Spectral method for the gravitational perturbations of black holes: Schwarzschild background case

Adrian Ka-Wai Chung, Pratik Wagle, and Nicolás Yunes

Phys. Rev. D 107, 124032 – Published 15 June 2023

Abstract

We develop a novel technique through spectral decompositions to study the gravitational perturbations of a black hole, without needing to decouple the linearized field equations into master equations and separate their radial and angular dependence. We first spectrally decompose the metric perturbation in a Legendre and Chebyshev basis for the angular and radial sectors respectively, using input from the asymptotic behavior of the perturbation at spatial infinity and at the black hole event horizon. This spectral decomposition allows us to then transform the linearized Einstein equations (a coupled set of partial differential equations) into a linear matrix equation. By solving the linear matrix equation for its generalized eigenvalues, we can estimate the complex quasinormal frequencies of the fundamental mode and various overtones of the gravitational perturbations simultaneously and to high accuracy. We apply this technique to perturbations of a nonspinning, Schwarzschild black hole in general relativity and find the complex quasinormal frequencies of two fundamental modes and their first two overtones. We demonstrate that the technique is robust and accurate, in the Schwarzschild case leading to relative fractional errors of $\leq 10^{- 10} - 10^{- 8}$ for the fundamental modes, $\leq 10^{- 7} - 10^{- 6}$ for their first overtones, $\leq 10^{- 7} - 10^{- 4}$ for their second overtones. This method can be applied to any black hole spacetime, irrespective of its Petrov type, making the numerical technique extremely powerful in the study of black hole ringdown in and outside general relativity.

Received 1 March 2023
Accepted 10 May 2023

DOI:https://doi.org/10.1103/PhysRevD.107.124032

Physics Subject Headings (PhySH)

Classical black holes

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Adrian Ka-Wai Chung ^*, Pratik Wagle ^†, and Nicolás Yunes

Illinois Center for Advanced Studies of the Universe and Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

^*akwchung@illinois.edu
^†wagle2@illinois.edu

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Issue

Vol. 107, Iss. 12 — 15 June 2023

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Images

Figure 1
Schematic illustration of the structure of the system of ordinary differential equations obtained by spectral decomposition of the linearized Einstein field equations. The vector $y$ is related to the amplitude of metric perturbations at a given angular position. The matrix on the left-hand side is the coefficient matrix $Q (r)$ of the $r$ derivatives of the system of ordinary differential equations. The null column (red rectangle) indicates the existence of algebraic variables, which are those whose $r$ derivatives are not contained in the differential equations. The null row (blue rectangle) indicates the existence of algebraic equations in different components of $y$ (green rectangle).
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Figure 2
The distribution of the eigenvalues in the complex plane for $4 \leq N \leq 25$ , where $N$ is the number of spectral functions used in the spectral decomposition (see Sec. 4b). Observe that the eigenvalues of the system of equations start to group together at certain points in the complex plane as $N$ is increased. For comparison, we have also shown the corresponding QNM frequency calculated with Leaver’s method [101], using black crosses. The labels near the crosses follow a $(n, l, m)$ notation, where $n$ is the principal mode number, $l$ is the azimuthal mode number and $m$ is the magnetic mode number. Since the QNM frequencies of the Schwarzschild BH do not depend on $m$ , we have left this quantity unspecified in the labels. An animated version of these plots is available in the Supplemental Material [143].
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Figure 3
The top, left panel shows the absolute difference between the QNM frequencies computed with adjacent $N$ s, $D (N) = | ω (N + 1) - ω (N) |$ , with the threshold $10^{- 3}$ denoted by the horizontal solid black line. The $N$ that minimizes $D (N)$ , $N_{opt}$ corresponds to the optimal truncation order and selects the optimal approximation $ω (N_{opt})$ to the QNM frequency (circled symbol). To gauge the accuracy of the spectral method, we compare the QNM frequencies computed using this spectral method at various $N$ [ $ω (spectral)$ ] to those computed through Leaver’s method [101] [ $ω (L)$ ]. The top, right panel shows the absolute error $E (N) = | ω (spectral) - ω (L) |$ as a function of $N$ , while the bottom panels show the relative fractional error in the real [ $Δ^{Re} = | 1 - ω^{Re} (spectral) / ω^{Re} (L) |$ ] and imaginary [ $Δ^{Im} = | 1 - ω^{Im} (spectral) / ω^{Im} (L) |$ right panel] parts. Observe that the QNM frequencies calculated with the spectral method are highly accurate for the fundamental mode and its overtones.
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Figure 4
Base-10 logarithm of the relative fractional differences between the real (left) and imaginary parts (right) of the QNM frequencies when setting $m = 0$ and $m = 2$ , both computed using our spectral method of at most $25 \times 25$ spectral functions. The relative fractional difference for different QNMs is between $10^{- 10}$ and $10^{- 4}$ , which is smaller than, or at worst approximately equal to, the numerical uncertainty of the $m = 2$ frequencies (green squares). Thus, effectively, the QNM frequencies computed by setting $m$ to different values in our spectral method are the same. Such $m$ independence of our results is a nontrivial verification of the correctness and robustness of the spectral method.
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Figure 5
Numerical uncertainty of the real (left) and imaginary parts (right) of different QNM frequencies [ $δ^{Re / Im}$ , see Eq. (67)] computed using the spectral method with at most $25 \times 25$ spectral functions and assuming different $ρ_{H}^{(i)}$ and $ρ_{\infty}^{(i)}$ boundary conditions. Observe that the accuracy of our series solution, computed with different boundary conditions, is approximately the same, indicating the robustness of the spectral method.
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Figure 6
Numerical uncertainty in the real (left) and imaginary parts (right) of different QNM frequencies by spectrally decomposing different sets of components of the linearized Einstein equations. In all cases, we use at most $25 \times 25$ spectral functions when computed the QNM frequencies. Observe that the accuracy of the QNM frequencies calculated is approximately independent of the choice of components of the linearized Einstein equations that we choose to solve.
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