Gravitational perturbations of the Kerr geometry: High-accuracy study

Abstract

We present results from a new code for computing gravitational perturbations of the Kerr geometry. This new code carefully maintains high precision to allow us to obtain high-accuracy solutions for the gravitational quasinormal modes of the Kerr space-time. Part of this new code is an implementation of a spectral method for solving the angular Teukolsky equation that, to our knowledge, has not been used before for determining quasinormal modes. We focus our attention on two main areas. First, we explore the behavior of these quasinormal modes in the extreme limit of Kerr, where the frequency of certain modes approaches accumulation points on the real axis. We compare our results with recent analytic predictions of the behavior of these modes near the accumulation points and find good agreement. Second, we explore the behavior of solutions of modes that approach the special frequency M ω = −2i in the Schwarzschild limit. Our high-accuracy methods allow us to more closely approach the Schwarzschild limit than was possible with previous numerical studies. Unlike previous work, we find excellent agreement with analytic predictions of the behavior near this special frequency. We include a detailed description of our methods, and make use of the theory of confluent Heun differential equations throughout. In particular, we make use of confluent Heun polynomials to help shed some light on the controversy of the existence, or not, of quasinormal and total-transmission modes at certain special frequencies in the Schwarzschild limit.

Post-Kerr black hole spectroscopy

“…As shown in Table I, the absolute error (defined as |y fit − y data |) remains below 0.13 in the interval a/M ∈ [0, 0.9999]. Our results may not be reliable in the near extremal limit (a/M ≈ 1), where the computation of QNMs in computationally challenging and multipole QNM branches exist [24,26,[76][77][78]. The QNM data tables (calculated using Leaver's continued fraction method) used to obtain β K are reliable below the theoretical upper bound on the dimensionless spin of astrophysical BHs (the so-called Thorne limit, a/M ≈ 0.998 [79], so our β K fits should be adequate for astrophysical applications of the present formalism.…”

qnm: A Python package for calculating Kerr quasinormal modes, separation constants, and spherical-spheroidal mixing coefficients

“…where min = max(|m|, |s|), and the coefficients C m (aω) are called the spherical-spheroidal mixing coefficients (we follow the conventions of Cook and Zalutskiy (2014), but compare Berti and Klein (2014)). When recast in this spectral form, the angular equation becomes very easy to solve via standard matrix eigenvector routines, see Cook and Zalutskiy (2014) for details. If one picks values for (s, , m, a, ω), then the separation constant A(aω) is returned as an eigenvalue, and a vector of mixing coefficients C m (aω) are returned as an eigenvector.…”

Scattering of massless bosonic fields by Kerr black holes: On-axis incidence

“…We obtain the spin-weighted spheroidal harmonics and their eigenvalues via spectral decomposition using the description outlined in Refs. [10,14,24,25], in which S sωlm is written as a sum of spin-weighted spherical harmonics,…”