14
2 Lorentz Transformations
Fig. 2.1 The light cone in two space and one time dimension
x
=
3
0
a
μ ν x
ν
= a
μ ν x
ν
.
(2.2)
Notice that in (2.2) we simply omitted the summation sign with the understanding
that repeated indices are to be summed over. This is the famous Einstein summation
convention which we will use henceforth; it makes the equations look much simpler.
The light cone equation (2.1) may be written in matrix notation as.
(ct, x, y, z)
⎛
⎜
⎜
⎝
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
ct
x
y
z
⎞
⎟
⎟
⎠ = 0
(2.3a)
It may also be written using summation indices, called tensor component notation,
as
x
μ g μν x
ν
= 0
(2.3b)
The array g μν defined in (2.3a) is called the Lorentz metric. Notice that the order in
which we write factors in (2.3b) is unimportant (see Exercise 2.2). In order that the
equation of the light cone be invariant we now demand that the quantity s
2
= x
μ g μν x
ν
be unchanged under the coordinate transformation (2.2); if it is zero in one frame it
is zero in all frames related by the transformation (2.2). Thus we write in the original
system and in the primed system,
s
= x
g μν x
= (a
μ α x
α
)g μν
a
ν β x
β
= x
α
a
μ α g μν a
ν β
x
β
,
s
2
= x
α g αβ x
β
,
(2.4)
and set them equal. Since the coordinates label an arbitrary event or point in the
4-space we find the following relation for the transformation.
g αβ = a
μ α g μν a
ν β
(2.5a)
2 Lorentz Transformations
Fig. 2.1 The light cone in two space and one time dimension
x
=
3
0
a
μ ν x
ν
= a
μ ν x
ν
.
(2.2)
Notice that in (2.2) we simply omitted the summation sign with the understanding
that repeated indices are to be summed over. This is the famous Einstein summation
convention which we will use henceforth; it makes the equations look much simpler.
The light cone equation (2.1) may be written in matrix notation as.
(ct, x, y, z)
⎛
⎜
⎜
⎝
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
ct
x
y
z
⎞
⎟
⎟
⎠ = 0
(2.3a)
It may also be written using summation indices, called tensor component notation,
as
x
μ g μν x
ν
= 0
(2.3b)
The array g μν defined in (2.3a) is called the Lorentz metric. Notice that the order in
which we write factors in (2.3b) is unimportant (see Exercise 2.2). In order that the
equation of the light cone be invariant we now demand that the quantity s
2
= x
μ g μν x
ν
be unchanged under the coordinate transformation (2.2); if it is zero in one frame it
is zero in all frames related by the transformation (2.2). Thus we write in the original
system and in the primed system,
s
= x
g μν x
= (a
μ α x
α
)g μν
a
ν β x
β
= x
α
a
μ α g μν a
ν β
x
β
,
s
2
= x
α g αβ x
β
,
(2.4)
and set them equal. Since the coordinates label an arbitrary event or point in the
4-space we find the following relation for the transformation.
g αβ = a
μ α g μν a
ν β
(2.5a)