If we set in Equation (6), with appropriate change of the source term, it becomes the perturbation equation for . Moreover, it describes the perturbation for a scalar field (), a neutrino field (), and an electromagnetic field () as well.

We decompose into the Fourier-harmonic components according to

Update The radial function and the angular function satisfy the Teukolsky equations with as The potential is given by where is the eigenvalue of and . The angular function is called the spin-weighted spheroidal harmonic, which is usually normalized as In the Schwarzschild limit, it reduces to the spin-weighted spherical harmonic with . In the Kerr case, however, no analytic formula for is known. The source term is given by We mention that for orbits of our interest, which are bounded, has support only in a compact range of .We solve the radial Teukolsky equation by using the Green function method. For this purpose, we define two kinds of homogeneous solutions of the radial Teukolsky equation:

where , and is the tortoise coordinate defined by where , and where we have fixed the integration constant.Combining with the Fourier mode , we see that has no outcoming wave from past horizon, while has no incoming wave at past infinity. Since these are the properties of waves causally generated by a source, a solution of the Teukolsky equation which has purely outgoing property at infinity and has purely ingoing property at the horizon is given by

where the Wronskian is given by Then, the asymptotic behavior at the horizon is while the asymptotic behavior at infinity isWe note that the homogeneous Teukolsky equation is invariant under the complex conjugation followed by the transformation and . Thus, we can set , where the bar denotes the complex conjugation.

We consider of a monopole particle of mass . The energy momentum tensor takes the form

where is a geodesic trajectory, and is the proper time along the geodesic. The geodesic equations in the Kerr geometry are given by Update where and , , and are the energy, the -component of the angular momentum, and the Carter constant of a test particle, respectively. These constants of motion are those measured in units of . That is, if expressed in the standard units, they become , , and .Using Equation (27), the tetrad components of the energy momentum tensor are expressed as

where and . Substituting Equation (10) into Equation (18) and performing integration by part, we obtain where and denotes for simplicity.For a source bounded in a finite range of , it is convenient to rewrite Equation (33) further as

where Inserting Equation (36) into Equation (25), we obtain as whereIn this paper, we focus on orbits which are either circular (with or without inclination) or eccentric but confined on the equatorial plane. In either case, the frequency spectrum of becomes discrete. Accordingly, in Equation (24) or (25) takes the form,

Then, in particular, at is obtained from Equation (13) as At infinity, is related to the two independent modes of gravitational waves and as From Equations (46) and (47), the luminosity averaged over , where is the characteristic time scale of the orbital motion (e.g., a period between the two consecutive apastrons), is given by In the same way, the time-averaged angular momentum flux is given byhttp://www.livingreviews.org/lrr-2003-6 |
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