Rigidity for extremal horizons
Summary
Rigidity for extremal horizons
Why this matters
Extremal horizons fall outside some of the standard rigidity machinery.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: stationary-reduction.
- 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
- Subextremal Kerr moduli $|a|<M$ (or stated KN/KdS extension); smallness measured in the stability topology on Cauchy data.
- 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
Extend black-hole rigidity and axisymmetry results to extremal stationary horizons where nondegeneracy assumptions fail.
What is already known
- Analytic stationary uniqueness theorems identify Kerr in the asymptotically flat vacuum class (Carter–Robinson–Mazur line).Regime: Real-analytic stationary vacuum.Classical baseline; smooth non-analytic uniqueness remains the sharp open gap for many formulations.
- Near-Kerr perturbative rigidity and Carter-type structures are studied in separability and hidden-symmetry programs.Regime: Perturbations of Kerr; operator commutators.Context for approximate operators and photon-region stability questions.
- Ernst reduction and harmonic-map formulations package stationary axisymmetric vacuum equations; sharp global uniqueness domains are formulation-dependent.Regime: 2D elliptic reductions.Explains why Ernst-domain questions must pin boundary data and function classes.
Progress summary: Partial progress exists in adjacent regimes;
What remains open
A complete answer must prove the existence of the needed symmetry or identify a counterexample under natural hypotheses.
Mathematical prerequisites
Degenerate horizon geometry; near-horizon expansion; unique continuation in degenerate settings; Killing field extension.
Completion criteria
A complete answer must prove the existence of the needed symmetry or identify a counterexample under natural hypotheses.
Implications if solved
Would clarify whether extremal stationary black holes are as rigid as their nonextremal counterparts.
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 Hamilton–Jacobi and Schrödinger separability and integrability of the Kerr metric — Carter (1968) Fourth constant / separability structure on exact Kerr; operator-algebra backdrop for rigidity questions.
- survey Black Uniqueness Theorems — Mazur (2001) Survey of stationary uniqueness and reduction routes (including Ernst-type formulations).
Unlocks (other problems list this one as a dependency)
- K-510 — Ernst equation on stationary axisymmetric vacuum exteriors — sharp uniqueness class for asymptotically flat Kerr (boundary-value formulation)
Related problems
Related by shared tags
- K-501 — Quantitative Mars–Simon tensor gap for near-Kerr stationary vacuum data
- K-502 — Horizon-data rigidity and effective reconstruction of Kerr parameters
- K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)
- K-510 — Ernst equation on stationary axisymmetric vacuum exteriors — sharp uniqueness class for asymptotically flat Kerr (boundary-value formulation)
- 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.