(a) Let $S$ with coordinates $(c t, x, y)$ and $S^{\prime}$ with coordinates $\left(c t^{\prime}, x^{\prime}, y^{\prime}\right)$ be inertial frames in Minkowski space with two spatial dimensions. $S^{\prime}$ moves with velocity $v$ along the $x$-axis of $S$ and they are related by the standard Lorentz transformation:

$$\left(\begin{array}{c} c t \\ x \\ y \end{array}\right)=\left(\begin{array}{ccc} \gamma & \gamma v / c & 0 \\ \gamma v / c & \gamma & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} c t^{\prime} \\ x^{\prime} \\ y^{\prime} \end{array}\right), \quad \text { where } \gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}} .$$

A photon is emitted at the spacetime origin. In $S^{\prime}$ it has frequency $\nu^{\prime}$ and propagates at angle $\theta^{\prime}$ to the $x^{\prime}$-axis.

Write down the 4 -momentum of the photon in the frame $S^{\prime}$.

Hence or otherwise find the frequency of the photon as seen in $S$. Show that it propagates at angle $\theta$ to the $x$-axis in $S$, where

$$\tan \theta=\frac{\tan \theta^{\prime}}{\gamma\left(1+\frac{v}{c} \sec \theta^{\prime}\right)}$$

A light source in $S^{\prime}$ emits photons uniformly in all directions in the $x^{\prime} y^{\prime}$-plane. Show that for large $v$, in $S$ half of the light is concentrated into a narrow cone whose semi-angle $\alpha$ is given by $\cos \alpha=v / c$.

(b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.

Two particles $A_{1}$ and $A_{2}$ of rest masses $m_{1}$ and $m_{2}$ move collinearly with uniform velocities $u_{1}$ and $u_{2}$ respectively, along the $x$-axis of a frame $S$. They collide, coalescing to form a single particle $A_{3}$.

Determine the velocity of the centre-of-mass frame of the system comprising $A_{1}$ and $A_{2}$.

Find the speed of $A_{3}$ in $S$ and show that its rest mass $m_{3}$ is given by

$$m_{3}^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1} \gamma_{2}\left(1-\frac{u_{1} u_{2}}{c^{2}}\right),$$

where $\gamma_{i}=\left(1-u_{i}^{2} / c^{2}\right)^{-1 / 2}$