Prove that Kerr is uniquely determined among stationary vacua by a finite set of multipole moments with stability.

Summary

Prove that Kerr is uniquely determined among stationary vacua by a finite set of multipole moments with stability.

Why this matters

Effective uniqueness via multipoles.

Exact scope

Background / setting

asymptotically flat general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: stationary-reduction.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

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

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Prove that Kerr is uniquely determined among stationary vacua by a finite set of multipole moments with stability.

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: Context: Effective uniqueness via multipoles.

What remains open

Prove that Kerr is uniquely determined among stationary vacua by a finite set of multipole moments with stability.

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 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).

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-508 — Stability or obstruction for approximate Killing–Yano tensors near Kerr
• K-619 — Prove uniqueness of stationary black holes with small deviations in asymptotic charges (effective inverse problems).
• K-306 — Hidden symmetries and approximate Carter-type operators under metric perturbation
• K-307 — Persistence of normally hyperbolic trapping for dynamical near-Kerr spacetimes
• K-617 — Prove a quantitative distance-to-Kerr estimate from a small invariant (Mars-Simon-type) with computable constants.
• K-618 — Prove global Kerr uniqueness without analyticity under minimal smoothness/decay hypotheses.
• K-626 — Prove a Kerr inverse problem: determine (M,a) from finitely many resonances with stability estimates.

Editorial / maintainer notes

Source manifest: N-097 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.