Sharp nonlinear Price law for curvature
Summary
Sharp nonlinear Price law for curvature
Why this matters
Sharp tails control how exterior decay feeds into interior instability and into the interpretation of ringdown signals.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: full-einstein.
- Linearity
- Includes or emphasizes nonlinear dynamics.
- Regularity
- Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
- 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
Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.
Problem statement
Prove optimal late-time inverse-power decay for gauge-invariant curvature quantities and relevant metric coefficients in nonlinear near-Kerr evolution.
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: Partial progress exists in adjacent regimes;
What remains open
A complete answer must state exact exponents, identify generic nonvanishing leading coefficients, and prove matching upper and lower bounds.
Mathematical prerequisites
Sharp $r^p$ hierarchies; resonance/low-frequency analysis; nonlinear bootstrap estimates; Bianchi equations; asymptotic mode coupling.
Completion criteria
A complete answer must state exact exponents, identify generic nonvanishing leading coefficients, and prove matching upper and lower bounds.
Implications if solved
Would connect linear tail predictions to the full Einstein dynamics and feed directly into SCC formulations.
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 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-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
- K-014 — Nonlinear decay versus resonance expansions
- K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
- K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.
Editorial / maintainer notes
Partial: substantial adjacent results or special cases exist, but the statement as written is not fully settled. : replace with a precise description of what is proved vs. conjectured.