\affiliation{International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India.}
\bigskip
\date{\today}
\maketitle
\section{Linearized theory}
\subsection{Tutorial 1}
\begin{enumerate}
\item Derive the Riemann tensor in linearized theory 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 $\xi^\mu$.
\item Let $\bar{h}_{\mu\nu}(\underline{x})=\mathrm{Re}\left[A_\mathrm{\mu\nu}\exp(i k_\alpha x^\alpha)\right]$ be a plane wave propagating in the $\underline{k}$ direction, in some arbitrary gauge in linearized theory. Work out the generating function $\xi^\mu$ that will generate a gauge transformation that will put $\bar{h}_{\mu\nu}$ in the TT gauge.
\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}
% \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
