In 1905, Albert Einstein showed that, as a consequence of his special theory of relativity, mass can be considered to be another form of energy. Thus the law of conservation of energy is really the law of conservation of mass-energy. In normal everyday interactions, the amount of mass that is transferred into other forms of energy (or vice versa) is such a tiny fraction of the total mass that it is beyond our sensory perceptions and measurement techniques. Thus, in a chemical reaction, for example, mass and energy truly seem to be separately conserved. In a nuclear reaction, however, the energy released is often about a million times greater than in a chemical reaction, and the change in mass can easily be measured.
Mass and energy are related by what is certainly the best-known equation in physics:
in which E is the energy equivalent (called mass energy) of mass m, and c is the speed of light.
Because energy is the integral of force with respect to distance, kinetic energy K can be represented by the following expression:
where F is the component of the applied force in the direction of the displacement ds and s is the distance over which the force acts. Using the relativistic form of the second law of motion
the expression for kinetic energy becomes
Integrating by parts
This result states that the kinetic energy of a body is equal to the increase in its mass consequent upon its relative motion multiplied by the square of the speed of light. This result may also be written as
If we interpret mc2as the total energy E of the body, we see that when the body is at rest and K=0, it still possesses the energy m0c2. Accordingly m0c2 is called the rest energy E0of a body whose mass at rest is m0. We therefore have
If the body is moving, its total energy is
and the proof is complete. Q.E.D.