Full asymptotically flat stability of the subextremal Kerr–Newman family
Summary
Prove linear and nonlinear stability for the full subextremal Kerr–Newman family; restricted linear results exist for weak charge and slow rotation, but the unrestricted coupled problem remains open.
Why this matters
Kerr–Newman is the natural stationary charged rotating family; settling its stability is the coupled-field analogue of vacuum Kerr stability.
Exact scope
- Background / setting
- Four-dimensional Einstein–Maxwell, asymptotically flat, subextremal Kerr–Newman as reference.
- Equation type
- Coupled Einstein–Maxwell (full nonlinear field equations).
- Linearity
- Target includes nonlinear stability; current partial results are linearized.
- Regularity
- Strong enough Sobolev/decay assumptions to close coupled estimates (as in cited linear work).
- Parameter regime
- Eventual goal is the full subextremal range $a^2+Q^2<M^2$. Existing theorem-level linear results cover restricted windows (e.g. weak charge and slow rotation).
- Asymptotics
- Asymptotically flat.
- Gauge / formulation
- Lorenz / EM gauge and harmonic-type coordinates as in the linearized theory; nonlinear gauge choice is part of a complete theorem.
Status explanation
Marked “open” because the unrestricted subextremal Kerr–Newman statement is not proved. Partial progress refers only to restricted linearized regimes; the UI must not suggest the full problem is nearly solved.
Problem statement
Prove linear and then nonlinear asymptotic stability of the full subextremal Kerr–Newman family ($a^2+Q^2<M^2$ in standard parameters) under coupled Einstein–Maxwell dynamics in an asymptotically flat setting—without imposing small-charge or slow-rotation restrictions once a complete theorem is available.
What is already known
- Linearized stability (decay to a linearized Kerr–Newman solution modulo gauge) for weakly charged, slowly rotating Kerr–Newman black holes.Regime: Linearized Einstein–Maxwell, stated smallness regime of arXiv:2301.08557.Confirms partial linearized control in an open subset of parameters; does not constitute full subextremal linear or any nonlinear theorem.
Progress summary: Theorem-level linearized stability is available in a weak-charge, slow-rotation window; full-parameter linear and all nonlinear stability remain open.
What remains open
Extend coupled linearized theory to the full subextremal parameter range and build a nonlinear modulated stability theorem comparable in scope to the vacuum Kerr program.
Mathematical prerequisites
Non-selfadjoint spectral theory; coupled gravito-EM perturbation equations; physical-space energy estimates; nonlinear modulation of mass, charge, and spin.
Completion criteria
Treat the full subextremal parameter range and produce decay to linearized or nonlinear Kerr–Newman modulo gauge and parameter shifts as claimed.
Implications if solved
Would give a coupled-field analogue of Kerr stability and reshape the Einstein–Maxwell scattering picture.
Formal verification suitability
FV: medium
Coupled Einstein–Maxwell perturbation theory; full nonlinear theorem remains large, but structured linearized subproblems exist.
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, L. (2023) Theorem-level partial linearized result; defines the precise restricted regime where “partial progress” is justified.
- survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022) Vacuum Kerr stability context; illustrates the scale of the nonlinear program the Einstein–Maxwell analogue would require.
Related problems
Related by shared tags
- 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-642 — Prove analogous Cauchy-horizon instability results for coupled gravito-electromagnetic perturbations on Kerr-Newman.
- K-509 — Charge–tail correspondence in the extremal limit
- K-001 — Full nonlinear stability of subextremal Kerr
- K-002 — Uniform nonlinear stability as $a \to M^-$