Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
Summary
Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
Why this matters
Consolidates extremal linear theory into a complete picture.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: full-einstein.
- Linearity
- linearized
- Regularity
- Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.
- 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
Partial results exist in adjacent regimes (see references); sharp alignment with this page’s exact target remains open.
Problem statement
Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
What is already known
- Named papers in the reference list establish partial or neighboring results under explicit hypotheses; treat those as the proved baseline.Regime: As stated in cited references (often restricted parameters or linearized settings).Orients readers to literature without equating it with the full title-length target.
Progress summary: Manifest rationale: Consolidates extremal linear theory into a complete picture.
What remains open
Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
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
- survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022) Program overview, gauge structure, and relation between linear tools and nonlinear stability.
- primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016) Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.
Related problems
Related by shared tags
- K-620 — Establish a rigorous nonlinear theory for extremal Kerr dynamics incorporating Aretakis charges.
- K-622 — Prove uniform (in kappa) energy/decay estimates for near-extremal Kerr approaching kappa to 0.
- K-657 — Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
- K-691 — Prove that event-horizon redshift estimates remain valid for near-extremal Kerr with uniform constants away from kappa=0.
- K-621 — Prove a definitive linear stability/instability dichotomy for extremal Kerr spin-2 with sharp norms.
Editorial / maintainer notes
Source manifest: N-056 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.