The detection of gravitational waves will be done by extracting gravitational wave signals from a noisy data stream. In developing the data analysis strategy, detailed knowledge of the gravitational waveforms will help us greatly to detect a signal, and to extract the physical information about its source. Thus, it has become a very important problem for theorists to predict with sufficiently good accuracy the waveforms from possible gravitational wave sources.

Gravitational waves are generated by dynamical astrophysical events, and they are expected to be strong enough to be detected when compact stars such as neutron stars (NS) or black holes (BH) are involved in such events. In particular, coalescing compact binaries are considered to be the most promising sources of gravitational radiation that can be detected by the ground-based laser interferometers. The last inspiral phase of a coalescing compact binary, in which the binary stars orbit each other for cycles, will be in the bandwidth of the interferometers, and this phase may not only be detectable: it could provide us with important astrophysical information about the system, if the theoretical templates are sufficiently accurate.

Unfortunately, it seems difficult to attain the sensitivity to detect NS-NS binary inspirals with the first generation of interferometric detectors. However, the coalescence of BH-NS/BH-BH binaries with a black hole mass of may be detected out to the distance of the VIRGO cluster if we are lucky enough. In any case, it will be necessary to wait for the next generation of interferometric detectors to see these coalescing events more frequently [73, 82].

To predict the waveforms, a conventional approach is to formulate the Einstein equations with respect to the flat Minkowski background and apply the post-Newtonian expansion to the resulting equations (see the Section 1.2).

In this paper, however, we review a different approach, namely the black hole perturbation approach. In this approach, binaries are assumed to consist of a massive black hole and a small compact star which is taken to be a point particle. Hence, its applicability is constrained to the case of binaries with large mass ratio. Nevertheless, there are several advantages here that cannot be overlooked.

Most importantly, the black hole perturbation equations take full account of general relativistic effects of the background spacetime and they are applicable to arbitrary orbits of a small mass star. In particular, if a numerical approach is taken, gravitational waves from highly relativistic orbits can be calculated. Then, if we can develop a method to calculate gravitational waves to a sufficiently high PN order analytically, it can give insight not only into how and when general relativistic effects become important, by comparing with numerical results, but it will also give us a knowledge, complementary to the conventional post-Newtonian approach, about as yet unknown higher-order PN terms or general relativistic spin effects.

Moreover, one of the main targets of LISA is to observe phenomena associated with the formation and evolution of supermassive black holes in galactic centers. In particular, a gravitational wave event of a compact star spiraling into such a supermassive black hole is indeed a case of application for the black hole perturbation theory.

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