1.1 The Trouble with Absolute Time
5
was that he abandoned Newton’s absolute time and developed the special theory of
relativity.
1.2 The Simplest Lorentz Transformation
Einstein’s 1905 approach to special relativity was based on the following two
postulates:
I. The analytical form of physical laws is the same in all inertial reference frames
as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
Postulate (I) is a criterion of elegance, while (II) was supported by experiments
done before 1905, such as that of Michelson and Morley, and is now verified to very
high accuracy.
We want to derive now a transformation of the space coordinates plus time, to
replace the Galilean transformation discussed above, but in which the velocity of
light is the same in both systems. This is called a Lorentz transformation; due to
its fundamental importance our derivation will be detailed and based on the most
elementary assumptions (Sard 1970).
To begin we modify the Galilean transformation (1.1) in as simple a way as we
can. First, we suppose that y and z are not changed, that is y
= y and z
= z (You
should think about this a little). We next assume that time may be different in the
two systems, and that the transformation is linear in x and t. That is we assume
ct
= a 11 ct + a 12 x, x
= a 21 ct + a 22 x.
(1.4a)
In equivalent matrix form,
ct
x
=
a 11 a 12
a 21 a 22
ct
x
, A(v) ≡
a 11 a 12
a 21 a 22
.
(1.4b)
The matrix elements a i j must, of course, depend only on the velocity v. The notable
property of this transformation is that time is allowed to be different in the two
systems, which is the fundamental break with classical ideas made by Einstein. It is
this which allows c to be a universal constant. The use of ct instead of t in (1.4a) is
for dimensional convenience, since ct and x both have dimensions of distance. There
are 4 parameters in the transformation matrix A, which we must determine. We will
make four physical demands based on the above two postulates that determine them
uniquely.
Demand 1. We can describe the origin of the system S
in terms of both coordinate
systems. In the primed coordinates it is given by x
= 0 and in the unprimed coordinates it is given by x = vt. This is simply the statement that S
moves at velocity
v relative to S. We use (1.4a) to express x
= 0 as
5
was that he abandoned Newton’s absolute time and developed the special theory of
relativity.
1.2 The Simplest Lorentz Transformation
Einstein’s 1905 approach to special relativity was based on the following two
postulates:
I. The analytical form of physical laws is the same in all inertial reference frames
as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
Postulate (I) is a criterion of elegance, while (II) was supported by experiments
done before 1905, such as that of Michelson and Morley, and is now verified to very
high accuracy.
We want to derive now a transformation of the space coordinates plus time, to
replace the Galilean transformation discussed above, but in which the velocity of
light is the same in both systems. This is called a Lorentz transformation; due to
its fundamental importance our derivation will be detailed and based on the most
elementary assumptions (Sard 1970).
To begin we modify the Galilean transformation (1.1) in as simple a way as we
can. First, we suppose that y and z are not changed, that is y
= y and z
= z (You
should think about this a little). We next assume that time may be different in the
two systems, and that the transformation is linear in x and t. That is we assume
ct
= a 11 ct + a 12 x, x
= a 21 ct + a 22 x.
(1.4a)
In equivalent matrix form,
ct
x
=
a 11 a 12
a 21 a 22
ct
x
, A(v) ≡
a 11 a 12
a 21 a 22
.
(1.4b)
The matrix elements a i j must, of course, depend only on the velocity v. The notable
property of this transformation is that time is allowed to be different in the two
systems, which is the fundamental break with classical ideas made by Einstein. It is
this which allows c to be a universal constant. The use of ct instead of t in (1.4a) is
for dimensional convenience, since ct and x both have dimensions of distance. There
are 4 parameters in the transformation matrix A, which we must determine. We will
make four physical demands based on the above two postulates that determine them
uniquely.
Demand 1. We can describe the origin of the system S
in terms of both coordinate
systems. In the primed coordinates it is given by x
= 0 and in the unprimed coordinates it is given by x = vt. This is simply the statement that S
moves at velocity
v relative to S. We use (1.4a) to express x
= 0 as