Derivation E=mc^2

HPS 0410

Einstein for Everyone

The World's Quickest Derivation of E = mc²

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

For the little bit of calculus behind this derivation, see this.

Consider a body that moves at very close to the speed of light. A constant force acts on it and, as a result, the force pumps energy and momentum into the body. That force cannot appreciably change the speed of the body because it is going just about as fast as it can. So all the increase of momentum = mass x velocity of the body is manifest as an increase of mass.

We want to show that in unit time the energy E gained by the body due to the action of the force is equal to mc², where m is the mass gained by the body.

Mass at c

We have two relations between energy, force and momentum from earlier discussion. Applying them to the case at hand and combining the two outcomes returns E=mc².

The first equation is:

Energy gained
= Force
x Distance through which force acts

The energy gained is labeled E. Since the body moves very close to c, the distance it moves in unit time is c or near enough.

The first equation is now

E = Force x c

The second equation is:

Momentum gained
= Force
x Time during which force acts

The unit time during which the force acts, the mass increases by an amount labeled m and the velocity stays constant at very close to c. Since momentum = mass x velocity, the momentum gained is m x c.

The second equation is now:

Force = m x c

Combining the two equations, we now have for energy gained E and mass gained m:

E = Force x c = (m x c) x c

Simplified, we have

E = mc²

We now see where the two c's in c²=cxc come from. One comes from the equation relating energy to distance; the second comes from the equation relating momentum to time.

The simplicity of this derivation comes with a price. It is limited to the case of the kinetic energy of body moving at very close to the speed of light. Further argumentation will be supplied to show that in all cases a mass m and energy E are related by Einstein's equation.

Back to main text E = mc²