Quasinormal modes of black holes

Gravitational perturbations of the curvature of Kerr black holes can be described by two components of the Weyl tensor: $\Psi_0$ and $\Psi_4$. Because these are components of the curvature tensor, they have dimensions of $[L^{-2}]$. Of particular interest is the quantity $\Psi_4$ since it is this term that is suitable for the study of outgoing waves in the radiative zone. The formalism for the study of perturbations of rotating black holes was developed originally by Teukolsky [30] who was able to separate the differential equation to obtain solutions of the form

$$(r-i\mu a)^4\Psi_4 = e^{-i\omega t} {}_{-2}R_{\ell m}(r) {}_{-2}S_{\ell m}(\mu) e^{im\beta}$$ (8.1.168)

where ${}_{-2}R_{\ell m}(r)$ is a solution to a radial differential equation, and  ${}_{-2}S_{\ell m}(\mu)$ is a spin-weighted spheroidal wave function (see [30], equations (4.9) and (4.10)). The black hole has mass $M$ and specific angular momentum $a=cJ/M$ (which has dimensions of length) where $J$ is the angular momentum of the spinning black hole. We shall often refer to the dimensionless angular momentum parameter, $\hat{a}=c^2a/GM=c^3J/GM^2$. For a Kerr black hole, $\hat{a}$ must be between zero (Schwarzschild limit) and one (extreme Kerr limit). The observer of the perturbation is located at radius $r$, inclination $\mu=\cos\iota$, and azimuth $\beta$ (see figure ). The perturbation itself has the spheroidal eigenvalues $\ell$ and $m$, and has a (complex) frequency $\omega$. The constants $G$ and $c$ are Newton's gravitational constant and the speed of light.

Figure: The polar angle, $\iota$, and the azimuthal angle, $\beta$, of the observer relative to the spin axis of a black hole and the (somewhat arbitrary) axis of perturbation.

The important physical quantities for the study of the gravitational waves arising from black hole perturbations can be recovered from the field $\Psi_4$. In particular, the ``$+$'' and ``$\times$'' polarizations of the strain induced by the gravity waves are found by [30]

$$h_+ - ih_\times = -\frac{2c^2}{\vert\omega\vert^2}\,\Psi_4\;.$$ (8.1.169)

The quantity  $h_+=h_{\hat\iota\hat\iota}$ is the metric perturbation that represents the linear polarization state along ${\mathbf{e}}_{\hat\iota}$ and  ${\mathbf{e}}_{\hat\beta}$, while the quantity  $h_\times=h_{\hat\iota\hat\beta}$ represents the linear polarization state along  ${\mathbf{e}}_{\hat\iota}\pm{\mathbf{e}}_{\hat\beta}$. The power radiated towards the observer (per unit solid angle) is

$$\frac{d^2E}{dt\,d\Omega} = \lim_{r\to\infty} \frac{c^7r^2}{4\pi G\vert\omega\vert^2} \, \vert\Psi_4\vert^2\;.$$ (8.1.170)

Thus, in order to compute the relevant information about gravitational waves emitted as perturbations to rotating black hole spacetimes, one needs to calculate the value of $\Psi_4$ at large radii from the black hole.

The quasinormal modes are resonant modes of the Teukolsky equation that describe purely outgoing radiation in the wave-zone and purely ingoing radiation at the event horizon. The quasi-normal modes are described by a spectrum of complex eigenvalues (which include the spectrum of eigenfrequencies $\omega_n$), and eigenfunctions ${}_{-2}R_{\ell m}(r)$ and  ${}_{-2}S_{\ell m}(\mu)$ for each value spheroidal mode $\ell$ and $m$. These eigenvalues and functions also depend on the mass and angular momentum of the black hole. We shall only consider the fundamental ($n=0$) mode since the harmonics of this mode have shorter lifetimes. For the same reason, we are most interested in the quadrupole ($\ell=2$ and $m=2$) mode. The observer is assumed to be at a large distance; in this case, one can approximate the perturbation as follows:

$$\Psi_4 \approx \frac{A}{r}\,e^{-i\omega t_{\mathrm{\scriptscriptstyle ret}}} {}_{-2}S_{\ell m}(\mu)e^{im\beta}.$$ (8.1.171)

Here $t_{\mathrm{\scriptstyle ret}}=t-r^\star/c$ represents the retarded time, where $r^\star$ is a ``tortoise'' radial parameter. For large radii, the tortoise radius behaves as $r-r_+\log(r/r_+)$ where $r_+$ is the ``radius'' of the black hole event horizon. Thus, we see that the tortoise radius is nearly equal to the distance of the objects surrounding the black hole, and we shall view it as the ``distance to the black hole.'' The parameter $A$ represents the amplitude of the perturbation, which has the dimensions of $[L^{-1}]$.

Given the asymptotic form of the perturbation in equation , we can integrate equation  over the entire sphere and the interval  $t_{\mathrm{\scriptstyle ret}}\in[0,\infty)$ to obtain an expression for the total energy radiated in terms of the amplitude $A$ of the perturbation. Thus, we can characterize the amplitude by the total amount of energy emitted: $A^2=4Gc^{-7}E\vert\omega\vert^2(-{\mathrm{Im}}\,\omega)$. The gravitational waveform is found to be

$$h_+ - ih_\times \approx - \frac{4c}{r} \biggl(\frac{-{\mat... ...\scriptscriptstyle ret}}} {}_{-2}S_{\ell m}(\mu) e^{im\beta}.$$ (8.1.172)

In order to simulate the quasinormal ringing of a black hole, it is necessary to determine the complex eigenvalues of the desired mode, and then to compute the spheroidal wave function  $S_{\ell m}(\mu)$. The routines to perform these computations are discussed in the following sections.

Rather than computing the actual gravitational strain waveforms at the detector, the routines will calculate the quantity $H_+-iH_\times=(c^2r/GM_\odot)(h_+-ih_\times)$; the normalization of these waveforms to the correct source distance is left to the calling routine. The distance normalization can be computed as follows:

$$\frac{c^2r}{GM_\odot} = \frac{r}{T_\odot c} = \biggl( \fra... ...) = 2.090\times10^{19}\biggl( \frac{r}{\mathrm{Mpc}} \biggr).$$ (8.1.173)

where $T_\odot=4.925491\,\mu{\mathrm{s}}$ is the mass of the sun expressed in seconds (see equation ). It will be convenient to write the time dependence of the strain as the complex function  ${\mathcal{H}}(t_{\mathrm{\scriptstyle ret}})$ so that $H_+-iH_\times={\mathcal{H}}(t_{\mathrm{\scriptstyle ret}}) {}_{-2}S_{\ell m}(\mu)e^{im\beta}$. The dimensionless eigenfrequency, $\hat{\omega}=GM\omega/c^3$, depends only on the mode and the dimensionless angular momentum of the black hole. In terms of this quantity, the function  ${\mathcal{H}}(t_{\mathrm{\scriptstyle ret}})$ is

$${\mathcal{H}}(t_{\mathrm{\scriptstyle ret}}) \approx -4 \ep... ...}{T_\odot}\biggr) \biggl(\frac{M}{M_\odot}\biggr)^{-1}\biggr]$$ (8.1.174)

where $\epsilon$ is the fractional mass loss due to the radiation in the excited quasinormal mode.