Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
Summary
Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
Why this matters
Enables machine-checked components of linear stability proofs.
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 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
**Open (formalization):** analytic equivalence between Teukolsky and Regge–Wheeler/Zerilli-type variables is classical in physics and PDE treatments; a **machine-checkable** proof covering Schwarzschild and a stated slowly-rotating Kerr window is not claimed complete here—`needs_review` until a pinned reference list matches the formal statement.
Problem statement
Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
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: Manifest rationale: Enables machine-checked components of linear stability proofs.
What remains open
Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
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: high
Formalization targets map naturally to proof-assistant-sized subtasks once scoped.
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-630 — Prove nonlinear stability of Schwarzschild without codimension restrictions (full moduli convergence to Kerr).
- K-002 — Uniform nonlinear stability as $a \to M^-$
- K-679 — Prove stability/instability of higher-dimensional Kerr (Myers-Perry) in PDE sense for small angular momentum.
- K-204 — Uniform estimates in the surface-gravity limit $\kappa \to 0$
- K-207 — Extremal tail asymptotics versus conserved charges
- K-209 — Codimension and modulation theory for extremal endstates
- 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-050 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.