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 circularly (right and left separately) polarized GWs.
 		\item for elliptically polarized GWs.
 		\item for elliptically polarized GWs.
 	\end{itemize}
 	\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 
 \item For GWs at large distances from the source ($T_{\mu\nu} = 0$) show that 
 \begin{equation}
 \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. 
 \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}
 \end{equation}
 Also, show that constructed energy-momentum tensor $t_{\mu\nu}$ of GWs is invariant under residual gauge transformation.
 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 
 \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: 
 \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}
 \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.  
 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}
 \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}
 \end{enumerate}