Quasinormal modes of the Schwarzschild black hole

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Numbers

$s$

$\ell$

$n$

$M\omega_{s\ell n}$

0.3736716844 + i * -0.08896231569

0.5994432884 + i * -0.09270304794

0.8091783775 + i * -0.09416396099

1.012295312 + i * -0.09487051608

1.212009821 + i * -0.09526584584

0.1104549391 + i * -0.1048957171

0.2929361333 + i * -0.09765998891

0.4836438722 + i * -0.09675877598

0.6753662325 + i * -0.09649962773

0.8674156417 + i * -0.09639169235

0.2482632642 + i * -0.09248771795

0.4575955116 + i * -0.09500442582

0.6568986705 + i * -0.09561621793

0.8530951930 + i * -0.09585993483

1.047912782 + i * -0.09598167203

Definition

For a Schwarzschild black hole of mass $M$, this table gives the dimensionless quasinormal-mode frequencies $M\omega_{s\ell n}$: the values of $\omega$ for which the spin-$s$ Regge-Wheeler equation has a solution that is purely ingoing at the horizon and purely outgoing at infinity [1].

Parameters

$s$

— spin of the perturbing field ($s\in\{0,1,2\}$)

$\ell$

— spherical-harmonic degree ($\ell\geq s$)

$n$

— overtone number ($n\geq0$)

Formulas

(1)

As $\ell\to\infty$ with $s$ and $n$ fixed, $\sqrt{27}\,M\omega_{s\ell n}=\ell+\tfrac12-i(n+\tfrac12)+O(\ell^{-1})$, independently of $s$ [2].

Comments

(2)

Perturbations use the time dependence $\exp(-i\omega t)$, so a damped mode has $\operatorname{Im}(\omega)<0$. The values $s=0,1,2$ correspond respectively to scalar, electromagnetic and gravitational perturbations. For fixed $s$ and $\ell$, the modes are ordered by increasing $|\operatorname{Im}\omega|$; $n=0$ is the least-damped mode.

(3)

The stored value is the dimensionless product $M\omega$ in geometric units $G=c=1$. Berti's Schwarzschild tables [7] give $2M\omega_R$ and $2M\omega_I$ with $\omega_I>0$, so a value read from there is twice this one and its imaginary part has the opposite sign.

(4)

For $s=2$, the polar Zerilli equation is isospectral to the axial Regge-Wheeler equation, so the table serves both. The qnm package [3] indexes these modes by the Teukolsky spin weight $-s$, so the gravitational modes of this table are its $s=-2$.

(5)

The Schwarzschild spectrum is independent of the azimuthal number $m$, so $m$ is not a parameter of this table.

Programs

(P1)

Python

from qnm.schwarzschild.tabulated import QNMDict
schwarzschild = QNMDict(init=True)
schwarzschild(-2, 2, 0)[0]

References

[1]

E. W. Leaver, An analytic representation for the quasi-normal modes of Kerr black holes, Proceedings of the Royal Society of London. Series A 402 (1985), 285-298. (doi)

[2]

Emanuele Berti, Vitor Cardoso and Andrei O. Starinets, Quasinormal modes of black holes and black branes, Classical and Quantum Gravity 26 (2009), 163001. (arXiv) (doi)

[3]

Leo C. Stein, qnm: A Python package for calculating Kerr quasinormal modes, separation constants, and spherical-spheroidal mixing coefficients, Journal of Open Source Software 4 (2019), 1683. (arXiv) (doi)

[4]

Gregory B. Cook and Maxim Zalutskiy, Gravitational perturbations of the Kerr geometry: High-accuracy study, Physical Review D 90 (2014), 124021. (arXiv) (doi)

[5]

Sam R. Dolan and Adrian C. Ottewill, On an expansion method for black hole quasinormal modes and Regge poles, Classical and Quantum Gravity 26 (2009), 225003. (arXiv) (doi)

Links

[6]

Black Hole Perturbation Toolkit: qnm

[7]

Emanuele Berti: Ringdown, quasinormal modes and light rings

Data properties

Entries are of type: complex number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (it holds $s=2$ for $2\leq\ell\leq6$, $s=1$ for $1\leq\ell\leq5$ and $s=0$ for $0\leq\ell\leq4$, in every case only the fundamental mode $n=0$)

How they were obtained:

The generator solves Leaver's continued-fraction equation [1] by Newton iteration on the real and imaginary parts. The right infinite tail is evaluated by capped modified Lentz iteration as in [4], and each stored entry is the complex rectangle spanned by two computations.

Most rows use 50 decimal working digits with tail tolerance $10^{-10}$ and 70 decimal working digits with tail tolerance $10^{-12}$. The scalar monopole row uses 60 decimal working digits with tail tolerance $10^{-10}$ and 80 decimal working digits with tail tolerance $10^{-12}$ because its continued-fraction tail converges more slowly. The generator uses the Dolan-Ottewill expansion [5] only for initial guesses, compares every stored row with the Black Hole Perturbation Toolkit qnm cache [6] [3], and compares the diagonal $s=\ell=2$ and $s=\ell=1$ rows with Berti's tabulation [7]. The continued-fraction residuals are used as numerical diagnostics rather than certified error bounds, so the table does not claim proven values.