Invariant mass of two photons

Problem

Challenge: invariant mass of two photons. Two photons, each of energy $E$, travel in opposite directions. Calculate the invariant mass of the system starting from $M^2 c^4 = E_{tot}^2 - (p_{tot}c)^2$. Discuss why the mass of a system is not additive.

Solution

Quantities for a single photon. Each photon has zero mass and satisfies $E = pc$, so it has momentum of magnitude $p = E/c$. The two photons travel in opposite directions: their momenta have opposite sign.

Total energy. The energies add (scalars):

$E_{tot} = E + E = 2E.$

Total momentum. The momenta are opposite vectors and cancel:

$p_{tot} = \frac{E}{c} - \frac{E}{c} = 0.$

Invariant mass. Substituting into the invariant:

$M^2 c^4 = E_{tot}^2 - (p_{tot}c)^2 = (2E)^2 - 0 = 4E^2.$

$M = \frac{2E}{c^2}.$

$\ev{M = \frac{2E}{c^2}}$

Mass is not additive. Each photon has zero mass, yet the system of the two has mass $M = 2E/c^2 \neq 0$. The mass of a system is not the sum of the masses of its components: it depends on the total energy and the total momentum taken together. Here, since the momenta cancel, all the energy contributes as the system’s rest mass (there exists a frame — the centre-of-momentum frame — in which the system as a whole is at rest, even though it is made of two photons that never are). It is the same principle by which most of a proton’s mass comes from the energy of its constituents, not from their masses.

Links

Topics: Special relativity Concepts: Mass-energy equivalence · Relativistic momentum Skills: Symbolic setup