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
=0,1,2,3; a single time-like component 0,
followed by the three space-like components 1,2 and 3:
x
= (
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
= (
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
−
2
)
−
½
as:
22
2
00
01
,
with
det(
)
1
10
µµ
νν
γβ
βγ
−
Λ=
Λ = −
=
−
(II.1)
Note the lower and upper indices: this facilitates the definition of a summation convention of same in-
dices:
3
01 2 3
0123 0
1
2
3
0
'
xx
x
x
x
x
x
t
x
y
z
ν µ
ν
µµµ
=
=Λ
≡
Λ
+Λ
∑
(II.2)
The invariance of the length of
x
is indeed satisfied:
t
’
2
−
x
’
2
−
y
’
2
−
z
’
2
= (
t
−
z
)
2
–
x
2
–
y
2
– (
−
t
+
z
)
2
= (
2
−
2
2
)
t
2
–
x
2
–
y
2
– (
2
−
2
2
)
z
2
=
t
2
–
x
2
–
y
2
–
z
2
.
This leads to a definition of an invariant four-dimensional dot-product:
x
x
= (
x
0
)
2
– (
x
1
)
2
– (
x
2
)
2
–
(
x
3
)
2
=
g
µν
x
x
, where the metric tensor
g
in special relativity is defined as:
1
000
00 10
00 0 1
gg
−
==
−
−
(II.3)
Other commonly used fourvectors, contravariant (= index high) or covariant (= index low), are:
(
,
)
(,
)
,
(
,)
,
(
,
(
,
)
,
(
,
)
pE
i
ii
j
A
tt
t
ρφ
∂∂
∂
=
=
−∇=∂
∂=
−
∇ ∂=
+
∇
=
=
∂
p
j
A
##
#
(II.4)