(ed.). Gravitational waves (IOP, 2001)(422s).

278 Gyroscopes and gravitational waveswhen φ = 0 it reduces to (15.31). Once the boosted frame ¯ f (U)â =B (lrs,U,u) f (u)â in LRS U is obtained, the components of the precession velocityturn out to beζ (fw,U, ¯ f (U)â) ˆ1 ≃− 1 2 h 23,tζ (fw,U, f ¯ˆ2 (U)â) ≃ α/2[cos φh 22,t − sin φh 23,t ]ζ (fw,U, f ¯ˆ3 (U)â) ≃ α/2[− sin φh 22,t + cos φh 23,t ], (15.37)again showing that, in the limit of small linear momentum α, the precessionmainly occurs about the direction of propagation of the wave.15.6 ConclusionsWe have operationally defined a tetrad frame adapted to a family of staticobservers in the background of a plane gravitational wave. Then we have usedthis family to study the precession angular velocity of a gyroscope moving alonga spacetime geodesic. The results show that, to first order in h and in thecase of non-relativistic motion (α ≪ 1), the observed gyroscopic precession ismainly induced by the cross-polarization only so a gyro appears to behave as apolarization filter.Assuming the form h 23 = h × sin(λ 2πcGW(t − x/c) + ψ) for the crosspolarization,with an obvious meaning for the symbols, the precession frequency(in conventional units) would be (gyro) (t) ≃− πch ×λ GW( )2πccos (t − x/c) + ψ . (15.38)λ GWThe values of the precession frequency are very small, as expected. With a typicalamplitude of 10 −21 at the Earth and a frequency of 10 3 Hz, we could hope for amaximum precession of the order of 10 −18 s −1 .Clearly the precession effect is larger with high-frequency gravitationalwaves or when the gyroscope is close enough to the wave source to allow fora higher value of h × . This situation is encountered by a spinning neutron star,say, in a compact binary system.The type of analysis we have considered, is most suitable to describe theinteraction of a moving gyroscope with a continuous flow of plane gravitationalwaves with metric form as in (15.18). These are expected to be emitted bysources like compact binaries. If we have an impulsive source, like a supernova,then one expects a burst of gravitational radiation which is better described by agravitational sandwich. This latter case is now under investigation.