Rigidity and uniqueness with matter: full Kerr–Newman regime
Summary
Complete stationary uniqueness/rigidity for Kerr–Newman without smallness caveats; partial linearized exterior progress exists (cf. K-008), but global stationary uniqueness in the smooth category remains incomplete.
Why this matters
Charged rotating solutions are physically central; their rigidity theory lags the vacuum Carter–Robinson picture in generality.
Exact scope
- Background / setting
- Stationary Einstein–Maxwell, asymptotically flat, subextremal Kerr–Newman as target family.
- Equation type
- Stationary coupled elliptic-type reduction; EM gauge issues included.
- Linearity
- Stationary nonlinear coupled system.
- Regularity
- Smooth / high Sobolev as in rigidity programs.
- Parameter regime
- Full subextremal $(M,a,Q)$ range is the goal; partial results may cover restricted sets.
- Asymptotics
- Asymptotically flat.
- Gauge / formulation
- Lorenz / generalized wave-map gauges as in linearized stability literature.
Status explanation
Partial: coupled linearized results exist; full stationary uniqueness without smallness remains open.
Problem statement
Prove the full stationary uniqueness and perturbative rigidity theory for asymptotically flat Einstein–Maxwell black-hole solutions in the subextremal Kerr–Newman class—without small-charge or slow-rotation restrictions once the sharp theorem is available.
What is already known
- Linearized decay/stability for weakly charged, slowly rotating Kerr–Newman (partial exterior linear result).Regime: Linearized Einstein–Maxwell as in arXiv:2301.08557.Partial progress on dynamics; stationary uniqueness is a distinct stationary elliptic problem.
Progress summary: Vacuum rigidity is more complete; Einstein–Maxwell stationary uniqueness in full parameters and partial linearized stability are both active areas.
What remains open
Global smooth stationary uniqueness/rigidity matching vacuum rigidity strength across parameters.
Mathematical prerequisites
Einstein–Maxwell stationary reduction; coupled invariant tensors; horizon regularity; unique continuation.
Completion criteria
Sharp hypothesis theorem classifying stationary solutions as Kerr–Newman in the smooth category.
Implications if solved
Completes charged rotating rigidity parallel to vacuum Kerr uniqueness.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
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) Partial linearized progress on coupled Einstein–Maxwell around Kerr–Newman; related but not identical to stationary uniqueness.
- survey Black Uniqueness Theorems — Mazur (2001) Classical uniqueness landscape including charged settings at a survey level.
Related problems
Related by shared tags
- K-687 — Prove a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
- K-501 — Quantitative Mars–Simon tensor gap for near-Kerr stationary vacuum data
- K-502 — Horizon-data rigidity and effective reconstruction of Kerr parameters
- K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)