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.

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|ds² = −dt² + [1+h+(t − z)] dx² + [1 − h+(t = z)]dy² +dz² (27.80)