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

Splitting formalism and test particle motion: a short review 269and references therein). The effects of a gravitational wave on a frame which isnot Fermi–Walker transported, are best appreciated by studying the precession ofa gyro at rest in that frame. Clearly it is essential to identify a class of frameswhich optimize the corresponding precession effect.In section 15.2 we give a short review of the observer dependent spacetimesplitting which enables one to describe in physical terms the motion of a testparticle as well as that of a test gyroscope. In section 15.3 we discuss thespacetime of a plane gravitational wave and confine our attention to the family ofstatic observers; we give an example of a tetrad frame adapted to these observerswith respect to which the precession of a gyroscope is induced by the crosspolarizationonly. In section 15.4 we discuss the non-trivial problem of how tofix, in an operational and non-ambiguous way, a frame of reference which is notFermi–Walker transported in the spacetime of a plane gravitational wave. Finally,in section 15.5 we calculate the precession of a gyroscope, in a general geodesicmotion, with respect to the above frame.In what follows, Greek indices run from 0–3, latin indices from 1–3.15.2 Splitting formalism and test particle motion: a shortreviewA given family of test observers, namely a congruence of timelike lines with unittangent vector field u (i.e. u · u =−1) induces a splitting of the spacetime intospace plus time through the orthogonal decomposition of the tangent space at eachpoint into the local time direction along u and the local rest space LRS u .Projection of spacetime tensor fields onto LRS u is accomplished using theprojection operatorP (u) = I + u ⊗ u (15.1)and yields a family of spatial tensors (belonging to LRS u ⊗ ··· ⊗ LRS u , i.e.for which any contraction with u vanishes). The collection of all the spatiallyprojected tensor fields, associated to a given spacetime tensor field, will bereferred to as the ‘measure’ of the spacetime tensor itself. For instance, themeasure of the unit volume four-form η αβµν , gives only one non-trivial spatialfield: ɛ (u) αβγ= uδ η δαβγ which can be used in turn to define the spatial crossproduct × u in LRS u .One can also spatially project the various derivative operators so that theresult of the derivative of any tensor field is itself a spatial tensor; examplesare: the spatial Lie derivative, £ (u) X = P (u)£ X for any vector field X, thespatial covariant derivative ∇ (u) α = P β(u) P (u) α ∇ β , the Lie temporal derivative,∇ (lie,u) = P (u) £ u , the Fermi–Walker temporal derivative, ∇ (fw,u) = P (u) ∇ u andseveral other natural derivatives for which a detailed discussion can be foundin [2].The measure of the covariant derivative of the four-velocity of the observers,gives rise to the kinematical coefficients of the observer congruence, namely the