Photon Box

An empty box of total mass M with perfectly reflecting walls is at rest in the lab frame. Then electromagnetic standing waves are introduced along the x direction, consisting of N photons, each of frequency v₀ (see Figure 1.1).

Figure 1.1

a) State what the rest mass of the system (box + photons) will be when the photons are present.

b) Show that this answer can be obtained by considering the momentum and/or energy of the box-plus-photon system in any inertial frame moving along the axis.

SOLUTION

a) Consider the initial state of the system. Write the 4-momentum of the box and photons as p^μ_box and p^μ_ph, respectively:

(1)

where we have used the fact that for a standing wave (which can be represented as the sum of traveling waves with opposite momenta) the total momentum is zero. Therefore, from (1), the 4-momentum of the system P^μ is given by

(2)

From (2), we can find the rest mass of the total system M (which is defined by M² c² = P^μ P_μ)

(3)

b) Transform the 4-momentum by going into a frame moving with velocity –v along the x axis. We have in this frame for energy ε' and momentum P'

where ε and P are the total energy and momentum in the rest frame, respectively. So

Therefore in the moving frame

and

We expect this to be true, of course, since mass is a relativistic invariant under a Lorentz transformation.

Another way to look at it is to consider a transformation of energy and momentum of the photons and the box separately. The frequencies of these photons will be Doppler shifted:

The energy of the photons

The energy of the box ε'_box = γM₀ c². The momentum of the photons

The momentum of the box

So the 4-momentum is the same as found above