Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
Summary
Hung–Kellerbauer–Luk prove **linearized** stability of slowly rotating Kerr–Newman under small charge/rotation hypotheses. The open program is to enlarge the parameter region to a sharp subextremal family ($a^2+Q^2<M^2$ in standard parameters) with uniform decay/Morawetz estimates for the coupled linearized Einstein–Maxwell system in a fixed gauge—without restricting to $|Q|\ll M$ and $|a|\ll M$ once a full theorem is available.
Why this matters
Extends Einstein-Maxwell stability program toward the full family.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: linearized-gravity.
- Linearity
- linearized
- Regularity
- Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.
- Parameter regime
- Subextremal Kerr–Newman moduli $a^2+Q^2<M^2$; linearized perturbations off the exact KN solution; enlarge beyond the weak-charge/slow-rotation window proved to date.
- 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
Partial results exist in adjacent regimes (see references); sharp alignment with this page’s exact target remains open.
Problem statement
Establish boundedness and quantitative decay for solutions of the linearized Einstein–Maxwell equations on fixed subextremal Kerr–Newman backgrounds across an explicitly stated enlargement of the Hung–Kellerbauer–Luk parameter window, closing toward the full subextremal moduli space.
What is already known
- Hung–Kellerbauer–Luk prove linear stability of **slowly rotating** Kerr–Newman black holes for small charge and angular momentum (arXiv:2301.08557).Regime: Linearized Einstein–Maxwell on fixed KN; restricted parameter window.Current unconditional linear theorem closest to the title target.
- Teukolsky-type linearized gravity on Kerr is controlled in the full subextremal $|a|<M$ range (Dafermos–Holzegel– Rodnianski–Teixeira da Costa program); incorporating **charge** and the Maxwell sector is the additional difficulty.Regime: Linearized gravity on Kerr (vacuum component).Explains which vacuum linear tools already exist before coupling to electromagnetism.
Progress summary: KN linear stability is proved in a weak-coupling/slow-rotation window; extending uniform estimates toward the full subextremal moduli space is the remaining PDE step.
What remains open
Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
Mathematical prerequisites
Match hypotheses to primary sources cited on this page; state minimal regularity, gauge class, and parameter windows in any claimed theorem.
Completion criteria
Prove a theorem or give a rigorous counterexample that matches the scoped statement under explicitly listed hypotheses.
Implications if solved
Impact depends on the solved formulation; sharpen once the statement is pinned to a literature-compatible theorem.
Formal verification suitability
FV: low
Global PDE or phenomenological target; lemma-level formalization may be possible after scoping.
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.
- primary Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: frequency space analysis — Dafermos, Holzegel, Rodnianski, Teixeira da Costa (2020) Full subextremal linearized gravitational master equations on Kerr (vacuum component of KN linearization).
- 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-007 — Einstein–Maxwell stability near Kerr
- K-008 — Full asymptotically flat stability of the subextremal Kerr–Newman family
- 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-658 — Prove linear stability/instability classification for extremal Kerr-Newman under coupled perturbations.
- K-308 — Rigidity and uniqueness with matter: full Kerr–Newman regime
- K-601 — Prove unconditional linear stability of Kerr (full subextremal range) in a fixed gauge, with full decay rates.
- K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
Editorial / maintainer notes
Source manifest: N-006 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.