### Emanuele Berti

Associate Professor, University of Mississippi

# Ringdown

The oscillation modes of black holes are called “quasinormal modes” (QNMs; “quasi” because they are damped by the emission of gravitational radiation). These quantities are of interest in gravitational-wave astronomy, to test the so-called “no-hair theorem” of general relativity, and in the context of the gauge-gravity duality. Here you can find data for:

• the QNM spectrum of Schwarzschild (anti-de Sitter) and Kerr black holes.
• the excitation factors of Kerr quasinormal modes.
• the spherical-spheroidal mixing coefficients between the angular functions that correspond to each mode (spin-weighted spheroidal harmonics) and the spin-weighted spherical harmonics normally used to decompose gravitational radiation.

You can also find Mathematica notebooks to compute some of these quantities. The data and routines are freely available, but please reference the original work(s). Important caveat: the calculations of Kerr QNM frequencies presented here are unreliable very close to the Kerr extremal limit (roughly, when $a/M\geq 0.999$). The near-extremal regime is discussed in arXiv:1212.3271 and arXiv:1307.8086; see also arXiv:1410.7698.

I would like to thank Oscar Dias, Mahdi Godazgar and Jorge Santos for providing Kerr-Newman quasinormal mode data from arXiv:1501.04625.

Schwarzschild QNM frequencies arXiv:0905.2975 $s=\ell=2$
Data format: gr-qc/0512160 $s=\ell=1$
$2M\omega_{\rm R}, 2M\omega_{\rm I}$, error, $n$   $s=\ell=0$
Kerr QNM frequencies ($s=-2$) arXiv:0905.2975 $\ell=2$
Data format: gr-qc/0512160 $\ell=3$
$a/M, M\omega_{\rm R}, M\omega_{\rm I}, {\rm Re}(A_{\ell m}), {\rm Im}(A_{\ell m})$   $\ell=4$
$\ell=5$
$\ell=6$
$\ell=7$
Kerr QNM frequencies ($s=-1$) arXiv:0905.2975 $\ell=1$
Data format: gr-qc/0512160 $\ell=2$
$a/M, M\omega_{\rm R}, M\omega_{\rm I}, {\rm Re}(A_{\ell m}), {\rm Im}(A_{\ell m})$   $\ell=3$
$\ell=4$
$\ell=5$
$\ell=6$
$\ell=7$
Kerr QNM frequencies ($s=0$) arXiv:0905.2975 $\ell=0$
Data format: gr-qc/0512160 $\ell=1$
$a/M, M\omega_{\rm R}, M\omega_{\rm I}, {\rm Re}(A_{\ell m}), {\rm Im}(A_{\ell m})$   $\ell=2$
$\ell=3$
$\ell=4$
$\ell=5$
$\ell=6$
$\ell=7$
Fits to Kerr QNM frequencies arXiv:0905.2975 dat
$\ell, m, n, f_1, f_2, f_3, q_1, q_2, q_3$ gr-qc/0512160
$M\omega_{\rm R}=f_1+f_2(1-a/M)^{f_3}$
$Q=q_1+q_2(1-a/M)^{q_3}$
Calculation of Kerr QNMs arXiv:0905.2975 Notebook
using Leaver’s method gr-qc/0512160
Calculation of Schwarzschild-AdS QNMs arXiv:0905.2975 Notebook
using power-series methods
Proca field on a Kerr background arXiv:1209.0465 Notebook
at second order in a slow-rotation expansion arXiv:1209.0773
Gravito-EM Kerr-Newman perturbations arXiv:1304.1160 Notebook
at first order in a slow-rotation expansion arXiv:1304.1160
Kerr-Newman gravitational QNMs arXiv:1501.04625 $\ell=m=2$
Data format: ($Q=a$)
$a/M=Q/M, M\omega_{\rm R}, M\omega_{\rm I}$
Kerr excitation factors arXiv:1305.4306 Notebook
Data format:   $s=0$
$a/M, M\omega_{\rm R}, M\omega_{\rm I}, {\rm Re}(A_{\ell m}), {\rm Im}(A_{\ell m})$   $s=1$
${\rm Re}[B_{\rm Teuk}], {\rm Im}[B_{\rm Teuk}], {\rm Re}[B_{\rm SN}], {\rm Im}[B_{\rm SN}]$   $s=2$
Spherical-spheroidal mixing coefficients arXiv:1408.1860 $\ell=2$
Gravitational case ($s=-2$)   $\ell=3$
Data format:   $\ell=4$
$a/M, {\rm Re}[\mu_{m\ell \ell' n'}], {\rm Im}[\mu_{m\ell \ell' n'}]$   $\ell=5$
$\ell=6$
$\ell=7$
Fits
Spherical-spheroidal mixing coefficients arXiv:1408.1860 $\ell=1$
Electromagnetic case ($s=-1$)   $\ell=2$
Data format:   $\ell=3$
$a/M, {\rm Re}[\mu_{m\ell \ell' n'}], {\rm Im}[\mu_{m\ell \ell' n'}]$   $\ell=4$
$\ell=5$
$\ell=6$
$\ell=7$
Spherical-spheroidal mixing coefficients arXiv:1408.1860 $\ell=0$
Scalar case ($s=0$)   $\ell=1$
Data format:   $\ell=2$
$a/M, {\rm Re}[\mu_{m\ell \ell' n'}], {\rm Im}[\mu_{m\ell \ell' n'}]$   $\ell=3$
$\ell=4$
$\ell=5$
$\ell=6$
$\ell=7$
Spheroidal-spheroidal mixing coefficients arXiv:1408.1860 $\ell=2$
Gravitational case ($s=-2$)   $\ell=3$
Data format:   $\ell=4$
$a/M, {\rm Re}[\alpha_{m\ell \ell' n n'}], {\rm Im}[\alpha_{m\ell \ell' n n'}]$   $\ell=5$
$\ell=6$
$\ell=7$
Fits