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K-509

Charge–tail correspondence in the extremal limit

Open Classical frontier Open in literature Mostly scoped Extremal / Near-Extremal Mixed FV: medium
Kerr–Newman Asymptotically flat Matter-coupled Einstein–MaxwellFull Einstein ExtremalNear-extremalExterior

Summary

Charge–tail correspondence in the extremal limit

Why this matters

Extremal limits mix long-range electromagnetic and gravitational effects; tails encode observables in idealized models.

Exact scope

Background / setting
Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
Equation type
PDE level: einstein-maxwell, 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

Relate leading-order late-time tails of coupled Maxwell or scalar fields on near-extremal Kerr–Newman backgrounds to conserved charges (electric/magnetic) and angular momentum in a quantitative theorem with explicit extremal limits.

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: Near-extremal and extremal decay literature exists for several equations; a sharp unified “charge–tail” correspondence statement is not catalogued as proved on this site. See .

What remains open

Theorems with explicit exponents and coefficients linking tail amplitudes to charge data, including convergence as extremality is approached.

Mathematical prerequisites

Einstein–Maxwell perturbations; Aretakis-type blow-up mechanisms; matched asymptotics; Price-law heuristics vs. theorem-level decay rates.

Scope / taxonomy note

Family nuance: Kerr–Newman and near-extremal families differ from vacuum Kerr; asymptotic tail rates couple to charge and spin.

Completion criteria

Theorems with explicit exponents and coefficients linking tail amplitudes to charge data, including convergence as extremality is approached.

Implications if solved

Unifies extremal decay phenomenology with conserved-charge algebra.

Formal verification suitability

FV: medium

Extremal limits often admit scaling symmetries and power-law expansions; coupled Maxwell tensor identities are partially algebraic, though tail–charge correspondence may still need PDE control.

See Formal verification for how this database uses these labels.

References

Depends on

Conceptual dependencies (not necessarily logical lemmas in a proof assistant).

  • K-201 — Nonlinear codimension-1 stability of extremal Kerr with horizon hair

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Related by shared tags

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

  • K-658 — Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.
  • K-007 — Einstein–Maxwell stability near Kerr
  • K-205 — Rigorous near-horizon scattering theory for NHEK
  • K-308 — Rigidity and uniqueness with matter: full Kerr–Newman regime
  • K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
  • K-201 — Nonlinear codimension-1 stability of extremal Kerr with horizon hair
  • K-202 — Full linear theory for extremal Kerr (spin 2)

Editorial / maintainer notes

Mixed math–physics; not asserted solved.

Last updated: 2026-04-06 · Last verified (editorial): 2026-04-06 (bulk-editorial-fixes) · Edit on GitHub →