Journal ArticleDOI

# A Matrix Method for Quasinormal Modes: Schwarzschild Black Holes in Asymptotically Flat and (Anti-) de Sitter Spacetimes

, Kai Lin2

AbstractIn this work, we study the quasinormal modes of Schwarzschild and Schwarzschild (Anti-) de Sitter black holes by a matrix method. The proposed method involves discretizing the master field equation and expressing it in form of a homogeneous system of linear algebraic equations. The resulting homogeneous matrix equation furnishes a non-standard eigenvalue problem, which can then be solved numerically to obtain the quasinormal frequencies. A key feature of the present approach is that the discretization of the wave function and its derivatives are made to be independent of any specific metric through coordinate transformation. In many cases, it can be carried out beforehand which in turn improves the efficiency and facilitates the numerical implementation. We also analyze the precision and efficiency of the present method as well as compare the results to those obtained by different approaches.

Topics: Schwarzschild radius (59%), Anti-de Sitter space (55%), Matrix method (53%), De Sitter universe (52%)

### 1. Introduction

• The second stage consists of a long period dominated by the quasinormal oscillations, where the amplitude of the oscillation decays exponentially in time.
• Mathematically, the QNMs are governed by the linearized equations of general relativity constraining perturbations around a black hole solution.
• Due to the difficulty in finding exact solutions to most problems of interest, various approximate methods have been proposed [8].
• In their approach, the master field equation is presented in terms of linear equations describing 3 N discretized points where the wave function is expanded up to Nth order for each of these points.

### 2. Matrix method and the eigenvalue problem for quasinormal modes

• Recently, the authors proposed a non-grid-based interpolation scheme which can be used to solve the eigenvalue problem [1].
• Based on the information about N scattered data point, Taylor series are carried out for the unknown eigenfunction up to Nth order for each discretized point.
• It was shown [1] that the boundary conditions can be implemented by properly replacing some of the above equations.
• In the case of asymptotically flat Schwarzschild spacetime below, the authors choose to replace the first and the last line in the matrix equation and implement the boundary condition by replacing equation (3.11) with equation (3.13).
• Now the authors apply the above method to investigate the master field equation of for QNM.

### 4. Quasinormal modes in Schwarzschild de Sitter black hole spacetime

• Now the authors are in the position to utilize the same numerical procedure to discretize the wave function in the interval ⩽ ⩽y0 1, and solve for the quasinormal frequencies.
• The obtained the quasinormal frequencies are presented in table 2 compared to those obtained by the WKB method.
• It is found that the results from the present method are consistent with those from the WKB method.

### 5. Quasinormal modes in in Schwarzschild anti-de Sitter black hole spacetime

• 10 Again, the eigenequation can be obtained following the same procedures as before.
• The calculated quasinormal frequencies are shown in table 3 and compared with the results obtained by the HH method.
• It is inferred from the results that the present method is in accordance with the HH method.

### 6. Discussions and outlooks

• The authors proposed a new interpolation scheme to discretize the master field equation  for the scalar quasinormal modes.
• It possesses flexibility and therefore the potential to explore some black hole metrics where the applications of other traditional methods become less straightforward.
• As an example, the quasinormal modes of a rotational black hole is characterized by, besides the quasinormal frequency ω, a second eigenvalue λ whose physical content is associated with the angular quantum number L. Its numerical solution, therefore, involves finding the two eigenvalues, ω and λ, simultaneously.
• A preliminary attempt [15] shows that the approach proposed in this work, on the other hand, can be applied straightforwardly in a more intuitive fashion.
• This procedure is identical for most calculations once the grid points are fixed, therefore the efficiency of the method can be significantly improved for such similar problems.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

Classical and Quantum Gravity
PAPER
A matrix method for quasinormal modes:
Schwarzschild black holes in asymptotically flat
and (anti-) de Sitter spacetimes
To cite this article: Kai Lin and Wei-Liang Qian 2017 Class. Quantum Grav. 34 095004
View the article online for updates and enhancements.
Related content
Black hole quasinormal modes using the
asymptotic iteration method
H T Cho, A S Cornell, Jason Doukas et al.
-
Fermion perturbations in string theory
black holes
Owen Pavel Fernández Piedra and
Jeferson de Oliveira
-
Quasinormal Modes of a Noncommutative-
Geometry-Inspired Schwarzschild Black
Hole
Jun Liang
-
Recent citations
Charged scalar fields around Einstein-
power-Maxwell black holes
Grigoris Panotopoulos
-
The matrix method for black hole
quasinormal modes
Kai Lin and Wei-Liang Qian
-
Quasinormal modes of the BTZ black hole
under scalar perturbations with a non-
minimal coupling: exact spectrum
Grigoris Panotopoulos
-

1
Classical and Quantum Gravity
A matrix method for quasinormal
modes: Schwarzschild black holes
inasymptotically at and (anti-)
de Sitter spacetimes
KaiLin
1
,
2
and Wei-LiangQian
2
,
3
1
Universidade Federal de Itajubá, Instituto de Física e Química, Itajubá, MG, Brazil
2
Escola de Engenharia de Lorena, Universidade de São Paulo, Lorena, SP, Brazil
3
Guaratinguetá, SP, Brazil
E-mail: lk314159@hotmail.com and wlqian@usp.br
Received 11 January 2017, revised 1 March 2017
Accepted for publication 13 March 2017
Published 31 March 2017
Abstract
In this work, we study the quasinormal modes of Schwarzschild and
Schwarzschild (Anti-) de Sitter black holes by a matrix method. The proposed
method involves discretizing the master eld equationand expressing it in the
form of a homogeneous system of linear algebraic equations. The resulting
homogeneous matrix equationfurnishes a non-standard eigenvalue problem,
which can then be solved numerically to obtain the quasinormal frequencies.
A key feature of the present approach is that the discretization of the wave
function and its derivatives is made to be independent of any specic metric
through coordinate transformation. In many cases, it can be carried out
beforehand, which in turn improves the efciency and facilitates the numerical
implementation. We also analyze the precision and efciency of the present
method as well as compare the results to those obtained by different approaches.
Keywords: quasinormal modes, black hole, Schwarzschild spacetime,
deSitter spacetime
(Some guresmay appear in colour only in the online journal)
1. Introduction
A black hole, long considered to be a physical as well as mathematical curiosity, is derived
in general relativity as a generic prediction. Through gravitational collapse, a stellar-mass
black hole can be formed at the end of the life cycle of a very massive star, when its gravity
K Lin and W-L Qian
A matrix method for quasinormal modes: Schwarzschild black holes in asymptotically ﬂat and (anti-) de Sitter spacetimes
Printed in the UK
095004
CQGRDG
34
Class. Quantum Grav.
CQG
1361-6382
10.1088/1361-6382/aa6643
Paper
9
1
13
Classical and Quantum Gravity
IOP
2017

8,011 citations