A single photon travelling through empty space is proposed to spontaneously convert…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Minkowski spacetime and relativistic dynamics, completed

Problem

A single photon travelling through empty space is proposed to spontaneously convert into an electron-positron pair,

\gamma \to e^- + e^+

, with no other particle involved. The electron and positron each have rest energy

m_ec^2 = 0.511\ \mathrm{MeV}

. ((a)) State the sign and value of the invariant square

p^\mu p_\mu

of the photon's four-momentum, and what class of four-vector this makes it. ((b)) The total four-momentum of the electron-positron pair,

P^\mu = p_-^\mu + p_+^\mu

, has invariant square

P^\mu P_\mu = -W^2/c^2

, where

W

is the invariant mass-energy of the pair;

W

is smallest, equal to

2m_ec^2

, when the two particles move with the same velocity. Give this minimum value of

W

in MeV, and say what class of four-vector

P^\mu

belongs to. ((c)) Using the classification of four-vectors and the fact that every inertial observer agrees on that classification, explain in one or two sentences why the decay

\gamma \to e^-+e^+

is kinematically forbidden for a photon travelling alone in vacuum.

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