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Let us now investigate whether we can write the components of
the electric and magnetic
fields as the components of some proper 4tensor. There
is an obvious problem here. How can we identify the components
of the magnetic
field, which is a pseudovector, with any of the components of a
proper4tensor? The former components transform differently under parity inversion
than the latter components.
Consider a proper3tensor whose
covariant components are written , and which is
antisymmetric:

(1467) 
This immediately implies that all of the diagonal components of the
tensor are zero. In fact, there are only three independent nonzero
components of such a tensor. Could we, perhaps, use these components to
represent the components of a pseudo3vector? Let us write

(1468) 
It is clear that transforms as a contravariant pseudo3vector.
It is easily seen that

(1469) 
where
, etc. In this manner, we can actually
write the components of a pseudo3vector as the components of an
antisymmetric
proper3tensor. In particular, we can write the components of the magnetic field
in terms of
an antisymmetric
proper magnetic field 3tensor which we shall denote .
Let us now examine Eqs. (1465) and (1466) more carefully. Recall that
and
. It follows that we can write
Eq. (1465) in the form

(1470) 
Likewise, Eq. (1466) can be written

(1471) 
Let us multiply this expression by
, making use of the
identity

(1472) 
We obtain

(1473) 
or

(1474) 
since
.
Let us define a proper4tensor whose covariant
components are given by

(1475) 
It is clear that this tensor is antisymmetric:

(1476) 
This implies that the tensor only possesses six independent nonzero
components. Maybe it can be used to specify the components of
and ?
Equations (1470) and (1475) yield

(1477) 
Likewise, Eqs. (1474) and (1475) imply that

(1478) 
Thus,
In other words, the completely spacelike components of the tensor specify
the components of the magnetic field, whereas the hybrid space and timelike
components specify the components of the electric field. The covariant
components
of the tensor can be written

(1481) 
Not surprisingly, is usually called the electromagnetic
field tensor.
The above expression, which appears in all standard textbooks, is very
misleading. Taken at face value, it is simply wrong! We cannot form
a proper4tensor from the components of a proper3vector and a pseudo3vector. The expression only makes sense if we interpret (say) as representing the component of the proper magnetic field 3tensor
The
contravariant components of the electromagnetic field tensor
are given by
or

(1484) 
Let us now consider two of Maxwell's equations:
Recall that the 4current is defined
.
The first of these equations can be written

(1487) 
since .
The second of these equations takes the form

(1488) 
Making use of Eq. (1472), the above expression reduces to

(1489) 
Equations (1487) and (1489) can be combined to give

(1490) 
This equation is consistent with the equation of charge
continuity,
, because of the antisymmetry of
the electromagnetic field tensor.
Next: The dual electromagnetic field
Up: Relativity and electromagnetism
Previous: Tensors and pseudotensors
Richard Fitzpatrick
20060202