Derivation of E=mc²

Introduction

The following is the thought experiment that Einstein and others used to show that the energy of a photon is E=mc². This formula was later extrapolated to include particles with mass.

Physical system

We have the following system: A box of inside length L from which a photon is emitted from the left side.

From classical physics, the momentum, p, of a light pulse (or photon) is its energy E divided by the speed of light c:

(1)

When the photon is emitted from the left side, the box recoils with a momentum in the opposite direction and keeps moving until the photon hits the right side canceling the box’s momentum.

(2)

(3)

The center of mass of the box and photon must be constant during the process.

(4)

(5)

Substitute equation 5 into equation 4 and use equations 1-3 as well:

(6)

Solve equation 6 for E:

(7)