16
2 Lorentz Transformations
We define another 4-component object with a lower index using the Lorentz
metric,
V α = g μν V
ν
,
(2.9)
which we call a covariant 4-vector. For example, the covariant position 4-vector is.
x μ = (ct, −x, −y, −z).
(2.10)
The operation in (2.10) is called lowering an index. An index may be raised similarly
with the inverse of the Lorentz metric, which we denote as g
μν ,
V
α
= g
αν V ν , g
αλ g λω = δ
α
ω .
(2.11)
You may easily verify that (2.10) and (2.11) are consistent. From the specific form
of the Lorentz metric it is easy to see that the inverse of the Lorentz metric is simply
the Lorentz metric itself, which is a convenient fact,
g
αλ
=
⎛
⎜
⎜
⎝
−1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
⎞
⎟
⎟
⎠ .
(2.12)
Since the two arrays in (2.12) and (2.3a) are the same the difference in index position
is at this point purely for notational convenience. This will not be true later in a more
general context. We also define for convenience a mixed index object, denoted by
g
τ α =
⎛
⎜
⎜
⎝
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
⎞
⎟
⎟
⎠ = δ
τ
α .
(2.13)
This is called the Kronecker delta, equivalent to the identity matrix. Since it is
symmetric the index order is irrelevant.
Let us now ask how covariant vectors transform as we go to a new coordinate
system, labeled with a bar. We find from above that in the new system.
¯
V α = g ατ ¯
V
τ
= g ατ
a
τ β V
β
= g ατ
a
τ β g
βλ V λ
= (g ατ a
τ β g
βλ
)V λ .
(2.14)
We therefore define a new array called b α
λ and rewrite (2.14) as.
¯
V α = b α
λ V λ , b α
λ
≡ g ατ a
τ β g
βλ
.
(2.15)