tutorial-3 added

tuturials/gw_tutorials.tex

@@ -83,6 +83,21 @@ and that the corresponding Ricci tensor can be written as
 
 \end{enumerate}	
 
+\subsection{Tutorial 3}
+\begin{enumerate}
+\item Consider a circle with test particles on it. Derive the effect of gravitational wave on the particles on the circle in a proper reference frame of a test particle at the centre of the circle for the following cases. 
+	\begin{itemize}
+		\item for $"+"$ polarized gravitational waves.
+		\item for $"\times"$ polarized gravitational waves.
+		\item for circularly (right and left separately) polarized gravitational waves.
+		\item for elliptically polarized gravitational waves.
+	\end{itemize}
+\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 $410\;\text{Mpc}$
+\end{enumerate}
 
 
 \bibliography{Lab}