Gravitational waves and matter

Problem 1: Particle motion in TT gauge

Consider a plane wave with "$+$" polarization.
1) Write down the equations of motion for a non-relativistic particle; what happens to particles at rest?
2) Find the geodesic deviation from the Raychaudhuri equation and show that the result is the same;

Problem 2: Proper distance variation

Find the variation of proper distance between two particles at rest in the TT frame in the presence of a plane gravitational wave with the "$+$" polarization. Show that a ring of test particles placed initially at rest in the plane orthogonal to the wave vector will be distorted into an ellipse with its axes directed along the polarization tensor's main axes and oscillating with the wave's frequency.

Problem 3: Particle motion in the proper frame

Derive the equation of motion and geodesic deviation in the proper frame of one of the particles.

Problem 4: Euclidean spatial coordinates

Consider the plain gravitational wave with "$+$" polarization, propagating in the $z$ direction $h_{\alpha \beta }(t,z)=e_{\alpha\beta}^{(+)}f(t-z)$ and introduce new coordinates in the $(x,y)$ plane at $z=0$: $X=\left( 1+\frac{1}{2}f \right)x, \quad Y=\left( 1-\frac{1}{2}f \right)y$ The coordinates $(x,y)$ of test particles do not change with time, but $(X,Y)$ do.
Show, that the distance between the particles can be calculated in the first order by the wave's amplitude in the $(X,Y)$ coordinates using the Euclidean metric

Problem 5: Metric far from a non-relativistic source

Show that far from an isolated non-relativistic system of small enough mass the spacial components of the metric perturbation in the long-wave approximation are $\bar{h}_{\alpha\beta}(t,\mathbf{r}) =\frac{2}{r}\frac{d^2 I_{\alpha\beta}(t_r)}{dt^2},$ where $t_{r}=t-r$ is retarded time and $I_{\alpha\beta}=\int d^3 x\; x^\alpha x^\beta T_{00}(t,\mathbf{x})$ is the second mass moment of the system.