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 $$P_{jk}=\delta_{ik}-\frac{1}{\nabla^2}\partial_j\partial_k$$
\begin{itemize}
\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.
\end{itemize}
\item In the class we discussed the expansion of a metric around a non-flat background: $g_{\mu\nu}(\underline{x})=\bar{g}_{\mu\nu}(\underline{x})+{h}_{\mu\nu}(\underline{x}),~~ |{h}_{\mu\nu}| \ll1,$
and that the corresponding Ricci tensor can be written as
