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

272 Gyroscopes and gravitational wavesaxes ē(U)â is the same as the precession, with respect to the axes e(u)â, of theboosted spinvector s (u) , which reads:[s (u) = B (lrs,U,u) S (U) = P (u) −γ]γ + 1 ν (U,u) ⊗ ν (U,u) P (U,u) S (U) . (15.12)Hence, from (15.9) and (15.11) and the acceleration-equals-force equationfor U; wefind:[ ]ds â(u)− γɛâ dτ ˆbĉ [ζ (fw,U,u) + ζ (sc,fw,U,u) ] ˆb sĉ(u) e(u)â = 0 (15.13)Uζ (fw,U,u) = 1γ + 1 ν (U,u) × u [F (G)(fw,U,u) − γ F (U,u)]where 1ζ (sc,fw,U,u) = 1 2 δâ ˆb D (fw,U,u)dτ (U,u)e(u)â × u e(u) ˆb(15.14)so thatζ (fw,U,ē(U)â) = γ B (lrs,u,U) [ζ (fw,U,u) + ζ (sc,fw,U,u) ]. (15.15)Finally, by rescaling equation (15.13) to the proper-time of u, one hasds â (u)− ɛâ dτ ˆbĉ [ζ (fw,U,u) + ζ (sc,fw,U,u) ] ˆb s (u)ĉ = 0 (15.16)(U,u)whereζ (fw,U,u) + ζ (sc,fw,U,u) ≡ ˜ζ (fw,u,e(u)â) = γ −1 B (lrs,U,u) ζ (fw,U,ē(U)â) (15.17)is the angular velocity precession of the gyroscope as measured by the observer uwith respect to the axes e(u)â.It is worth mentioning here that while the observer U, who is comovingwith the gyro’s centre of mass, measures the precession (15.10) along his ownworldline, the observer’s u can only compare the instantaneous measurementsof ˜ζ (fw,u,e(u)â) in (15.17), made by each of them along the gyro’s worldline.Evidently either type of measurements requires the tetrad frames to beoperationally well defined. This will be discussed in the following section.15.3 The spacetime metricThe metric of a plane monochromatic gravitational wave, elliptically polarizedand propagating along a direction which we fix as the coordinate x direction, canbe written in the ‘TT’ gauge as [5]:ds 2 =−dt 2 + dx 2 + (1 − h 22 ) dy 2 + (1 + h 22 ) dz 2 − 2h 23 dy dz (15.18)1 This notation for the Fermi–Walker relative angular velocity ζ (fw,U,u) and the Fermi–Walker spacecurvaturerelative angular velocity ζ (sc,fw,U,u) has been introduced in [2].