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

**Partial / strong literature-supported reduction:** nonzero real-frequency mode stability and the algebraically-special Teukolsky-Starobinsky degeneracy are well supported by primary literature. The decisive missing piece is still the consolidated zero-frequency and normalization-dependent threshold catalog requested by this page.

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.
• Real-axis mode stability for separated Teukolsky radial ODEs rules out nontrivial outgoing modes for admissible real frequencies, and gives quantitative Wronskian nonvanishing on compact real-frequency sets away from the expected zero-frequency and threshold exclusions.
  Regime: Exact Kerr; Teukolsky spins; real nonzero frequencies.
  Settles the nonzero real-frequency outgoing-mode obstruction and supplies much of the Wronskian control needed for the catalog away from threshold endpoints.
• Algebraically special frequencies are identified by vanishing of the radial Teukolsky-Starobinsky constant; the Teukolsky-Starobinsky identities show these are transform degeneracies rather than physical outgoing radial modes.
  Regime: Exact Kerr radial Teukolsky ODEs.
  Strongly supports the algebraically-special part of the K-505 catalog.
• Full-subextremal frequency-space estimates for spin ±1 and ±2 Teukolsky equations provide uniform separated-ODE estimates in the separation parameters, later upgraded to physical-space boundedness and decay.
  Regime: Subextremal Kerr, spin ±1 and ±2 Teukolsky equations.
  Supplies uniform frequency-space technology relevant to compact-subextremal threshold packaging, though it is not itself a mode-wise zero-frequency catalog.
• Low-energy Fredholm and resolvent analyses for Teukolsky equations give substantial control near zero frequency, including invertibility and mapping estimates in transformed frameworks.
  Regime: Full subextremal Kerr, low-energy Teukolsky resolvent.
  Provides serious evidence for the zero-frequency part, but still needs translation into the explicit separated-ODE Frobenius/connection catalog requested here.

Progress summary: Nonzero real-frequency mode stability and quantitative Wronskian nonvanishing for Teukolsky modes are established in the literature, and algebraically special frequencies are understood as Teukolsky-Starobinsky transform degeneracies rather than outgoing resonances. The remaining open packaging is a complete mode-wise threshold catalog, especially at zero frequency, including stationary/gauge exceptions and uniform compact-subextremal parameter dependence.

What remains open

Prove or cite a single theorem giving the zero-frequency Frobenius/connection catalog for each spin and angular branch, identify all non-radiative stationary/gauge/charge exceptions, translate low-energy Fredholm resolvent results into separated-ODE language, compare Teukolsky and Regge-Wheeler-type normalizations, and package compact-subextremal $(M,a)$ uniformity across all threshold families.

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.
• primary Mode Stability for the Teukolsky Equation on Extremal and Subextremal Kerr Spacetimes — Teixeira da Costa, Rita (2020)
  Proves real-axis and upper-half-plane mode stability, defines the Wronskian criterion, treats algebraically special frequencies, and gives quantitative Wronskian bounds away from threshold exclusions.
• primary Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a|<M: frequency space analysis — Shlapentokh-Rothman, Yakov; Teixeira da Costa, Rita (2020)
  Develops full-subextremal fixed-frequency estimates for separated Teukolsky ODEs, uniform in separation parameters.
• primary Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range |a|<M: physical space analysis — Shlapentokh-Rothman, Yakov; Teixeira da Costa, Rita (2023)
  Upgrades the frequency-space ODE estimates to physical-space boundedness and decay for spin ±1 and ±2 Teukolsky equations on full-subextremal Kerr.
• primary Optimal decay for solutions of the Teukolsky equation on the Kerr metric for the full subextremal range |a| < M — Millet, Pascal (2023)
  Provides low-energy Fredholm/resolvent machinery and Price-law asymptotics; important input for the zero-frequency catalog, but not yet a substitute for the explicit separated-ODE threshold theorem.

Depends on

Conceptual dependencies (not necessarily logical lemmas in a proof assistant).

• K-303 — Quantitative Kerr characterization via the Mars–Simon tensor

Related problems

Related by shared tags

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

• K-504 — Quantitative stability of the photon region and spherical null geodesics under near-Kerr perturbations
• K-503 — Uniqueness questions for Carter-type symmetry operators on Kerr
• 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