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

270 Gyroscopes and gravitational wavesacceleration, vorticity expansionand the spatial dual of the vorticity fielda(u) α =∇ (fw,u)u α ,γ δω (u)αβ = P (u) α P (u) β ∇ [γ u δ] ,γ δθ (u)αβ = P (u) α P (u) β ∇ (γ u δ) , (15.2)ω (u) α = 1 2 ɛ (u) αβγ ω (u) βγ . (15.3)When dealing with different families of test observers, say u and U, themixed projection map P (U,u) = P (u) P (U) from LRS U to LRS u (and the analogouscompositions of two or more projectors) will be useful. Let l U be the world lineof a nonzero rest mass test particle with U as its unit timelike tangent vector. Theorthogonal decomposition of U relative to the family of test observers u, identifiesits relative velocity ν (U,u) = ν ˆν (U,u) where ν =‖ν (U,u) ‖=‖ν (u,U) ‖ and ˆν (U,u) isthe unit spatial vector, so thatU = γ [u + ν ˆν (U,u) ]. (15.4)Here γ = (1 − ν 2 ) −1/2 is the local relative Lorentz factor. If the four accelerationof the particlea (U) =∇ U U =D Udτ Uis non-vanishing, then its projection onto LRS u , leads to the acceleration-equalsforceequation:P (U,u) a (U) ≡ γ F (U,u)where F (U,u) is the spatial force acting on the particle as seen by the observer u.In a similar way, one defines a spatial gravitoinertial forceF (G)(fw,U,u) =−γ −1 DuP (u)dτ UDu=−P (u)dτ (U,u)= γ [g (u) + ν( 1 2 ˆν (U,u) × u H (u) − θ (u) Lˆν (U,u) )], (15.5)where τ U is a proper time parametrization for U and τ (U,u) = ∫ l Uγ dτ U is thecorresponding Cattaneo relative standard time parametrization; g (u) =−a (u) andH (u) = 2ω (u) are, respectively, the electric- and magnetic-like components ofthe gravitoinertial force. This terminology is justified by the Lorentz form of thegravitoinertial force which appears in the last of equations (15.5).If we define p (U,u) = γν (U,u) , E (U,u) = γ andD (fw,U,u)dτ (U,u)= γ −1 P (u)Ddτ U=∇ (fw,u) + ν α (U,u) ∇(u) α (15.6)