The Kasner (vacuum) cosmological model is defined by the line element

$$d s^{2}=-c^{2} d t^{2}+t^{2 p_{1}} d x^{2}+t^{2 p_{2}} d y^{2}+t^{2 p_{3}} d z^{2} \quad \text { with } \quad t>0$$

where $p_{1}, p_{2}, p_{3}$ are constants with $p_{1}+p_{2}+p_{3}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1$ and $0<p_{1}<1$. Show that $p_{2} p_{3}<0$.

Write down four equations that determine the null geodesics of the Kasner model.

If $k^{a}$ is the tangent vector to the trajectory of a photon and $u^{a}$ is the four-velocity of a comoving observer (i.e., an observer at rest in the $(t, x, y, z)$ coordinate system above), what is the physical interpretation of $k_{a} u^{a}$ ?

Let $O$ be a comoving observer at the origin, $x=y=z=0$, and let $S$ be a comoving source of photons located on one of the spatial coordinate axes.

(i) Show that photons emitted by $S$ and observed by $O$ can be either redshifted or blue-shifted, depending on the location of $S$.

(ii) Given any fixed time $t=T$, show that there are locations for $S$ on each coordinate axis from which no photons reach $O$ for $t \leqslant T$.

Now suppose that $p_{1}=1$ and $p_{2}=p_{3}=0$. Does the property in (ii) still hold?