(a) Define the 4-momentum $\mathbf{P}$ of a particle of rest mass $m$ and 3 -velocity $\mathbf{v}$, and the 4-momentum of a photon of frequency $\nu$ (having zero rest mass) moving in the direction of the unit vector $e$.

Show that if $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ are timelike future-pointing 4-vectors then $\mathbf{P}_{1} \cdot \mathbf{P}_{2} \geqslant 0$ (where the dot denotes the Lorentz-invariant scalar product). Hence or otherwise show that the law of conservation of 4 -momentum forbids a photon to spontaneously decay into an electron-positron pair. [Electrons and positrons have equal rest masses $m>0$.]

(b) In the laboratory frame an electron travelling with velocity u collides with a positron at rest. They annihilate, producing two photons of frequencies $\nu_{1}$ and $\nu_{2}$ that move off at angles $\theta_{1}$ and $\theta_{2}$ to $\mathbf{u}$, in the directions of the unit vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that

$$\frac{1+\cos \left(\theta_{1}+\theta_{2}\right)}{\cos \theta_{1}+\cos \theta_{2}}=\sqrt{\frac{\gamma-1}{\gamma+1}}$$

where $\gamma=\left(1-\frac{u^{2}}{c^{2}}\right)^{-1 / 2}$