Prove explicit sharp late-time asymptotics for scalar field along the event horizon in Kerr (full range).

Summary

Prove explicit sharp late-time asymptotics for scalar field along the event horizon in Kerr (full range).

Why this matters

Needed for sharp interior instability thresholds.

Exact scope

Background / setting

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

Equation type

PDE level: scalar-wave.

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 explicit sharp late-time asymptotics for scalar field along the event horizon in Kerr (full range).

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: Needed for sharp interior instability thresholds.

What remains open

Prove explicit sharp late-time asymptotics for scalar field along the event horizon in Kerr (full range).

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

Global PDE or phenomenological target; lemma-level formalization may be possible after scoping.

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-009 — Einstein–massive Klein–Gordon near Kerr: classification of stable and unstable regimes
• K-628 — Derive rigorous late-time tail constants for scalar wave on Kerr in full subextremal range.
• K-639 — Classify superradiant instability windows for massive scalar fields on Kerr in fully rigorous parameter inequalities.
• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-011 — Spin fields on dynamical near-Kerr backgrounds
• K-013 — Formation plus relaxation to Kerr

Editorial / maintainer notes

Source manifest: N-055 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.