Merge branch 'master' of gitlab.icts.res.in:ajith/gwcourse2023

tuturials/gw_tutorials.tex

@@ -96,18 +96,32 @@ and that the corresponding Ricci tensor can be written as
 		\item for circularly (right and left separately) polarized GWs.
 		\item for elliptically polarized GWs.
 	\end{itemize}
+%<<<<<<< HEAD
+
+\emph{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 $$\frac{D^2n^j}{D\tau^2} + R_{0j0k}n^k=0$$ where $n^j = x_B^j-x_A^j$ and use $h_{jk}^{TT} = \mathrm{Re} [A_0 e^{i(kz-\omega t)}{\mathrm{e}_{\boldmath{p}}}_{jk}]$, $\mathrm{e}_{\boldmath{p}}$ is the polarization tensor. Initially $x_B^j = x_B^j(0)$, $h_{ij} = 0$.
+
 \item For GWs at large distances from the source ($T_{\mu\nu} = 0$) show that 
 \begin{equation}
 \left<R^{(2)}_{\mu\nu}\right> = \frac{1}{4}\left<\partial_{\mu}h_{\alpha\beta}\partial_{\nu}h^{\alpha\beta}\right>, ~~~ \mathrm{and} ~~~ \left<R^{(2)}\right>=0. 
 \end{equation}
 Also, show that constructed energy-momentum tensor $t_{\mu\nu}$ of GWs is invariant under residual gauge transformation.
+
 \item Derive that momentum transported (per unit time) by outward propagating GW at a large distance $r$ from the source, over a solid angle $d\Omega$ is 
-$$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>$$
+$$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>.$$
+
 \item Estimate the total energy energy radiated by the binary black hole merger that produced the  GW event GW150914, assuming that the distance to the source is $r \simeq 400\;\text{Mpc}$ and assuming isotropic emission of GWs. \emph{Hint:} Download the whitened data $h(t)$ from the LIGO Hanford detector at the time of the GW150914 event (you may use \href{https://www.gw-openscience.org/s/events/GW150914/GW150914_tutorial.html}{this} notebook). Compute the Fourier transform $\tilde{h}(f)$ using \texttt{numpy.fft}.  The radiated energy is: 
 \begin{equation}
 E = \int_0^\infty \frac{dE}{df} \simeq \frac{4 \pi r^2 c^3}{2G} \int_{10 Hz}^{300 Hz} f^2 \tilde{h}(f)^2 df.  
 \end{equation}
 
+%=======
+%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 $$\frac{D^2n^j}{D\tau^2} + R_{0j0k}n^k=0$$ 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 
+%$$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}\partial^k h^{TT}_{IJ}\right>$$
+%\item Estimate the energy 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 $29M_{\odot}$.
+%>>>>>>> d6d01cbcc3d1671ee7f22483482305d8e28fcf9f
 \end{enumerate}