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## Quasinormal modes of black holes

Gravitational perturbations of the curvature of Kerr black holes can be described by two components of the Weyl tensor: and . Because these are components of the curvature tensor, they have dimensions of . Of particular interest is the quantity  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
 (8.1.168)

where is a solution to a radial differential equation, and  is a spin-weighted spheroidal wave function (see [30], equations (4.9) and (4.10)). The black hole has mass  and specific angular momentum  (which has dimensions of length) where is the angular momentum of the spinning black hole. We shall often refer to the dimensionless angular momentum parameter, . For a Kerr black hole, must be between zero (Schwarzschild limit) and one (extreme Kerr limit). The observer of the perturbation is located at radius , inclination , and azimuth  (see figure ). The perturbation itself has the spheroidal eigenvalues and , and has a (complex) frequency . The constants and  are Newton's gravitational constant and the speed of light.

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

 (8.1.169)

The quantity  is the metric perturbation that represents the linear polarization state along and  , while the quantity  represents the linear polarization state along  . The power radiated towards the observer (per unit solid angle) is
 (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  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 ), and eigenfunctions and  for each value spheroidal mode and . These eigenvalues and functions also depend on the mass and angular momentum of the black hole. We shall only consider the fundamental () mode since the harmonics of this mode have shorter lifetimes. For the same reason, we are most interested in the quadrupole ( and ) mode. The observer is assumed to be at a large distance; in this case, one can approximate the perturbation as follows:

 (8.1.171)

Here represents the retarded time, where  is a tortoise'' radial parameter. For large radii, the tortoise radius behaves as where  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  represents the amplitude of the perturbation, which has the dimensions of .

Given the asymptotic form of the perturbation in equation , we can integrate equation  over the entire sphere and the interval  to obtain an expression for the total energy radiated in terms of the amplitude  of the perturbation. Thus, we can characterize the amplitude by the total amount of energy emitted: . The gravitational waveform is found to be

 (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  . 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 ; the normalization of these waveforms to the correct source distance is left to the calling routine. The distance normalization can be computed as follows:

 (8.1.173)

where 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  so that . The dimensionless eigenfrequency, , depends only on the mode and the dimensionless angular momentum of the black hole. In terms of this quantity, the function  is
 (8.1.174)

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

Next: Function: qn_eigenvalues() Up: GRASP Routines: Black hole Previous: GRASP Routines: Black hole   Contents
Bruce Allen 2000-11-19