where and are ``beam-pattern'' factors, that depend on the direction of the source and on a polarization angle , and and are gravitational waveforms corresponding to the two polarizations of the gravitational wave (for a review, see ). In a source coordinate system in which the x - y plane is the plane of the sky and the z -direction points toward the detector, these two modes are given by
where represent transverse-traceless (TT) projections of the calculated waveform of Eq. (51), given by
where is a unit vector pointing toward the detector. The beam pattern factors depend on the orientation and nature of the detector. For a wave approaching along the laboratory z -direction, and for a mass whose location on the x - y plane makes an angle with the x axis, the beam pattern factors are given by and . For a resonant cylinder oriented along the laboratory z axis, and for source direction , they are given by , (the angle measures the relative orientation of the laboratory and source x -axes). For a laser interferometer with one arm along the laboratory x -axis, the other along the y -axis, and with defined as the differential displacement along the two arms, the beam pattern functions are and .
The waveforms and depend on the nature and evolution of the source. For example, for a binary system in a circular orbit, with an inclination i relative to the plane of the sky, and the x -axis oriented along the major axis of the projected orbit, the quadrupole approximation of Eq. (53) gives
where is the orbital phase.
|The Confrontation between General Relativity and
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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