hints added to tutorial 3

tuturials/gw_tutorials.tex

@@ -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 circularly (right and left separately) polarized gravitational waves.
 		\item for elliptically polarized gravitational waves.
 		\item for elliptically polarized gravitational waves.
 	\end{itemize}
 	\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 $$\left<R^{(2)}_{\mu\nu}\right> = \frac{1}{4}\left<\partial_{\mu}h_{\alpha\beta}\partial_{\nu}h^{\alpha\beta}\right>,$$
 \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.
 $$\left<R^{(2)}\right>=0$$ 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 
 \item Derive that momentum transported (per unit time) by outward propagating GW is