Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
Summary
Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
Why this matters
Core unknown in extremal nonlinear regime.
Exact scope
- Background / setting
- asymptotically flat 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
- 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
Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).
Problem statement
Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
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: Context: Core unknown in extremal nonlinear regime.
What remains open
Prove nonlinear evolution of near-horizon conserved quantities in extremal Kerr produces curvature singularity (or not).
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
- 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-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
Source manifest: N-057 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.