0\end{aligned}$$ is exactly the same as the gauge freedom that is left after specifying the Lorentz condition and imposing traceless condition on the metric perturbation in GR and can be used to transform $\theta_{\mu\nu}$ to the transverse–traceless (TT) form [@MTW]. We call the obtained gauge the [*conformal gauge*]{} since it allows to represent a gravitational wave as the sum $$\begin{aligned}
\label{eq:zeta:res}
h_{\mu\nu}(t,{\bf x}) & = & A_{\mu\nu}(t,{\bf x}) + \Phi(t,{\bf x})\eta_{\mu\nu},\end{aligned}$$ of the TT wave $A_{\mu\nu}$ satisfying $A_{\mu0}=0$, $A^{ij}{}_{,j}=0$, $A^{i}{}_{i}=0$ and the scalar wave conformal to the Minkowski metric, $\Phi(t,{\bf x})\eta_{\mu\nu}$. For the plane wave propagating in the $-z$ direction (\[eq:zeta:res\]) simplifies to $$\begin{aligned}
\label{eq:c:z}
h_{\mu\nu}(t,z) & = & A^{+}(t+z)\epsilon^{+}_{\mu\nu} + A^{\times}(t+z)\epsilon^{\times}_{\mu\nu} + \Phi(t,z)\epsilon^{s}_{\mu\nu},\end{aligned}$$ with the polarization tensors $$\begin{aligned}
\label{eq:polar:pc}
\epsilon^{s} & = &
\left(
\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}
\right),
\qquad
\epsilon^{+} = \left(
\begin{array}{cccc}
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\right),
\qquad
\epsilon^{\times} = 