Natural polynomials for Kerr quasi-normal modes

TL;DR

This work introduces canonical confluent Heun polynomials that are both complete and orthogonal to a radial inner product, enabling Teukolsky's radial equation for Kerr QNMs to be represented as a symmetric, tridiagonal matrix eigenproblem. The approach clarifies the radial problem's polynomial/non-polynomial duality, links to confluent Heun and Pollaczek–Jacobi polynomials, and yields accurate numerical eigenvalues and radial functions. The Kerr radial functions can be efficiently represented using Schwarzschild radial bases, suggesting practical simplifications for modeling Kerr perturbations and informing time-dependent Kerr perturbation theory. The results offer a principled framework for understanding Kerr QNMs' radial structure and motivate further exploration of completeness and analytic properties across confluent Heun-type equations.

Abstract

We present a polynomial basis that exactly tridiagonalizes Teukolsky's radial equation for quasi-normal modes. These polynomials naturally emerge from the radial problem, and they are canonical in that they possess key features of classical polynomials. Our canonical polynomials may be constructed using various methods, the simplest of which is the Gram-Schmidt process. In contrast with other polynomial bases, our polynomials allow for Teukolsky's radial equation to be represented as a simple matrix eigenvalue equation. We expect that our polynomials will be useful for better understanding the Kerr quasinormal modes' properties, particularly their prospective spatial completeness and orthogonality. We show that our polynomials are closely related to the confluent Heun and Pollaczek-Jacobi type polynomials. Consequently, our construction of polynomials may be used to tridiagonalize other instances of the confluent Heun equation. We apply our polynomials to a series of simple examples, including: (1) the high accuracy numerical computation of radial eigenvalues, (2) the evaluation and validation of quasinormal mode solutions to Teukolsky's radial equation, and (3) the use of Schwarzschild radial functions to represent those of Kerr. Along the way, a potentially new concept, polynomial/non-polynomial duality, is encountered and applied to show that some quasinormal mode separation constants are well approximated by confluent Heun polynomial eigenvalues. We briefly discuss the implications of our results on various topics, including the prospective spatial completeness of Kerr quasinormal modes.

Figures (7)

• Figure 1: Distributions of radial (black dots) and angular eigenvalues (open circles) for $s=-2$, black hole (BH) dimensionless spin $a/M=0.7$, and the $(\ell,m,n)=(4,1,3)$ Quasi-Normal Mode (QNM) with $M\tilde{\omega}_{413}=0.8545-0.6305i$. Since both eigenvalue distributions have been computed with a Quasi-Normal Mode (QNM) frequency, they have one coincident value corresponding to $(\ell',n')=(\ell,n)=(4,3)$. Radial egenvalues shown have been computed using the results of the current work, and verified using Leaver's method leaver85.
• Figure 2: The first five canonical confluent Heun polynomials, i.e. $u_p(\xi)$ with $p\in\{0,1,2,3,4\}$. Polynomials are constructed using field spin weight $s=-2$, black hole (BH) spin $a/M=0.7$, and Kerr Quasi-Normal Mode (QNM) frequency $M\tilde{\omega}_{220}=0.5326-0.0808i$. For ease of presentation, polynomials have been divided by their value at $\xi=1$ (i.e. spatial infinity). Top panel: Real part of each polynomial. Bottom panel: imaginary part of each polynomial. Related discussion is provided in Sec. \ref{['s6']}.
• Figure 3: Matrices related to canonical confluent Heun polynomials, and representations of Teukolsky's radial operator for $(s,\ell,m,n)=(-2,2,2,6)$, $a/M=0.7$, and $M\tilde{\omega}_{{\ell m n}}=0.4239-1.0954i$. All color bars correspond to the $\log_{10}$ of the absolute values of quantities noted in annotation, e.g. for panel (a), $\log_{10}| {\langle {u_p} \, | \, {u_q} \rangle} |$ is visualized. Tiles (in purple) away from each panel's tridiagonal band are exactly zero. See Sec. \ref{['s6']} for related discussion.
• Figure 4: Example of confluent Heun polynomial/non-polynomial duality for polynomial order $p=6$, and physical parameters $(s,\ell,m,n)=(-2,2,2,6)$, $a/M=0.7$, and $M\tilde{\omega}_{{\ell m n}}=0.4239-1.0954i$. All color bars correspond to the $\log_{10}$ of the absolute values of quantities noted in annotation. Matrix elements of zeo value are colored purple. Panel $(a)$: Matrix representation of the radial operator (Eq. \ref{['p53']}). Panel $(b)$: Matrix representation of the radial operator's eigenvectors (Eq. \ref{['p22a-4']}). In panels $(a)$ and $(b)$, vertical and horizontal white lines delineate polynomial and non-polynomial sectors. Panel $(c)$: The radial operator's polynomial eigenvalues (filled circles) and the first $13$ non-polynomial eigenvalues (open circles).
• Figure 5: Radial eigenvalue distributions for select quasinormal mode cases, all with black hole (BH) dimensionless spin $a/M=0.7$, and angular indices $(\ell,m)=(2,2)$. In all panels, radial eigenvalues, $A_{n'}$, are noted with open circles, and the known Quasi-Normal Mode (QNM) eigenvalue, $A_{\ell m n}$, is shown as a red filled circle (c.f. Fig. \ref{['F1']}). Left to right: Radial eigenvalues for the respective $n \in \{0,3,6,12\}$ Quasi-Normal Mode (QNM) overtones. Respective absolute differences between known Quasi-Normal Mode (QNM) eigenvalues (Refs. positive:2020Stein:2019mop) and those in open circles (i.e. $|A_{\ell m n}-A_{n}|$) are $\{2.72\,\times 10^{-9},4.72\,\times 10^{-9},1.24\,\times 10^{-9},6.28\,\times 10^{-9}\}$. Eigenvalues for the confluent Heun polynomials with order $[\text{Re}\; {p_\star^+} ]$ (Eq. \ref{['near-int']}) are shown by black triangles. Center right ($n=6$): See panel $(c)$ of Fig. \ref{['F4']} for comparison.