Rigorous near-horizon scattering theory for NHEK

Summary

Rigorous near-horizon scattering theory for NHEK

Why this matters

The near-horizon extremal geometry is the natural local model controlling many extremal phenomena.

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

Construct a mathematically precise scattering theory for the near-horizon extremal Kerr limit and prove convergence from rescaled near-extremal Kerr dynamics.

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 operator and solution convergence under rescaling and define a genuine scattering theory in the limit geometry.

Mathematical prerequisites

Geometric blow-up limits; near-horizon analysis; scattering on non-asymptotically-flat geometries; matched expansions.

Completion criteria

A complete answer must prove operator and solution convergence under rescaling and define a genuine scattering theory in the limit geometry.

Implications if solved

Would isolate the intrinsic throat dynamics responsible for extremal instabilities.

Formal verification suitability

FV: high

Stationary, algebraic, ODE/separable, or finite-dimensional substatements admit clearer formalization boundaries.

See Formal verification for how this database uses these labels.

References

• 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.
• 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-623 — Build a rigorous scattering theory for NHEK matching to global Kerr exterior at near-extremality.
• K-109 — Near-extremal interior scaling laws
• K-509 — Charge–tail correspondence in the extremal limit

Editorial / maintainer notes

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