4
1 A Brief Stroll in Special Relativity
Fig. 1.1 Reference frames and coordinate systems in relative motion in the x direction
If a body moves with velocity u in the x direction in system S then it will have a
velocity in system S
given by differentiating this with respect to the absolute time,
u
=
dx
dt
=
dx
dt
− v = u − v, u = u
+ v.
(1.2)
That is the velocities u
and v simply add to give u
+ v. You may easily convince
yourself that the general vector expression for the addition of velocities must be
u = =
u
+ +
v.
(1.3)
The invariance of Newton’s second law under the transformation is evident since the
relative velocity is a constant and the acceleration is then the same in both systems.
This is the basis of Galilean invariance: Newton’s laws and the behavior of mechanical
systems are the same in all uniformly moving reference frames.
At the end of the nineteenth century Maxwell’s electromagnetism was generally
accepted, partly because it predicted the correct velocity for light, c = 1/
√√ μ 0 ε 0
= 2.9979 × 10
8 m/s, and it even predicted the existence of radio waves. But this
implied an interesting fact, that the velocity of light, according to (1.3), should be
different in different frames. Thus Maxwell’s equations should somehow be different
in different frames, either in the value of μ 0 ε 0 or in their mathematical structure. The
conventional viewpoint was that the equations were valid and c had the indicated
value in one special frame, that in which the supposed medium that supported light
waves, the luminiferous ether, was at rest.
This viewpoint was apparently self-consistent. The problem came when experimenters searched for evidence of the ether and of the velocity of the earth through
the ether and did not find it. The best known such experiment was that of Michelson
and Morley, which we will not discuss here since it is discussed in many books
(Taylor 1963). A number of phenomenological explanations were proposed to explain
the failure to observe effects of the ether but were largely forgotten when Einstein
presented his explanation in terms of the theory of special relativity.
Einstein’s approach was to assume that the Maxwell equations were valid and that
the speed of light was the same in all inertial systems, and then to rethink the whole
question of space and time, based on the constancy of the speed of light. The result
1 A Brief Stroll in Special Relativity
Fig. 1.1 Reference frames and coordinate systems in relative motion in the x direction
If a body moves with velocity u in the x direction in system S then it will have a
velocity in system S
given by differentiating this with respect to the absolute time,
u
=
dx
dt
=
dx
dt
− v = u − v, u = u
+ v.
(1.2)
That is the velocities u
and v simply add to give u
+ v. You may easily convince
yourself that the general vector expression for the addition of velocities must be
u = =
u
+ +
v.
(1.3)
The invariance of Newton’s second law under the transformation is evident since the
relative velocity is a constant and the acceleration is then the same in both systems.
This is the basis of Galilean invariance: Newton’s laws and the behavior of mechanical
systems are the same in all uniformly moving reference frames.
At the end of the nineteenth century Maxwell’s electromagnetism was generally
accepted, partly because it predicted the correct velocity for light, c = 1/
√√ μ 0 ε 0
= 2.9979 × 10
8 m/s, and it even predicted the existence of radio waves. But this
implied an interesting fact, that the velocity of light, according to (1.3), should be
different in different frames. Thus Maxwell’s equations should somehow be different
in different frames, either in the value of μ 0 ε 0 or in their mathematical structure. The
conventional viewpoint was that the equations were valid and c had the indicated
value in one special frame, that in which the supposed medium that supported light
waves, the luminiferous ether, was at rest.
This viewpoint was apparently self-consistent. The problem came when experimenters searched for evidence of the ether and of the velocity of the earth through
the ether and did not find it. The best known such experiment was that of Michelson
and Morley, which we will not discuss here since it is discussed in many books
(Taylor 1963). A number of phenomenological explanations were proposed to explain
the failure to observe effects of the ether but were largely forgotten when Einstein
presented his explanation in terms of the theory of special relativity.
Einstein’s approach was to assume that the Maxwell equations were valid and that
the speed of light was the same in all inertial systems, and then to rethink the whole
question of space and time, based on the constancy of the speed of light. The result