(a) Define the 4-momentum of a particle of rest mass and 3 -velocity , and the 4-momentum of a photon of frequency (having zero rest mass) moving in the direction of the unit vector .
Show that if and are timelike future-pointing 4-vectors then (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 .]
(b) In the laboratory frame an electron travelling with velocity u collides with a positron at rest. They annihilate, producing two photons of frequencies and that move off at angles and to , in the directions of the unit vectors and respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that