372
D. Special Relativity: Invariance and Covariance
where, in practice, the summation sign on repeated ‘upstairs’ and ‘downstairs’
indices is always omitted. We shall often shorten the expression ‘A
μ A μ ’ even
2
2
further, to ‘A
2 ’; thus p = E
2
− p = m
2 . The ‘downstairs’ version of ∂
μ is
∂ μ = (∂
0 , ∇). Then ∂ μ ∂
μ = ∂
2 = ❗. ‘Lowering’ and ‘raising’ indices is effected
μν
00
11
22
33
by the metric tensor g or g μν , where g = g 00 = 1, g = g = g =
g 11 = g 22 = g 33 = −1, all other components vanishing. Thus if A μ = g μν A
ν
then A 0 = A
0 , A 1 = −A
1 , etc.
In the same way, the scalar product A · B of two 4-vectors is
A · B = A
μ B μ = A
0 B
0
− A · B
(D.3)
and this is also invariant under Lorentz transformations. For example, the
invariant four-dimensional divergence of a 4-vector j
μ = (ρ, j) is
∂
μ j μ = ∂
0 ρ − (−∇) · j = ∂
0 ρ + ∇ · j = ∂ μ j
μ
(D.4)
since the spatial part of ∂
μ is −∇.
Because the Lorentz transformation is linear, it immediately follows that
the sum (or difference) of two 4-vectors is also a 4-vector. In a reaction of the
type ‘1 + 2 → 3 + 4 + · · · N ’ we express the conservation of both energy and
momentum as one ‘4-momentum conservation equation’:
μ
μ
μ
μ
μ
p 1 + p = p 3 + p 4 + · · · p N .
(D.5)
2
In practice, the 4-vector index on all the p’s is conventionally omitted in
conservation equations such as (D.5), but it is nevertheless important to remember, in that case, that it is actually four equations, one for the energy
components and a further three for the momentum components. Further, it
follows that quantities such as (p 1 +p 2 )
2 , (p 1 −p 3 )
2 are invariant under Lorentz
transformations.
We may also consider products of the form A
μ B
ν , where A and B are
4-vectors. As μ and ν each run over their four possible values (0, 1, 2, 3)
16 different ‘components’ are generated (A
0 B
0 , A
0 B
1 , . . . , A
3 B
3 ). Under a
Lorentz transformation, the components of A and B will transform into definite linear combinations of themselves, as in the particular case of (D.1). It
follows that the 16 components of A
μ B
ν will also transform into well-defined
linear combinations of themselves (try it for A
0 B
1 and (D.1)). Thus we have
constructed a new object whose 16 components transform by a well-defined
linear transformation law under a Lorentz transformation, as did the components of a 4-vector. This new quantity, defined by its transformation law, is
called a tensor – or more precisely a ‘contravariant second-rank tensor’, the
‘contravariant’ referring to the fact that both indices are upstairs, the ‘second
rank’ meaning that it has two indices. An important example of such a tensor
is provided by ∂
μ A
ν (x) − ∂
ν A
μ (x), which is the electromagnetic field strength
tensor F
μν , introduced in chapter 2. More generally we can consider tenB
μν
sors
which are not literally formed by ‘multiplying’ two vectors together,
but which transform in just the same way; and we can introduce third- and
D. Special Relativity: Invariance and Covariance
where, in practice, the summation sign on repeated ‘upstairs’ and ‘downstairs’
indices is always omitted. We shall often shorten the expression ‘A
μ A μ ’ even
2
2
further, to ‘A
2 ’; thus p = E
2
− p = m
2 . The ‘downstairs’ version of ∂
μ is
∂ μ = (∂
0 , ∇). Then ∂ μ ∂
μ = ∂
2 = ❗. ‘Lowering’ and ‘raising’ indices is effected
μν
00
11
22
33
by the metric tensor g or g μν , where g = g 00 = 1, g = g = g =
g 11 = g 22 = g 33 = −1, all other components vanishing. Thus if A μ = g μν A
ν
then A 0 = A
0 , A 1 = −A
1 , etc.
In the same way, the scalar product A · B of two 4-vectors is
A · B = A
μ B μ = A
0 B
0
− A · B
(D.3)
and this is also invariant under Lorentz transformations. For example, the
invariant four-dimensional divergence of a 4-vector j
μ = (ρ, j) is
∂
μ j μ = ∂
0 ρ − (−∇) · j = ∂
0 ρ + ∇ · j = ∂ μ j
μ
(D.4)
since the spatial part of ∂
μ is −∇.
Because the Lorentz transformation is linear, it immediately follows that
the sum (or difference) of two 4-vectors is also a 4-vector. In a reaction of the
type ‘1 + 2 → 3 + 4 + · · · N ’ we express the conservation of both energy and
momentum as one ‘4-momentum conservation equation’:
μ
μ
μ
μ
μ
p 1 + p = p 3 + p 4 + · · · p N .
(D.5)
2
In practice, the 4-vector index on all the p’s is conventionally omitted in
conservation equations such as (D.5), but it is nevertheless important to remember, in that case, that it is actually four equations, one for the energy
components and a further three for the momentum components. Further, it
follows that quantities such as (p 1 +p 2 )
2 , (p 1 −p 3 )
2 are invariant under Lorentz
transformations.
We may also consider products of the form A
μ B
ν , where A and B are
4-vectors. As μ and ν each run over their four possible values (0, 1, 2, 3)
16 different ‘components’ are generated (A
0 B
0 , A
0 B
1 , . . . , A
3 B
3 ). Under a
Lorentz transformation, the components of A and B will transform into definite linear combinations of themselves, as in the particular case of (D.1). It
follows that the 16 components of A
μ B
ν will also transform into well-defined
linear combinations of themselves (try it for A
0 B
1 and (D.1)). Thus we have
constructed a new object whose 16 components transform by a well-defined
linear transformation law under a Lorentz transformation, as did the components of a 4-vector. This new quantity, defined by its transformation law, is
called a tensor – or more precisely a ‘contravariant second-rank tensor’, the
‘contravariant’ referring to the fact that both indices are upstairs, the ‘second
rank’ meaning that it has two indices. An important example of such a tensor
is provided by ∂
μ A
ν (x) − ∂
ν A
μ (x), which is the electromagnetic field strength
tensor F
μν , introduced in chapter 2. More generally we can consider tenB
μν
sors
which are not literally formed by ‘multiplying’ two vectors together,
but which transform in just the same way; and we can introduce third- and