Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2012

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

Show that if P1\mathbf{P}_{1} and P2\mathbf{P}_{2} are timelike future-pointing 4-vectors then P1P20\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>0m>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 ν1\nu_{1} and ν2\nu_{2} that move off at angles θ1\theta_{1} and θ2\theta_{2} to u\mathbf{u}, in the directions of the unit vectors e1\mathbf{e}_{1} and e2\mathbf{e}_{2} respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that

1+cos(θ1+θ2)cosθ1+cosθ2=γ1γ+1\frac{1+\cos \left(\theta_{1}+\theta_{2}\right)}{\cos \theta_{1}+\cos \theta_{2}}=\sqrt{\frac{\gamma-1}{\gamma+1}}

where γ=(1u2c2)1/2\gamma=\left(1-\frac{u^{2}}{c^{2}}\right)^{-1 / 2}

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