5 Stellar System Tests of 4 Strong Gravity and Gravitational 4.4 Equations of motion and

4.5 Gravitational wave detection 

A gravitational wave detector can be modelled as a body of mass M at a distance L from a fiducial laboratory point, connected to the point by a spring of resonant frequency tex2html_wrap_inline4777 and quality factor Q . From the equation of geodesic deviation, the infinitesimal displacement tex2html_wrap_inline4355 of the mass along the line of separation from its equilibrium position satisfies the equation of motion


where tex2html_wrap_inline5151 and tex2html_wrap_inline5153 are ``beam-pattern'' factors, that depend on the direction of the source tex2html_wrap_inline5155 and on a polarization angle tex2html_wrap_inline5157, and tex2html_wrap_inline5159 and tex2html_wrap_inline5161 are gravitational waveforms corresponding to the two polarizations of the gravitational wave (for a review, see [126]). 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 tex2html_wrap_inline5167 represent transverse-traceless (TT) projections of the calculated waveform of Eq. (51Popup Equation), given by


where tex2html_wrap_inline5169 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 tex2html_wrap_inline4521 with the x axis, the beam pattern factors are given by tex2html_wrap_inline5179 and tex2html_wrap_inline5181 . For a resonant cylinder oriented along the laboratory z axis, and for source direction tex2html_wrap_inline5155, they are given by tex2html_wrap_inline5187, tex2html_wrap_inline5189 (the angle tex2html_wrap_inline5157 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 tex2html_wrap_inline4355 defined as the differential displacement along the two arms, the beam pattern functions are tex2html_wrap_inline5201 and tex2html_wrap_inline5203 .

The waveforms tex2html_wrap_inline5159 and tex2html_wrap_inline5161 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. (53Popup Equation) gives


where tex2html_wrap_inline5213 is the orbital phase.

5 Stellar System Tests of 4 Strong Gravity and Gravitational 4.4 Equations of motion and

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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