@@ -92,7 +92,7 @@ and that the corresponding Ricci tensor can be written as
\item for circularly (right and left separately) polarized gravitational waves.
\item for elliptically polarized gravitational waves.
\end{itemize}
hint: Use geodesic deviation equation in the proper frame at the centre of the circle. For a GW propagating in the z direction, we can write $$\frac{D^2n^j}{D\tau^2}+ R_{0jok}n^k=0$$ where $n^j = x_B^j-x_A^j$ and use $h_{jk}^{TT}= Re[A_0 e^{i(kz-\omega t)}{e_{\boldmath{p}}}_{jk}]$, $e_{\boldmath{p}}$ is the polarization tensor.
hint: Use geodesic deviation equation in the proper frame at the centre of the circle. For a GW propagating in the z direction, we can write $$\frac{D^2n^j}{D\tau^2}+ R_{0jok}n^k=0$$ where $n^j = x_B^j-x_A^j$ and use $h_{jk}^{TT}= Re[A_0 e^{i(kz-\omega t)}{e_{\boldmath{p}}}_{jk}]$, $e_{\boldmath{p}}$ is the polarization tensor. Initially $x_B^j = x_B^j(0)$, $h_{ij}=0$
\item For GWs at large distances show that $$\left<R^{(2)}_{\mu\nu}\right> =\frac{1}{4}\left<\partial_{\mu}h_{\alpha\beta}\partial_{\nu}h^{\alpha\beta}\right>,$$
$$\left<R^{(2)}\right>=0$$ and show that constructed energy-momentum tensor of GW ($t_{\mu\nu}$) is invariant under residual gauge transformation.
\item Derive that momentum transported (per unit time) by outward propagating GW is
