Elements of Modern Physics
6
transformations (1.2) be discarded. He found that space and time are related in
an intimate manner and should be treated on an equal basis. Their relation has a
far-reaching influence on the laws of physics. We begin the discussion of
Einstein’s results with a formal statement of the postulates of the special theory
of relativity.
1. The laws of nature are of the same form in all inertial frames of reference.
2. The speed of light is the same in all inertial frames of reference, and is
independent of the motion of the source.
It is implicit in the first postulate that, since the coordinates of the different
inertial frames are related, the laws of nature written in the various inertial
frames can be deduced from one another. It also follows that the Galilean
transformations (1.2) relating the coordinates of the inertial frames, cannot be
right since they would imply that the speed of light is different in different inertial
frames, in contradiction to the second postulate. Hence, a more general relation
between the coordinates must be obtained, which incorporates the information
that the speed of light is the same in all inertial frames.
1.5 LORENTZ TRANSFORMATIONS
In deriving the transformation equations consistent with the postulates of the
special theory of relativity, it was assumed that space is homogeneous, i.e., that
all points in space and time are equivalent. This means that the separation between
space-time points should remain invariant under translations which implies that
the relations between the coordinates of different inertial frames should be linear.
Let us consider again the inertial frames F and F ′ mentioned in Sec. 1.2,
whose axes coincide at time t = t′ = 0, and F ′ moves with velocity v along the
x-axis with respect to F. It is assumed that the y-axis is perpendicular to the
x′-axis since otherwise the inclinations of the positive and negative y-axis with
respect to the x-axis would be different, violating the rotational (or alternatively,
left-right) symmetry about the direction of relative velocity. It is also assumed
that the y- and z-axes are orthogonal to each other in either of the frames of
reference. Finally, since the lengths of two rods, which are at rest in frames
F and F′ respectively, and which are perpendicular to the x-axis, can be compared
while they are passing each other,
y′ = y
z′ = z
(1.12)
in order that the relations between F and F′ be reciprocal. For the transformation
of the x-coordinate, it is noted that the origin of F ′ travels with velocity v with
respect to frame F, which implies that
x′ = α(x – vt)
(1.13)
6
transformations (1.2) be discarded. He found that space and time are related in
an intimate manner and should be treated on an equal basis. Their relation has a
far-reaching influence on the laws of physics. We begin the discussion of
Einstein’s results with a formal statement of the postulates of the special theory
of relativity.
1. The laws of nature are of the same form in all inertial frames of reference.
2. The speed of light is the same in all inertial frames of reference, and is
independent of the motion of the source.
It is implicit in the first postulate that, since the coordinates of the different
inertial frames are related, the laws of nature written in the various inertial
frames can be deduced from one another. It also follows that the Galilean
transformations (1.2) relating the coordinates of the inertial frames, cannot be
right since they would imply that the speed of light is different in different inertial
frames, in contradiction to the second postulate. Hence, a more general relation
between the coordinates must be obtained, which incorporates the information
that the speed of light is the same in all inertial frames.
1.5 LORENTZ TRANSFORMATIONS
In deriving the transformation equations consistent with the postulates of the
special theory of relativity, it was assumed that space is homogeneous, i.e., that
all points in space and time are equivalent. This means that the separation between
space-time points should remain invariant under translations which implies that
the relations between the coordinates of different inertial frames should be linear.
Let us consider again the inertial frames F and F ′ mentioned in Sec. 1.2,
whose axes coincide at time t = t′ = 0, and F ′ moves with velocity v along the
x-axis with respect to F. It is assumed that the y-axis is perpendicular to the
x′-axis since otherwise the inclinations of the positive and negative y-axis with
respect to the x-axis would be different, violating the rotational (or alternatively,
left-right) symmetry about the direction of relative velocity. It is also assumed
that the y- and z-axes are orthogonal to each other in either of the frames of
reference. Finally, since the lengths of two rods, which are at rest in frames
F and F′ respectively, and which are perpendicular to the x-axis, can be compared
while they are passing each other,
y′ = y
z′ = z
(1.12)
in order that the relations between F and F′ be reciprocal. For the transformation
of the x-coordinate, it is noted that the origin of F ′ travels with velocity v with
respect to frame F, which implies that
x′ = α(x – vt)
(1.13)