Properties of natural polynomials for Schwarzschild and Kerr black holes

TL;DR

This paper explores the mathematical properties of polynomials associated with black hole quasi-normal modes, revealing their structure and potential for improved understanding of gravitational wave signals.

Contribution

It introduces a class of natural polynomials for Schwarzschild and Kerr black holes, characterizing their recurrence relations, differential equations, and identifying them as Pollaczek-Jacobi polynomials.

Findings

Polynomials are identified as Pollaczek-Jacobi with complex parameters.

Recurrence relation peaks at the physical overtone index for Schwarzschild.

Provides analytic properties like ladder operators and differential equations.

Abstract

The quasi-normal modes of black holes play various important roles in gravitational wave theory, signal modeling, and data analysis; however, there remain open questions about their mathematical properties. Aspects of classical polynomial theory have been proposed as a framework to investigate quasinormal mode orthogonality and completeness. We have recently presented a class of polynomials that are "natural" to quasi-normal modes in that they are restricted by the quasi-normal mode boundary conditions, and exactly tridiagonalize Teukolsky's radial equation. In turn, these polynomials may be useful for better understanding the vector space properties of quasi-normal mode solutions to that equation. Here, we provide an overview of these polynomials' analytic properties: their 3-term recurrence relation, ladder operators and governing differential equation. We demonstrate that the natural polynomials for Schwarzschild and Kerr black holes are Pollaczek-Jacobi polynomials with complex valued parameters. Along the way, we observe a novel property that is particular to Schwarzschild: the polynomials' 3-term recurrence relation always peaks at the physical overtone index. This work supports the broader application of these polynomials, as well as their extension to black hole spacetimes beyond Schwarzschild and Kerr.