Nonlinear fate of Aretakis instability

Summary

Nonlinear fate of Aretakis instability

Why this matters

The key physical question is whether the linear Aretakis effect survives, saturates, or resolves nonlinearly.

Exact scope

Background / setting

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

Equation type

PDE level: full-einstein.

Linearity

Primarily stationary or linearized reductions unless the statement says otherwise.

Regularity

Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.

Parameter regime

Extremal or near-extremal Kerr-type parameters; quantify smallness of $|1-|a|/M|$ or surface gravity $κ$ in any claim.

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

Determine whether generic small perturbations of extremal Kerr settle to subextremal Kerr, remain extremal with horizon hair, or develop a stronger nonlinear instability.

What is already known

• Aretakis instability and conserved charges on extremal horizons are established for scalar test fields; spin-2 and nonlinear extremal dynamics are much less complete.
  Regime: Extremal horizons; often linear scalar.
  Shows qualitative difference from subextremal decay.
• Subextremal nonlinear Kerr stability is known for small $|a|/M$; uniformity as $|a|\to M$ is not a corollary.
  Regime: Nonlinear vacuum, restricted subextremal window.
  Separates near-extremal uniformity from existing subextremal theorems.
• Near-horizon NHEK limits capture extremal mode structure but matching to global Kerr is an open PDE bridge.
  Regime: Near-horizon scaling limits.
  Clarifies what NHEK analyses do and do not imply globally.

Progress summary: Partial progress exists in adjacent regimes;

What remains open

A complete answer must prove one of the possible nonlinear outcomes for a generic class of perturbations and quantify the transition mechanism.

Mathematical prerequisites

Nonlinear continuation near degenerate horizons; parameter drift; horizon-local blow-up criteria; conserved-charge dynamics.

Completion criteria

A complete answer must prove one of the possible nonlinear outcomes for a generic class of perturbations and quantify the transition mechanism.

Implications if solved

Would tell us whether extremality is dynamically fragile or structurally persistent.

Formal verification suitability

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

References

• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.
• 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.

Related problems

Related by shared tags

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

• K-620 — Establish a rigorous nonlinear theory for extremal Kerr dynamics incorporating Aretakis charges.
• K-622 — Prove uniform (in kappa) energy/decay estimates for near-extremal Kerr approaching kappa to 0.
• K-656 — Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
• K-657 — Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
• K-691 — Prove that event-horizon redshift estimates remain valid for near-extremal Kerr with uniform constants away from kappa=0.
• K-621 — Prove a definitive linear stability/instability dichotomy for extremal Kerr spin-2 with sharp norms.

Editorial / maintainer notes

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