Find the change of variable that goes from the TT coordinates to the proper detector coordinates, neglecting the low-frequency terms and working at linear order in [math]\displaystyle{ h_{\mu\nu} }[/math].
Show that the geodesic deviation [math]\displaystyle{ \xi_i(\tau) }[/math] remains constant in the TT coordinate system.
We worked out the response of an interferometer w.r.t. a [math]\displaystyle{ + }[/math] gravitational wave, aligned with the arms, and coming in axis normal to the plane of the interferometer. What happens if the interferometer is rotated by an angle [math]\displaystyle{ \theta, }[/math] around the [math]\displaystyle{ z }[/math] axis? What happends if it stays in the same position, but the source is at an angle [math]\displaystyle{ \phi }[/math] w.r.t. the [math]\displaystyle{ z }[/math] axis and in the [math]\displaystyle{ z x }[/math] plane?
In the TT gauge, calculate the coordinate speed [math]\displaystyle{ dx\over dt }[/math] of a photon moving in the [math]\displaystyle{ x }[/math] direction in the presence of [math]\displaystyle{ h_+ }[/math] moving in the [math]\displaystyle{ z }[/math] direction. Can it be larger than [math]\displaystyle{ c }[/math]?
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