Exact Solutions of Teukolsky Master Equation with Continuous Spectrum

Weak gravitational, electromagnetic, neutrino and scalar fields, considered as perturbations on Kerr background satisfy Teukolsky Master Equation. The two non-trivial equations obtained after separating the variables are the polar angle equation and the radial equation. We solve them by transforming each one into the form of a confluent Heun equation. The transformation depends on a set of parameters, which can be chosen in a such a way, so the resulting equations have simple polynomial solutions for neutrino, electromagnetic, and gravitational perturbations, provided some additional conditions are satisfied. Remarkably there exists a class of solutions for which these additional conditions are the same for the two different equations for $|s|=1/2$ and $|s|=1$. As a result the additional conditions fix the dependence of the separation constant on the angular frequency but the frequency itself remains unconstrained and belongs to a continuous spectrum.

💡 Research Summary

The paper addresses the long‑standing problem of finding exact analytic solutions to the Teukolsky Master Equation (TME) for perturbative fields on a rotating (Kerr) black‑hole background. The TME governs weak spin‑s fields (s = 0, ±½, ±1, ±2) and, after separation of variables, reduces to two coupled ordinary differential equations: an angular (θ‑) equation and a radial (r‑) equation. Both equations possess three regular singular points (including the horizon and infinity) and are not solvable in terms of elementary special functions.

The authors’ key methodological step is to map each of these equations onto the confluent Heun equation, a second‑order linear ODE with one irregular and two regular singularities. By introducing a set of transformation parameters (α, β, γ, δ, η) that depend on the physical quantities—spin weight s, Kerr rotation parameter a, azimuthal number m, frequency ω, and the separation constant λ—they rewrite the angular and radial equations in Heun form. This mapping brings the problem into a framework where the solution can be expressed as a power‑series expansion whose coefficients obey a three‑term recurrence relation.

A crucial observation is that the Heun series terminates (i.e., becomes a finite‑degree polynomial) when a specific “termination condition” is satisfied. This condition can be expressed as a relation among the Heun parameters that forces one of the recurrence coefficients to vanish at a finite order N. Physically, such polynomial solutions correspond to modes that are regular on the entire domain (no logarithmic divergences at the horizon or at infinity).

The paper focuses on the cases |s| = ½ (neutrino) and |s| = 1 (electromagnetic) and shows that the same termination condition applies simultaneously to both the angular and radial Heun equations. Explicitly, the condition leads to a functional dependence of the separation constant λ on the frequency ω of the form

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