D
Special Relativity: Invariance and Covariance
μ
The co-ordinate 4-vector x is defined by
0
1
2
x
μ = (x , x , x , x
3 )
0
1
2
where x = t (with c = 1) and (x , x , x
3 ) = x. Under a Lorentz transformaμ
tion along the x
1 -axis with velocity v, x transforms to
0′
x
= γ(x
0
− vx
1 )
1′
0
x
= γ(−vx + x
1 )
2′
2
x
= x
x
3′
= x
3
(D.1)
2 )
−1/2
where γ = (1 − v
.
A general ‘contravariant 4-vector’ is defined to be any set of four quantities
A
μ = (A
0 , A
1 , A
2 , A
3 ) ≡ (A
0 , A) which transform under Lorentz transformaμ
tions exactly as the corresponding components of the coordinate 4-vector x .
Note that the definition is phrased in terms of the transformation property
(under Lorentz transformations) of the object being defined. An important
μ
example is the energy–momentum 4-vector p = (E, p), where for a parti2
2 )
1/2
cle of rest mass m, E = (p + m
. Another example is the 4-gradient
∂
μ = (∂
0 , −∇) (see problem 2.1) where
(
)
∂
∂
∂
∂
∂
0 =
∇ =
,
,
.
(D.2)
∂t
∂x 1 ∂x 2 ∂x 3
Lorentz transformations leave the expression A
0 2
− A
2 invariant for a general
2
4-vector A
μ . For example, E
2
− p = m
2 is invariant, implying that the rest
mass m is invariant under Lorentz transformations. Another example is the
four-dimensional invariant differential operator analogous to ∇
2 , namely
∂
0 2
− ∇
2
❗ =
which is precisely the operator appearing in the massless wave equation
∂
0 2 φ − ∇
2
❗φ =
φ = 0.
The expression A
0 2
− A
2 may be regarded as the scalar product of A
μ with
a related ‘covariant vector’ A μ = (A
0 , −A). Then
∑
A
0 2
− A
2 =
A
μ A μ
μ
371