(i) What is an affine parameter $\lambda$ of a timelike or null geodesic? Prove that for a timelike geodesic one may take $\lambda$ to be proper time $\tau$. The metric

$$d s^{2}=-d t^{2}+a^{2}(t) d \mathbf{x}^{2}$$

with $\dot{a}(t)>0$ represents an expanding universe. Calculate the Christoffel symbols.

(ii) Obtain the law of spatial momentum conservation for a particle of rest mass $m$ in the form

$$m a^{2} \frac{d \mathbf{x}}{d \tau}=\mathbf{p}=\text { constant }$$

Assuming that the energy $E=m d t / d \tau$, derive an expression for $E$ in terms of $m, \mathbf{p}$ and $a(t)$ and show that the energy is not conserved but rather that it decreases with time. In particular, show that if the particle is moving extremely relativistically then the energy decreases as $a^{-1}(t)$, and if it is moving non-relativistically then the kinetic energy, $E-m$, decreases as $a^{-2}(t)$.

Show that the frequency $\omega_{e}$ of a photon emitted at time $t_{e}$ will be observed at time $t_{o}$ to have frequency

$$\omega_{o}=\omega_{e} \frac{a\left(t_{e}\right)}{a\left(t_{o}\right)}$$