Nonlinear asymptotic completeness near Kerr
Summary
Nonlinear asymptotic completeness near Kerr
Why this matters
This is the cleanest theorem-level formulation of relaxation to Kerr plus radiation.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: spectral-operator.
- Linearity
- Includes or emphasizes nonlinear dynamics.
- Regularity
- Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
- Parameter regime
- Subextremal Kerr moduli $|a|<M$ (or stated KN/KdS extension); smallness measured in the stability topology on Cauchy data.
- Asymptotics
- asymptotically-flat
- Gauge / formulation
- State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).
Status explanation
Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.
Problem statement
Construct a nonlinear scattering map between near-Kerr initial data and asymptotic radiation data at null infinity plus final Kerr parameters.
What is already known
- Nonlinear stability of vacuum Kerr is proved for sufficiently small $|a|/M$ (Klainerman–Szeftel).Regime: Nonlinear Einstein vacuum, asymptotically flat, small angular momentum per unit mass.Strongest unconditional nonlinear theorem toward the full subextremal conjecture.
- Linearized Teukolsky/wave decay and mode stability on fixed subextremal Kerr are developed in depth (microlocal and physical-space methods).Regime: Linearized gravity and scalar waves on exact Kerr.Standard toolbox; not equivalent to nonlinear stability for all parameters.
- Sharp Price-law exponents and nonlinear tail matching are understood in restricted settings (e.g. linearized models, Schwarzschild); sharp nonlinear Kerr curvature tails are not settled.Regime: Late-time asymptotics; mixed linear vs nonlinear literature.Locates what “sharp Price law” demands beyond integrated decay.
Progress summary: Partial progress exists in adjacent regimes;
What remains open
A complete answer must define the forward and inverse scattering maps in explicit function spaces and prove existence, uniqueness, and continuous dependence modulo symmetries.
Mathematical prerequisites
Hyperboloidal or conformal compactification techniques; nonlinear scattering theory; Bondi framework; constraint propagation; inverse problems for radiation fields.
Completion criteria
A complete answer must define the forward and inverse scattering maps in explicit function spaces and prove existence, uniqueness, and continuous dependence modulo symmetries.
Implications if solved
Would make the Kerr endstate picture mathematically precise and usable for waveform extraction and uniqueness questions.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
See Formal verification for how this database uses these labels.
References
- primary Nonlinear stability of Kerr for small angular momentum (program) — Klainerman, Szeftel (2021) Theorem-level nonlinear stability of vacuum Kerr in a small $|a|/M$ regime; benchmark for full subextremal conjectures.
- survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022) Program overview, gauge structure, and relation between linear tools and nonlinear stability.
Related problems
Related by shared tags
- K-014 — Nonlinear decay versus resonance expansions
- K-659 — Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
- K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
- K-009 — Einstein–massive Klein–Gordon near Kerr: classification of stable and unstable regimes
- K-011 — Spin fields on dynamical near-Kerr backgrounds
Editorial / maintainer notes
Partial: substantial adjacent results or special cases exist, but the statement as written is not fully settled. : replace with a precise description of what is proved vs. conjectured.