Threshold phenomena in separated Teukolsky-type ODEs on Kerr (zero modes, algebraically special limits, superradiant edges)
Summary
Classify threshold behavior (zero-frequency modes, algebraically special / repeated-root regimes, superradiant frequency boundaries) for separated radial/angular ODEs on Kerr with uniform $(M,a)$ control.
Why this matters
Threshold modes control low-frequency resolvent behavior and appear in stability and scattering constructions.
Exact scope
- Background / setting
- Exact Kerr or agreed near-Kerr ODE coefficients after separation.
- Equation type
- Coupled radial/angular ODEs from spin-weighted master equations.
- Linearity
- Linearized field equations on a fixed background.
- Regularity
- Analytic coefficients in the ODE domain away from singular points; Frobenius analysis at singularities.
- Parameter regime
- Compact subsets of subextremal $(M,a)$ excluding extremality unless a separate limit is explicitly formulated.
- Asymptotics
- Asymptotically flat Kerr boundary conditions encoded in the ODEs.
- Gauge / formulation
- Teukolsky vs RW–Zerilli gauge choices recorded explicitly when comparing thresholds.
Status explanation
“Threshold” is disambiguated on-page (zero-frequency / algebraic / superradiant edge); not a vague umbrella phrase.
Problem statement
For the separated ODEs arising from Teukolsky or Regge–Wheeler–type reductions on Kerr, classify threshold resonant phenomena including (i) **zero-frequency** limits affecting low-energy resolvents, (ii) **algebraically special / repeated-root** regimes in the radial ODE, and (iii) edges of the **superradiant** frequency window—each with uniform bounds as parameters vary in compact subintervals of subextremality. The catalog should be checkable mode-by-mode and include uniformity statements in $(M,a)$.
What is already known
- Separation of the Teukolsky master equation into radial and angular ODEs on Kerr (classical derivation).Regime: Linearized perturbations of exact Kerr.Foundational ODE setup; this entry targets **uniform threshold classification**, not the original separation itself.
Progress summary: Mode-wise QNM literature is large; a single uniform threshold classification matching this phrasing is the open packaging.
What remains open
Consolidated theorem catalog with proofs of uniformity and explicit handling of each threshold family named above.
Mathematical prerequisites
ODE spectral theory; Frobenius methods; connection formulas; Teukolsky equations; asymptotic analysis of eigenvalue branches.
Completion criteria
Complete catalog with checkable mode-wise proofs and $(M,a)$ uniformity.
Implications if solved
Strengthens QNM and scattering foundations used across clusters.
Formal verification suitability
FV: high
Separated radial/angular ODEs with polynomial coefficients; threshold behavior is finite-dimensional spectral-edge analysis.
See Formal verification for how this database uses these labels.
References
- primary Perturbations of a rotating black hole. I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations — Teukolsky, S. A. (1973) Original separated ODE framework whose threshold structure this entry seeks to classify uniformly.
- primary Perturbations of a rotating black hole. II. Dynamical stability of the Kerr metric — Press, Teukolsky (1973) Companion stability analysis using the master equations; historical anchor for mode-wise threshold discussions.
Depends on
- K-303 — Quantitative Kerr characterization via the Mars–Simon tensor
Related problems
Related by shared tags
- K-504 — Quantitative stability of the photon region and spherical null geodesics under near-Kerr perturbations
- K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)
- K-102 — Derive the interior theorem directly from exterior data
- K-103 — Vacuum curvature blow-up rates on the Kerr Cauchy horizon
- K-105 — Critical horizon-decay exponent controlling extendibility
- K-106 — Genericity of lower bounds for linearized-gravity interior instability
- K-107 — Scattering map to the Cauchy horizon for linearized gravity