Extremal interior SCC and Cauchy-horizon regularity
Summary
Extremal interior SCC and Cauchy-horizon regularity
Why this matters
The extremal interior is much less understood than the subextremal one and may not fit the usual weak-null-singularity paradigm.
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
Determine the interior stability or instability picture for extremal Kerr and identify the correct extendibility threshold across its inner horizon structure.
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 specify the relevant extremal horizon geometry, the extension class, and the precise blow-up or continuity mechanism.
Mathematical prerequisites
Degenerate interior analysis; coupling of horizon and interior asymptotics; extremal transport estimates.
Completion criteria
A complete answer must specify the relevant extremal horizon geometry, the extension class, and the precise blow-up or continuity mechanism.
Implications if solved
Would complete the extremal analogue of SCC.
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
Editorial / maintainer notes
Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.