## How Einstein got to E=mc^2

In 1865 James Clerk Maxwell had unified electricity and magnetism by developing his equations of electromagnetism. It was soon realized that these equations supported wave-like solutions in a region free of electrical charges or currents, otherwise known as vacuums.  Later experiments identified light as having electromagnetic properties and Maxwell’s equations predicted that light waves should propagate at a finite speed c (about 300,000 km/s).  With his Newtonian ideas of absolute space and time firmly entrenched, most physicists thought that this speed was correct only in one special frame, absolute rest, and it was thought that electromagnetic waves were supported by an unseen medium called the ether, which is at rest in this frame.

Let an object in a rest frame simultaneously emit two light waves with the same energy E/2 in opposite directions (now having equal but opposite momenta), the object remains at rest, but its energy decreases by E.  By the Doppler effect, in another frame, which is moving at the velocity v in one of those directions, the object will appear to lose energy equal to

The difference in energy loss as viewed from the two frames must therefore appear as a difference in kinetic energy seen by the second observing frame.  Hence, if v/c is very small, in the second frame (the one in motion) the object loses an amount of kinetic energy given by

Since the kinetic energy of an object with mass M moving with speed v is given by (1/2)Mv2 (for v/c≪1), this means that the object has lost an amount of mass given by E/c2.  In other words, a loss of energy of E is equivalent to the loss in mass of E/c2.  This implies equivalence between the mass and energy content of any object.  It turns out that for a particle of mass M, this quantity is equal to M2c2.  After implementing the Lorentz invariant (and if the frame in which the particle has zero momentum), then the equation E=mc2 is recovered.

Vasant Natarajan and Dipitman Sen, “The Special Theory of Relativity,” Resonance (April 2005): 32-33, 41-42.