updated

tuturials/gw_tutorials.tex

@@ -78,7 +78,10 @@ For a Newtonian source $T_{00} \gg |T_{0j}|, \; T_{00} \gg |T_{jk}|$.  Then comp
    	\item Considering the gauge invariant linearized field equations, show that $$\nabla^2 h^T= -16\pi T^{00} ~~~ \mathrm{and} ~~~\nabla^2 \tilde{h}_{0k} = -16\pi T_{0k},$$ so $h^T$ and $\tilde{h}_{0k}$ must vanish for waves in empty space.
    \end{itemize}
  
-\item In the class we discussed the expansion of a metric around a non-flat background: $g_{\mu \nu}(\underline{x}) = \bar{g}_{\mu \nu}(\underline{x}) + {h}_{\mu \nu}(\underline{x}),~~  |{h}_{\mu \nu}| \ll 1,$ 
+\item In the class we discussed the expansion of a metric around a non-flat background: 
+\begin{equation}
+g_{\mu \nu}(\underline{x}) = \bar{g}_{\mu \nu}(\underline{x}) + {h}_{\mu \nu}(\underline{x}),~~  |{h}_{\mu \nu}| \ll 1,
+\end{equation}
 and that the corresponding Ricci tensor can be written as 
 	$R_{\mu \nu} = \bar{R}_{\mu \nu} + {R}^{(1)}_{\mu \nu} + {R}^{(2)}_{\mu \nu} + \dots$,
 		where $\bar{R}_{\mu \nu}$ is computed from $\bar{g}_{\mu \nu}$ only, ${R}^{(1)}_{\mu \nu}$ is linear in $h_{\mu\nu}$ and ${R}^{(2)}_{\mu \nu}$ is quadratic in $h_{\mu\nu}$. Compute the explicit expressions of $\bar{R}_{\mu \nu}$, ${R}^{(1)}_{\mu \nu}$ and ${R}^{(2)}_{\mu \nu}$.
@@ -111,7 +114,7 @@ $$\frac{dP_k}{dt} = -\frac{c^3}{32\pi G} r^2\int d\Omega \left<\dot{h}^{TT}_{ij}
 
 \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.  
+E = \int_0^\infty \frac{dE}{df} \simeq \frac{4 \pi r^2 c^3}{2G} \int_{10 \mathrm{Hz}}^{300 \mathrm{Hz}} f^2 \tilde{h}(f)^2 df.  
 \end{equation}
 
 %=======
@@ -124,6 +127,40 @@ E = \int_0^\infty \frac{dE}{df} \simeq \frac{4 \pi r^2 c^3}{2G} \int_{10 Hz}^{30
 %>>>>>>> d6d01cbcc3d1671ee7f22483482305d8e28fcf9f
 \end{enumerate}
 
+\subsection{Tutorial 4}
+\begin{enumerate}
+\item We saw in the class that, in the short wavelength approximation ($\lambdabar \ll L_B$), in the leading order, the trace-reversed metric perturbation $\bar{h}_{\mu\nu}$ satisfies the following equations: 
+\begin{eqnarray} 
+\bar{D}^\rho\bar{D}_\rho \bar{h}_{\mu\nu} & = & 0, ~~~  \mathrm{(wave~ equation~ in~ background~ spacetime)} \label{eq:wave_eqn_curved}\\ 
+\bar{D}^\nu \bar{h}_{\mu\nu} &=& 0. ~~~  \mathrm{(``Lorentz~gauge''~ in~ background~ spacetime)} \label{eq:lorentz_curved}
+\end{eqnarray} 
+Show that, in the geometric optics approximation (slowly changing amplitude and polarisation, rapidly changing phase), GWs satisfy the following properties: 
+\begin{eqnarray} 
+k^\nu \, e_{\mu\nu} & = &  0 ~~~~ \mathrm{(polarisation~tensor} ~e_{\mu\nu}~ \mathrm{is~orthogonal~to~the~wave~vector}) \label{eq:k_ortho_pol}\\ 
+k^\rho k_\rho & = & 0 ~~~~ \mathrm{(wave~vector~is~null)} \label{eq:k_null}\\ 
+k^\mu \bar{D}_\mu k_\nu & = & 0 ~~~~ \mathrm{(wave~vector~satisfies~the~geodesic~equation~of~the~background~spacetime)} \label{eq:k_gedesic} \\ 
+\bar{D}_\mu (A^2 k^\mu) & = & 0  ~~~~ \mathrm{(``number~of~gravitons''~is~conserved)} \label{eq:graviton_conserv}\\ 
+k^\rho \bar{D}_\rho  \, e_\mathrm{\mu\nu} &  =  & 0  ~~~~ \mathrm{(polarisations~are~parallel~transported)}. \label{eq:pol_parallel}
+\end{eqnarray} 
+\\
+\emph{Hint:} Start from the ansatz 
+\begin{equation}
+\bar{h}_{\mu\nu}(\underline{x}) = [A_{\mu\nu}(\underline{x}) +\epsilon  B_{\mu\nu}(\underline{x}) + \dots ] e^{i \theta(\underline{x})/\epsilon}
+\label{eq:h_ansatz}
+\end{equation}
+and the definitions 
+\begin{equation}
+A_{\mu\nu} \equiv A \, e_{\mu\nu}, ~~ e^{\mu\nu} e_{\mu\nu}^* = 1, ~~ k_\nu \equiv \partial_\nu. 
+\end{equation}
+\begin{itemize}
+\item Use Eq.\eqref{eq:h_ansatz} in Eq.\eqref{eq:lorentz_curved} and truncate to leading order in $\epsilon$. This will give Eq.\eqref{eq:k_ortho_pol}.
+\item Use Eq.\eqref{eq:h_ansatz} in Eq.\eqref{eq:wave_eqn_curved} and truncate to leading order in $\epsilon$. This will give Eq.\eqref{eq:k_null}.
+\item Take a covariant derivative of Eq.\eqref{eq:k_null}. This will give Eq.\eqref{eq:k_gedesic}.
+\item Use Eq.\eqref{eq:h_ansatz} in Eq.\eqref{eq:wave_eqn_curved} and consider the next-to-leading order term in $\epsilon$. Use the nullity of $k^\mu$. This can be used to derive Eq.\eqref{eq:graviton_conserv} and Eq.\eqref{eq:pol_parallel}.
+\end{itemize}
+
+\end{enumerate}
+
 
 \bibliography{Lab}
 \end{document}