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

18 Elements of gravitational waves’+’0.2h/2-0.2t’x’Figure 2.1. Illustration of two linear polarizations and the associated wave amplitude.where h + = e xx exp[−iω(t − z)]. Such a metric produces opposite effects onproper distance at the two transverse axes, contracting one while expanding theother.If e xx = 0 we have pure ⊗ polarization h × which can be obtained from theprevious case by a simple 45 ◦ rotation, as in figure 2.1. Since the wave equationand TT conditions are linear, a general wave will be a linear combination of thesetwo polarization tensors. A circular polarization basis would be:e R = 1 √2(e + + ie × ), e L = 1 √2(e + − ie × ), (2.11)where e + , e × are the two linear polarization tensors and e R and e Lare polarizations that rotate in the right-handed and left-handed directions,respectively. It is important to understand that, for circular polarization,the polarization pattern rotates around the central position, but test particlesthemselves rotate only in small circles relative to the central position.Now we compute the effects of a wave in the TT gauge on a particle at rest inthe flat background metric η αβ before the passage of the gravitational wave. Thegeodesic equationd 2 x µdτ 2 + dx α dx βƔµ αβdτ dτ = 0implies in this case:d 2 x idτ 2 =−Ɣi 00 =− 1 2 (2h i0,0 − h 00,i ) = 0, (2.12)so that the particle does not move. The TT gauge, to first order in h αβ , representsa coordinate system that is comoving with freely-falling particles. Becauseh 0α = 0, TT time is proper time on the clock of freely-falling particles at rest.Tidal forces show the action of the wave independently of the coordinates.Let us consider the equation of geodesic deviation, which governs the separationof two neighbouring freely-falling test particles A and B. If the particles are