HPS 0410 | Einstein for Everyone |

The World's
Quickest Derivation of E = mc^{2}

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 uniform 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 mc^{2}, where m is the mass gained by the body.

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=mc^{2}.

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 = mc^{2}

We now see where the two c's in c^{2}=cxc come
from. One comes from the equation relating energy to distance; the second
comes from the equation relating momentum to time.

This derivation is for the special case at hand and further
argumentation is needed to show that *in all cases* a mass m and energy
E are related by Einstein's equation.

Back to main text E =
mc^{2}

Copyright John D. Norton. January 2001; July 2006; January 22, 2015.