It is often convenient to introduce the proper reference frame of the detector with coordinates . Because the motion of this frame with respect to the SSB is non-relativistic, we can assume that the transformation between the SSB-related coordinates and the detector’s proper reference frame coordinates has the formorthogonal, then , and Eq. (19) simplifies to [31, 54, 68, 78] for more details.
For a ground-based laser-interferometric detector, the long-wavelength approximation can be employed (however, see [27, 115, 114] for a discussion of importance of high-frequency corrections, which modify the interferometer response function computed within the long-wavelength approximation). In the case of an interferometer in a standard Michelson and equal-arm configuration (such configurations can be represented by Figure 1 with the particle 1 corresponding to the corner station of the interferometer and with ), the observed relative frequency shift is equal to the difference of the round-trip frequency shifts in the two detector’s arms :response function of the interferometric detector to a plane gravitational wave in the long-wavelength approximation. To get Eqs. (24) – (25) directly from Eqs. (22) – (23) one should assume that the quantities , , and [entering Eq. (22)] do not depend on time . But the formulae (24) – (25) can also be used in the case when those quantities are time dependent, provided the velocities of the detector’s parts with respect to the SSB are non-relativistic. The error we make in such cases is on the order of , where is a typical value of the velocities . Thus, the response function of the Earth-based interferometric detector equals
From Eqs. (26) and (3) it follows that the response function is a linear combination of the two wave polarizations and , so it can be written asbeam-pattern functions. They depend on the location of the detector on Earth, the position of the gravitational-wave source in the sky, and the polarization angle of the wave (this angle describes the orientation, with respect to the detector, of the axes relative to which the plus and cross polarizations of the wave are defined, see, e.g., Figure 9.2 in ). Derivation of the explicit formulae for the interferometric beam patterns and can be found, e.g., in Appendix C of .
In the long-wavelength approximation, the response function of the interferometric detector can be derived directly from the equation of geodesic deviation . Then the response is defined as the relative difference between the wave-induced changes of the proper lengths of the two arms, i.e., , where and are the instantaneous values of the proper lengths of the interferometer’s arms and is the unperturbed proper length of these arms.
In the case of an Earth-based resonant-bar detector the long-wavelength approximation is very accurate and the dimensionless detector’s response function can be defined as , where is the wave-induced change of the proper length of the bar. The response function can be computed in terms of the detector’s proper-reference-frame quantities from the formula (see, e.g., Section 9.5.2 in )bar-pattern functions and different from the interferometric beam-pattern functions. Derivation of the explicit form of the bar patterns can be found, e.g., in Appendix C of .
Living Rev. Relativity 15, (2012), 4
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