Consider a particle that is at rest in the TT coordinate system of the gravitational-wave metric (27.80) before the gravitational wave arrives. In the text it is shown that the particle’s 4-velocity has u^x = u^y = 0 as the wave passes. Show that u^z = 0 and u^t = 1 as the wave passes, so the components of the particle’s 4-velocity are unaffected by the passing gravitational wave, and the particle remains at rest (constant x, y, and z) in the TT coordinate system.

|ds = dt + [1+h+(t z)] dx + [1 h+(t = z)]dy +dz (27.80)