MASS & ENERGY > Origins Special Relativity is a theory of space and time in which all physical laws (including electromagnetism) have the same form for any inertial observer. Einstein's Relativity principle generalizes Galilean relativity, which only applies to mechanical phenomena. It entails the departure from the very ideas of an ‘absolute’ time and an ‘absolute’ space. Not only velocity, but also length and time intervals (including zero-time intervals, i.e. the simultaneity of two events) become frame-dependent notions in Special Relativity.

In order to ensure the equivalence of any two inertial frames when electromagnetic phenomena are considered, the transformation rules connecting their space-time coordinates have to be modified with respect to the Galilean principle of velocity composition. This modification requires in turn a reformulation of Newtonian mechanics. If one seeks:

•  To have energy and momentum conservation principles analogous to those of Newtonian mechanics, governing the dynamics of elastic and inelastic collisions between particles

•  To recover Newtonian dynamics in the limit v<<c, where the classical theory has proved empirically adequate

then the pre-relativistic mass that appears in Newton's laws of motion has to be replaced by a velocity-dependent quantity, namely the quantity defined in equation (2).

For ordinary velocities (v<<c), eq. (2) becomes:



The second term of this expression is just the classical kinetic energy divided by c². Therefore, mass changes with velocity as kinetic energy does. This observation, together with other considerations suggested by relativistic dynamics, leads to the following equation.



This equation states the equivalence of mass and energy. It reduces to equation (1) if negative energy solutions are discarded.

Equation (4) establishes the conservation law that governs processes of creation and annihilation of particles. A formal account of these processes is provided by quantum field theory. Quantum field theory generalizes quantum mechanics to the relativistic regime (v ~ c), in which the number of particles that can be found in a given portion of space becomes itself an observable. Like any other quantum observable, the particle number can, in general, be predicted only statistically. Furthermore, its determination is incompatible with the attribution of a precise value to other physical observables. More generally, the very possibility of interpreting a physical situation as if there were n particles (where n is given integer) depends on the experimental context, as is the case for any physical properties in ordinary quantum mechanics (see the section on uncertainty).

Anti-particles are connected to the formal existence of negative energy solutions of equation (4). In general, the solutions of relativistic equations which represent particle motion propagate in a four-dimensional formal structure called Minkowski space-time. Within such a structure, equation (4) only allows for negative energy solutions running backward in time (whereas the positive energy solutions run forward). From the viewpoint of an observer running forward in time, however, an elementary particle associated with negative energy and moving backward in time appears to be acting like a positive energy particle traveling forward in time except for the fact that it reacts to electromagnetic field as if its electric charge had opposite sign.