Nonlinear QNMs from full Einstein evolution

Summary

Nonlinear QNMs from full Einstein evolution

Why this matters

Observational ringdown analysis increasingly depends on knowing when linear mode models are justified and when nonlinear corrections matter.

Exact scope

Background / setting

Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.

Equation type

PDE level: spectral-operator, full-einstein.

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 (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.

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

Derive theorem-level nonlinear ringdown expansions directly from near-Kerr Einstein dynamics, including quadratic or higher corrections to linear QNM behavior.

What is already known

• Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.
  Regime: Linear waves; fixed background.
  Standard input for QNM expansions and superradiance discussions.
• Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).
  Regime: Nonlinear vacuum, restricted parameters.
  Closest nonlinear analogue for exterior stability conjectures.
• Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.
  Regime: Spectral theory on Kerr.
  Distinguishes partial mode stability from full expansion/completeness.

Progress summary: Partial progress exists in adjacent regimes;

What remains open

A complete answer must bound the difference between the full Einstein waveform and a stated nonlinear QNM model over an explicit time interval.

Mathematical prerequisites

Second-order perturbation theory; nonlinear normal forms; matched asymptotics; control of nonlinear waveform remainders.

Scope / taxonomy note

Completion criteria

A complete answer must bound the difference between the full Einstein waveform and a stated nonlinear QNM model over an explicit time interval.

Implications if solved

Would bridge rigorous PDE theory and high-precision black-hole spectroscopy.

Formal verification suitability

FV: medium

Bridges full Einstein evolution with QNM-style expansions; some linearized spectral pieces are structured, but nonlinear remainder control is a poor first formalization target compared to pure ODE or stationary rigidity problems.

See Formal verification for how this database uses these labels.

References

• primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016)
  Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.

Depends on

Conceptual dependencies (not necessarily logical lemmas in a proof assistant).

• K-001 — Full nonlinear stability of subextremal Kerr
• K-401 — QNM completeness for Kerr ringdown expansions

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-506 — High-frequency Kerr quasinormal-mode laws with explicit remainder bounds
• K-507 — Finite-data inverse resonance recovery of Kerr parameters
• K-603 — Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).

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.