\item Derive the Riemann tensor in linearized theory and show that its components are invariant under gauge transformations.
\item Derive the Riemann tensor in linearized theory ($g_{\mu\nu}=\eta_{\mu\nu}+ h_{\mu\nu}, ~~ |h_{\mu\nu}| \ll1$) and show that its components are invariant under gauge transformations.
\item\label{Riemann-tensor} Derive the Ricci and Einstein tensors from the Riemann tensor.
\item Show that the trace-reversed metric perturbation $\bar{h}_{\alpha\beta}$ transforms in the following way, under a gauge transformation generated by a vector $\xi^\mu$.
\begin{equation}
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@@ -64,14 +64,18 @@
For a Newtonian source $T_{00}\gg |T_{0j}|, \; T_{00}\gg |T_{jk}|$. Then compute $R_{0j0k}$ from it and using Eq.\eqref{TT-riemann} infer $h^\mathrm{TT}_{jk}$, and then compute other Riemann tensor components in original gauge and TT gauge and show that they disagree.
\item Repeat the same for a rotating spherical star, whose metric components are given by
$h_{00}=\frac{2M}{r}$, $h_{jk}=\frac{2M}{r}\delta_{jk}$, $h_{0k}=-2\epsilon_{klm}\frac{S^l x^m}{r^3}$, where $r =(x^2+y^2+z^2)^{\frac{1}{2}}$, $M$ is the mass and $\vec{S}$ the angular momentum. Show that this can not be put into TT gauge.
\item Let's define a general projection operator in direction independent form $$P_{jk}=\delta_{ik}-\frac{1}{\nabla^2}\partial_j\partial_k$$
\item Let us define a general projection operator in direction independent form
\item show that under a gauge transformation the TT part change in $h_{\mu\nu}$ is zero $\delta h_{jk}^{TT}= P_{jl}P_{km}\delta h_{lm}=0$. Then verify this formula for plane wave.
\item Let us define $h^T_{ij}=\frac{1}{2}P_{jk}(P_{lm}h_{lm})$ and $h^T =\text{Tr}(h^T_{ij})$, Using direction independent projection operator show that $$h^T =\frac{1}{\nabla^2}\left(h_{kk,ll}-h_{kl,kl}\right)$$
\item Verify gauge invariance of $h^T$.\\
hint: show the invariance of $\left(h_{kk,ll}-h_{kl,kl}\right)\;$ using $\delta h_{ij}=\xi_{i,j}+\xi_{j,i}$
\item Similarly show that the quantities $\tilde{h}_{0k}$ defined by $$\tilde{h}_{0k}=\bar{h}_{0k}-\frac{1}{\nabla^2}(\bar{h}^{\mu}_{0,\mu k}+\bar{k}_{k l,l0})$$ are gauge invariant.
\item Considering the gauge invariant linearized field equations, show that $$\nabla^2 h^T=-16\pi T^{00}$$$$\nabla^2\tilde{h}_{0k}=-16\pi T_{0k}$$, so $h^T$ and $\tilde{h}_{0k}$ must vanish for waves in empty space.
\item Show that under a gauge transformation, change in the TT part of $h_{\mu\nu}$ is zero. That is, $\delta h_{jk}^\mathrm{TT}= P_{jl}P_{km}\delta h_{lm}=0$. Then verify this formula for plane wave.
\item Let us define $h^\mathrm{T}_{ij}=\frac{1}{2}P_{jk}(P_{lm}h_{lm})$ and $h^\mathrm{T}=\text{Tr}(h^T_{ij})$, Using direction independent projection operator defined in Eq.\eqref{eq:proj_oper} show that $$h^\mathrm{T}=\frac{1}{\nabla^2}\left(\partial_l \partial_l h_{kk}-\partial_k \partial_l h_{kl}\right)$$
\item Verify gauge invariance of $h^T$. \emph{Hint:} Show the invariance of $\left(\partial_l \partial_l h_{kk}-\partial_k \partial_l h_{kl}\right)\;$ using $\delta h_{ij}=\xi_{i,j}+\xi_{j,i}$
\item Similarly, show that the quantities $\tilde{h}_{0k}$ defined by $$\tilde{h}_{0k}=\bar{h}_{0k}-\frac{1}{\nabla^2}(\partial_\mu\partial_k \bar{h}^{\mu}_{0}+\partial_l \partial_0\bar{k}_{k l})$$ are gauge invariant.
\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: $g_{\mu\nu}(\underline{x})=\bar{g}_{\mu\nu}(\underline{x})+{h}_{\mu\nu}(\underline{x}),~~ |{h}_{\mu\nu}| \ll1,$
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@@ -85,18 +89,25 @@ and that the corresponding Ricci tensor can be written as
\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.
\item Consider a circle made of test particles. Derive the effect of gravitational waves (GWs) on the particles in the proper reference frame of the test particle at the centre of the circle. Consider 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.
\item for $``+"$ polarized GWs.
\item for $``\times"$ polarized GWs.
\item for circularly (right and left separately) polarized GWs.
\item for elliptically polarized GWs.
\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
\item For GWs at large distances from the source ($T_{\mu\nu}=0$) show that
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 Estimate the energy carried by the first detected GW event "GW150914". Distance to the source is $410\;\text{Mpc}$
\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 $r \simeq400\;\text{Mpc}$ 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:
