= Solution
Use $c=1$, background <Minkowski spacetime> <metric tensor> $\eta_{ab}=\operatorname{diag}(1,-1,-1,-1)$, and $g_{ab}=\eta_{ab}+h_{ab}$ with small <metric perturbation>. All indices and <derivatives> in this <linear> approximation use $\eta$. The <linearized inverse metric> is $\eta^{ab}-h^{ab}$, and the <linearized Levi-Civita connection> is
$$
\Gamma^a{}_{bc}=\frac12\eta^{ad}
(\partial_bh_{cd}+\partial_ch_{bd}-\partial_dh_{bc}).
$$
Define the <trace-reversed metric perturbation> $\bar h_{ab}=h_{ab}-\eta_{ab}h/2$, with $h=\eta^{ab}h_{ab}$.
An infinitesimal coordinate change $x'^a=x^a+\xi^a$ gives the <linearized coordinate gauge transformation>
$$
h'_{ab}=h_{ab}-\partial_a\xi_b-\partial_b\xi_a,\qquad
\bar h'_{ab}=\bar h_{ab}-\partial_a\xi_b-\partial_b\xi_a
+\eta_{ab}\partial_c\xi^c.
$$
The <metric perturbation> therefore has redundant components. The <gauge invariance of the linearized Riemann tensor> follows because its change contains third <derivatives> of $\xi$ cancelling by commutation of <partial derivatives>. Conversely, <flat linearized metric perturbations are locally pure gauge> on a <contractible> patch. Thus linearized <curvature>, rather than a particular coordinate value of $h$, detects the physical disturbance. Gauge freedom can also be restricted by global <topology> or <boundary conditions>.
Keeping the paper's <curvature sign convention>, its <linearized Ricci tensor and scalar> are
$$
R^{(1)}_{ab}=\frac12\left[
\Box h_{ab}+\partial_a\partial_bh
-\partial_a\partial^ch_{bc}-\partial_b\partial^ch_{ac}\right],
\qquad R^{(1)}=\Box h-\partial_a\partial_bh^{ab}.
$$
These are the negatives of the alternative frequently used <curvature> convention. Put $v_b=\partial^a\bar h_{ab}$. The <Linearized Einstein equations> in vacuum are
$$
G^{(1)}_{ab}=\frac12\left[
\Box\bar h_{ab}-\partial_av_b-\partial_bv_a
+\eta_{ab}\partial^cv_c\right]=0,\qquad
\Box=\partial_t^2-\nabla^2.
$$
Under a gauge change $v_b\mapsto v_b-\Box\xi_b$, so solving $\Box\xi_b=v_b$ imposes the <Lorenz gauge in linearized gravity>. The field equations then become $\Box\bar h_{ab}=0$. The remaining <residual gauge symmetry of linearized gravity> has $\Box\xi_b=0$; Lorenz gauge does not exhaust the coordinate freedom.
A <Fourier mode> $\bar h_{ab}=A_{ab}e^{ik_cx^c}$ obeys $k^ak_a=0$ and $k^aA_{ab}=0$. Hence the <plane gravitational waves in linearized gravity> propagate on the background <light cones>. Four transversality conditions and four residual gauge amplitudes leave $10-4-4=2$ physical degrees of freedom. The <explicit plane-wave reduction to transverse-traceless gauge> can set $h_{0a}=0$, spatial <trace> zero and $k^ih_{ij}=0$. For a wave in the $z$ direction, write its physical spatial strain as $H_{ij}=-h_{ij}$, so the spatial <metric tensor> is $-(\delta_{ij}+H_{ij})$:
$$
H_{ij}(t-z)=
\begin{pmatrix}
H_+&H_\times&0\\
H_\times&-H_+&0\\
0&0&0
\end{pmatrix}.
$$
There are \b[two <gravitational wave polarizations>], <plus polarization> and <cross polarization>. Rotating transverse axes through $\theta$ rotates their amplitude pair through $2\theta$. Nonconstant profiles with nonzero second <derivative> have nonzero linearized <Riemann curvature tensor> and cannot be removed as pure gauge.
The <TT coordinate separation and measured separation> distinction explains a detector's response. Freely falling particles initially at rest can keep fixed TT coordinates, since $\Gamma^a{}_{00}=0$, while their proper separation changes. For a short arm along a unit vector $n$,
$$
\frac{\delta L}{L}=\frac12H_{ij}n^in^j,\qquad
\frac{d^2\delta L}{dt^2}=\frac12\ddot H_{ij}n^in^jL.
$$
This is the tidal response described by <geodesic deviation>; coordinate motion by itself is not the measured signal.
Finally, gravitational-wave energy is second order in the perturbation, so it is absent from a first-order vacuum equation. In a short-wavelength averaging regime the <averaged stress-energy of transverse gravitational waves> is
$$
t^{\rm GW}_{ab}=\frac{1}{32\pi G}
\left\langle\partial_aH_{ij}\partial_bH_{ij}\right\rangle.
$$
For a plane wave its <energy flux> is $(16\pi G)^{-1}\langle\dot H_+^2+\dot H_\times^2\rangle$. The averaging scale and weak-field assumptions matter: this does not assign a coordinate-independent local gravitational <energy density> to an arbitrary first-order perturbation.
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