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

The spacetime metric 273with h AB = h AB (t − x), (A, B = 2, 3). Let u denote the tangent vector field ofa family of geodesic, non-rotating and expanding observers defined byu ♭ =−dt, u = ∂ t . (15.19)One can adapt to these observers an infinite number of spatial frames, by rotatingany given one arbitrarily. For example, consider the following u-frame {e ˆα }={eˆ0 = u, e â = e(u)â} with its dual {ω ˆα }={ωˆ0 =−u ♭ ,ωâ = ω(u)â}u = ∂ te(u)ˆ1 = ∂ xe(u)ˆ2 = (1 − h 22) −1/2 ∂ y ≃ (1 + 1 2 h 22)∂ ye(u)ˆ3 = (1 − h2 22 − h2 23 )−1/2 [(1 − h 22 ) −1/2 h 23 ∂ y + (1 − h 22 ) 1/2 ∂ z ]≃ h 23 ∂ y + (1 − 1 2 h 22)∂ z (15.20)−u ♭ = dtω(u)ˆ1 = dxω(u)ˆ2 = (1 − h 22 ) 1/2 [dy − h 23ω(u)ˆ3 =(1 − h222− h 2 231 − h 22) 1/2dz ≃]dz ≃1 − h 22(1 − 1 )2 h 22 dy − h 23 dz(1 + 1 )2 h 22 dz,where ≃ denotes the corresponding weak-field limit. Any other spatial frame{ẽ(u)â}, adapted to the observers (15.19), can be obtained from this one by aspatial rotation Rẽ(u)â = e(u) ˆb R ˆbâ. (15.21)Among all the possible frames, there exists only one with respect to which thelocal compass of inertia experiences no precession. This frame is Fermi–Walkertransported along u, namely it satisfies the conditionP (u)Ddτ uẽ(u)â ≡∇ (fw,u) ẽ(u)â = 0. (15.22)A Fermi–Walker frame is the most natural of the u-frames; its spatialdirections, in fact, are fixed by three mutually orthogonal axes of small sizecomoving gyroscopes. However, if a metric perturbation causes a dragging ofthe local compass of inertia, the only way to detect and measure it, is to select aframe which is not Fermi–Walker transported. In this frame, in fact, a gyroscopewould be seen to precess and indeed its precession contains all the informationsabout gravitational dragging. Nonetheless, it is quite non-trivial to identify, inan operational way, a frame which is not Fermi–Walker transported along theobserver’s worldline when it is acted upon by a gravitational wave.