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

Splitting formalism and test particle motion: a short review 271the latter being the measure of the (rescaled) absolute derivative along U, the(3+1) version of the equation of motion of the particle and of the energy theorem,acquires the Newtonian formD (fw,U,u)dτ (U,u)p (U,u) = F (U,u) + F (G)(fw,U,u) ,dE (U,u)dτ (U,u)= [F (U,u) + F (G)(fw,U,u) ] ·u ν (U,u) . (15.7)Let us now consider the motion of a test gyroscope. As it is well known,the spin vector S (U) of a gyroscope carried by an observer U, is Fermi–Walkertransported along his worldline (i.e. S (U) does not precess with respect to spatialaxes which are Fermi–Walker dragged along U), namely:D (fw,U)dτ US (U) = P (U)Ddτ US (U) = 0. (15.8)Suppose that we have chosen a spatial triad ē(U)â which is adapted to the observerU and is not a Fermi–Walker frame. The observer U will then see the spin S(U)of the gyroscope to precess with respect to these axes according to the law:[ ]dS â(U)− ɛâ dτ ˆbĉ ζ ˆb(fw,U,ē(U)â) Sĉ(U) ē(U)â = 0 (15.9)Uwhereζ â(fw,U,ē(U)â) ≡ ɛâ ˆbĉē(U)ˆb ·∇ (fw,U)ē(U)ĉ (15.10)is the precession rate vector.However, we may want the gyroscope to be analysed by a different observer,u say, who is not comoving with the gyro’s centre of mass. In this case we needa smooth family of these observers, each one intersecting the gyro’s worldline atany of its spacetime points where he measures the instantaneous precession ofthe spin vector relative to a suitably defined frame, adapted to u. Of course, werequire that the observer’s u are synchronized so that their measurements can becompared. The results of these measurements are described by a smooth and atleast once differentiable function of the proper-time of u.Let {e(u)â} be a field of spatial triads adapted to u; then the restriction of{e(u)â} to the worldline l U of the gyroscope, allows one to define on l U a fieldof tetrad frames, adapted to U, given by {(U, ē(U)â}, where:ē(U)â = B (lrs,u,U) e(u)â, (15.11)B (lrs,u,U) = P (U) B (u,U) P (u) : LRS u → LRS U being the boost map between therest spaces of the observers U and u; this map has been studied extensively in[2–4]. Since the boost is an isometry, the precession of S (U) with respect to the