Scattering theory for linearized gravity on Kerr
Summary
Scattering theory for linearized gravity on Kerr
Why this matters
Decay results are stronger when organized into a full scattering theory.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: full-einstein.
- Linearity
- Primarily stationary or linearized reductions unless the statement says otherwise.
- Regularity
- Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
- Parameter regime
- Subextremal Kerr (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.
- 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
Construct a complete gauge-invariant scattering theory for linearized vacuum Einstein perturbations on Kerr, paralleling the scalar-wave case.
What is already known
- Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.Regime: Linear waves; fixed background.Standard input for QNM expansions and superradiance discussions.
- Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).Regime: Nonlinear vacuum, restricted parameters.Closest nonlinear analogue for exterior stability conjectures.
- Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.Regime: Spectral theory on Kerr.Distinguishes partial mode stability from full expansion/completeness.
Progress summary: Partial progress exists in adjacent regimes;
What remains open
A complete answer must define the scattering states, wave operators, and asymptotic completeness modulo stationary/gauge modes.
Mathematical prerequisites
Gauge fixing; Teukolsky-to-metric reconstruction; horizon and infinity radiation fields; asymptotic completeness for constrained systems.
Completion criteria
A complete answer must define the scattering states, wave operators, and asymptotic completeness modulo stationary/gauge modes.
Implications if solved
Would give the definitive linearized-gravity input for both exterior stability and interior genericity problems.
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 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.
- 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-507 — Finite-data inverse resonance recovery of Kerr parameters
- K-603 — Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).
- K-660 — Establish a full nonlinear ringdown plus tail decomposition for near-Kerr vacuum spacetimes.
- K-662 — Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).
- K-683 — Prove a sharp characterization of the Kerr trapped set as a normally hyperbolic invariant manifold uniformly in a/M.
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.