Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.

Summary

Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.

Why this matters

Charged extremal rotations combine two difficult degeneracies.

Exact scope

Background / setting

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

Equation type

PDE level: linearized-gravity.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

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

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.

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: Context: Charged extremal rotations combine two difficult degeneracies.

What remains open

Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.

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

• 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-509 — Charge–tail correspondence in the extremal limit
• K-606 — Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
• K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
• K-007 — Einstein–Maxwell stability near Kerr
• K-008 — Full asymptotically flat stability of the subextremal Kerr–Newman family
• K-308 — Rigidity and uniqueness with matter: full Kerr–Newman regime
• K-621 — Prove a definitive linear stability/instability dichotomy for extremal Kerr spin-2 with sharp norms.
• K-642 — Prove analogous Cauchy-horizon instability results for coupled gravito-electromagnetic perturbations on Kerr-Newman.

Editorial / maintainer notes

Source manifest: N-058 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.