I am trying to see that in linearized gravity, choosing the transverse-traceless gauge is actually a valid choice to make. More specifically, I am trying to explicitly show what most textbooks just mention in passing, e.g. Maggiore:

Where the [itex]\xi^\mu[/itex] encode the gauge transformation [itex]x^\mu \rightarrow x^\mu + \xi^\mu (x)[/itex].

None of the books I have do this explicitly, or they do it by substituting plane waves as solutions, so I tried my hand at it myself and while it seems so far correct that one can choose the transformation as such, I'm unable to prove so far that this choice is compatible with the requirement [itex]\partial^2 \xi^\mu = 0[/itex].

Does anyone have a reference where this is done by just examining the transformation behavior of [itex]\overline{h}_{\mu\nu}[/itex]? Or is there some reason why it doesn't make sense to do this?

As an additional question, I read that going to the TT gauge also implies "choosing a Lorentz frame." What is the significance of this statement and where does it enter?