I guess you are referring to the contrast between imposed TT condition yielding null result (1.52), and subsequent, finite proper distance results in (1.53), (1.54).Q-reeus,
This is authentic GW text..
http://www.nikhef.nl/pub/services/biblio/theses_pdf/thesis_T_G_F_Li.pdf
for you its a cakewalk, can you pin point the problem ? It covers everything so you can easily pin point.
Well as mentioned, zero coordinate relative disiplacements in former case is owing to gauge choice imposing zero relative motion! The later proper distance calculations assume the shear strains as locally acting, but that falls over when global consistency is required. Local can't be if global can't be.
I do not wish to go around in circles here. The global inconsistency of GR GW's is the Achilles heel.
You evidently refer to 1.4.3 Ring of test masses p13. That is just the standard treatment of a local patch. Thought you had me there? Nope.PS : It covers hoop bead too.