Prove quantitative stability of the Carter constant for geodesics under small metric perturbations.
Summary
Prove quantitative stability of the Carter constant for geodesics under small metric perturbations.
Why this matters
Geodesic integrability robustness.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: null-geodesics.
- 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 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
Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).
Problem statement
Prove quantitative stability of the Carter constant for geodesics under small metric perturbations.
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: Context: Geodesic integrability robustness.
What remains open
Prove quantitative stability of the Carter constant for geodesics under small metric perturbations.
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 Nonlinear stability of Kerr for small angular momentum (program) — Klainerman, Szeftel (2021) Theorem-level nonlinear stability of vacuum Kerr in a small $|a|/M$ regime; benchmark for full subextremal conjectures.
- 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-649 — Formalize in a proof assistant: the exact Kerr separability structure and Carter constant derivation.
- K-695 — Formalize in a proof assistant: Carter separability and associated conserved quantities for Kerr geodesics.
- K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
- K-009 — Einstein–massive Klein–Gordon near Kerr: classification of stable and unstable regimes
- K-011 — Spin fields on dynamical near-Kerr backgrounds
- K-013 — Formation plus relaxation to Kerr
Editorial / maintainer notes
Source manifest: N-085 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.