Full linear theory for extremal Kerr (spin 2)
Summary
Full linear theory for extremal Kerr (spin 2)
Why this matters
The linear theory is the mandatory first step before any nonlinear extremal theorem.
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
- Extremal or near-extremal Kerr-type parameters; quantify smallness of $|1-|a|/M|$ or surface gravity $κ$ in any claim.
- 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
Develop a complete boundedness, decay, and instability theory for linearized vacuum Einstein perturbations of extremal Kerr.
What is already known
- Aretakis instability and conserved charges on extremal horizons are established for scalar test fields; spin-2 and nonlinear extremal dynamics are much less complete.Regime: Extremal horizons; often linear scalar.Shows qualitative difference from subextremal decay.
- Subextremal nonlinear Kerr stability is known for small $|a|/M$; uniformity as $|a|\to M$ is not a corollary.Regime: Nonlinear vacuum, restricted subextremal window.Separates near-extremal uniformity from existing subextremal theorems.
- Near-horizon NHEK limits capture extremal mode structure but matching to global Kerr is an open PDE bridge.Regime: Near-horizon scaling limits.Clarifies what NHEK analyses do and do not imply globally.
Progress summary: Partial progress exists in adjacent regimes;
What remains open
A complete answer must separate decay away from the horizon from conserved or growing horizon quantities and handle gauge issues explicitly.
Mathematical prerequisites
Degenerate vector-field method; zero-energy analysis; extremal Teukolsky theory; gauge-invariant reconstruction.
Completion criteria
A complete answer must separate decay away from the horizon from conserved or growing horizon quantities and handle gauge issues explicitly.
Implications if solved
Would become the linear backbone of extremal Kerr dynamics.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
See Formal verification for how this database uses these labels.
References
- survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022) Program overview, gauge structure, and relation between linear tools and nonlinear stability.
- 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.
Related problems
Related by shared tags
- K-620 — Establish a rigorous nonlinear theory for extremal Kerr dynamics incorporating Aretakis charges.
- K-622 — Prove uniform (in kappa) energy/decay estimates for near-extremal Kerr approaching kappa to 0.
- K-656 — Prove decay/growth dichotomy for extremal Kerr perturbations with explicit identification of conserved charges.
- K-657 — Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
- K-691 — Prove that event-horizon redshift estimates remain valid for near-extremal Kerr with uniform constants away from kappa=0.
- K-621 — Prove a definitive linear stability/instability dichotomy for extremal Kerr spin-2 with sharp norms.
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.