Prove a fully rigorous PDE theorem for nonlinear superradiant instability of Kerr in a confining setting (mirror/AdS).
Summary
Prove a fully rigorous PDE theorem for nonlinear superradiant instability of Kerr in a confining setting (mirror/AdS).
Why this matters
Would formalize famous black hole bomb phenomenon.
Exact scope
- Background / setting
- anti de sitter general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: full-einstein.
- Linearity
- both linearized and fully nonlinear aspects
- Regularity
- Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.
- Parameter regime
- Subextremal Kerr (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.
- Asymptotics
- anti de sitter
- 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 a fully rigorous PDE theorem for nonlinear superradiant instability of Kerr in a confining setting (mirror/AdS).
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: Context: Would formalize famous black hole bomb phenomenon.
What remains open
Prove a fully rigorous PDE theorem for nonlinear superradiant instability of Kerr in a confining setting (mirror/AdS).
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 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-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.
- K-404 — Zero-frequency structure and tail universality
- K-506 — High-frequency Kerr quasinormal-mode laws with explicit remainder bounds
Editorial / maintainer notes
Source manifest: N-088 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.