Einstein–Maxwell stability near Kerr
Summary
Einstein–Maxwell stability near Kerr
Why this matters
Realistic black-hole models often involve coupled fields, and the Einstein–Maxwell system is the next natural extension of the vacuum problem.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: maxwell, einstein-maxwell.
- Linearity
- Includes or emphasizes nonlinear dynamics.
- Regularity
- Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
- 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
Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.
Problem statement
Prove asymptotic stability for coupled gravitational and electromagnetic perturbations near a rotating black-hole endstate.
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: Partial progress exists in adjacent regimes;
What remains open
A complete answer must identify the final stationary family, derive quantitative decay for the coupled fields, and prove orbital stability in a precise topology.
Mathematical prerequisites
Coupled hyperbolic systems; gravito-electromagnetic gauge fixing; Teukolsky-type equations with coupling; conserved fluxes for charge and angular momentum.
Completion criteria
A complete answer must identify the final stationary family, derive quantitative decay for the coupled fields, and prove orbital stability in a precise topology.
Implications if solved
Would open a path toward the full Kerr–Newman stability program.
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 Linear stability of slowly rotating Kerr–Newman black holes — Hung, Kellerbauer, Luk (2023) Linearized Einstein–Maxwell decay on weakly charged, slowly rotating Kerr–Newman.
- 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
- K-606 — Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
- K-642 — Prove analogous Cauchy-horizon instability results for coupled gravito-electromagnetic perturbations on Kerr-Newman.
- K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
- K-509 — Charge–tail correspondence in the extremal limit
- K-629 — Prove robust decay estimates for Maxwell/Dirac fields on Kerr without small-a restriction, in sharp norms.
- K-658 — Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.
Editorial / maintainer notes
Partial: substantial adjacent results or special cases exist, but the statement as written is not fully settled. : replace with a precise description of what is proved vs. conjectured.