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

Precession of a gyroscope in geodesic motion 277from (15.11), will be seen by U as the corresponding axes with the sameorientation which are ‘momentarily at rest’. The orientation of the spin vectorS with respect to the axes ¯λ(U)â is also the orientation of S with respect to themoving axes λ(u)â.The velocity of spin precession then corresponds to the spatial dual ofthe Fermi–Walker structure functions of ¯λ(U)â, namely C (fw) (U, ¯λ(U)â) ˆbâ ,according to relation (15.23).Confining our attention to the weak-field approximation, the components ofthe precession velocity with respect to the triad ¯λ(U)â areζ (fw,U,¯λ(U)â) ˆ1 ≃ − 1 2 h 23,tζ (fw,U,¯λ(U)â) ˆ2 ≃ α/2h 22,tζ (fw,U,¯λ(U)â) ˆ3 ≃ α/2h 23,t . (15.34)We observe that in the limit of small linear momentum, α ≪ 1, thedominant precession is in the direction of wave propagation e(u) ˆ1(to zerothorder, ¯λ(U)ˆ1 ≃ e(u)ˆ1 ) and is induced by the cross-polarization only. (Note thatthe precession in the direction of propagation of the wave does not depend onα.) In this case we can conclude that the gyro can act as a polarization filter forgravitational waves.In the opposite limit of large linear momentum, α ≫ 1, the precession vectorlies mainly in the plane orthogonal to the propagation direction and is contributedlikewise by both polarizations. Indeed the measurement of the precession inducedby a plane gravitational wave, of a gyroscope set in relativistic motion, wouldenable one to identify the local direction of propagation of the wave. A similarsituation will be encountered in the rest frame of u, where from (15.17), (15.28)and (15.29) we have:˜ζ ˆ1 = γ −1(fw,u,λ(u)â) (U,u) B (lrs,U,u)ζ ˆ1 (fw,U,¯λ(U)â) ≃− 1 2˜ζ ˆ2 = γ −1(fw,u,λ(u)â) (U,u) B (lrs,U,u)ζ ˆ2 (fw,U,¯λ(U)â) ≃ 1 2˜ζ ˆ3 = γ −1(fw,u,λ(u)â) (U,u) B (lrs,U,u)ζ ˆ3 (fw,U,¯λ(U)â) ≃ 1 21√ h 23,t1 + α 2α√ h 22,t1 + α 2α√ h 23,t. (15.35)1 + α 2Finally, let us note that results (15.35) are only slightly modified afterrotating the frame (15.31) by a constant angle φ around the propagation directionof the wave. In fact, in this case, the new spatial u-frame becomesf (u)ˆ1 = λ(u)ˆ1f (u)ˆ2 = cos φλ(u)ˆ2 + sin φλ(u)ˆ3f (u)ˆ3 =−sin φλ(u)ˆ2 + cos φλ(u)ˆ3 ; (15.36)