2.2 Four-Vectors and Tensors
17
We call any quantity that transforms as in (2.15) a covariant 4-vector; it is consistent
with the definition in (2.9). Note how similar the transformation law is to that for a
contravariant 4-vector in (2.8). Note also that the index positions in the transformation
matrices are relevant in (2.8) and (2.15).
Example 2.2 For the elementary Lorentz transformation in (2.6) we may
calculate the array b α
λ to be.
1 0
0 −1
γ −βγ
−βγ γ
1 0
0 −1
=
γ βγ
βγ γ
.
(2.16)
Here we have again suppressed the irrelevant y and z coordinates.
There is an important orthogonality relation between the transformation arrays a
α τ
and b α
λ that follows from the definition (2.15). From the definition of the Lorentz
group in (2.5a), and using (2.11) and (2.15) we obtain.
a
μ α b μ
τ
= a
μ α
g μν a
ν β g
βτ
= (a
μ α g μν a
ν β )g
βτ
= g αβ g
βτ
= δ
τ
α
(2.17)
In matrix notation we may express this as
A
T B = I, B =
A
T
−1
(2.18)
From this it is also easy to see that.
a
α ω b λ
ω
= δ
α λ
(2.19)
Equations (2.17) and (2.19) will be very useful, and provide a preview of how similar
things work in the general theory. We may also now give an elegant alternative form
for the Lorentz group definition, using (2.5a) and (2.19),
g μν = b μ
λ b ν
ω g λω .
(2.20)
We will return to this shortly and interpret its meaning and see why it is elegant.
Example 2.3 We may show that the 4-vector inner product V
μ V μ is invariant
using the transformation properties and the orthogonality relation (2.17).
V
μ V μ = (a
μ α V
α
)
b μ
β V β
= V
α
(a
μ α b μ
β
)V β = V
α
δ
β
α V β = V
α V α . (2.21)
Such invariant quantities are of great importance throughout relativity theory.