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

Solutions to exercisesChapter 2Exercise (a)1. Let us take the form of the wave to beh TT jk = e jk⊕ h +(t − ˆn · ˆx)where e jk⊕ is the polarization tensor for the ⊕ polarization, and where ˆn is theunit vector in the direction of travel of the wave. We will let h + be an arbitraryfunction of its phase argument.If the wave travels in the x–z plane at an angle θ to the z-direction, then theunit vector in our coordinates isˆn i = (sin θ,0, cos θ).We need to calculate the polarization tensor’s components in x, y, z coordinates.We do this by rotating the ⊕ polarization tensor from its TT form in coordinatesparallel to the wavefront to its form in our coordinates. This requires a simplerotation around the y-axis. The transformation matrix is:( cos θ 0) sin θ j ′ k = 0 1 0 .− sin θ 0 cos θThe polarization tensor in our coordinates (primed indices) becomes:e j ′ k ′ = j ′ l k′ m e lm(cos 2 )θ 0 − sin θ cos θ= 0 −1 0 .− sin θ cos θ 0 sin 2 θNotice that the new polarization tensor is again traceless.The gravitational wave will be, at an arbitrary time t and position (x, z) inour (x, z)-plane,84h TT j ′ k ′ = e j ′ k ′ h + (t − x sin θ − z cos θ).