Charge–tail correspondence in the extremal limit

Summary

A plausible Kerr–Newman extremal-limit tail problem, held for bibliography repair: existing references support adjacent Aretakis, charged-field, and linear Kerr–Newman facts, not yet the full stated correspondence.

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

**Needs reference repair:** the target is scientifically plausible, but this page should not advertise the charge–tail correspondence as a literature-grounded open theorem until maintainers pin primary references that directly support the Kerr–Newman tail/charge formulation. Current references support adjacent ingredients only.

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. The site has not yet catalogued a primary source proving or explicitly posing the sharp unified Kerr–Newman “charge–tail” correspondence stated here.

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

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

• primary The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes — He (2023)
  Proves linear stability for weakly charged slowly rotating Kerr–Newman under coupled gravitational and electromagnetic perturbations; adjacent support, not a tail–charge correspondence theorem.
• primary Horizon Instability of Extremal Black Holes — Aretakis (2012)
  Baseline extremal-horizon conservation and instability mechanism motivating charge/tail questions.
• primary Horizon instability of extremal Reissner-Nordström black holes to charged perturbations — Zimmerman (2017)
  Charged scalar perturbations and near-extremal limit behavior in the Reissner–Nordström model.

Depends on

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

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

Related problems

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-642 — Prove analogous Cauchy-horizon instability results for coupled gravito-electromagnetic perturbations on Kerr-Newman.
• K-201 — Nonlinear codimension-1 stability of extremal Kerr with horizon hair

Editorial / maintainer notes

Mixed math–physics; not asserted solved.