Part II Kinematics
39
10/23/2000
Part II Kinematics
1
Special Relativity
Before going any further, we present a quick overview of notations for the kinematics of spe-
cial relativity.
1.1
Four Vectors and Lorentz Transformations
Four-vectors are denoted by
x
µ
, with the greek index
µ
=0,1,2,3; a single time-like component 0,
followed by the three space-like components 1,2 and 3:
x
µ
= (
x
0
,
x
1
,
x
2
,
x
3
). The space-like indices 1,2,
and 3 are habitually represented by lower case roman characters: e.g.
x
i
=
x
= (
x
1
,
x
2
,
x
3
). In (special)
relativity a space-time distance (four-length)
t
2
−
x
2
−
y
2
−
z
2
is constant and invariant under Lorentz trans-
formations: i.e.
t
’
2
−
x
’
2
−
y
’
2
−
z
’
2
=
t
2
−
x
2
−
y
2
−
z
2
, a direct consequence of the constancy of the speed of
light for all observers.
The space-time fourvector coordinate is denoted by
x
µ
= (
x
0
,
x
1
,
x
2
,
x
3
) = (
x
0
,
x
i
) = (
t
,
x
) = (
t
,
x
,
y
,
z
).
The space-time components mix between themselves by Lorentz transformations from one inertial sys-
tem to another. A Lorentz transformation from an (unprimed) system to another (primed) system mov-
ing at relative velocity
β//
x
with respect to the first, can be expressed in terms of speed
β
and relativis-
tic factor
γ
≡
(1
−
β
2
)
−
½
as:
2
2
2
0
0
0
1
0
0
,
with
det(
)
1
0
0
1
0
0
0
µ
µ
ν
ν
γ
βγ
γ
β γ
βγ
γ
−
Λ
=
Λ
=
−
=
−
(II.1)
Note the lower and upper indices: this facilitates the definition of a summation convention of same in-
dices:
3
0
1
2
3
0
1
2
3
0
1
2
3
0
'
x
x
x
x
x
x
x
t
x
y
z
µ
µ
ν
µ
ν
µ
µ
µ
µ
µ
µ
µ
µ
ν
ν
ν
=
= Λ
≡
Λ
= Λ
+ Λ
+ Λ
+ Λ
= Λ
+ Λ
+ Λ
+ Λ
∑
(II.2)
The invariance of the length of
x
µ
is indeed satisfied:
t
’
2
−
x
’
2
−
y
’
2
−
z
’
2
= (
γ
t
−
βγ
z
)
2
–
x
2
–
y
2
– (
−
βγ
t
+
γ
z
)
2
= (
γ
2
−
β
2
γ
2
)
t
2
–
x
2
–
y
2
– (
γ
2
−
β
2
γ
2
)
z
2
=
t
2
–
x
2
–
y
2
–
z
2
.
This leads to a definition of an invariant four-dimensional dot-product:
x
µ
x
µ
= (
x
0
)
2
– (
x
1
)
2
– (
x
2
)
2
–
(
x
3
)
2
=
g
µν
x
ν
x
µ
, where the metric tensor
g
µν
in special relativity is defined as:
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
g
g
µν
µν
−
=
=
−
−
(II.3)
Other commonly used fourvectors, contravariant (= index high) or covariant (= index low), are:
(
,
)
(
,
)
,
(
,
),
(
,
),
(
, ),
( ,
)
p
E
i
i
i
j
A
t
t
t
µ
µ
µ
µ
µ
µ
ρ
φ
∂
∂
∂
=
=
− ∇ = ∂
∂
=
−∇
∂
=
+∇
=
=
∂
∂
∂
p
j
A
#
#
#
(II.4)