Prove sharp stability of Kerr under small violations of vacuum Einstein (numerical truncation errors).

Summary

Prove sharp stability of Kerr under small violations of vacuum Einstein (numerical truncation errors).

Why this matters

Bridges rigorous GR and numerical relativity.

Exact scope

Background / setting

asymptotically flat general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: full-einstein.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

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

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Prove sharp stability of Kerr under small violations of vacuum Einstein (numerical truncation errors).

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: Context: Bridges rigorous GR and numerical relativity.

What remains open

Prove sharp stability of Kerr under small violations of vacuum Einstein (numerical truncation errors).

Mathematical prerequisites

Match hypotheses to primary sources cited on this page; state minimal regularity, gauge class, and parameter windows in any claimed theorem.

Completion criteria

Prove a theorem or give a rigorous counterexample that matches the scoped statement under explicitly listed hypotheses.

Implications if solved

Impact depends on the solved formulation; sharpen once the statement is pinned to a literature-compatible theorem.

Formal verification suitability

FV: low

Speculative or open-ended; formalization boundary unclear.

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

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

• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-013 — Formation plus relaxation to Kerr
• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.
• K-605 — Prove sharp nonlinear Price-law tails for curvature in near-Kerr vacuum.
• K-616 — Prove robust control of trapping geometry under dynamical near-Kerr perturbations in a sharp topology.

Editorial / maintainer notes

Source manifest: N-094 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.