Wave equation

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Problem 1: Lorenz (Hilbert, harmonic) gauge

The Lorenz${}^{*}$ gauge conditions are \begin{equation} \label{LorenzGauge} \partial_{\mu}\bar{h}^{\mu\nu}=0. \end{equation} Write down the Einstein's equations in the Lorenz gauge.

${}^{*}$By analogy with electromagnetism; note the spelling: Lorenz, not Lorentz.

Problem 2: Lorenz frame

Lorenz frame is the coordinate frame in which the Lorenz gauge conditions (\ref{LorenzGauge}) are satisfied.
1) find the coordinate transformation $x^\mu \to {x'}^{\mu}=x^\mu +\xi^\mu$ from a given frame to the Lorenz frame;
2) is the Lorenz frame unique? what is the remaining freedom for the choice of $\xi^\mu$?

Problem 3: Wave equation from action

Obtain the Einstein's equation in the Lorenz gauge in the first order by $h$ directly from the action

Problem 4: Vacuum equations in the transverse gauge

Simplify the vacuum Einstein's equations in the transverse gauge (without the Lorenz gauge conditions), assuming vanishing boundary conditions.

The only difference of the wave equation for gravitational perturbations from the one for the electromagnetic field is in its tensorial nature. The Green's function for the wave equation is known (see e.g. here or here), and the retarded solution, which is usually thought of as the one physically relevant, is \begin{equation} \bar{h}_{\mu\nu}(t,\mathbf{x})=\frac{1}{4\pi}\int \frac{d^{3}x'}{|\mathbf{x}-\mathbf{x'}|}\cdot \frac{16\pi G}{c^4}T_{\mu\nu} \Big(t-\frac{|\mathbf{x}-\mathbf{x'}|}{c}, \mathbf{x'}\Big). \end{equation}