Inverse scattering for Kerr parameters
Summary
Inverse scattering for Kerr parameters
Why this matters
The forward scattering map on Kerr is rich enough that one expects inverse geometric information to be hidden in it.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: spectral-operator.
- 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 (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
Determine whether scattering data for a natural field equation uniquely and stably determine the Kerr mass and spin.
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 specify the data set, prove uniqueness of $(M,a)$, and give quantitative stability estimates.
Mathematical prerequisites
Inverse scattering; geometric optics; spectral invariants; uniqueness and stability estimates for parameter recovery.
Completion criteria
A complete answer must specify the data set, prove uniqueness of $(M,a)$, and give quantitative stability estimates.
Implications if solved
Would turn black-hole scattering into a geometric inverse problem with direct interpretive value.
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.
Unlocks (other problems list this one as a dependency)
- K-507 — Finite-data inverse resonance recovery of Kerr parameters
Related problems
Related by shared tags
- K-504 — Quantitative stability of the photon region and spherical null geodesics under near-Kerr perturbations
- K-505 — Threshold phenomena in separated Teukolsky-type ODEs on Kerr (zero modes, algebraically special limits, superradiant edges)
- K-107 — Scattering map to the Cauchy horizon for linearized gravity
- K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)
- K-507 — Finite-data inverse resonance recovery of Kerr parameters
- K-102 — Derive the interior theorem directly from exterior data
Editorial / maintainer notes
Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.