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

274 Gyroscopes and gravitational wavesFrame (15.20) is clearly Fermi–Walker transported in the absence ofgravitational waves (h 22 and h 23 being time independent), but it is not so whenthey are present. The Fermi rotation of the frame, in this case, is described by the(antisymmetric) angular velocity spatial tensor [6]:C (u) ˆbâ = e(u) ˆb ·u ∇ (fw,u) e(u)â, (15.23)hence, a gyroscope carried by the observer u will precess with respect to frame(15.20) with an angular velocity tensor which has only one independent nonzerocomponent, namely:C =−[h 23,t(1 − h 22 ) + h 22,t h 23 ](u)ˆ3ˆ2√≃− 12(1 − h 22 ) 1 − h 2 22 − h2 2 h 23,t. (15.24)23However, frame e(u)â cannot be operationally defined, so result (15.24) is of littlephysical significance although it shows the existence of frames which respond toone state of polarization only, at least to first order in h AB . We are, therefore,motivated to search for ‘frames’ that can be fixed from a viable experimental setup.15.4 Searching for an operational frameLet us consider the timelike geodesics of the metric (15.18). These are well known[7]; the four-velocity of a general such geodesic can be written asU g = 12E [(1 + f + E 2 )∂ t + (1 + f − E 2 )∂ x ]1+1 − h 2 22 − {[α(1 + h 22 ) + βh 23 ]∂ y + [β(1 − h 22 ) + αh 23 ]∂ z },h2 23(15.25)where α, β and E are Killing constants and f = g AB U A U B is equal to1f =1 − h 2 22 − [α 2 (1 + h 22 ) + β 2 (1 − h 22 ) + 2αβh 23 ]h2 23≃ α 2 (1 + h 22 ) + β 2 (1 − h 22 ) + 2αβh 23 . (15.26)If u = ∂ t is the family of observers who make the mesurements and {e(u) â}is an adapted spatial frame, then the relative velocity ν (Ug ,u)â of U g with respectto u is defined by the relationU g = γ (Ug ,u)[u + ν (Ug ,u)âe(u)â], (15.27)