World's Quickest Derivation of E = mc2 (not written to satisfy classical physicists like Newton)
Adapted from a derivation posted on the internet by John D. Norton
Department of History and Philosophy of Science, University of Pittsburgh
Consider a body of mass m moving at a velocity very close to c. A constant force F acts on the body in the same direction as its motion for a duration set at unit time interval delta T (=1), and as a result, the force increases both the energy and momentum of the body. The force cannot increase its speed because it is already c, so all of the increase of momentum = mass x velocity of the body manifests here mathematically as an increase in mass. This thought experiment dispenses with any references to center of mass for Newton's benefit.
Side note to particle physicists: Of course, mass = rest mass + kinetic energy, expressed in units of energy. No one expects that a particle changing relativistic speed actually gains or loses inertial mass. The Higgs field pushes back to keep speeds below c for objects that have mass, including itself (by giving mass to them). But it would be impractical to use (delta P) = m x (delta v) because c can't change, and (delta v) would then be zero. By the same token, a single photon has no defined energy or even existence relative to an observer who knows nothing of its origin, but a pair of photons do. This is the raison d'etre for any observer-based version of relativity, expressed without the obligatory math. This principle works for non-virtual particles also.
A basic rule in quantum physics is: If you look, it's a particle; if you don't look, it's a wave.
Note the choice of unit time in this derivation (as opposed to unit c, popular among classical physicists including Einstein himself).
We want to show that in unit time interval delta T, the energy E gained due to the action of the Force applied is equal to mc^2, where m is the mass gained by the relativistic projectile, and c is the speed of light. For this example, E = delta E, and m = delta m.
We have two relations involving Energy, Force, and Momentum over the unit time interval delta T (=1):
The first relation is about the change in energy in unit time interval delta T, (or 'work' in classical physics):
delta E = E = Force x (distance through which force acts) = Force x (c x delta T), or E = Force x c (1-1)
The second relation is about the change in momentum P in unit time interval delta T (or 'impulse' in classical physics):
delta P = (delta m) x c = m x c = (Force) x (delta T), or m x c = Force (1-2)
Which means: delta E = E = Force x c = (m x c) x c
(1-3)
In other words (QED): E = mc^2 (0-0)
We now see that c^2 = c x c derives from relating energy to distance and momentum to time in the same expression.
