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D. Special Relativity: Invariance and Covariance
higher-rank tensors similarly, which can also be ‘mixed’, with some upstairs
and some downstairs indices.
We now state a very useful and important fact. Suppose we ‘dot’ a downstairs 4-vector A μ into a contravariant second-rank tensor B
μν , via the operation A μ B
μν , where as always a sum on the repeated index μ is understood.
Then this quantity transforms as a 4-vector, via its ‘loose’ index ν. This is
B
μν
obvious if B
μν is actually a product such as
= C
μ D
ν , since then we have
A μ B
μν = (A · C)D
ν , and (A · C) is an invariant, which leaves the 4-vector D
μ
as the only ‘transforming’ object left. But even if B
μν is not such a product,
it transforms under Lorentz transformations in exactly the same way as if it
were, and this leads to the same result. An example is provided by the quantity ∂ μ F
μν which enters on the left-hand side of the Maxwell equations in the
form (2.18).
This example brings us conveniently to the remaining concept we need to
introduce here, which is the important one of ‘covariance’. Referring to (2.18),
we note that it has the form of an equality between two quantities (∂ μ F
μν on
the left, j
ν on the right) each of which transforms in the same way under
em
Lorentz transformations – namely as a contravariant 4-vector. One says that
(2.18) is ‘Lorentz covariant’, the word ‘covariant’ here meaning precisely that
both sides transform in the same way (i.e. consistently) under Lorentz transformations. Confusingly enough, this use of the word ‘covariant’ is evidently
quite different from the one encountered previously in an expression such as
‘a covariant 4-vector’, where it just meant a 4-vector with a downstairs index.
This new meaning of ‘covariant’ is actually much better captured by an alternative name for the same thing, which is ‘form invariant’, as we will shortly
see.
Why is this idea so important? Consider the (special) relativity principle,
which states that the laws of physics should be the same in all inertial frames.
The way in which this physical requirement is implemented mathematically
is precisely via the notion of covariance under Lorentz transformations. For,
consider how a law will typically be expressed. Relative to one inertial frame,
we set up a coordinate system and describe the phenomena in question in
terms of suitable coordinates, and such other quantities (forces, fields, etc) as
may be necessary. We write the relevant law mathematically as equations relating these quantities, all referred to our chosen frame and coordinate system.
What the relativity principle requires is that these relationships – these equations – must have the same form when the quantities in them are referred to
a different inertial frame. Note that we must say ‘have the same form’, rather
than ‘be identical to’, since we know very well that coordinates, at least, are
not identical in two different inertial frames (cf (D.1)). This is why the term
‘form invariant’ is a more helpful one than ‘covariant’ in this context, but the
latter is more commonly used.
A more elementary example may be helpful. Consider Newton’s law in the
simple form F = mr ¨. This equation is ‘covariant under rotations’, meaning
that it preserves the same form under a rotation of the coordinate system –
D. Special Relativity: Invariance and Covariance
higher-rank tensors similarly, which can also be ‘mixed’, with some upstairs
and some downstairs indices.
We now state a very useful and important fact. Suppose we ‘dot’ a downstairs 4-vector A μ into a contravariant second-rank tensor B
μν , via the operation A μ B
μν , where as always a sum on the repeated index μ is understood.
Then this quantity transforms as a 4-vector, via its ‘loose’ index ν. This is
B
μν
obvious if B
μν is actually a product such as
= C
μ D
ν , since then we have
A μ B
μν = (A · C)D
ν , and (A · C) is an invariant, which leaves the 4-vector D
μ
as the only ‘transforming’ object left. But even if B
μν is not such a product,
it transforms under Lorentz transformations in exactly the same way as if it
were, and this leads to the same result. An example is provided by the quantity ∂ μ F
μν which enters on the left-hand side of the Maxwell equations in the
form (2.18).
This example brings us conveniently to the remaining concept we need to
introduce here, which is the important one of ‘covariance’. Referring to (2.18),
we note that it has the form of an equality between two quantities (∂ μ F
μν on
the left, j
ν on the right) each of which transforms in the same way under
em
Lorentz transformations – namely as a contravariant 4-vector. One says that
(2.18) is ‘Lorentz covariant’, the word ‘covariant’ here meaning precisely that
both sides transform in the same way (i.e. consistently) under Lorentz transformations. Confusingly enough, this use of the word ‘covariant’ is evidently
quite different from the one encountered previously in an expression such as
‘a covariant 4-vector’, where it just meant a 4-vector with a downstairs index.
This new meaning of ‘covariant’ is actually much better captured by an alternative name for the same thing, which is ‘form invariant’, as we will shortly
see.
Why is this idea so important? Consider the (special) relativity principle,
which states that the laws of physics should be the same in all inertial frames.
The way in which this physical requirement is implemented mathematically
is precisely via the notion of covariance under Lorentz transformations. For,
consider how a law will typically be expressed. Relative to one inertial frame,
we set up a coordinate system and describe the phenomena in question in
terms of suitable coordinates, and such other quantities (forces, fields, etc) as
may be necessary. We write the relevant law mathematically as equations relating these quantities, all referred to our chosen frame and coordinate system.
What the relativity principle requires is that these relationships – these equations – must have the same form when the quantities in them are referred to
a different inertial frame. Note that we must say ‘have the same form’, rather
than ‘be identical to’, since we know very well that coordinates, at least, are
not identical in two different inertial frames (cf (D.1)). This is why the term
‘form invariant’ is a more helpful one than ‘covariant’ in this context, but the
latter is more commonly used.
A more elementary example may be helpful. Consider Newton’s law in the
simple form F = mr ¨. This equation is ‘covariant under rotations’, meaning
that it preserves the same form under a rotation of the coordinate system –