[Paper Review] Matrix method for perturbed black hole metric with discontinuity

Shuifa Shen, Wei‐Liang Qian+3 more|arXiv (Cornell University)|Mar 27, 2022

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper proposes a modified matrix method to compute quasinormal modes (QNMs) of black holes with discontinuous effective potentials, a class of perturbations known to induce structural instability in the QNM spectrum. By adapting the matrix method to handle discontinuities through boundary condition enforcement at the horizon and spatial boundaries, the approach achieves efficient and precise computation of low-lying modes, offering a robust alternative to traditional methods like the continued fraction method.

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.

Motivation & Objective

• To address the challenge of computing quasinormal modes (QNMs) in black hole metrics featuring discontinuous effective potentials, which disrupt standard numerical methods.
• To generalize the matrix method for QNMs to handle discontinuities arising from physical scenarios such as thin shells, dark matter halos, or exotic compact objects.
• To demonstrate that the modified matrix method maintains high precision and computational efficiency even in the presence of discontinuities.
• To provide a reliable alternative to the continued fraction method, which often fails under discontinuous potential conditions.
• To support future studies on QNM structural instability, black hole echoes, and GW spectroscopy in realistic astrophysical environments.

Proposed method

• The matrix method is adapted by modifying the first and last rows of the system matrix to enforce ingoing wave boundary conditions at the black hole horizon and spatial boundary.
• The effective potential is discretized on a non-uniform grid, and the Schrödinger-like wave equation is transformed into a generalized eigenvalue problem.
• The wave function is expressed in terms of a transformed variable to preserve the ingoing condition at the horizon, ensuring correct physical boundary behavior.
• The method uses a collocation approach with Chebyshev spectral methods to achieve high accuracy, particularly for low-lying modes.
• The implementation is validated by comparing results with other established methods, including the continued fraction method, for benchmark cases.
• The code is made publicly available as supplemental material on arXiv, enabling reproducibility and extension to other systems.

Experimental results

Research questions

• RQ1 How can quasinormal modes be accurately computed in black hole metrics with discontinuous effective potentials, a scenario that invalidates standard perturbative and iterative methods?
• RQ2 What modifications to the matrix method are necessary to preserve numerical stability and accuracy when discontinuities are present in the potential?
• RQ3 How does the performance and precision of the modified matrix method compare to the continued fraction method in the presence of discontinuities?
• RQ4 To what extent do discontinuities in the effective potential lead to structural instability in the quasinormal mode spectrum?
• RQ5 Can the modified matrix method serve as a reliable computational tool for studying QNMs in realistic astrophysical environments with thin shells or matter discontinuities?

Key findings

• The modified matrix method successfully computes low-lying quasinormal modes with high precision, even when the effective potential contains discontinuities.
• The method achieves convergence and stability in a few seconds per mode, demonstrating computational efficiency for practical applications.
• The results from the modified matrix method show excellent agreement with those from the continued fraction method in benchmark cases, validating its accuracy.
• The method is robust across different potential forms and boundary conditions, indicating broad applicability to various physical scenarios involving discontinuities.
• The presence of discontinuities leads to structural instability in the high-overtone QNM spectrum, with modes stretching along the real axis, consistent with prior theoretical expectations.
• The approach provides a viable alternative to the continued fraction method, especially where the latter fails due to divergent recurrence relations under discontinuous potentials.