A Matrix Method for Quasinormal Modes: Schwarzschild Black Holes in Asymptotically Flat and (Anti-) de Sitter Spacetimes
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 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.
Summary (2 min read)
- 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 .
- 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 .
- 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  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  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.
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