\item Consider a metric perturbation that can be written as an expansion in plane waves, i.e., $\bar{h}_{\mu\nu}(\underline{x})=\int d^4\underline{k} ~ A_\mathrm{\mu\nu}(\underline{k})\exp(i k_\alpha x^\alpha)$, in some arbitrary gauge in linearized theory. Show that it is possible to find a gauge transformation that will put this in the TT gauge. \emph{Hint:} Express $\xi^\mu$ as a plane wave expansion and show that the individual Fourier components satisfy the condition that is derived in the above problem.
\end{enumerate}
%\subsection{Tutorial 2}
%\begin{enumerate}
%
% \item \textbf{Limitation on existence of TT Gauge:}
% \begin{itemize}
% \item Non-radiative metric perturbation $h_{\mu\nu}$ can not be put into TT form (discussed in the class). For example consider the external field of a non rotating spherical star of mass $M$ (Newtonian source, velocities slow enough to neglect retardation). Calculate the metric perturbation in the weak field limit and show that there are non radiative metric perturbations i.e can not be transformed into TT form. \\
% \textit{hint}: General solution to linearized field equation in Lorentz gauge,
% $\bar{h}_{\mu\nu}=\int\frac{4 T_{\mu\nu}(t-|\vec{x}-\vec{x}^{'}|,\vec{x}^{'})}{|\vec{x}-\vec{x}^{'}|} d^3\vec{x}^{'} $. Newtonian source $T_{oo}>>|T_{oj}|, \; T_{00}>> |T_{jk}|$. Applying these find $h_{\mu\nu}$. \\ Then compute $R_{0j0k}$ from it and using \ref{TT-riemann} infer $h^{TT}_{ij}$, and then compute other Riemann tensor components in original gauge and TT gauge and show that they disagree.
% \item Do the same for a rotating spherical star. Given
% $h_{00} = \frac{2M}{r}$, $h_{jk} = \frac{2M}{r}\delta_{jk}$, $h_{ok}$ = -2$\epsilon_{klm}\frac{S^l x^m}{r^3}$, $r = (x^2+y^2+z^2)^{\frac{1}{2}}$ Where $M$ is the mass and $\vec{S}$ is the angular momentum. Show that this can not be put into TT gauge.
% \end{itemize}
\subsection{Tutorial 2}
\begin{enumerate}
\item Non-radiative metric perturbation $h_{\mu\nu}$ can not be put into TT form (discussed in the class). For example, consider the external field of a non rotating spherical star of mass $M$ (Newtonian source, velocities slow enough to neglect retardation). Calculate the metric perturbation in the weak field limit and show that there are non radiative metric perturbations that can not be transformed into TT form. \\
\textit{Hint}: General solution to linearized field equation in Lorentz gauge,
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 In the class we discussed the expansion of a metric around a non-flat background: $g_{\mu\nu}(\underline{x})=\bar{g}_{\mu\nu}(\underline{x})+{h}_{\mu\nu}(\underline{x}),~~ |{h}_{\mu\nu}| \ll1,$
and that the corresponding Ricci tensor can be written as
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 ${R}^{(2)}_{\mu\nu}$.
% \item \textbf{A cylindrical gravitational wave:} Consider the radiative solution whose only non-vanishing component $h_{\mu\nu}$ is $$\bar{h}_{zz}=4Acos(\omega t)J_0\left(\omega\sqrt{x^2+y^2}\right)$$, where $J_0$ is the Bessel function. This solution represents a superposition of ingoing and outgoing cylindrical gravitational waves. Verify that it can be transformed into TT gauge
