Black-Hole Stability, Decay, and Scattering

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Black-hole stability is a ladder of logically distinct results. Boundedness of a test scalar does not imply decay; mode stability does not imply a time-domain estimate; decay of curvature does not reconstruct a metric; and a linear theorem does not control nonlinear errors. This chapter develops those rungs in their proper order for Schwarzschild and subextremal Kerr exteriors, ending with radiation-field scattering and a dated comparison of nonlinear Kerr theorems.

Useful preparation includes Analysis for Mathematical Relativity, Partial Differential Equations and Green Functions, and the geometric orientation in Black-Hole Geometry and Mechanics. Readers needing the perturbative variables can first consult Black-Hole Perturbation Theory; the underlying field equations are reviewed in The Einstein Equation.

A hierarchy of stability claims

The background is a four-dimensional vacuum Schwarzschild or Kerr exterior. For Kerr, physical mass $M$ , specific angular momentum $a=J/M$ , and $\lvert a\rvert\lt GM$ are stated whenever the spin range matters. The scalar wave equation tests the dispersive geometry without introducing gravitational gauge. Linearized gravity then adds constraints, pure-gauge solutions, and stationary variations of mass and angular momentum. Nonlinear stability adds motion of the geometry, quasilinear error terms, and a modulation problem for the final Kerr parameters.

The proof tasks can be compared without conflating their conclusions:

Claim	What must be controlled	What remains to prove
Mode stability	Separated solutions with unstable frequency	Uniform frequency estimates and superpositions
Energy boundedness	A specified energy at each time	Escape of energy from compact regions
Integrated local-energy decay	Time-integrated energy, with trapping losses stated	Quantitative rates, including the far region
Quantitative decay	Time-weighted energies or pointwise fields	Metric reconstruction, constraints, and stationary sectors
Nonlinear stability	Small constrained developments modulo gauge and parameters	Sharp tails, stronger asymptotic regularity, or scattering unless included in the theorem

These are distinct proof tasks, rather than sets related by strict inclusion. For example, a degenerate integrated estimate needs additional estimates before it controls a nondegenerate energy. Scattering also requires a well-posed inverse map, which decay alone does not construct. The horizon-degenerate scalar forward and inverse maps are constructed in Dafermos, Rodnianski, and Shlapentokh-Rothman 2018, §§2.3.3–2.3.5.

The recurring analytic obstacles

Three geometric effects organize the chapter. At a nonextremal horizon, a redshift multiplier recovers the derivative missed by stationary energy. Trapped null geodesics force a degeneration or derivative loss in integrated local-energy estimates. In rotating Kerr, superradiance makes stationary horizon flux indefinite for part of the frequency spectrum. The full- subextremal scalar estimates resolve all three by combining physical-space currents with quantitative frequency control; see Dafermos, Rodnianski, and Shlapentokh-Rothman 2016, Main Theorem and §§5–13.

For metric perturbations, a fourth obstacle appears. Teukolsky curvature variables do not determine the stationary Kerr and pure-gauge sectors. A stability theorem must extract those kernels, reconstruct the remaining connection and metric components, and control its gauge. Schwarzschild’s linear theorem carries out these steps in Dafermos, Holzegel, and Rodnianski 2019, Main Theorem and §§7–13. At the nonlinear level, these choices become modulation parameters coupled to a bootstrap or Nash–Moser iteration.

Contents in theorem order

Scalar-Wave Boundedness and Decay derives the redshift, Morawetz, and far-field hierarchy on black-hole exteriors.
Linear Stability of Schwarzschild separates radiative decay from pure gauge and the linearized Kerr remainder.
Mode Stability and Decay on Kerr explains why absence of growing separated modes is necessary but insufficient for time-domain decay.
Linearized Gravity on Kerr connects Teukolsky estimates to metric reconstruction, charge completion, and gauge control.
Nonlinear Stability of Kerr compares the slow-rotation theorem program with the full-subextremal claim of Hintz 2026, arXiv v2, Theorem 1.1 at the 2026-08-12 evidence cutoff.
Black-Hole Scattering Theory constructs bounded forward and inverse radiation maps in the correct horizon-degenerate energy spaces.

This order is deliberate. Readers should not use a later conclusion while silently dropping an earlier distinction. In particular, the recent nonlinear preprint does not turn scalar scattering into tensor scattering, and a metric decay theorem does not imply a quasinormal-mode completeness statement.

Reader routes

For a PDE route, begin with scalar energy currents, continue to mode stability, and finish with scattering. This route emphasizes norms, trapping, superradiance, and asymptotic completeness.

For a gravitational-stability route, move from Schwarzschild linear stability to Kerr reconstruction and then to nonlinear Kerr. This route emphasizes gauge, constraints, stationary kernels, parameter modulation, and exact data classes. The nonlinear page is dated because theorem status can change; the earlier scalar and Schwarzschild pages state stable peer-reviewed results.

For a quick theorem-comparison route, read mode stability first, then the Kerr linearized-gravity method, and finally the nonlinear evidence matrix. At every stage ask: What is the field? What norm is controlled? Which spin range and boundary conditions are assumed? What finite-dimensional remainder survives?

For post-cutoff changes in parameter range, data class, or proof status, use the dated Nonlinear Stability of Kerr Black Holes assessment. This chapter retains the theorem statements and analytic mechanisms against which that record is compared.

Chapter review

Classify each statement as mode stability, boundedness, decay, linear metric stability, nonlinear metric stability, or scattering: (i) no separated solution grows like $e^{\gamma t}$ with $\gamma>0$ ; (ii) every small vacuum perturbation approaches a modulated Kerr metric; (iii) past horizon and null- infinity radiation determine future radiation by a bounded isomorphism; (iv) a normalized Schwarzschild perturbation approaches linearized Kerr.

Solution

Statement (i) is mode stability. It does not supply uniform resolvent bounds. Statement (ii) is nonlinear metric stability and requires a data class, gauge, spin range, and modulation norm. Statement (iii) is scattering and must name the degenerate horizon energy in which the inverse is bounded. Statement (iv) is linear metric stability: the radiative part decays after pure gauge is controlled, while mass and angular-momentum variations remain. None of the four statements may be substituted for another.

References

Dafermos, Mihalis, Gustav Holzegel, and Igor Rodnianski. “The Linear Stability of the Schwarzschild Solution to Gravitational Perturbations.” Acta Mathematica 222 (2019): 1–214. DOI.
Dafermos, Mihalis, Igor Rodnianski, and Yakov Shlapentokh-Rothman. “A Scattering Theory for the Wave Equation on Kerr Black Hole Exteriors.” Annales Scientifiques de l’École Normale Supérieure 51 (2018): 371–486. DOI.
Dafermos, Mihalis, Igor Rodnianski, and Yakov Shlapentokh-Rothman. “Decay for Solutions of the Wave Equation on Kerr Exterior Spacetimes III: The Full Subextremal Case $|a|<M$ .” Annals of Mathematics 183 (2016): 787–913. arXiv:1402.7034.
Hintz, Peter. “Nonlinear Stability of Subextremal Kerr Black Holes.” arXiv:2606.28253v2 (revised 2026-08-03). Versioned arXiv.