hints added to tutorial 3

tuturials/gw_tutorials.tex

@@ -92,11 +92,12 @@ 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.
 \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 
 $$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>$$
-\item Estimate the energy carried by the first detected GW event "GW150914". Distance to the source is $410\;\text{Mpc}$
+\item Estimate the energy in GW for the first detected GW event "GW150914". Distance to the source is $410\;\text{Mpc}$ and in the source frame initial black hole masses are $36 M_{\odot}$ and $29M_{\odot}$.
 \end{enumerate}