374
D. Special Relativity: Invariance and Covariance
and this in turn means that the physics it expresses is independent of the
orientation of our coordinate axes. The ‘same form’ in this case is of course
′
′
′
just F = mr ¨ . We emphasize again that the components of F are not the
′
same as those of F , nor are the components of r ¨ the same as those of r ¨;
′
′
but the relationship between F and r ¨ is exactly the same as the relationship
between F and r ¨, and that is what is required.
′
It is important to understand why this deceptively simple result (‘F =
′
mr ¨ ’) has been obtained. The reason is that we have assumed (or asserted)
that ‘force’ is in fact to be represented mathematically as a 3-vector quantity.
Once we have said that, the rest follows. More formally, the transformation
′
law of the components of r is r i = R ij r j (sum on j understood), where the
matrix of transformation coefficients R is ‘orthogonal’ (RR
T = R
T
R = I),
2
′ 2
which ensures that the length (squared) of r is invariant , r = r . To say
that ‘force is a 3-vector’ then implies that the components of F transform
′
by the same set of coefficients R ij : F i = R ij F j . Thus starting from the
law F j = mr ¨ j which relates the components in one frame, by multiplying
′
′
both sides of the equation by R ij and summing over j we arrive at F = mr ¨ i ,
i
which states precisely that the components in the primed frame bear the same
relationship to each other as the components in the unprimed frame did. This
is the property of covariance under rotations, and it ensures that the physics
embodied in the law is the same for all systems which differ from one another
only by a rotation.
In just the same way, if we can write equations of physics as equalities
between quantities which transform in the same way (i.e. ‘are covariant’) under
Lorentz transformations, we will guarantee that these laws obey the relativity
principle. This is indeed the case in the Lorentz covariant formulation of
Maxwell’s equations, given in (2.18), which we now repeat here: ∂ μ F
μν = j
ν .
em
To check covariance, we follow essentially the same steps as in the case of
Newton’s equations, except that the transformations being considered are
Lorentz transformations. Inserting the expression (2.19) for F
μν , the equation
can be written as (∂ μ ∂
μ )A
ν
− ∂
ν (∂ μ A
μ ) = j
ν . The two quantities enclosed
em
in parentheses are actually invariants, as was mentioned earlier. This means
′ ∂
′ μ
′ A
′ μ
that ∂ μ ∂
μ is equal to ∂ μ
, and similarly ∂ μ A
μ = ∂ μ
, so that we can
write the equation as (∂ μ
′ ∂
′ μ )A
ν
− ∂
ν (∂ μ
′ A
′ μ ) = j
ν . It is now clear that if
em
we apply a Lorentz transformation to both sides, A
ν and ∂
ν will become A
′ ν
and ∂
′ ν respectively, while j
ν
will become j
′ ν , since all these quantities
em
em
are 4-vectors, transforming the same way (as the 3-vectors did in the Newton
case). Thus we obtain just the same form of equation, written in terms of the
‘primed frame’ quantities, and this is the essence of (Lorentz transformation)
covariance.
Actually, the detailed ‘check’ that we have just performed is really unnecessary. All that is required for covariance is that (once again!) both sides of
equations transform the same way. That this is true of (2.18) can be seen ‘by
inspection’, once we understand the significance (for instance) of the fact that
the μ indices are ‘dotted’ so as to form an invariant. This example should
D. Special Relativity: Invariance and Covariance
and this in turn means that the physics it expresses is independent of the
orientation of our coordinate axes. The ‘same form’ in this case is of course
′
′
′
just F = mr ¨ . We emphasize again that the components of F are not the
′
same as those of F , nor are the components of r ¨ the same as those of r ¨;
′
′
but the relationship between F and r ¨ is exactly the same as the relationship
between F and r ¨, and that is what is required.
′
It is important to understand why this deceptively simple result (‘F =
′
mr ¨ ’) has been obtained. The reason is that we have assumed (or asserted)
that ‘force’ is in fact to be represented mathematically as a 3-vector quantity.
Once we have said that, the rest follows. More formally, the transformation
′
law of the components of r is r i = R ij r j (sum on j understood), where the
matrix of transformation coefficients R is ‘orthogonal’ (RR
T = R
T
R = I),
2
′ 2
which ensures that the length (squared) of r is invariant , r = r . To say
that ‘force is a 3-vector’ then implies that the components of F transform
′
by the same set of coefficients R ij : F i = R ij F j . Thus starting from the
law F j = mr ¨ j which relates the components in one frame, by multiplying
′
′
both sides of the equation by R ij and summing over j we arrive at F = mr ¨ i ,
i
which states precisely that the components in the primed frame bear the same
relationship to each other as the components in the unprimed frame did. This
is the property of covariance under rotations, and it ensures that the physics
embodied in the law is the same for all systems which differ from one another
only by a rotation.
In just the same way, if we can write equations of physics as equalities
between quantities which transform in the same way (i.e. ‘are covariant’) under
Lorentz transformations, we will guarantee that these laws obey the relativity
principle. This is indeed the case in the Lorentz covariant formulation of
Maxwell’s equations, given in (2.18), which we now repeat here: ∂ μ F
μν = j
ν .
em
To check covariance, we follow essentially the same steps as in the case of
Newton’s equations, except that the transformations being considered are
Lorentz transformations. Inserting the expression (2.19) for F
μν , the equation
can be written as (∂ μ ∂
μ )A
ν
− ∂
ν (∂ μ A
μ ) = j
ν . The two quantities enclosed
em
in parentheses are actually invariants, as was mentioned earlier. This means
′ ∂
′ μ
′ A
′ μ
that ∂ μ ∂
μ is equal to ∂ μ
, and similarly ∂ μ A
μ = ∂ μ
, so that we can
write the equation as (∂ μ
′ ∂
′ μ )A
ν
− ∂
ν (∂ μ
′ A
′ μ ) = j
ν . It is now clear that if
em
we apply a Lorentz transformation to both sides, A
ν and ∂
ν will become A
′ ν
and ∂
′ ν respectively, while j
ν
will become j
′ ν , since all these quantities
em
em
are 4-vectors, transforming the same way (as the 3-vectors did in the Newton
case). Thus we obtain just the same form of equation, written in terms of the
‘primed frame’ quantities, and this is the essence of (Lorentz transformation)
covariance.
Actually, the detailed ‘check’ that we have just performed is really unnecessary. All that is required for covariance is that (once again!) both sides of
equations transform the same way. That this is true of (2.18) can be seen ‘by
inspection’, once we understand the significance (for instance) of the fact that
the μ indices are ‘dotted’ so as to form an invariant. This example should