Extremal tail asymptotics versus conserved charges

Summary

Extremal tail asymptotics versus conserved charges

Why this matters

In extremal geometry, the horizon itself stores conserved information that changes the tail law.

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

Prove sharp late-time asymptotics for waves and spin fields on extremal Kerr in terms of both infinity charges and horizon Aretakis charges.

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 give leading terms, error bounds, and genericity of nonzero coefficients in the exterior and on the horizon.

Mathematical prerequisites

Low-frequency asymptotics; time inversion; charge extraction; mode coupling in degenerate horizon settings.

Completion criteria

A complete answer must give leading terms, error bounds, and genericity of nonzero coefficients in the exterior and on the horizon.

Implications if solved

Would quantify exactly how extremal tails differ from subextremal Price laws.

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-002 — Uniform nonlinear stability as $a \to M^-$
• K-679 — Prove stability/instability of higher-dimensional Kerr (Myers-Perry) in PDE sense for small angular momentum.

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.