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
,
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