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We have seen that it is possible to write the components of
the electric and magnetic fields as the components of a proper4tensor.
Is it also possible to write the components of these fields as the components
of some pseudo4tensor? It is obvious that we cannot identify
the components of
the proper3vector with any of the components of a
pseudotensor. However, we can represent the components of
in terms of those of an antisymmetric pseudo3tensor
by writing

(1491) 
It is easily demonstrated that

(1492) 
in a righthanded coordinate system.
Consider the dual electromagnetic field tensor, ,
which is defined

(1493) 
This tensor is clearly an antisymmetric pseudo4tensor. We have

(1494) 
plus

(1495) 
where use has been made of
. The above
expression yields

(1496) 
It follows that
or

(1499) 
The above expression is, again, slightly misleading, since
stands for the component of the pseudo3tensor ,
and not for an element of the proper3vector . Of course,
in this case, really does represent
the first element of the pseudo3vector
.
Note that the elements of are obtained from those of
by making the transformation
and
.
The covariant elements of the dual electromagnetic field tensor are
given by
or

(1502) 
The elements of are obtained from those of
by making the transformation
and
.
Let us now consider the two Maxwell equations
The first of these equations can be written

(1505) 
since . The second equation takes the form

(1506) 
or

(1507) 
Equations (1505) and (1507) can be combined to give

(1508) 
Thus, we conclude that Maxwell's equations for the electromagnetic fields
are equivalent to the following pair of 4tensor equations:
It is obvious from the form of these equations that the laws of electromagnetism
are invariant under translations, rotations, special Lorentz transformations,
parity inversions, or any combination of these transformations.
Next: Transformation of fields
Up: Relativity and electromagnetism
Previous: The electromagnetic field tensor
Richard Fitzpatrick
20060202