= Explicit plane-wave reduction to transverse-traceless gauge
{title2=$H_+=(H_{11}-H_{22})/2,\quad H_\times=H_{12}$}
For a <plane gravitational wave in linearized gravity> with $k_\mu=(-\omega,0,0,\omega)$, <Lorenz gauge in linearized gravity> gives $H_{3\nu}=-H_{0\nu}$. Under $\delta H_{\mu\nu}=i(k_\mu X_\nu+k_\nu X_\mu-\eta_{\mu\nu}k^\rho X_\rho)$, choose $X_i=H_{0i}/(i\omega)$ for $i=1,2$, $X_3-X_0=iH_{00}/\omega$, and $X_0+X_3=(H_{11}+H_{22})/(2i\omega)$. This <residual gauge symmetry of linearized gravity> removes all time and longitudinal entries and the transverse trace. The remaining amplitudes $H_+,H_\times$ are the two <gravitational wave polarizations>. The formula uses the plus-sign convention for $\delta h=2\partial_{(\mu}\xi_{\nu)}$; reversing the definition of $\xi$ reverses the gauge change.
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