Matrix method for perturbed black hole metric with discontinuity

Abstract

Recent studies based on the notion of black hole pseudospectrum indicated substantial instability of the fundamental and high-overtone quasinormal modes. Besides its theoretical novelty, the details about the migration of the quasinormal mode spectrum due to specific perturbations may furnish valuable information on the properties of associated gravitational waves in a more realistic context. This work generalizes the matrix method for black hole quasinormal modes to cope with a specific class of perturbations to the metric featured by discontinuity, which is known to be intimately connected with the quasinormal mode structural instability. In practice, the presence of discontinuity poses a difficulty so that many well-known approaches for quasinormal modes cannot be straightforwardly applied. By comparing with other methods, we show that the modified matrix method is efficient, which can be used to solve for the low-lying modes with reasonable precision. Therefore, it might serve as an alternative gadget for relevant studies.

High-order matrix method with delimited expansion domain

“…It is also noted that even though, in principle, equation ( 15) might introduce additional irrelevant roots, it does not pose a serious problem, as long as they stay away from the low-lying quasinormal frequencies. Furthermore, it was proposed [59] that the above algorithm can be adapted for the effective potential possessing discontinuity. To proceed, one assigns the point of discontinuity to one of the grid points x = x c .…”

“…The matrix method [54][55][56][57][58][59] is an approach that reformulates the QNM problem into a matrix equation for the complex frequencies. The approach is reminiscent of the continued fraction method, and their main difference resides in the choice of the grid points where the expansion of the waveform is performed [54].…”

“…It has been generalized to deal with dynamic black hole spacetimes [58]. More recently, the original method was generalized [59] to handle effective potentials containing discontinuity. These features indicate that the method is a promising alternative in the toolkit for black hole QNMs.…”

An implementation of the matrix method using Chebyshev grid

“…Proposed by some of us, the matrix method [26][27][28][29][30][31][32] is an approach that turns the problem of solving the master equation for the quasinormal frequencies into a non-standard matrix eigenvalue problem. Like the well-known continued fraction method [22], Taylor expansion is utilized to rewrite the wave function.…”

“…It can be adapted to different boundary conditions [29] and dynamic black hole spacetimes [30]. More recently, the method was generalized [31] to the potentials containing discontinuity and aimed to the higher orders [32]. The approach provides reasonable accuracy and efficiency and has been utilized in various studies [6,[34][35][36][37][38][39][40][41][42][43][44][45][46].…”