Formation plus relaxation to Kerr

Summary

Formation plus relaxation to Kerr

Why this matters

This would turn the standard numerical-relativity narrative into a theorem.

Exact scope

Background / setting

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

Equation type

PDE level: 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 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

Prove a global theorem in which generic asymptotically flat vacuum data form a black hole and the exterior subsequently relaxes to Kerr with quantitative radiation control.

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 specify an open class of initial data, prove black-hole formation, establish complete null infinity, and show quantitative Kerr asymptotics outside the hole.

Mathematical prerequisites

Trapped-surface formation; global causal geometry; continuation criteria; matching formation and relaxation regimes.

Completion criteria

A complete answer must specify an open class of initial data, prove black-hole formation, establish complete null infinity, and show quantitative Kerr asymptotics outside the hole.

Implications if solved

Would unite collapse theory and Kerr stability into one coherent mathematical picture.

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

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-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.
• K-632 — Quantify how much Kerrness can be certified from finite-radius curvature invariants (numerical-relativity certification).

Editorial / maintainer notes

Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.