Special Relativity


Galilean invariance, which is embodied in Newton's mechanics, for example, was considered intuitively obvious and observationally successful for several centuries. Among other things it implies that velocities add; i.e., if A observes B moving with speed v1 and B observes C moving with speed v2 in the same direction, then A observes C moving with speed v12 = v1 + v2 in that direction.
A contradiction arose as a consequence of the development of a very successful theory of electricity and magnetism in the nineteenth century, which is embodied in a set of differential equations known as Maxwell's equation. One of the implications of Maxwell's equations is the existence of various waves, such as radio waves, all of which are just light waves at various different frequencies. The equations imply that all of these waves travel with the speed of light (about 300,000 km/sec) regardless of the motion of the observer. This is a violation of Galilean invariance.
Einstein resolved this paradox by recognizing that Galilean invariance is just an approximation, valid for speeds much smaller than the speed of light. The principles that apply for all speeds are embodied in the special theory of relativity. In this theory the rule for the addition of velocities is

,

where c denotes the speed of light.


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