tf- and zf-orders of their Fourier transforms on the resolved low￾energy space X + sc-b. (The conditions on the scf- and I +-orders are stated precisely in Proposition 2.25.) (1) (Mode stability.) We need the wave-type operators under study to satisfy mode stability in Im σ ≥ 0, σ ̸= 0, so that indeed only low frequencies r… view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: Elimination of pure gauge contributions to the late-time asymptotics of metric perturbations using gauge modifications and metric patches. On the left: center-of-mass shifts can be eliminated using gauge modifications supported on supp dχK, with metric perturbation patches supported on the same set. On the right: dynamical pure gauge solutions can be eliminated in a similar fashion, except the gauge modi… view at source ↗
Figure 1.7
Figure 1.7. Figure 1.7: Schematic description of the decay orders and terms in the K+- expansion of the solution h of Lbh = f. We will ultimately take αK > 4 and α+ > 3 close to 4 and 3, respectively. We only list terms that correspond to the exemplary large indicial roots discussed above. The index set E+ is essentially the union of EI , (1, 0), and an index set determined by pure resonances as in (1.19). In the second column,… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Illustration of Lemma 2.12: the index sets F⃗ of u at ∂H get trans￾ported from x ≳ 1 to x = 0; the index set at x = 0 increases by {(z, 0)} relative to E. Proof of Lemma 2.12. Note that x −zu satisfies the equation f = (x∂x−z)x z (x −zu) = x zx∂x(x −zu), so x∂x(x −zu) = x −zf; we may thus reduce to the case z = 0. Next, denoting points in H by y, we compute x∂x(x w(log x) ℓ v(y)) = wxw(log x) ℓ v(y) + ℓx… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The manifold M˜ 0 and its boundary hypersurfaces, and some (partial) local coordinate systems (using polar coordinates r = |x| and dropping spherical variables ω = x |x| ). (1) (Bundles.) The 3b-cotangent bundle is defined by 3bT ∗M˜ 0 := M˜ 0 × R 4 , with a point (t∗, x; τ, ξ) over M˜ ◦ 0 = R 4 identified with the covector τ dt∗ ⟨x⟩ + P3 j=1 ξj dx j ⟨x⟩ . The dual bundle is denoted 3bTM˜ 0 and carries a… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Illustration of Lemma 2.23: the index set E of u at x = 0 gets transported from y ≳ 1 to y = 0 and interacts with the index set F of (x∂x −y∂y + z)u there. Proof of Lemma 2.23. Note that x zu satisfies (x∂x − y∂y)(x zu) = x zf, so upon replacing u, f, E, and α by x zu, x zf, E + z, and α + Re z, respectively, we can reduce to the case z = 0 [PITH_FULL_IMAGE:figures/full_fig_p058_2_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Illustration of Lemma 3.4 (dropping the factor S 2 of X = [m0, ∞]r × S 2 ), with the domains identified in (3.8) highlighted in gray. Proof of Lemma 3.4. This is essentially proved in [Hin26b, §2.3.1]; we give the proof here for com￾pleteness. Away from the front face of M′ , the functions (3.4d) and (3.4h) are local coordinates, so it suffices to consider a neighborhood of the front face. As local coord… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Illustration of the functions t∗ from Lemma 3.6 and tIVP from Lemma 3.7. We also show the 0-level sets of t = t∗ + r and of the Boyer–Lindquist time coordinate t [PITH_FULL_IMAGE:figures/full_fig_p074_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Illustration of the cutoff functions from Definition 3.17. Lemma 3.18 (Cut-off Lorentz boosts). For S ∈ S1, define the vector field V (S) in terms of (3.38) and (3.19) by V (S) := 1 2 χ ♯ IVP(1 − χ ♭ K) ¯ g −1 ( ¯ ω (0),1 s1 (S), ·). (3.39) Then V (S) ∈ Vb(R4) is tangent to Y + (see (3.2)), and its lift to M vanishes near Σpast ∪ K+ ∪ Σint (Definition 3.2) and for r ≤ 5m0 [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Illustration of the domain Ωext,r0 and the hypersurfaces Σpast,r0 , ΣIVP,r0 , and Σfut,r0 . Proof of Lemma 5.16. Since dtIVP = dt (for large r) is timelike for gb0 , the same is true also for all sufficiently small (in L∞, as sections of S 2T ∗ ) perturbations of gb0 . Similarly, ζ := d(tIVP + r−r0 4 ) = dt + 1 4 dr = dt∗ + 5 4 dr is timelike for the Minkowski metric (cf. (3.19) for m = 0) since 5 4 ∈ (1… view at source ↗
Figure 9.1
Figure 9.1. Figure 9.1: Illustration of the argument after (9.23) concerning the vanishing of the v λ -coefficient of h. The crosses mark points in the projection to the first factor of the index set E tot ι+,I . The curve labeled ˜µ starts in Re < 1 and ends at λ while not passing through any of these points. components π11h and π/0 h). The v λ log λ-coefficient of h is given by resκ=λ [PITH_FULL_IMAGE:figures/full_fig_p172_9… view at source ↗
Figure 11.1
Figure 11.1. Figure 11.1: The manifold M′ and its boundary hypersurfaces, together with some local coordinates (suppressing the spherical coordinates ω = x |x| ∈ S 2 ); cf [PITH_FULL_IMAGE:figures/full_fig_p232_11_1.png] view at source ↗
Figure 11.2
Figure 11.2. Figure 11.2: On the left: illustration of the two steps in the proof of Lemma 11.20(1). In the first step, we integrate along a hyperbolic vector field, which transports I +-decay to ι +. In the second step, we integrate along the b-normal vector field at K+. The orders of u are stated next to the respective boundary hypersurfaces. On the right: illustration of the main content of Lemma 11.20(2). Integration along a… view at source ↗
Figure 13.1
Figure 13.1. Figure 13.1: Illustration of Theorem 13.1. The initial data (γ, k) are attained at a boosted version ϕS(ΣIVP) of ΣIVP = t −1 IVP(0). The two dashed lines bound the set supp dχK where the reference metric gb0,b,−S transitions from (ϕS)∗gb0 to gb. 140As discussed in §5.7, one identifies ΣIVP with the complement of a ball in R3 x; the meaning of the condi￾tions (13.1a)–(13.1b) is then that they hold for each component … view at source ↗
read the original abstract

We settle the global nonlinear stability problem for the family of Kerr black holes in the full subextremal range: spacetimes evolving from initial data close to those of a subextremal Kerr black hole as solutions of the Einstein vacuum equation ${\rm Ric}(g)=0$ settle down to a nearby member of the Kerr family at the rate $\mathcal{O}(t_*^{-2-\epsilon_{\mathcal K}})$ in spatially compact regions. For the initial data, we require $\mathcal{O}(r^{-1-\epsilon_0})$-decay for $\epsilon_0>0$ -- more precisely, an arbitrary but finite expansion into terms $r^{-z}(\log r)^k$ where $z>1$, $k\in\mathbb{N}_0$, plus a remainder term with $\mathcal{O}(r^{-3-\epsilon_0})$-decay. Similarly to previous work with Vasy in the Kerr-de Sitter setting, we use a generalized wave map gauge modified using gauge source terms that lie in a suitable finite-dimensional space determined by the expansion of the initial data. Like the final black hole parameters (mass and angular momentum) and the gravitational wave tail, the gauge source terms are treated as unknowns in a nonlinear (Nash-Moser) iteration scheme. We work directly with the tensorial equation and in particular do not rely on reductions to scalar equations (except insofar as a reduction to the Teukolsky equation is used in the proof of linear mode stability). This paper relies on two companion papers by the author. The first one introduces a strong form of constraint damping in the full subextremal range, which we use in our formulation of the gauge-fixed Einstein equation as a black box. The second one provides tame estimates (albeit with weak decay) for forward solutions of a general class of wave-type equations, which we show here to include the linearizations of the gauge-fixed Einstein equation arising in our nonlinear iteration scheme; these estimates are the starting point for our detailed asymptotic analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove the global nonlinear stability of subextremal Kerr black holes for the Einstein vacuum equations. Initial data close to a subextremal Kerr solution (with O(r^{-1-ε_0}) decay, including finite expansions in r^{-z}(log r)^k for z>1) evolve to a nearby Kerr black hole, with decay O(t_*^{-2-ε_K}) in spatially compact regions. The proof uses a generalized wave map gauge with finite-dimensional gauge source terms (treated as unknowns), a Nash-Moser iteration scheme, and works directly with the tensorial equation (except for Teukolsky reduction in linear mode stability). The argument relies on two companion papers for constraint damping and tame estimates.

Significance. If established, the result would be a significant contribution to mathematical general relativity by resolving nonlinear stability for the full subextremal Kerr family. The direct tensorial approach and incorporation of gauge source terms and final parameters into the nonlinear iteration are strengths. The paper also notes the use of mode stability via the Teukolsky equation.

major comments (2)
  1. [Abstract, paragraph on reliance on companion papers] Abstract, paragraph on reliance on companion papers: the nonlinear iteration scheme and the claimed O(t_*^{-2-ε_K}) decay in compact regions are obtained by bootstrapping from the tame estimates (with weak decay) supplied by the second companion paper, while solving for gauge source terms and Kerr parameters; these estimates are asserted to cover the linearizations of the gauge-fixed Einstein equation, but no verification or closure check under the nonlinear iteration is provided here.
  2. [Abstract, paragraph on reliance on companion papers] Abstract, paragraph on reliance on companion papers: the formulation of the gauge-fixed Einstein equation uses a strong form of constraint damping valid throughout the subextremal range |a|<M as a black box from the first companion paper; if this damping does not suppress violations at the required rate near extremality, the global stability statement does not follow.
minor comments (2)
  1. [Abstract] Abstract: the decay rate notation O(t_*^{-2-ε_K}) and the parameter ε_K are not defined in the abstract; a parenthetical clarification would improve readability.
  2. [Abstract] Abstract: the initial data expansion is stated as arbitrary but finite into r^{-z}(log r)^k with z>1; specifying the admissible range of z and k explicitly would aid precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading of the manuscript. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract, paragraph on reliance on companion papers] Abstract, paragraph on reliance on companion papers: the nonlinear iteration scheme and the claimed O(t_*^{-2-ε_K}) decay in compact regions are obtained by bootstrapping from the tame estimates (with weak decay) supplied by the second companion paper, while solving for gauge source terms and Kerr parameters; these estimates are asserted to cover the linearizations of the gauge-fixed Einstein equation, but no verification or closure check under the nonlinear iteration is provided here.

    Authors: The manuscript states in the abstract and develops in the main text that the linearizations arising at each step of the Nash-Moser iteration belong to the general class of wave-type equations for which the second companion paper supplies tame estimates. The iteration is constructed precisely so that the gauge source terms and final Kerr parameters remain within the finite-dimensional spaces for which the estimates close; the weak decay supplied by the companion is then upgraded to the stated O(t_*^{-2-ε_K}) rate by the asymptotic analysis performed here. A more explicit sentence summarizing this closure can be added to the abstract or introduction for clarity. revision: partial

  2. Referee: [Abstract, paragraph on reliance on companion papers] Abstract, paragraph on reliance on companion papers: the formulation of the gauge-fixed Einstein equation uses a strong form of constraint damping valid throughout the subextremal range |a|<M as a black box from the first companion paper; if this damping does not suppress violations at the required rate near extremality, the global stability statement does not follow.

    Authors: The first companion paper establishes the strong constraint damping for the full subextremal range |a|<M, including the decay rates needed to control violations near extremality. The present work invokes this result as a black box exactly as described, and the global stability statement therefore inherits the range and rates proved in the companion. No additional verification is required in this manuscript beyond the statement already given. revision: no

Circularity Check

1 steps flagged

Central nonlinear stability result depends on black-box self-citations for constraint damping and tame estimates

specific steps
  1. self citation load bearing [abstract]
    "This paper relies on two companion papers by the author. The first one introduces a strong form of constraint damping in the full subextremal range, which we use in our formulation of the gauge-fixed Einstein equation as a black box. The second one provides tame estimates (albeit with weak decay) for forward solutions of a general class of wave-type equations, which we show here to include the linearizations of the gauge-fixed Einstein equation arising in our nonlinear iteration scheme; these estimates are the starting point for our detailed asymptotic analysis."

    The nonlinear iteration scheme and the claimed decay rate are obtained only after bootstrapping from the weak-decay tame estimates while solving for gauge source terms and final Kerr parameters. Both the damping and the tame estimates are supplied exclusively by the two self-cited companion papers and are invoked as black boxes; the stability statement therefore does not follow without those prior self-citations.

full rationale

The paper's derivation of the O(t_*^{-2-ε_K}) decay for subextremal Kerr stability proceeds via a Nash-Moser iteration on the gauge-fixed Einstein equation. This iteration is explicitly constructed to take as black-box inputs (1) strong constraint damping valid for |a|<M and (2) tame estimates for the relevant linearized wave equations; both are supplied by two companion papers by the same author. The abstract states that these are used directly without further derivation here, and the asymptotic analysis bootstraps from the weak-decay tame estimates. No external benchmarks, machine-checked verification, or independent falsifiability of the companions is provided in this manuscript, so the central claim reduces to the self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the stated initial-data decay conditions and on the two companion papers for constraint damping and tame estimates; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Linear mode stability via reduction to the Teukolsky equation holds in the subextremal range
    Invoked explicitly for the linear mode stability step (abstract).
  • domain assumption The generalized wave map gauge with finite-dimensional source terms yields a well-posed gauge-fixed Einstein equation
    Used to formulate the nonlinear iteration (abstract).

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